Scientific Representation: Paradoxes of Perspective

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Scientiﬁc Representation: Paradoxes of Perspective

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Scientiﬁc Representation: Paradoxes of Perspective Bas C. van Fraassen

CLARENDON PRESS ·

OXFORD

1

Great Clarendon Street, Oxford   Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With ofﬁces in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York  Bas C. van Fraassen 2008 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2008 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available Typeset by Laserwords Private Limited, Chennai, India Printed in Great Britain on acid-free paper by Biddles Ltd, King’s Lynn, Norfolk ISBN 978–0–19–927822–0 10 9 8 7 6 5 4 3 2 1

for Janine Blanc Peschard and the memory of Dina Landman van Fraassen

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Preface When I began to rewrite the Locke Lectures I gave in Oxford in 2001 I found the comments I received leading me into new paths, some quite unexpected. My main concern had been, and remained, the possibilities for an empiricist version of structuralism in philosophy of science. But how I came to conceive of those possibilities was altered ﬁrst of all by closer contact with the revivals of transcendentalist and neo-Kantian thought, and secondly by the lively and growing interest I encountered in technology, instruments, and experimental practices both in contrast with, and complementary to, philosophical reﬂection on scientiﬁc theories. Not all of my debts will appear explicitly in the text. I am thankful especially for the great good fortune I had to meet Isabelle Peschard, my main interlocutor on both these subjects. My thanks also to the writings, helpful discussion, and correspondence of Michel Bitbol, Michael Friedman, Alan Richardson, and Thomas Ryckman for more insight into the neo-Kantian tradition, and to Mieke Boon, Nancy Cartwright, Margaret Morrison, and Hans Radder for discussions of instruments and experimentation. These debts are in addition to many debts, accumulated with respect to structural realism and surrounding topics, to Jeffrey Bub, Jeremy Butterﬁeld, Otávio Bueno, Chris Fuchs, Hans Halvorson, James Ladyman, Bradley Monton, Carlo Rovelli, and Simon Saunders. On the subject of representation in the sciences, as will be quite clear, I have substantial debts to Ronald Giere and Paul Teller. In addition, Bradley Monton and Paul Teller went through an early manuscript version with a ﬁne tooth comb and made many valuable detailed comments on the text. I am painfully aware that I owe more debts to more people than I can relate here. But my greater debt, beyond words, is to Isabelle Peschard. The main addition to the locke lectures comes in Part Two, on measurement as representation, which is also the more technical part of the book. While in any logical ordering this material precedes the third and fourth parts, the less patient reader may wish to skip ahead to them.

viii



The generous sabbatical and leave policy of Princeton University, a splendid year at the Center for Advanced Study in the Behavioral Sciences in Stanford, and the hospitality enjoyed at All Souls and Magdalen Colleges in Oxford, the CREA (Centre de Recherche en Epistémologie Appliquée) in Paris, and the University of Twente in the Netherlands, as well as Senior Scholar Award SES-0549002 from the National Science Foundation, made this work possible. My special thanks to Ralph Walker, Jean Petitot, and Philip Brey for making me welcome at their institutions in Oxford, Paris, and Twente respectively. Bas C. van Fraassen

List of Figures 2.1. Reﬂection and Refraction

44

3.1. Perspective Altimetry

62

3.2. Window and Checkerboard

63

3.3. D¨urer, The Draughtsmen of the Lute

65

3.4. Frame of Reference . Perspective

67

3.5. Speed in Perspective

68

3.6. Cross Ratio Invariance

73

4.1. Geometry of the Rainbow

102

4.2. Image Categories

104

6.1. Measurement Schema

147

6.2. Coherence of Measurement

153

8.1. Adequacy as Symmetry

196

10.1. Putnam’s Paradox

230

12.1. Copernicus’s Model of Retrograde Motion

288

13.1. Failure of Supervenience

293

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Contents Preface List of Figures Introduction: the ‘picture theory of science’

vii ix 1

Part I. Representation . Representation of, Representation As The value of distortion How does a representation represent? What’s in a photo? What is a representation then? Appearance to the intellect: illumination as embedding In conclusion . Imaging, Picturing, and Scaling Modes of representation What distinguishes a picture? Mathematical imagery, distortion through abstraction Scale models and virtuous distortion Conclusion about imaging and scaling . Pictorial Perspective and the Indexical Pictorial perspective and the Art of Measuring Perspective versus Descartes’s frames of reference Mapping and perspectival self-location What is in a map? Visual perspective and the metaphor Concluding empiricist postscript

11 12 15 20 22 29 30 33 33 36 39 49 56 59 60 66 75 82 84 86

xii



Part II. Windows, Engines, and Measurement . A Window on the Invisible World (?) Instrumentation’s diversity of roles Engines of creation: engendering new phenomena The microscope’s public hallucinations Objections to this view of ‘observation by instruments’ Experimentation’s diversity of roles . The Problem of Coordination Coordination: a historical context The problem of coordination reconceived Mach on the history of the thermometer Poincaré’s analysis of time measurement Observables coordinated: two morals . Measurement as Representation: . The Physical Correlate Physical conditions of possibility for measurement General theory of measurement What is not measurement

93 94 100 101 105 111 115 116 121 125 130 137 141 141 147 156

. Measurement as Representation: . Information

157

What is measurement—number-assigning? The scale as logical space Data models and surface models The over-arching concept for measurement What is a measurement outcome? Relating the views ‘from above’ and ‘from within’

158 164 166 172 179 184

Part III. Structure and Perspective . From the Bildtheorie of Science to Paradox

191

The Bildtheorie controversy Representation: the problem for structuralism

191 204

 . The Longest Journey: Bertrand Russell Prolegomena to Russell’s conversion to structuralism Russell’s structuralist turn Conclusion

xiii 213 213 217 223

. Carnap’s Lost World and Putnam’s Paradox

225

Carnap: Der Logische Aufbau der Welt Putnam’s Paradox Staying with Putnam: the Paradox dissolved

225 229 232

. An Empiricist Structuralism What could be an empiricist structuralism? The fundamental remaining problem for a structuralist view of science The two main dangers for an empiricist The problem in concrete setting revisited and dissolved Return to our epistemological question

237 237 239 244 253 261

Part IV. Appearance and Reality . Appearance vs. Reality in the Sciences Appearance and reality: the real and unreal problem Appearance versus reality at the birth of modern science Three putative completeness criteria Appearance vs. reality: A deeper Criterion Phenomena versus appearances Three-faceted representation . Rejecting the Appearance from Reality Criterion The supervenience of mind challenge The Great Leibnizian Escape move The quantum mechanics challenge Exploring the case of quantum mechanics Supervenience? An empiricist view

269 270 270 276 280 283 288 291 292 296 297 300 304 304

xiv



APPENDICES Appendix to CH . Models and theories as representations Appendix to CH . Quantum peculiarities: fuzzy observables

309 312

Appendix to CH . Surface models and their embeddings

315

Appendix to CH . Retreat (?) from The Scientiﬁc Image

317

Notes to Appendices Bibliography Notes Index

320 322 345 399

Introduction: the ‘picture theory of science’ The Bildtheorie—‘picture theory of science’—formed the frame for much discussion and controversy among physicists in the decades around the year 1900. In retrospect, it had close connections with philosophers’ attempts to forge what we would now call a structuralist view of science. The debates and projects of those years foreshadowed the debates some half a century later, closer to our time, over scientiﬁc realism and structural realism. To understand science we need to approach it from many directions. I will focus on one aspect that I take to be central to the scientiﬁc enterprise: representation of the empirical phenomena, by means of artifacts, both physical and mathematical. The position that success in this respect is the deﬁning aim of the empirical sciences is an empiricist theme from which I will not depart. In fact, this focus on representation ﬁts the empiricist theme very well. While we are now, it seems to me, much more demanding in what we expect of philosophical accounts of science, I will take the Bildtheorie as exemplar and inspiration. But there will be some major differences between the view of science that I shall advocate and what was then known under that name. First of all, the Bildtheorie view was phrased in terms of mental pictures. Thus Boltzmann formulates the goal of physical theory as ‘‘constructing a picture of the external world that exists purely internally’’ (Boltzmann 1905, 77; 1974, 33) and called the product of scientiﬁc theorizing an inner picture or mental construction (‘‘inneres Bild’’, ‘‘gedankliche Konstruction’’; Boltzmann 1974, 106; 165–6.). In an Encyclopedia Britannica article he spelled it out:

2  On this view our thoughts stand to things in the same relation as models to the objects they represent. The essence of the process is the attachment of one concept having a deﬁnite content to each thing, but without implying complete similarity between thing and thought; for naturally we can know but little of the resemblance of our thoughts to the things to which we attach them. (Boltzmann 1974, 214.)

I will have no truck with mental representation, in any sense.1 The view Boltzmann expresses here, a view in philosophy of mind or language, has nothing to contribute to our understanding of scientiﬁc representation—not to mention that it threw some of the discussion then back into the Cartesian problem of the external world, to no good purpose. Scientiﬁc representation is—as Boltzmann’s own examples in that article amply show—by means of artifacts both concrete (graphs, scale models, computer monitor displays, and the like) and abstract (mathematical models, needed especially when the inﬁnite on inﬁnitesimal play a role). It is on these artifacts, their use, and the characteristics that are germane to the roles they play in this use, that we must focus. The reservations about how the represented must be like its representation, which Boltzmann expresses about thought, pertain equally to these artifacts, and are pertinent to any view of how a science relates to its domain of application.2 Secondly, it is not only to our understanding of theories and their models that representation is relevant. The achievement of theoretical representation is mediated by measurement and experimentation, in the course of which many forms of representation are involved as well. Scientiﬁc representation is not exhausted by a study of the role of theory or theoretical models. To complete our understanding of scientiﬁc representation we must equally approach measurement, its instrumental character and its role. I will argue that measuring, just as well as theorizing, is representing. The representing in question also need not be, and in general is not, a case of mimesis; rather, measuring locates the target in a theoretically constructed logical space. In this respect I shall make common cause with views currently found in philosophy of technology. Thirdly, the analysis of measurement as well as of the conditions of use for theoretical models can be completed only through a reﬂection on indexicality. Since at least the time of Poincar´e, Einstein, and Bohr it is a commonplace that a measurement outcome does not display what the measured entity is like, but what it ‘looks like’ in the measurement set-up.



3

That point does not go nearly far enough. It serves, however, to announce the introduction of relationality, perspective, intensionality, intentionality, and the essential indexical into the discussion of science, though it stops far short of presenting their full role. Debates in philosophy of science take place in the context of much wider tensions and oppositions in epistemology and metaphysics. When, in Part Three, I come to the theme of structuralism I will begin with protagonists in the historical Bildtheorie debate, and show how it was already implicitly challenged by an apparent paradox at the very heart of its conception of science. I will then take the message about the role of indexicality in scientiﬁc representation to the paradoxes that beset, bedeviled, or otherwise preoccupied Bertrand Russell, Rudolf Carnap, Hilary Putnam, and David Lewis. For my part I will propose an empiricist structuralism, in contrast to structural realism, as a view of science that can stand up to these challenges. Then, in the last part, I will address the troubling relations that the appearances bear to the world’s scientiﬁc image. This, so far, is the general outline of what I am setting out to do. But it may help to add here something about my own starting point. I try to be an empiricist, and as I understand that tradition (what it is, and what it could be in days to come) it involves a common sense realism in which reference to observable phenomena is unproblematic: rocks, seas, stars, persons, bicycles . . . . Empiricism also involves certain philosophical attitudes: to take the empirical sciences as a paradigm of rational inquiry, and to resist the demands for further explanation that lead to metaphysical extensions of the sciences. There is within these constraints a good deal of leeway for different sorts of empiricist positions. For my part, speciﬁcally, I add a certain view of science, that the basic aim—equivalently, the base-line criterion of success—is empirical adequacy rather than overall truth, and that acceptance of a scientiﬁc theory has a pragmatic dimension (to guide action and research) but need involve no more belief than that the theory is empirically adequate.3 While this will undoubtedly shape my discussion, I have tried to write as much as possible of this book in a way that does not trade on the differences between this view of science (‘constructive empiricism’) and its contraries (‘scientiﬁc realisms’). What scientiﬁc representation is and how it works is everyone’s concern, and there we may ﬁnd a large area where more general philosophical differences need make no difference.4

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PA RT I

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Representation Aristotle was trading on a typical ambiguity when he wrote ‘‘Tragedy is a representation of a serious, complete action which has magnitude, in embellished speech . . . ; represented by people acting and not by narration. . . .’’ (Poetics 49b25). We can read his statement as describing either the poet’s activity or the poet’s product. It is in the activity of representation that representations are produced. This is not an accidental equivocation. We lose our topic altogether if we attempt to ask ‘‘what is a representation?’’ and tacitly take just one or the other aspect into account; for in fact we cannot understand either in isolation. That applies to scientiﬁc representation as well. There is a vast and recently rapidly increasing literature on representation both in general and in philosophy of science. Let me express at once my accord with the approach advocated by Mauricio Suarez: I propose that we adopt from the start a deﬂationary attitude and strategy towards scientiﬁc representation, in analogy to deﬂationary or minimalist conceptions of truth, or contextualist analyses of knowledge. [. . .] Representation is not the kind of notion that requires a theory to elucidate it: there are no necessary and sufﬁcient conditions for it. We can at best aim to describe its most general features. . . .1

This does require some decisions about what we can and cannot take for granted, to be used in this description, as already understood. While not trying to deﬁne representation or to reduce it to something else, we will have to place it in a context where we know our way around. The ﬁrst question to broach is this: how, or to what extent, is representation related to resemblance or likeness-making? This venerable question occupies the ﬁrst two chapters. Not all, but certainly many forms of representation do trade on likeness, likeness in some respects, selective likeness. That is not what makes them representations; it is part of what deﬁnes them as the sort of representation they are, and may ﬁgure in what constitutes success. But even these tend to trade equally on unlikeness, distortion, addition. A representation is made with a purpose or goal in mind, governed by criteria of adequacy pertaining to that goal, which guide its means, medium, and selectivity. Hence there is even in those cases no general valid inference

8   :  from what the representation is like to what the represented is like overall. Not surprisingly, empiricist views of science will differ from scientiﬁc realist views on where they locate the selective likeness and unlikeness. The second question concerns perspective. The third chapter will examine and elaborate on the enigmatic but now oft-seen contention that scientiﬁc representation is perspectival. In their general use, the words ‘‘perspective’’ and ‘‘perspectival’’ are largely metaphorical. The literal use appears only when we assert, for example, that artists in the Renaissance began to draw and paint in perspective.2 As I shall elaborate, the pertinent point about this technique is Albrecht Dürer’s: drawing in perspective is a measurement technique.3 The art of perspectival drawing is an art of measuring. It is a technique for rendering a systematically selective likeness, yielding information in desired respects, and it provides an initial paradigm for measurement in general. We are perennially plagued by the shifting uses and senses of even our most common terms. In the long history of tension between physics and astronomy before the modern period, ‘‘saving the phenomena’’ refers to the appearances of the celestial bodies and their motions to the astronomer, that is, in the outcomes of the astronomer’s measurements. Those celestial bodies and their motions are one and all observable, unlike e.g. the postulated crystalline spheres. But when Kant takes on this terminology of ‘‘appearance’’ and ‘‘phenomenon’’, he entwines their meanings so deeply in his transcendentalist philosophy that we ﬁnd ourselves as it were in a different language. While I have not done so before, I will here make a terminological distinction between these two words, though neither will have Kant’s meaning. Phenomena will be observable entities (objects, events, processes). Thus ‘‘observable phenomenon’’ is redundant in my usage. Appearances will be the contents of observation or measurement outcomes. The celestial motions of concern to the ancient or medieval astronomer were all phenomena, in my sense, but Copernicus insisted that they had confused what those phenomena are like with how they appear to the earthly observer. Thus he distinguished, in my terms, the appearances (in the measurements made by the astronomers) from the phenomena (which they observed and measured). Regimenting the terminology in this way, or in any way at all, will chafe on some common usage. But it will align with other usage no less common. For example, combustion, St. Elmo’s ﬁre, lightning, and the aurora borealis

  : 

9

are all commonly named as phenomena; and whereas Giorgione was so called because of his size, Mars is called the Red Planet not because of its color but because of its reddish appearance as seen or photographed from the Earth.4 With this regimentation we will have available a distinct terminology to honor the insight that what measurement shows is not directly what the measured is like but how it appears in that particular measurement set-up. There are other techniques besides perspectival drawing, such as those of the cartographer, that can provide a paradigm for our conception of measurement and modelling. Perspective comes in there as well, though in a quite different way, when we examine the conditions for the possibility of use of these representations, for prediction and manipulation in practice. The lessons drawn from these seminal examples illuminate not only the painter’s art but the construction and use of models in science and technology. Much of Part I will come by way of prolegomenon to further study, marshalling telling cases to illuminate representation—and what will be its use? Firstly, to remove the blinders that could focus us naïvely on the idea that what is represented is simply like what is presented in the representation. Equally, to save us from the opposite error, of assuming a total independence of the represented from the content of its representation. Likeness in contextually selective fashion is important to scientiﬁc practice. The world, the world that our science is of , is the world depicted in science, and what is depicted there, is the content of its theoretical representations; but there is less to this equation than meets the eye—and thereby hangs a tale. . . .

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1 Representation Of, Representation As The most naïve view of representation might perhaps be put something like this: ‘‘A represents B if and only if A appreciably resembles B’’. Vestiges of this view, with assorted reﬁnements, persist in most writing on representation. Yet more error could hardly be compressed into so short a formula. (Goodman 1976: 3–4)

Nelson Goodman was quite right to say so. The budget of examples and counter-examples that prove it—we will look at some below—have largely been with us since almost the beginning of philosophy. But there must be a reason if the idea he disparages, that resemblance is crucial to representation, is so persistently seductive. That perfect likeness is an ideal pursued in visual imagery, at least, has much historical support. Pliny described the painter Zeuxis’ grapes as so lifelike that birds tried to eat them—while Zeuxis in turn was fooled by Parrhasios who painted a curtain with such trompe l’oeil perfection that Zeuxis asked him to pull the curtain aside in order to show his painting.1 We can cite the art of our time as well: hyperrealism in recent painting, such as Donald Jacot’s or Jacques Bodin’s, is surely admired in part for excellence of that kind. So while representation cannot be equated with the presentation of a likeness, and resemblance to what is represented is not crucial to representation as such, resemblance does play a role inviting our attention.

12  : 

The value of distortion2 That even visual, pictorial or plastic, representation is not a matter of producing accurate ‘copies’ exactly like their originals is already clear in Plato’s dialogues. Socrates and his young students once had a visitor from Elea whom they invited to take over Socrates’ usual role in their ongoing seminar. The visitor agreed, choosing Theaetetus as interlocutor, and soon has him tied in knots on the subject of representation. There are real things, says the Eleatic Stranger, and there are images of real things—some are artifacts like paintings or sculptures, others natural like dreams, shadows, or reﬂections (Sophist 265e–266d). Aren’t these images copies of what they are images of? Theatetus had already expressed a view on copy-making: ‘‘What in the world would we say a copy is, sir, except something that’s made similar to a true thing and is another thing that’s like it?’’ (Sophist 239d–240a). But does that apply to images in general? The Stranger reminds him that a sculptor may need to distort in order to represent something successfully. To make an exact likeness with respect to shape would require preserving the proportions of length, breadth, and depth of the original, and the colors of the parts as well. Those who sculpt or draw very large works don’t do that: ‘‘If they reproduced the true proportions . . . , the upper parts would appear smaller than they should, and the lower parts would appear larger, because we see the upper parts from farther away. . . .’’ (Sophist 235d–236a). We may wonder whether Plato is referring to actual examples generally discussed in Athens. There is a story, though its provenance not entirely clear, related by Ernst Gombrich, concerning two sculptors of the ﬁfth century : The Athenians intending to consecrate an excellent image of Minerva upon a high pillar, set Phidias and Alcamenes to work, meaning to chuse the better of the two. Aclamenes being nothing at all skilled in Geometry and in the Optickes made the goddesse wonderfull faire to the eye of them that saw her hard by. Phidias on the contrary . . . did consider that the whole shape of his image should change according to the height of the appointed place, and therefore made her lips wide open, her nose somewhat out of order and all the rest accordingly . . . when these two images were afterwards brought to light and compared, Phidias was in great danger to have been stoned by the whole multitude, until the statues were at

 ,  

13

length set on high. For Alcamenes his sweet and diligent strokes being drowned, and Phidias his disﬁgured and distorted hardnesse being vanished by the height of the place, made Alcamenes to be laughed at, and Phidias to be much more esteemed. (Gombrich 1960: 191)

As Roger Shepard (1990) has studied and richly illustrated in our own time, this point is general, and does not just apply to sculptors of images to be seen from below. Even in an ordinary drawing, if you want two differently oriented parallelograms to look congruent, you have to make one larger than the other. It seems then that distortion, inﬁdelity, lack of resemblance in some respect, may in general be crucial to the success of a representation.3 This does not rule out that resemblance in some other respect may be required. Yet even when that is the case—and it may be a special case—the choice of those respects in which resemblance or a speciﬁc kind of distortion is required, and those for which just anything at all will do, will have to be seen as crucial as well. There must be a cautionary tale here for how we are to understand scientiﬁc representation. It may be natural to take a successful representation to be a likeness of what it represents—but much hinges here on what the criteria of success were when it was made. One sort of success is precisely what Copernicus was taken to have as against Ptolemy: that his theory displayed the real structure of the cosmos.4 In general though, can we infer from success of a representation, in respects that we can directly appreciate, to the conclusion that it bears a structural resemblance to what is represented? The examples of how distortion may be crucial to successful representation (in view of the purpose of the representing) should certainly give us pause. But now we are running ahead of the story. Caricature and misrepresentation Successful representation may require deliberate departures from resemblance. It does not follow that likeness will always be irrelevant to successful representation. Certainly the Eleatic Stranger’s example does not show that, since even the sculptor of statues placed on high must ensure resemblance in some respect. But now consider another side to the role of distortion. Misrepresentation is a species of representation after all: a caricature of Mrs. Thatcher may

14  :  misrepresent her as draconian, but it certainly does represent her, and not her sister or her pet dragon or whatever else she may have. Yet even if we take the caricature to represent her because of some carefully introduced resemblance there, we can declare it a misrepresentation by insisting that it represents her as something she is not. A caricature may represent a rather tall man as short (as a well known cartoon depicts Supreme Court Justice Clarence Thomas as very small compared to the chair he occupies), but it represents that man, and not someone that it resembles more as to height. A caricature misrepresents on purpose, to convey a message that is clear enough in context but is to be gleaned in a quite indirect fashion. My two examples are both visual representations, but they are not visually accurate, nor does their visual inaccuracy serve to produce a visually accurate appearance to a properly placed eye (as in the Sophist’s sculptor’s case), and yet, neither is their inaccuracy accidental.5 So distortion—departure from resemblance—which may be crucial to accurate representation in certain cases, is in other cases the vehicle of effective misrepresentation. Resemblance in some particular respect may be the vehicle of reference: we recognize the caricature as being of Mrs. Thatcher because of resemblance in certain respects. It may also be the means of attribution or misattribution of some characteristic: we take the caricature to represent her as draconian because of some likeness to a dragon, which is actually an unlikeness to her. She is represented, and she is represented as thus or so: the drawing is of her, and depicts her as thus or so.6 But a list of likenesses and unlikenesses does not tell us this much—why, for example, is this not a caricature of a dragon as Thatcherian? Let us look at another drawing, say, Spott’s drawing of Bismarck as a peacock.7 Is this drawing a misrepresentation? There we broach a question of truth that certainly is not settled in terms of visual accuracy or inaccuracy—even if both reference and attribution were effected by selective uses of resemblance and non-resemblance. The judgment whether this is an illuminating caricature that conveys a truth about him, or amounts to a falsehood, a lie, a misrepresentation, is not settled by geometrical relations between the line drawing and Bismarck’s appearance to the eye. If we just focus on resemblance in some respect as the core notion in representation then it is at best puzzling that distortion might be needed for effective representation. But if the resemblances are just a means to an

 ,  

15

end there is no puzzle. The sculptor wants the object he makes to have a certain use, and he chooses the way in which the proportions of the object are related to those of the original—the ways in which they are like and the ways in which they are unlike —so as make that use possible. There have (of course!) been efforts aimed at naturalizing representation, such as Fred Dretske’s account of information-bearing as correlation, but these tend to founder speciﬁcally on the issue of misrepresentation.8 Misrepresentation is a species of representation. If the relationship ‘X represents Y’ were to lie in a resemblance or correlation or other such structural relation between the two, what would misrepresentation be? Suppose I say that the caricature that depicts Mrs. T as draconian misrepresents her. Then my assertion has as ﬁrst part that it does represent Mrs. T as draconian, but as second part not only that it is unlike her with respect to shape, but that it depicts her as something she is not. To say that it misrepresents her with respect to shape is to say that rather than resembling her it depicts her as resembling (being like) something which, according to me, she is not like. To say that it is a caricature, however, is to say that it purveys an interpretative attribute, something that the picture can convey only by drawing on a social context, not just on what is ‘in’ the picture taken by itself. So what is accuracy? The evaluation of a representation as accurate or inaccurate is highly context-dependent. A subway map, for example, is typically not to scale, but only shows topological structure. Relative to its typical use and our typical need, it is accurate; with a change in use or need, it would at once have to be classiﬁed as inaccurate. Similarly, in one political context, or relative to a certain kind of evaluation, a caricature may rightly be judged to be accurate, in another misleading or blatantly false.

How does a representation represent? We confront here the general question of how an item such as a picture can correctly represent, misrepresent, caricature, ﬂatter, or revile its subject. Note well that the answer will not be also an answer to the question of what representation is. The question here addressed arises rather if we take it for granted that something is playing the representational role, and want to know just how it plays that role.

16  :  Nelson Goodman’s Languages of Art, the twentieth century’s seminal text on the matter, characterizes even pictures as being like statements, depicting a subject (the referent) as thus or so (as with a predicate).9 By drawing the of/as distinction I was already more or less following him in this view. That something is a picture of Bismarck does not imply that Bismarck was in every respect the way he looks in the picture, it does depict Bismarck as thus or so. Goodman translates this into: Depiction is [also] predication. Since pictures denote things they are, in that respect, like names; but since they depict those things as thus or so, they are also predicates applied to what they denote. The content of this picture is the same sort of thing—or should be talked about in the same sort of way—as the content of a predicate or a sentence applying a predicate to the subject of that painting. To say that X represents Y as F, that is taken to have as guiding example something like ‘‘Snow is white’’ represents snow as colored Putting these two points together it seems quite apt to liken a picture to a sentence, rather than to either a name or a predicate. For in a sentence we typically also see a subject to which a predicate is being applied—the subject may be real enough, but it may or may not be correctly characterized. If the sentence is complex, we may distinguish respects in which the predication is accurate and ways in which it is not, just as we do with a painting. Denotation, as Goodman understands it, is of something real, and the relation is extensional. Pictorial predication need meet neither of the conditions placed on denotation. If the painting depicts Thackeray as the author of Waverley we cannot infer that Waverly exists, nor that it had a single author if it does. But in addition, if Scott is the author of Waverley we cannot even say that the painting depicts Thackeray as Scott, except as a joke. Neither ‘‘predicates’’ nor ‘‘depicts as’’ is an extensional term.10 Goodman’s most controversial thesis is that denotation and predication are also in the case of pictures entirely independent of each other. ‘‘The denotation of a picture no more determines its kind’’, he writes, ‘‘than the kind of picture determines its denotation’’.11 (He uses the ‘‘kind’’ terminology to distinguish pictures that predicate differently.) This thesis is controversial since it is hard to accept that a picture could fail to convey anything correct or true about something and still be a picture of that thing.12 Instead, we would typically think that we can identify what it is a

 ,  

17

picture of by looking at how things are depicted in it! But there are limiting cases where this can be drawn into doubt.13 Undeniably, though, Goodman has brought into the limelight the strong analogy: a picture is a picture of something, and depicts that something as thus or so, and so is in that respect similar to how a verbal description is a description of something, and describes that something as being thus or so. We need now to see how we can go beyond this analogy. One way in which Goodman did bring in resemblance was through the intricate notion of exempliﬁcation. If a hardware store clerk or interior decorator shows you color swatches or fabric samples, those exemplify the property of interest in the following sense: they both have and refer to that property (Goodman 1976: 45–68 ; Elgin 1996: 171–83). Obviously their use is to represent to you what your wall or ﬂoor will look like if you choose the corresponding paint or carpet. Representation by exempliﬁcation involves likeness, but much more than likeness. In these examples visual resemblance plays a role: in fact, the color swatch has the same color, and in that crucial respect resembles, the paint that it represents to the customer. This only works if the relevant visual resemblance is highlighted in some way, has a status unlike that of the many other resemblances—that is the point of taking exempliﬁcation of the relevant property, rather than mere instantiation, as the vehicle by which representation is achieved.14 The relevant visual resemblance must be highlighted in some way in that context, so as to bestow that status—here we have strayed from semantics into pragmatics. But do not equate even this beautifully articulated relationship with picturing or imaging! As Goodman later pointed out, the word ‘‘word’’ both refers to and has the property of being a word, so it exempliﬁes that property, but is not a picture or image. (Goodman 1987–8: 419) Asymmetry of representation There is an asymmetry in representation that resemblance does not have. This is a much repeated point, made to show that resemblance is not the right criterion for representation. Resemblance could not be the crucial clue to representation, it is said, for even if representation did require resemblance to its target, the target would then resemble its representation but not represent it. While resemblance is indeed not the right criterion, the argument from asymmetry is not all that strong.

18  :  First of all, we do tend to use terms like ‘‘resemble’’ and ‘‘looks like’’ and such asymmetrically in certain contexts, because the subject of a statement tends to select the focus or contrast. Of Rosemary’s baby they said ‘‘He has his father’s eyes’’. Hard to think of their saying about the father ‘‘He has his baby’s eyes’’. The retort may be that literally the noticed resemblance goes both ways; but ‘‘literally’’ may here just mean ‘‘if you ignore the context’’. Probably we can ﬁnd a context in which someone may, without oddity, assert not that a given picture is an exact likeness of me but that I am its exact likeness, or something to that effect. But that would not show that resembles is symmetric in general, let alone in every context in which it comes in. That literally resemblance must go both ways—that literally speaking it is both reﬂexive and symmetric, while representation is neither—is most likely based on the simple idea that to resemble (in some respect) is to have a property (of the pertinent kind) in common. But that is too simplistic a construal anyway. The production of a photo involves a ‘collapsing’ of shades of color and of three-dimensional spatial structure into two dimensions. If that counts as pertinent resemblance, then this is a relation of homomorphism rather than isomorphism, yet central to modeling. That A is a homomorphic image of B certainly does not entail that B is that of A.15 But this contrary point too is weaker than it seems. If A is a homomorphic image of B then there is a reduction of B, modulo some equivalence relation, to which A is isomorphic—so we might say that in this respect B resembles A just as A resembles B. So there is no strong argument, as far as I can see, based on any clear asymmetry to banish resemblance from our topic, nor one to make it relevant to representation in general. What does remain, as needs to be emphasized, is that certain modes or forms of representation (but not all) do trade on selective (and not arbitrary) resemblances for their effect, efﬁcacy, and usefulness, and that this typically goes in one direction only. Resemblance in discord with representation In examples of picturing, of visual representation, resemblance tends to spring to the eye. Both reference and attribution can be achieved in other ways, however, even in visual representation, rather than by means of carefully selected resemblances. And conversely, even there, what is represented, and how it is represented, cannot in general be deduced simply by

 ,  

19

attending to resemblances and non-resemblances. To show this, Goodman mentions the painting of the Duke of Wellington which everyone agreed resembled the Duke’s brother much better. Socrates argues the point by means of a look at the extreme case, where the resemblance is greatest, in his discussion with Cratylus about verbal representation: I quite agree with you [Cratylus] that words should as far as possible resemble things, but I fear that this dragging in of resemblance . . . is a shabby thing, which has to be supplemented by the mechanical aid of convention with a view to correctness. (Cratylus 435c)

Socrates’ thought experiment leading to this remark has a quite contemporary ring, if we replace gods (as is usual now) with mad scientists. In his discussion with Cratylus, correctness and accuracy are being characterized in terms of greater resemblance: [ . . . ] in pictures you may either give all the appropriate colors and ﬁgures, or you may not give them all—some may be wanting—or there may be too many or too much—may there not? [ . . . ] And he who gives all gives a perfect picture or ﬁgure, and he who takes away or adds also gives a picture or ﬁgure, but not a good one. (Cratylus 431c)

Here the ‘‘mere resemblance’’ view of representation appears on Socrates’ lips, somewhat surprisingly; but Socrates himself will soon put us right on this. When Cratylus draws the parallel with language too simplemindedly, the discussion immediately shifts into a subtler gear. Socrates replies, in partial contradiction to the above: [ . . . ] I should say rather that the image, if expressing in every point the entire reality, would no longer be an image. Let us suppose the existence of two objects. One of them shall be Cratylus, and the other the image of Cratylus, and we will suppose, further, that some god makes not only a representation such as a painter would make of your outward form and color, but also creates an inward organization like yours, having the same warmth and softness, and into this infuses motion, and soul, and mind, such as you have, and in a word copies all your qualities, and places them by you in another form. Would you say this was Cratylus and the image of Cratylus, or that there were two Cratyluses? Cratylus I should say there were two Cratyluses.

20  :  Socrates Then you see, my friend, that we must ﬁnd some other principle of truth in images. . . . Do you not perceive that images are very far from having qualities which are the exact counterpart of the realities which they represent? (Cratylus 432a–d)

That Cratylus does not grant that the copy, made by the god to duplicate Cratylus entirely, is an image of Cratylus shows at the very least that resemblance is not sufﬁcient to make for representation. But the example shows much more—we need to explore it in detail.

What’s in a photo? Just to assert of something that it is a representation, or that it represents, or that it represents something, is woefully elliptic and invites obscurity and confusion. Our full locution must in the general case be at least of form ‘‘X represents Y as F’’, as in ‘‘the caricature represents (depicts) Mrs. Thatcher as draconian’’.16 Here X is a representation, Y its referent, and F a predicate that X depicts Y as instantiating. The ‘‘as . . .’’ locution is fascinating, a difﬁcult topic in the analysis of language, and we will need to carefully distinguish how its intensionality is connected with the relationality and intentionality of representation. But this form too is still too brief to allow all needed distinctions. Why didn’t Cratylus agree that the god would have made an image of him? The god would, in Socrates’ example, have made a perfect replica, more perfect than any statue made by a human sculptor. The replica would certainly, if properly displayed, have created the appearance of Cratylus’ being there. So as far as the later classiﬁcation in the Sophist goes, the god would have acted both as likeness-maker and as appearance-creator. Yet Cratylus demurs: the god would not have made an image of him but would have made ‘another Cratylus’. What are we to make of this intuition? A more contemporary example will show the inherent ambiguity in play here about what represents what, with what, and to whom. Imagine: I have acquired a famous photograph of the Eiffel Tower, Au Pont de l’Alma by Doisneau. It hangs on my wall, but I scan it and print the scanned image. This print is an image too—what does it represent? The Eiffel Tower seen from the Pont de l’Alma, or the famous photograph?

 ,  

21

There is no single immediately and obviously right answer; there couldn’t be. It depends on what I do with the thing. If I send it to you from Paris as a postcard, with the single note ‘‘Wish you were here!’’, then it is itself a photo of the Eiffel Tower. If I insert it into the book I am writing about photography, then it represents the famous photo by Doisneau. There are still other possibilities.17 In other words, if it is an image of something at all then what it is an image of depends on the use, on what I use it to represent. So the question what does it represent? must in this case be taken as elliptic for what is it being used to represent? This term ‘‘use’’ can assimilate ‘‘make’’ and ‘‘take’’: the caricaturist made the caricature to depict Mrs. Thatcher as draconian while I, seeing the caricature, take it to depict Mrs. Thatcher as draconian, and display it to you so as to depict her to you as draconian. There may also be a disparity: the boy in William Golding’s Free Fall made drawings of hills and forests but his teacher takes the drawings as pornographic. All of this falls under ‘‘use’’ in the general sense in which we say that in pragmatics we are meant to study the relation not just of symbols to things, but the three-term relation between symbol, user, and thing. Our fuller locution—shortened in different ways depending on what is taken for granted in context—must be ‘‘Z uses X to depict Y as F’’.18 I have put this in individualistic terms: you or I can use something to represent something, though of course to communicate or convey anything at all that way—be it factual information, feeling, intention, or command—we must let each other in on how something is being represented. (As Goodman put it, the most probative question is not what is art? but when is art?) Since communication presupposes community to some signiﬁcant extent, this will be possible only in a context where some modes of representation are already held and understood in common. Within discussions in which the institutionalization of relevant symbolry is already taken for granted, the point about use may be put in terms of function or role. And although these terms make sense also for an individual creation of a symbol, the role will generally be one speciﬁable within a manifold of roles, a ‘system of representation’, a ‘language of art’. Thus Ned Block writes What any representation represents, and how it represents . . . depends on the system of representation within which it functions. (Block 1983: 511)

22  :  My point is not just that what represents what is relative to a system of representation. Rather my point is that you can’t tell for sure whether you are looking at a representation at all just by looking. . . . One has to determine how the thing functions. (Ibid.: 512)19

Relativity to systems, ‘languages’, recalls Goodman of course; and this applies very well to pictures, as is well enough understood when different styles of representation are studied.20 This notion of system, or of function if understood as a role in a system, sounds still quite impersonal, but we must understand it in terms of pragmatics, referring to contexts of use, broadly construed. The contextsensitivity does not go away when (in different ways) Nelson Goodman and Ned Block say that you need to know which system of representation the item belongs to. For since the same item will in different contexts belong to different such systems, we then need the relevant contextual factors which determine that. What is really to be emphasized here is the way in which individuals and groups, though relying on some pre-existing communally understood form of representation (a universal qualiﬁcation of all communal activity), create new representations and new modes of representation. When Descartes created his method of coordinates, it is not as if he was just using an already extant way of representing spatial shapes and motion. But it is true that in his initiative, to use known numerical equations in this way, he bestowed a role on already familiar equations that they had not had before. Unlike a moment’s poetic depiction quickly lost to history, this act engendered a mode of representation fundamental to all subsequent science.

What is a representation then? Look back now at Socrates, Cratylus, and the god they imagine. Did the god make an image of Cratylus or did he not make a representation of anything, but a clone? That depends. Cratylus was too hasty in his response! Did this god go on to display what he made to the Olympic throng as a perfect image of Greek manhood? Or did he display it as an example of his prowess at creature-making? Or did he do neither, but press the replica into personal service, since he couldn’t have Cratylus himself?

 ,  

23

What is represented, and how it is represented, is not determined by the colors, lines, shapes in the representing object alone. Whether or not A represents B, and whether or not it represents the represented item as C, depends largely, and sometimes only, on the way in which A is being used. ‘‘Use’’ must here be understood to encompass many contextual factors: the intention of the creator, the coding conventions extant in the community, the way in which an audience or viewer takes it, the ways in which the representing object is displayed, and so forth. To understand representation we must therefore look to the practice of representing, to how representation is a matter of use; and this involves attention ﬁrst of all to the users in a broad sense of ‘‘use’’. That is the main thing to be concluded both from our discussion of caricature and misrepresentation, and from the Cratylus and Eiffel Tower photo examples. There is no representation except in the sense that some things are used, made, or taken, to represent some things as thus or so. I do not advocate a theory of representation, and this could not possibly be offered as such since that would be circular. But if I did, I think this would be its Hauptsatz.21 What does this exclude from the category of representations? That depends of course on precisely what ‘‘used, made, or taken’’ means. And that in turn depends on what is required if this Hauptsatz is to solve or dissolve puzzles about representation. (For example, the puzzles that result if one begins with the thought that resemblances—likenesses, correlations—will determine, by themselves, what the representor represents.) What we can conclude, at least, is that use, in the appropriate sense, must determine the selection of likenesses and unlikenesses which may, in their different ways, play a role in determining what the thing is a representation of, and how it represents that. Moreover, the selection cannot be mute: in the pertinent context, this selection and the precise role it plays, the selection must be salient, so the use must be such as to highlight that selection.22 The use is what bestows the relevant role or function on the item used. There are uses of the terms ‘‘use’’, ‘‘make’’, ‘‘take’’ which imply no intentionality: the car uses gasoline, the tornado took the life of my neighbor, the Ice Age glaciers made these valleys. But our puzzles about what representation is do not disappear unless ‘‘use’’ and its cognates are understood here in the sense in which they presuppose intentional activity.

24  :  That said, I will just write ‘‘use’’ for use, make, or take understood in this sense. If that Hauptsatz is understood in this sense, then it places some immediate limits on the range of representation. Firstly, at least if taken entirely literally, it has no room for the notion of mental images or mental representations, whether taken to be brain states or something more ephemeral—for no such things, if they exist at all, are used or put to use, or taken in one way or another.23 At least, not in the relevant sense: we can conceive of a brain surgeon bestowing a representational role on the patients’ brain states, but not of a person bestowing roles on his or her own brain states—or, presumably, on whatever could count as mental states.24 Secondly, this conception leaves no room for ‘representation in nature’, in the sense of ‘naturally produced’ representations that have nothing to do with conscious or cognitive activity or communication. The Eleatic Stranger gave a whole series of examples of copies, but it is not clear that they are all images in the sense of representations: Things in dreams, and appearances that arise by themselves during the day. They’re shadows when darkness appears in ﬁrelight, and they’re reﬂections when a thing’s own light and the light of something else come together around bright, smooth surfaces and produce an appearance that looks the reverse of the way the thing looks from straight ahead. [ . . . ] And what about human expertise? We say housebuilding makes a house itself and drawing makes a different one, like a human dream made for people who are awake. (Sophist 266c–d)

Most shadows and reﬂections that occur in nature are not being used or taken, let alone made, by anyone to do anything. (The Balinese shadow puppet theater is an exception.) So by our Hauptsatz, they are generally not representations. Nor is the track left in sand by an ant in some desert long, long ago in a galaxy far, far away. . . . not even if it has the shape of our word ‘‘Coca Cola’’.25 A black mark on a rock does not refer, represent, or mean anything unless it has a role, or has bestowed on it a role, in some practice—no matter whether it is a simple stroke or a complex pattern. Nor is it sufﬁcient that it has the sort of shape, coloring, etc. that would place it in a certain role if encountered or produced in a certain cultural context, by persons belonging or assimilated there, if in fact it does not bear any relation to such a context.

 ,  

25

But a natural object can represent, just as it can play other roles, namely if we bestow such a role on it. Imagine I am using a stone, found on the ground, to hammer in a tent peg. I am using it as a hammer—it is my hammer now, I have bestowed the hammer role on it. The hammers we buy, in contrast, are manufactured precisely to play this role—they are manufactured artifacts. The stone was not made for that, and it is not an object that I created, constructed, or assembled. Nevertheless it is now a physical object with a function—that is to say, an artifact. There is an analogy here to ‘objets trouv´es’, natural objects ‘made’, without physical modiﬁcation, into works of art. All of this applies mutatis mutandis when I use the stone to represent, for example, a certain stateman’s heart: I bestow a role on the stone for it to play, I give the stone a function for it to serve. What is in a photo? What is in a picture? This question has the misleading form of ‘‘What is in a box?’’ We won’t get much further by taking this form at face value and giving an answer with the correlative form, such as ‘‘What is in the representation is its content’’. That is just a verbal answer, conveying nothing by itself. To call an object a picture at all is to relate it to use. As an analogous example we can think of Herbert Mead’s reﬂections on the teacup (McCarthy 1984). If there were no people there would be no teacups, even if there were teacup-shaped objects. For ‘‘there are teacups’’ implies ‘‘there are things used to drink tea from’’ which in turn implies ‘‘there are tea-drinkers’’. By ignoring the contextuality of representation, the fact that we are dealing with about-ness, and that what the representation is about is a function of its use, we could land ourselves in useless metaphysical byways. If we were to ask ‘‘What is in a picture?’’ while taking the picture simply to be the physical object and with no relation to anything that can bestow meaning, the answer would have to be ‘‘Nothing!’’ The notion of use, the emphasis on the pragmatics rather than syntax or semantics of representation in general, I will give pride of place in the understanding of scientiﬁc representation. But does that exclude too much? That a particular person at a speciﬁc time uses or takes or presents something to represent something else is a very local event. Could it really be a general condition on representation that something so speciﬁc has to happen? In a comment on similar ‘‘intentional’’ views of what constitutes representation, Mauricio Suarez suggests that it will hamstring the idea that theories represent:

26  :  on the intentional account of representation a theory cannot ‘represent’ a phenomenon that hasn’t been observed. For a theory cannot be intended for a phenomenon that hasn’t yet been established. (Suarez 1999: 82)26

The objection, if valid, would not just apply to a theory or a model, it would imply that nothing can be intended (let alone used) to represent something that has not entered our acquaintance, or something that we do not know to exist. The objection is presumably not that there can’t be a representation of something that we have not already encountered. A meteorological model, found for example on a weather forecasting website, does in fact provide us with a representation—more or less accurate, or not accurate at all—of the weather in the next ﬁve or ten days. Is the objection then that we cannot be said to use or take or present this meteorological model to represent the coming week’s weather? But we do use it, and the viewers so take it. So could the objection rather be that this model could have been or provided such a representation although it could not have been—to use Suarez’s exact words—intended for the actual meteorological phenomena, which are still in the future? It does not seem so, it seems that it was intended precisely to represent those (as yet unknown) phenomena. Relation, intention, intension Representation is a relational notion, if we go by the form of assertions that attribute this status. ‘‘Tragedy is a representation of an action. . . .’’ Here is a painting of a picnic on the grass, the statue over there represents justice, that graph depicts the growth of a bacteria colony. In each case we have a subject term, a relational predicate, and a term for the second relatum. But we are not dealing with something as simple as a relation of physical contact or impact or proximity. First of all, the second relatum may not be real. In fact, to say that something is a painting of a picnic does not at all imply that there is a real picnic which that painting depicts. ‘‘The Mona Lisa is a portrait of Mary Magdalene’’: this assertion purports to mention two real things and a relationship between them. Perhaps it does. But the important point is that the form by itself cannot reveal this. For the assertion may be true still if it turns out that Mary Magdalene was not a real historical character. Even if she was, the Mona Lisa could at best depict how Da Vinci imagined her to be, which we can’t necessarily equate with depicting her.27

 ,  

27

The ‘‘of’’ that marks the relation of representing object to what is represented is like the one familiar from Brentano’s characterization of mental acts as ‘directed’, intentional. Intentionality we also see in semantic discourse when we say that ‘‘Zeus’’ is the name of a god, for example. Representation is intentional in the sense of relating to epistemic intention, in the sense of being about something, in just the way that reference (by someone) and predication (by someone) are. But just as thought can be directed in this sense at what is not present, not experienced, not known, or even non-existent, so can any use of something to represent something. By so using it, the user bestows a role, the role of representing such and such as thus or so. If for example I draw a graph and present it as representing the rate of bacterial growth under certain conditions, then by virtue of that very act, what the graph represents is the bacterial growth rate under those conditions—period. It is equally apt to say that I represented that growth rate as thus or so—and it would be apt to say that if, instead of drawing the graph, I had displayed the equation of which the graphed function is a solution. And so forth, mutatis mutandis, for the case in which I display a function that has such graphs as output for inputs about ambient conditions of bacteria colonies, or state a theory that describes a family of such functions with a further free variable for the type of bacteria. . . . Given this intentionality it is perhaps not surprising that in the case of representation, the relations can change with context of use. The very same object or shape can be used to represent different things in different contexts, and in other contexts not represent at all.28 The expression ‘‘A represents B’’ when used all by itself is misleading. It is easy to get into confusions when the relational character of a term is suppressed. To illustrate: every woman is a daughter and every daughter is a woman, so why is being a daughter not the same as being a woman? Precisely because to be a daughter is to be daughter of someone. Analogously, to represent something is to represent something as thus or so. The complexities appear in force when we extend these assertions to the threeplace relation ‘‘A represents B as C’’. Simone de Beauvoir depicted herself as a dutiful daughter, but not as a dutiful woman.29 All and only creatures with hearts are creatures with kidneys—yet to represent something as having a heart is not the same as representing it as having kidneys. And so forth. This ‘opacity’, the resistance to substitutivity of identity, is a mark not only of the intentionality of thought, but of intensionality in discourse.30

28  :  We see this in modal contexts: 9 = the number of planets, but it does not follow that the number of planets is necessarily greater than 7. Also in oratio obliqua: the mathematics teacher who taught us that 7 < 9 did not tell us that 7 is smaller than the number of planets.31 Representation of an object as the evening star is an activity that is intentional, in the way that mental acts are traditionally said to be, precisely because to do so is not the same as representing it as the morning star—even though that is the very same object. So assertions to the effect that something represents, are intensional. This is primarily a point about language, but is closely related to the point that representation itself (the activity) is intentional, both in Brentano’s sense and in the common sense of the term. Ordinary discourse does not mark the distinctions we are making here, or not very well. In analytic philosophy language has been regimented to some extent to do so. Thus ‘‘He said of Mrs. Thatcher that she was draconian’’ asserts a relation of the speaker to the real Mrs. Thatcher, while ‘‘He said that Mrs. Thatcher was draconian’’ does not, in this regimented form of discourse. If Mrs. Thatcher was the Prime Minister, the ﬁrst sentence implies ‘‘He said of the Prime Minister that she was draconian’’ but the second does not imply that. The role here given to the ‘‘of’’ locution marks an artiﬁcial verbal distinction (even if not without roots in prior general usage), but such artiﬁce can be useful when confusion threatens.32 Where do the intensionality and intentionality come from? To understand this is as important for representation in science as in the arts. The answer lies of course in our Hauptsatz, that there is no representation except in the sense that some things are used, made, or taken, to represent some things as thus or so. But even this does not sufﬁce by itself, for it does not make explicit what all is involved in the use, by way of value, purpose, aim, and yes, intention. Ronald Giere spells this out concisely for scientiﬁc representation and one further contextual factor, purpose: If we think of representation as a relationship, it should be a relationship with more than two components. One component should be the agents, the scientists who do the representing. Because scientists are intentional agents with goals and purposes I propose explicitly to provide a space for purposes in my understanding of representational practices in science. So we are looking at a relationship with roughly the following form: S uses X to represent W for purposes P. (Giere 2006: 60)

 ,  

29

But the point is quite general. The spelling out can only go so far, because the notion of representing has (suffers from?) variable polyadicity: for every such speciﬁcation we add there will be another one.

Appearance to the intellect: illumination as embedding We have been concentrating mainly on appearance to the senses, but when viewing a painting or movie, reading a story, or watching a computer simulation, we may well ask ‘‘I can see what is happening, but what is really going on?’’ Quite often however we do not even get to this question: our active, agile imaginations have already supplied, assumed, or conjectured a pattern behind the displayed events. On other occasions we press to ﬁnd or construct a model in which the random or puzzling appearance is sublimated in a well-behaved structure, of which it is the surface. In such a case we arrive at a representation of the original as embedded in a larger whole, and thereby made to satisfy certain demands of the intellect. There certainly are examples of this in the visual arts. A picture of an apparently levitating woman may be immediately recognizable as of the Virgin Mary because the ﬁgure’s feet are on a crescent moon and she is surrounded by jubilant angels. The embedding structure presents the story in which the event is embedded. But something similar can be said of the event depicted in Jacques-Louis David’s Oath of the Horatii, though it does not draw on such pervasively common knowledge. The lamentations of the seated women in the background show how much more is going on than the event depicted taken in itself; they help to give meaning to what is happening through an embedding in a larger story.33 But the most straightforward examples come from the history of modeling in mechanics. Think for a moment of a swinging pendulum. Determinism requires that if the same state occurs at different times, it is always followed by the same succeeding states. At its lowest point, the bob has its maximum speed—but location and speed can be the same at two different moments, and the location a moment later be different. If we keep only those two factors in our description, we have an apparent violation of determinism. The remedy is simple: enlarge the description by replacing speed by velocity, which is speed + direction. The velocity at

30  :  the lowest point is not always the same; when it is, so are the immediately succeeding locations. All these features are still in the domain of kinematics. Descartes’s famous aim for mechanics was that it should be deterministic but only have ‘quantities of extension’, that is, kinematic parameters. When Leibniz and Newton each came up with counter-examples, they also introduced new, non-kinematic (dynamic) parameters to ‘‘ﬁll out the picture’’. The apparently indeterministic kinematic behavior is embedded in a model that has additional parameters—such as masses and forces—which is deterministic after all. These examples provide the pattern for ‘hidden variable’ interpretations of apparent indeterminism. Reichenbach argued that such added parameters might not correspond to anything real, and that physics could forego satisfying the demand for determinism. But as a good empiricist, he offered this as methodological advice: neither demand them nor ban them from modeling. The touch stone would be the usefulness of such models for empirical prediction. For us, the point here is simply that, for a given purpose, the best representation might well be one that embeds its target in a larger structure. So we can add addition of ‘surplus structure’ to distortion and the trading on selective unlikenesses, to our catalogue of means for representation achieved by departures from mimesis. Hence, again, there is no universally valid inference from what the best representation is like to what the represented is like.

In conclusion A scientiﬁc, technical, or artistic representation is an artifact. As such, it is both an object or event in nature, that we can regard purely through the physicist’s or chemist’s or mathematician’s eyes. But it is at the same time something constituted as a cultural object, through its role or function, bestowed upon it in practice. Just what the representation is, or what is represented and how, is not determined entirely—and often enough, hardly at all—either by what is ‘in’ the natural object or by its physical or structural relations to other things. When resemblance is the vehicle of representation, for example, the representation relation derives from selective resemblances and selective

 ,  

31

non-resemblance, but what the selections are must be somehow highlighted. If the selection or the highlighting is indicated by signs placed in the artifact itself, these too need to be meaningful to play their role, and so the task of identiﬁcation is pushed back but reappears as essentially unchanged. Thus what determines the representation relationship, with all its polyadicity, can at best be a relation of what is in it to factors neither in the artifact itself nor in what is being represented. In the examples and puzzles here examined, the extra factors characterized use, practice, and context, and these form the proper basis for generalization there. That is not the end of the matter, representation is not to be subjected to deﬁnition: it is inexhaustible as a subject.

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2 Imaging, Picturing, and Scaling Despite the arguments against taking resemblance to be the clue to representation, there are many cases where what the representation refers to, or what it attributes, is conveyed effectively by displaying a salient resemblance.1 Examples come readily from the visual and plastic arts, but there are relevant representational techniques commonly used in the sciences as well. Resemblance comes in, not when we are answering the question What is representation?, but rather when we address How does this or that representation represent, and how does it succeed? In the case of representation by pictures, scale models, diagrams, and maps, and many other examples, the initial answer to the latter is By selective resemblance and selective (even systematic) non-resemblance. To ﬁx terminology that I have used informally so far let us reserve the terms imaging and imagery for those cases.

Modes of representation Everything resembles everything else in many ways, so an effective use of resemblance must always be selective. This can only be effective if the selection itself is understood or conveyed. That point applies more generally: even if resemblance is not the vehicle, whatever features of the representing entity are instrumental to the representing must be somehow highlighted there. It is not sufﬁcient for the representor just to have them! Secondly, even if we recognize a role played by a resemblance, this resemblance need not be with respect to any visually or even perceptually detectable features. Additionally, resemblance need not consist in sameness of properties, but can also be at higher levels. This was a theme emphasized and elaborated especially by Wilfrid Sellars in his discussion of theories

34  :  and models.2 Resemblance may consist in having properties in common, or instead in having properties that have properties in common with relevant properties in what is represented. That is, the representor may have properties which form a structure resembling a structure formed by the properties of the represented, and so on up the hierarchy of types. We should honor these distinct categories with distinct names. Representation in general has under it the special case of representation that trades for its success on some (speciﬁc) resemblance, or on multiple resemblances. For this special case I will use the terms image, imaging, and imagery—which in turn has under it the sort of imaging that manages to represent in part by trading on some visually detectable resemblance—visual imagery.3 When Galileo introduced his primary qualities as the properties to be solely considered as primitive in scientiﬁc description, he both narrowed and extrapolated from this base of visually detectable resemblances. The list that became more or less standard in his century comprised just the quantities of spatial and temporal extension and their combinations—hence of space, time, and motion, the kinematic quantities—but were not qualiﬁed by the limitations of human perception. So let us set imaging which manages to represent in part by trading on resemblances with respect to kinematic quantities side by side with visual imagery. Call it kinematic imagery. Trading on resemblance is a very broad category, so we may come across other sorts still besides visual and kinematic imagery. These as well may admit of both simple and ‘higher level’ (structural) varieties. Overlapping these categories of representation that trade on selective resemblances lies still a further salient case, which shares some crucial features found in visual perspective, a development which in art we associate speciﬁcally with Renaissance painting. Perspective involves (as we shall explore further below) such features as occlusion, marginal distortion, texture-fading. For cases of imagery in which such features of perspective are present I’ll use the terms picture and picturing—these can include cases of kinematic and visual but perhaps also still further forms of imagery. Obviously none of these categories are hereby precisely deﬁned, we will have to explore all of them further. But there are two caveats we must emphasize. The ﬁrst, already noted several times now, relates to the level of resemblance, and the second to the reality or non-reality of what is represented.

, ,  

35

Resemblance, as I said, can be higher order: the spatial structure of a set of letters on a page may be the same as the temporal structure of a set of events named by those letters. The use of visual or kinematic imagery to depict things that are not visual or kinematic is rife, and not excluded by our notion of imagery. For the resemblance of some structure to a visually recognizable structure may be precisely on the level of structure, not on the level of features that only visible things can have. This point may threaten to trivialize the notion of imagery. For it does not take much by way of intellectual gymnastics to ﬁnd some minimal relevant resemblance in any two things at all. But the threat does not seem to me very serious, for actualizing it would quickly turn into something no one could really take seriously.4 Does the word ‘‘animal’’ resemble anything that is not a word, does it resemble anything having to do with animals? Perhaps so, but it will not be anything that plays a role in the representation of animals by that word. The Cratylus’ attempt to see verbal representation as hinging on resemblances of not just words but syllables and even letters to what is represented, I take to be a choice bit of Socratic irony, that reduces the entire idea to absurdity. Second caveat: a representation may be of something non-existent, non-actual. So a representation that trades on resemblance may in fact not resemble any real thing, any person living or dead, or actually occurring event. Le D´ejeuner sur l’herbe is immediately seen as a picture of a picnic because of a certain kind of resemblance to real scenes of the picnic-kind, but it is still a painting of something that did not happen.5 The Judgment of Paris, whether by Rubens, Renoir, or as Nazi propaganda by Ivo Saliger, trades for its success on resemblances. Not resemblance to Paris or to the goddesses, who may not exist, nor even to any real scene that ever happened, but on resemblances to human judgments, bodies, clothes, and the settings of such human scenes. So to call something an image, a visual image, or a picture will not imply anything about the reality of what it depicts.6 This is a point obviously of some importance to an empiricist view of scientiﬁc modeling as representation. That an image trades on resemblance, on any level, does not imply that it resembles what it represents, nor that there is something that it resembles, nor even that there exists something that it represents. Fundamental to the understanding of representation in all contexts is this fact, that images which represent something unreal have their importance, their role, their effect in the context in which they function.

36  : 

What distinguishes a picture? At ﬁrst blush, it is easy to see what is in a picture—such as that familiar one, of a picnic, in curiously dishabill´e condition. What is in Le D´ejeuner sur l’herbe? A picnic. But how does that distinguish the picture from a verbal expression or icon? After all, what does the phrase Le D´ejeuner sur l’herbe describe? A picnic. Despite the sameness in answer to these questions, there is a difference. As I have regimented the terms, above, picturing is a case of imagery, that is, representation that trades on resemblance, distinguished by bearing hallmarks of perspective. So it is the latter we should now try to isolate. One popular response is that a picture can’t help but be speciﬁc. A sentence that says that George is in the garden need have no information about whether he is standing, sitting, or lying down. But a picture of George in the garden, it is said, can’t very well be neutral on that. Dominic Lopes calls this a ‘‘myth of speciﬁcity’’.7 In fact a picture need not be determinate in any particular way. A picture of a written word might not be of any speciﬁc word; the word might be illegible in the picture. A picture of George in the garden might just show his hand emerging out of some shrubbery, without revealing his posture. A few brush strokes can sufﬁce to depict George as in the garden, in tears, or in love. But there are two ways in which pictures are peculiar in the way they represent. Both ways have to do with what is accessible to vision. The ﬁrst relates to resemblance, the second does not. First of all, as Dominic Lopes emphasizes: pictures are unlike other sorts of representation precisely because there is a way in which they do literally ‘‘look like’’ what they depict. Pictures are physical objects; among their physical properties there is a privileged set of visual properties, those by which pictures represent their subjects—such as line, shading, color, and visible texture. But note how this point is to be qualiﬁed. The line, shading, color, and texture in the picture may not match, and in fact generally must not match, those of the represented scene. This is essentially the same point as the Eleatic Stranger’s about large statues on high pedestals: to create a life-likeness, to create the right appearance, distortion is needed. That applies to color as well as shape or proportion. The technique of chiaroscuro was accordingly developed in the Renaissance along with perspective. This

, ,  

37

is only the beginning of the systematic mis-matching necessary in a picture to make it ‘look like’ the original. In addition, as we have seen, on many counts the resemblances and non-resemblances can easily give wrong clues both to what is depicted and what is attributed to it. A picture can misrepresent its subject, attributing properties to it that it does not have, whether by accident or on purpose. To that extent the idea that pictures represent because they ‘‘just look like’’ their subjects is indeed a mistake. All representation is selective and the selectivity is crucial to what is depicted, but for pictures, the selection is subject to very speciﬁc constraints. The selection is not a choice simply to render some aspects and be non-committal as to others. That sort of choice is also made when we describe something in words. The crucial difference—here we come to the second distinguishing point—appears when we notice that in picturing, the selection of one aspect may force the picture not to include certain others. There is ﬁrst of all occlusion, which is closely related to perspective: to depict a situation from a given point entails that some objects will be in front of, and hence hide, certain other objects. Thus revealing what things are like from one angle is incompatible with simultaneously revealing the values of certain other parameters. This point has been exploited well in technical description of painting. But the topic has also been explored as crucial to machine design and drawing (cf. Rothbart 2003: 242–4). John Hyman introduced the terms ‘‘occlusion shape’’ and ‘‘relative occlusion sizes’’: an occlusion shape of an object is its outline, relative to a line of sight.8 It is the shape of an object as seen from a particular point of view. The occlusion size of an object is the area that the object occludes from view from a particular viewpoint. It depends on the actual size of the object and its distance from the observer. As the observer moves, the occlusion shapes and the occlusion sizes of the objects around him change. But this is not something subjective. These terms concern the shapes and sizes that are projected from a particular viewpoint on a plane perpendicular to the line of sight. As Hyman points out, we may be mistaken about them and our mistake ‘‘can be corrected by measurement and geometrical calculation’’. Dominic Lopes develops this notion so as to distinguish picturing from other modes of representation. Revealing what things are like from one angle is incompatible with simultaneously revealing the values of certain other parameters. Besides ‘‘occlusion’’ there are also:

38  :  Grain: more distant objects, textures are not as ﬁnely depicted as near; and in fact there is a minimum to the ﬁne-grainedness of a given picture. Angle: even with multiple views, there is a limit to the number of angles and distances from which an object can be depicted Marginal distortion: this derives from the limitation of the ‘view’, which can take in only a circumscribed range, and picturing reaches its limits of reproduction near the limits of that range. Marginal distortion is clearest in the pinhole camera picture—that ordinary photos do not show this, is precisely because the camera, like the Sophist’s sculpture of large statues, adjusts the ‘copy’ so as to create a more ‘faithful’ appearance. Angle and marginal distortion are topics that belong under the heading of speciﬁcally spatial perspective, and we will discuss that further below. These characteristics are, for example, the ones drawn on in Ronald Giere’s account of astronomical observation, to argue that what is gained from the instruments is perspectival.9 All four of the characteristics mentioned are plausibly grouped under the heading of perspectivity, but we cannot hope for an explicit deﬁnition here. The notion of perspective is undoubtedly what Wittgenstein called a cluster-concept, so that no speciﬁc set of characteristics is both necessary and sufﬁcient for its application.10 Lopes introduces an important further characteristic to the cluster. He adapts here some terminology from Ned Block and Daniel Dennett.11 A representation is committal with respect to some property F if it represents its subject as having that property, also if it represents it as not having that property, and not otherwise. If it simply does not go into the matter of whether the subject has that property, then it is inexplicitly non-committal with respect to F. But ﬁnally, a representation is explicitly non-committal with regard to this property if it represents its subject as having some property (or properties) that preclude the representation from being committal in that respect. It is crucial to the notion of picturing that being committal in one respect will preclude being committal in some other respect—in the sense that it will force being explicitly non-committal. This point elaborates on perspectivity: perspectives can be ‘‘ ‘aufgehoben’ in a higher unity’’, if I may use an expression from a much earlier time, but they cannot be simply combined as parts of a third perspective.

, ,  

39

Does this allow us to differentiate picturing from describing? Given a particular style of representation, it may not be possible to add more to a picture and still let it remain a single picture as opposed to a gallery. But what about collage? And what about cubism? Or ‘impossible’ pictures, like Escher’s? Lopes maintains, and this seems correct, that although different styles differ in precisely what they make possible in this respect, the same point about the explicitly non-committal will apply mutatis mutandis. There is still in each case a choice to represent the subject as thus or so, and this precludes representing it as having certain other properties, which could have been selected for depiction in another picture in the same style.12 What is still not obvious, in the catalogue of characteristics we have now gone through, is that they capture the notion of perspective. When we think of a picture as being drawn from one point of view (the location of the eye and direction of vision), we are attending to its alternatives: thinking of it as set in a ‘horizon’ of other perspectives on the same objects. Occlusion is connected with this only if we have a sense that it can be varied so that other objects come to light, other objects are occluded. Similarly with what the picture is non-committal about: this is connected with perspective only if we can imagine a shift in what is excluded and included. It would be a mistake to concentrate on what is actually in a picture, taken by itself, if we want to say what it is for an image to be perspectival.13 Thus, as to the hallmarks of perspective: the characteristics listed will not sufﬁce if applied ‘piecemeal’. The content of the picture must be related to a ‘horizon’ of alternatives that we can think of as coming from ‘different points of view’, if these characteristics are to count as marks of perspectivity—and the explicit or implicit reference to such a horizons of alternatives is what is most important in the concept of perspectivity.14

Mathematical imagery, distortion through abstraction Visual imagery and kinematic imagery are so-called because of the category of features with respect to which they trade on selective resemblance. Mathematical imagery, on the other hand, is so-called ﬁrst of all because it is imagery—i.e. representation trading on selective resemblance—and

40  :  secondly because the representor is a mathematical object. While there is no implication therein of perspectivity, mathematical imaging too involves in general necessary or inevitable distortion, in both simple and subtle ways. Of course the story is apocryphal, that a professional gambler funded a mathematician to analyze horse-racing, and was thoroughly unhappy with the report which began ‘‘Let each horse be a perfect sphere, rolling along a Euclidean straight line . . .’’. But is that so far from real examples of mathematical modeling? Consider the example in dimensional analysis used to model the motion of a cloud of small, electrically charged oil droplets in air under the inﬂuence of an electric and a gravitational ﬁeld—reminiscent of Millikan’s famous experiment to measure the charge of the electron—which begins with the simplifying assumptions • the oil in the droplets is in thermodynamic equilibrium with the oil vapor in the air, and no further evaporation or condensation occurs • the oil droplets are so small that surface tension effects dominate the distortion of the droplets by the forces acting on it • the electric charge is distributed with spherical symmetry over each droplet and so forth, such as that the droplets’ acceleration is negligible. . . . If we are to understand mathematical modeling in general, we had better see such simplifying assumptions—granted to be most likely false in fact—as the norm. The question, though, is whether this is just a matter of human limitations, inessential to mathematical modeling as such, or whether distortion of any sort is inevitable in principle. Mathematical statuary It will have been obvious that the criteria for distinguishing pictures from other visual representations do not imply two-dimensionality. The word ‘‘picture’’ no doubt connotes, in common usage, representation on a plane surface. But a statue, as we saw at once in the Sophist example, is subject to the sort of distortion practiced in painting to produce a visually faithful image. There is necessary occlusion and the statue is explicitly non-committal with respect to certain features that we don’t even think about in the case of painting. That is easily seen when we compare, say, the Venus of Milo with the Anatomical Venus in Florence’s Museo della Specola.

, ,  

41

If we are to bring these concepts to the study of scientiﬁc representation we must look to how they can be applied more generally beyond painting, drawing, photography, holography, or sculpture. Descartes’s analytic geometry, Newton’s and Leibniz’s differential and integral calculus, and the subsequent developments in descriptive geometry and analysis provide, on an abstract level, resources for representation so perfect that they tend to engender oblivion to the distortions on which they trade—and oblivion as well to the necessary sacriﬁces of perfection in practice. To counteract this, let us begin with a Cartesian dream of abstract perfection, and then consider how abstraction itself blurs the real. When Galileo said that the Book of Nature is written in the language of mathematics, he was referring to geometry and geometric ﬁgures; shortly afterward Descartes founded analytic geometry in which these ﬁgures can be equally represented by numerical functions. That was an enormous step, in which our spatially structured world came to be represented algebraically—and one might equally say that almost all its qualities were so to speak ‘spatialized’. Let me illustrate this by introducing the notion of a mathematical statue.15 Here is a man, we’ll call him Kurtz, who stands at the precise intersection of the Equator and the Greenwich meridian. He is of a certain height, no more than 2 meters from head to toe. Our task: to construct a statue of this man, as accurately as possible with respect to size and shape. No plaster concoction will do him great justice. Let us deﬁne a function K as follows. Its domain is the set of triples of real numbers x,y,z. The value of K equals always 1 or 0, with this condition: K(x,y,z) = 1 if and only if Kurtz’s body occupies a region including latitude x, longitude y and distance z meters from the center of the Earth. This function has value 1 on a region that precisely ﬁts Kurtz’s body. In analytic geometry, this function describes a solid, three-dimensional ﬁgure, and indeed, within analytic geometry there is not much difference between ﬁgure and function. I offer this to you as a statue, invisible to be sure, but more accurate with respect to size and shape than any plaster or bronze could be. Such mathematical statues are the objects on which the new scientists of the modern era practiced their craft. Of course they are much more

42  :  versatile than I have indicated yet.16 Thermodynamic study lets in a fourth parameter: K(x,y,z,T) = 1 if and only if Kurtz’s body occupies a region including latitude x, longitude y and distance z meters from the center of the Earth, and T is the temperature in degrees Kelvin at that point. No Earthly museum contains a statue with this internal temperature correspondence to Mr. Kurtz! Kinematics lets in still more: K(x,y,z,T,t) = 1 if and only if at time t, Kurtz’s body occupies a region including latitude x, longitude y and distance z meters from the center of the Earth and T is the temperature in degrees Kelvin at that point. With time come trajectories; with Newton we add in masses and forces. Now the mathematical statuary can be thought of as ﬁgures in, or functions deﬁned on, higher dimensional spaces—conﬁguration spaces and phase spaces. . . . Trouble at the interface Of course we were idealizing! Who would think that there is an objective, sharp division between the geometric points inside and those outside a human body? But this idealizing ﬁction is the be-all and end-all of the seventeenthcentury geometric representation of nature, continued in the next several centuries of rational mechanics.17 Yet it also leads inevitably to its own limits, where retrenchment from the idealization becomes imperative. Indeed, this process in which deliberate idealization brings us to its own limits, and thereby deﬁnes a new problematic for the scientist, imparts a new impetus for scientiﬁc progress. As illustration let us look back to geometric optics in its simplest form.18 There light is treated as a set of rays, emanating from a source, propagating through transparent media according to three simple principles: • the law of rectilinear propagation, that light rays propagating through a homogeneous transparent medium propagate in straight lines • the law of reﬂection, which governs the rebound of light rays from reﬂecting surfaces

, ,  

43

• the law of refraction, concerning the behavior of light rays as they traverse a sharp boundary between two different transparent media (e.g., air and glass). Hero of Alexandria established the law of reﬂection on the basis of a principle of economy in nature: that light will always follow the shortest path as it moves from surface to surface.19 Proposition . If the light is unobstructed, it will travel in a straight line.20 Proposition . (Law of reﬂection) If light is reﬂected from a surface (such as water or a mirror), the angle of reﬂection will equal the angle of incidence. By convention, the angle of incidence is taken to be the angle between the ray and the normal, i.e. the line perpendicular to the surface at the point of incidence; similarly for the angle of reﬂection. Hero’s principle of economy, in which economy of action is identiﬁed as following the shortest path, is ﬁne when the light is traveling through the same homogeneous medium all the way. When the ray travels through different media, say air and glass or air and water, it will follow the shortest path in each but change direction when it moves from one into another. This refraction depends on the density of the media. Suppose light strikes a water/air surface at an angle. Again we draw the normal, i.e. the perpendicular line at the point of incidence. The empirically ascertained ﬁndings were: The light entering the denser medium is refracted toward the normal; if entering from the denser medium into the less dense, it is refracted away from the normal, to the same extent. Ptolemy, in the second century  treated this phenomenon systematically, but his work was lost to the Middle Ages until it became available, via Arab scholars, in the twelfth century. The Arab mathematician Alhazen also discussed refraction systematically and stated the above ca. 1100. The quantitative description we are about to present was found in the seventeenth century. Let R be chosen so as to be as far from the point of incidence as Q. Draw perpendiculars from Q and R to the surface (assumed ﬂat), to meet that surface in Q and R .

44  :  Proposition . The ratio Q P:PR is a constant, independent of the location of P, and depending only on the nature of the two media—their Refractive Index. This Proposition is called Snell’s (or Snel’s) law after its discoverer in the seventeenth century.21

Figure 2.1. Reﬂection and refraction

But now let us look at a range of phenomena that lies just where reﬂection and refraction compete. (Today it is easy to see this illustrated on the internet by computer simulation.22 ) Light is refracted if it strikes the surface at a shallow enough angle; but it is reﬂected, if it arrives at a sufﬁciently steep angle. What happens when a light source is moved so as to change the angle of incidence? Precisely where does the one phenomenon end, or the other begin? In the diagram, let QP equal PR, but consider various values for the angle QPQ . Let Q move downward toward the water surface; the angle of refraction away from the normal becomes larger, R moves up and to the left in our diagram. But noticing that the distance Q P is always larger than R P, what happens when Q and Q coincide? What happens then as the source moves still further? The only answer within this theory is that the light is at some point (at the critical angle) no longer refracted but completely reﬂected,

, ,  

45

and that when that happens there is suddenly a big jump to a distance below the surface—a singularity, a discontinuity in nature! Perhaps even a contradiction in the theory given the traditional principle accepted, at the time when Willebrord Snel formulated his law, that ‘nature makes no leaps’. Is there really such an enormous discontinuity in nature at this point? At the level of observation open to a swimmer or ﬁsh, this phenomenon can be found. But extrapolation to what happens ‘in the small’ is not valid; this is just the point where the model gives out, where the idealization reaches its limit of admissibility. The phenomena for which geometric optics works do include the more easily observable ones studied early in the history of optics. But there is no inﬁnitely thin precise demarcation between water and air, and in any case, ignoring the wave character of light will only yield adequate results even for the observable phenomena in a limited range. Inﬁnitely perfectible idealization? The assumption of continuity in all natural processes is no longer in force. That does not nullify the lesson illustrated by the above ‘trouble at the interface’, however. Agreed, we cannot demonstrate that in principle, as a matter of logic, mathematical modeling must inevitably be a distortion of what is modeled, although models actually constructed cannot have the perfection reachable in principle. But on the other hand, the conviction that perfect modeling is possible in principle—what Paul Teller calls the ‘‘perfect model model’’—does not have an a priori justiﬁcation either!23 One conviction which supports the ‘perfect model model’ is that however vague our ordinary language is, there is absolutely no vagueness in mathematics. This support too loses its plausibility, however, if we look not just to pure mathematics but to assertions made by way of application. There the fascinatingly creative changes in mathematics itself belie the idea. Since the scientiﬁc description of the world is couched throughout in mathematical language, we can put it this way: the scientiﬁc image itself harbors vagueness and ambiguity, at each historical moment of its development—but this only comes to light in retrospect. Consider this beer glass on the table: each has a shape. What that shape is, precisely, we do not know. In the heady early days of the mathematization of our world picture it could be assumed that this shape is [described by] a precise function of the spatial coordinates. The edge of this table could be thought of as a straight line, hence [described by] a function

46  :  of form y = ax + b. But of course, the edge of a table is not perfectly straight. . . . If eventually the table, the beer glass, and their environment are re-conceived as assemblies of classical particles, they still occupy precise regions of space. These regions can be similarly represented by functions on the spatial coordinates. It may be a bit arbitrary exactly which particle assembly is the glass at any given time—but upon any such arbitrary, admissible choice, the table and glass have a deﬁnite shape.24 So now these objects can be represented by a mathematical model in the same way but more accurately than before —though now deﬁnitely only ‘in principle’, not in practice! The shape is accurately modeled by a suitable mathematical function; what that function is, we do not know, but there must be such a function. We are speaking here of the continuum of classical mathematics which has equal use for the representation of each primary quality: length, duration, shape, size, number, mass, velocity, what have you. The equation of the primary quality shape with geometric shape is in effect the assertion that a certain representation is completely adequate. But now we must ask: what exactly is this representation? Not only the question as to what shape the glass has, but that question is continually answered differently. In the nineteenth century, mathematics developed to the point where it was sensible to ask: is this shape an analytic function?25 Or is it only ‘smooth’, i.e. inﬁnitely differentiable? There is no question but that, as a reconstruction of the world picture of Galileo, Descartes, and Newton, we can choose either option. They had not said that every physical magnitude in nature is an analytic function, but they had not conceived of any alternative. Nothing would have been lost from the subject as developed at that time if we thought of the functions then discussed in this way, the functions describing the primary qualities of real physical things, as all analytic—nor if we thought of them as not necessarily analytic. Nor is there any kind of experimental evidence to cite in favor of one or the other option. The description was open, indeterminate in that respect. To see how far such nineteenth-century questions are beyond those that arose in the seventeenth century, reﬂect on what Descartes created when he created analytic geometry. When Pascal took issue with Descartes, it was because he felt the need for the existence of points given only as limits of inﬁnite sequences, while Descartes was willing, within mathematics, to countenance only ﬁnitary constructions.26 By the time the mathematicians

, ,  

47

could ask whether all functions are analytic or even continuous, and could contemplate negative answers thereto, that controversy was long past. To go even further, after the development of measure theory Birkhoff and von Neumann pointed out that when classical mechanics solves problems about systems with given precise conﬁgurations, we can construe it as using conveniently simpliﬁed descriptions. More realistic, they suggested, would be the description that results if we transform the precise descriptions by identifying regions that differ only by sets of measure zero.27 Their reasons for thinking of that as more realistic may or may not be cogent, but it sufﬁces here to note the conceptual possibility. That is, after the time of Lebesgue we can look back to the older description of nature and we have the new option for how to conceive mathematically of the shapes of things.28 You will realize that I am simply giving examples of how, in many ways, we must in retrospect look upon the scientiﬁc image inherited from the older generation as open, vague, ambiguous in the light of our new understanding (that is: in the light of alternatives not previously conceived). What is the shape of this beer glass really? What was it in the Galilean, Cartesian, Newtonian scientiﬁc image? In each case the presupposition that it was one item in a certain class gives way to (i) the conditional that this was so if that shape was correctly represented by some item in that class, and (ii) the realization that there are other candidates. As it is for shape, so it is for each primary quality, represented by the mathematics of the continuum. Indeed, we need to cast our net more widely still, if we want to ﬁnd all the ways in which we could now understand the scientiﬁc image fashioned in the seventeenth century. There is no such thing as the classical continuum, if that is meant to be the continuum on which the classical (= modern, 17th century) scientiﬁc image was erected originally. Cantor, Brouwer, and Weyl had equal right to regard it as erected on their continua, which are very different. Of course, today we will use ‘‘the classical continuum’’ to refer to the subject of real number theory as it now exists in main stream mathematics. That is the politics of linguistic usage. But even in what we now call classical mathematics, recall that we have the option of saying that quotient constructions are more accurate (and the simple use of real numbers merely a convenient artiﬁce), as Birkhoff and von Neumann suggested. What would you like the shape of the beer glass to be?29

48  :  So, what is the shape of the beer glass in the scientiﬁc image? What is meant by the assertion that its shape is one of the surfaces in the mathematical representation of nature? The openness of scientiﬁc description here come to light is irremediable. Of course, every time we outline a range of alternatives for ourselves, we can ascend our private throne—are we not all kings and pontiffs in realms of the mind?—and assert that one of these alternatives is the one true story of the world. When the range of alternatives is reﬁned by new conceptual developments—or simply by having our attention drawn upward by logical reﬂection—we can choose a new option and make yet another declaration ex cathedra. Arbitrary perhaps, but as deﬁnite as can be, by choice. What we cannot pretend is to be non-arbitrary, or to close our text once and for all. Yet the form of understanding is always one of presumed objectivity and univocity. The scientiﬁc image is as replete with uncashed and ultimately uncashable promissory notes as the manifest image. Any practical context brings its own standards of appropriate precision, so it is neither proper nor practical to keep this open-endedness constantly salient, but to acknowledge that is not to deny it. Distortion by statistical abstraction (Simpson’s paradox) Abstraction just removes some factors or parameters, and leaves the relations among the remainder intact, isn’t that so? Just think of a set of premises: remove some, and the implications of the remainder, taken by itself, is precisely what it was before the removal. Think of a color photo; remove the color so that it becomes a photo in black and white: all features that are independent of color, such as relative size and angles, remain the same. So why ever think that abstraction can distort? Simpson’s paradox in statistics gives the lie to this rhetorical question. Here is an example: the civil servants in a given city claimed that lighting and ventilation were seriously affecting their well-being and productivity. The city hired a statistician who showed conclusively by means of sampling that the productivity among workers in ill-lit and ill-ventilated spaces was no less than that among workers in general (or in better lit, better ventilated spaces)—the productivity level was the same in both groups. So the complaint was concluded to be baseless. Eventually a new study was done, and the second statistician broke the data down by looking separately at women and at men. She showed clearly

, ,  

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that among women, the productivity was less for workers in ill-lit and ill-ventilated spaces than elsewhere. She also showed that among men, the productivity was less for workers in ill-lit and ill-ventilated spaces than elsewhere! So relevance of working conditions did not show up until there was a subdivision by this third factor (gender). How is this possible? That is precisely Simpson’s paradox: correlations can be washed out, or on the other hand brought to light, by averaging in different ways. Here is the solution to the puzzle: under all conditions the women were more productive than men working under the same conditions, but the women were predominantly working in poor conditions.30 The ﬁrst statistician abstracted from gender, and by this very means produced a ‘picture’ which was misleading, and even conveyed a falsehood. The appearance for him, that is, the outcome in his measurement set-up, has to be assessed as ‘‘how it looks in that set-up’’ or ‘‘relative to that set-up’’. His abstracting from gender would have been ﬁne if that factor had been irrelevant to level of productivity under various conditions. On the other hand, we say now that it was relevant only because at a lesser level of abstraction in this respect, the correlation is different. (Mind you, there is no guarantee that subdividing further, by other features besides gender, won’t undo the correlation again!) In this particular case we can say that by abstraction he produced a distorted picture of the reality, a picture in which lighting and ventilation were irrelevant to productivity. Abstraction is harmless only under very strict conditions of pertinence.

Scale models and virtuous distortion The nearest to three-dimensional pictorial representation in use in science is surely the scale model31 —can it be conceived of in the terms proper to picturing? As we will see, scaling is not simple reduction or increase in size in all dimensions. In that sense, useful scaling trades not just on the obvious resemblances in shape but on distortion, both resemblance and non-resemblance being selective in a way dictated by the purpose at hand. The scaling cannot be ‘proportionate’ in all respects. The pertinent question is whether there will be a sufﬁciently accurate resemblance in all relevant respects for the purpose at hand.

50  :  A presumption that this is always possible, and that the relevance will be transparently perceivable, has had a strong grip on the physical imagination. The idea of testing a hypothesis at a different scale tends to be immediately convincing. Think for example of the experiment proposed by Galileo concerning buoyancy. The conventional wisdom, which he disputed, was that a ship will ride higher in the water in the open sea than in port, due to the amount of water below it. This is difﬁcult to test directly because of the choppy waves on the high seas. So Galileo proposed to place a small vessel in a shallow tank and load it with lead pellets until the addition of just one more pellet would make it sink—and then to repeat this procedure in a much larger body of (quiet) water.32 Doesn’t this proposal design a deﬁnitive test of the hypothesis?33 To take a more critical attitude, we must recognize that for any concept there are boundary cases where guidance by our usage so far dwindles and eventually gives out. Scale modeling displays the characteristics of picturing, by relying on selective resemblance to achieve its aim, but in a way that is subject to inevitable occlusion or distortion. Scaling as picturing A scale model represents, and yields information about what it is a model of, by selective resemblance. Are there such necessary limitations in that case, analogous to occlusion, marginal distortion; is scaling explicitly non-committal? Consider a scale model of an airplane. It has the same shape overall, but with the proportions reduced by a multiplicative factor, say 0.0001. Will it ﬂy? Not necessarily. If it is to ﬂy, to mention just one factor, it must have something to propel it; but its size limits necessarily what that can be. For example, the relation to air resistance will be quite different at this scale: the air, after all, has not been similarly scaled down in any way! There are other reasons, as we will see below—‘the same shape’ is a deceptively simple concept.34 Scale models can be produced for the sheer aesthetic pleasure of it, but more typically they serve in studies meant to design the very things of which they are meant to be the scaled down versions. This use and its subtleties were brought out clearly in the Second Day of Galileo’s Two New Sciences.35 His calculations involved an error, but his principles were

, ,  

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correct. In modern terms we can summarize his conclusions easily for a cylindrical beam with constant density. Its strength decreases with its cross-sectional area, which is proportional to the square of its radius. But the mass is proportional to its volume, that is, to the cube of its radius. So the strength to mass ratio of such beams with the same density becomes N times less when the beam’s size is increased by a factor of N. Beyond a certain point, the mass can no longer be supported, and the structure collapses under its own weight. As Galileo observes, large ships taken out of water are in danger of breaking for just that reason, and he gives examples for optimal bodily structure: . . . nor can nature produce trees of extraordinary size because the branches would break down under their own weight; so also it would be impossible to build up the bony structures of men, horses, or other animals so as to hold together and perform their normal functions if these animals were to be increased enormously in height. . .

We can observe conversely that if a reduced structure is to remain feasibly like its original, some other features besides its size must be scaled as well, and not proportionately but appropriately.36 Principles of Similitude and Approximation Roughly speaking, a scale model of X is an object which is structurally similar to X but suitably smaller. ‘‘Similar’’ and ‘‘structurally’’ have their usual context dependence as much as does ‘‘smaller’’—in any particular case, the goal implicit in ‘‘suitable’’ will determine the contextual parameters for each. And still roughly speaking: there are two assumptions in force when conclusions about the target are drawn from characteristics of a scale model. The ﬁrst is that structurally similar objects will display the same behavior in structurally similar circumstances. This was glamorized by Richard Tolman in 1914 as his Principle of Similitude, more accurately called a principle of dimensional homogeneity, rather poetically expressed on a cosmic scale: The fundamental entities out of which the physical universe is constructed are of such a nature that from them a miniature universe could be constructed exactly similar in every respect to the present universe. (Tolman 1914, 1915)

52  :  So phrased it is a thesis in ontology; on the methodological side it could perhaps correspond to something like ‘‘All laws of physics are to be, and all measurable effects are to be conceived of as, invariant under scale transformations of any kind’’. Amazingly this principle, which occasioned a good deal of response in the literature at the time, appears here in this evangelical form decades after the advent of Planck’s quantum, also, almost ten years after Einstein’s study of the photoelectric effect, and several years after Bohr’s model of the atom. By this time it is certainly surprising to see the conviction that scale is essentially irrelevant to physical modeling. But Tolman is trying to capture a correct principle of scale invariance, though one that needs considerably more sophisticated formulation. If we doubt Tolman’s principle, then inferences from scale models are just inferences from false assumptions. But there are useful ﬁctions! The second principle in force is a Principle of Approximation. The centrality of this idea in applied science was highlighted by Reichenbach.37 Think of how Newton proceeds to deduce the laws of motion for our solar system. Keeping his basic laws of mechanics as foundation, he adds the law of gravitation to describe this universe, then he adds that there is one sun and six planets to describe our solar system, and ﬁnally he adds that there is one moon to describe the more immediate gravitational forces on our planet Earth. Newton demonstrates that from a very idealized, simpliﬁed description of the solar system, something approximating the known phenomena follows. Well, so what? What’s the point of deriving true conclusions from a false premise? This is the precise point where we notice this deep assumption at work: if certain conditions follow from the ideal case, then approximately those conditions will follow from an approximation to the ideal case. Mutatis mutandis, this assumption is in force when we have recourse to any model that we do not presume to be more than approximately similar to what it represents, but especially in the case of laboratory simulations. However, we cannot take this blithely as a context-independent methodological principle, given its phrasing with this sort of generality. The approximation to the ideal case, just as well as the similarity of a scale model to its original, is similarity in certain respects, with other aspects ignored as irrelevant for all practical purposes—which means, for the purposes at hand

, ,  

53

in the context of application. Whether that ignoring will be vindicated is an empirical question; we can’t very well decide it on principle. In fact, Reichenbach shows by illustration in statistical mechanics that even small departures in approximation can have widely divergent consequences—a point now popularly familiar from Chaos theory. Fine, but the study of scale models, and in general studies that seem to be inspired by these rough and ready ‘principles’ of similarity and approximation, are often useful, practical, and truly vindicated. So what are the facts of the matter, the real constraints on such modeling? Dimensions and invariance We can glean these from the critiques of Tolman published in a seminal paper by E. Buckingham and a critical article by Percy W. Bridgman, the physicist famous for originating operationalism as a philosophy of science.38 The applications that Tolman outlined for his principle, both authors argue to be derivable in dimensional analysis, a technique whose development started with Fourier. I’ll explain some of the basic ideas and then display the example of the screw propeller, which Buckingham analyzed in detail. Fourier had extended the geometrical notion of dimension to the now familiar general concept of physical dimensions, so that not just length, area, and volume but also mass, force, temperature, charge and the like are included: [E]very undetermined magnitude or constant has one dimension proper to itself, and . . . the terms of one and the same equation could not be compared, if they had not the same exponent of dimension. We have introduced this consideration into the theory of heat, in order to make our deﬁnitions more exact, and to serve to verify the analysis; it is derived from primary notions on quantities; for which reason, in geometry and mechanics, it is the equivalent of the fundamental lemmas which the Greeks have left us without proof. In the analytical theory of heat, every equation (E) expresses a necessary relation between the existing magnitudes [length] x, [time] t, [temperature] v, [capacity for heat] c, [surface conducibility] h, [speciﬁc conducibility] K. This relation depends in no respect on the choice of the unit of length, which from its very nature is contingent, that is to say, if we took a different unit to measure the linear dimensions, the equation (E) would still be the same.39

This same passage, which introduces the general conception, also introduces the idea of ‘‘dimensional homogeneity’’ and the importance of invariance

54  :  under scale transformations for the fundamental equations of physical theory.40 An equation must be dimensionally homogeneous to make sense: the dimension of the quantity on the left must be the same as that on the right. That is just the common place that you can’t add apples and oranges except in the sense that you can take them both as fruits and count them that way. In more complicated cases, this homogeneity has to be checked. Take the equation of the distance covered s calculated from time t, velocity v, and acceleration a: s = vt + (1/2)at2 Does that make sense? Here distance has the dimension of length, call it L; velocity has the dimension of length divided by time (T) and acceleration the dimension that has velocity divided by time. We must ﬁrst check that on the right-hand side we are not trying to add apples and oranges, but rather things of the same sort. We check this by replacing each of the parameters by its dimension alone, while multiplying replacements of the terms with each other: (L/T)T (L/T)(1/T). T.T and then treat those dimensions algebraically the same as numbers. The result, after cancelling the Ts against each other, is L in both cases. Thus we are adding two like quantities, and the quantity denoted by the right side of the equation has dimension L. But the dimension of s is also L, so we have the required match. Without detailed scrutiny this may look like a calculation by rules of thumb far removed from rigor, but I will leave the detailed justiﬁcation of dimensional analysis techniques to other sources. The second requirement, recall, is that of invariance under scale transformations. To achieve invariance under such transformations—rewriting equations stated in terms of certain quantities in terms of others—is precisely served in dimensional analysis by the search for ‘‘dimensionless’’ quantities as sole constituents for the fundamental equations.41 As illustration we can begin with the familiar cgs system of units in mechanics: centimeter for length, gram for mass, and second for time. A different scale belonging to the same class of systems of units is one deﬁned by multiplying each unit by a positive number. This is the form of a scale

, ,  

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transformation. If we choose the numbers 100, 1000, 1 for this role, we deﬁne the MKS system, with the units meter, kilogram, second—thus producing another system belonging to the same class of systems of units. The basic invariance requirement is now that to be signiﬁcant, an equation must have the same form regardless of which member of the class of systems of units is chosen. This requirement was obviously respected well before Fourier, let alone before the subject of dimensionless analysis matured. Newton’s famous F ≈ ma does not depend on its validity on a particular choice of units, and would not be famous if it did.42 A dimensionless number —more accurately speaking, a dimensionless parameter —of the class is a quantity that has the same value in every system of units in the class. That is, it is an invariant of the set of admissible transformations, which are precisely the scale transformations. The Hauptsatz of dimensionless analysis, prominent in Buckingham’s article, says in part that it is always possible to shift to a dimensionless representation.43 The screw propeller Susan Sterrett points out the relevance of how the Wright brothers and their colleagues in the ﬁeld were frustrated when they tried to extrapolate the behavior of children’s ﬂying toys to a larger scale. Making the object ‘the same but larger’ ruined its capacity to ﬂy—why? Wasn’t the toy a scale model for their construction? The fact is that here, as much as in the examples the Sophist pointed to, what counts as pertinent resemblance is not at all obvious in the way that ‘looking alike’ is obvious. Buckingham, partly in service of his critique of Tolman, analyzed the case in detail. In the studies of the screw propeller, which had of course been started for ships but were then crucial to the development of the airplane, both rough and ready ‘principles’ can be seen in a rigorous form. The thrust F of the propeller of given shape and immersion is taken to depend only on the diameter D, the speed of advance S, the number N of revolutions per unit time, the density and viscosity of the liquid, and the acceleration due to gravity. So suppose that a smaller propeller is meant to be a good scale model of a large one with respect to thrust —in contrast with some other effects, here regarded as ignorable side effects (noise, shape of the wake, . . .). Then the equation which expresses F in terms of those other quantities must be the same for both cases, provided that the ratios that specify the shape and

56  :  immersion of the propeller stay the same. Any set of kinds of quantity that furnish the basic units for this dynamics can be changed in any ratios whatsoever without affecting this. How does this serve to guide a practical study? The obvious thing to do is to make the smaller propeller geometrically similar to the original, to immerse it similarly, and to construct the propellers so that the ‘angle of attack’ of the blades on the water is the same. Is that enough to ensure that we can get information about the thrust of the original large propeller from the behavior of its scale model? It sufﬁces only if one can completely control similarity in the effects of gravity, density, and viscosity. It is easy to see that for extremely small propellers those effects will be signiﬁcant, and for larger but still small ones the difference is after all only a matter of degree. So in practice . . . one has to resort to some approximation. And here special conditions can be experimentally investigated to see what can and what cannot be ignored at various scales. For example, the pertinent mechanical behavior in very turbulent motion does not vary much with the viscosity of the ﬂuid. And similarity in the effects of gravity will be approached when the ratio between the two speeds of advance approximates the ratio of the squares of the two diameters. Deep immersion will also prevent signiﬁcant effects due to disturbance of the liquid surface. (Notice though that these are all matters of degree, and the purpose at hand may require a speciﬁc level of accuracy.) With all of this supposed under sufﬁcient control, the ratio of the thrust of the small propeller to that of the large one will be (DS/D S )2 , where the primes indicate the diameter and speed of advance of the large propeller, and not (DS/D S ), that is, not the ratio by which this product was altered in construction.

Conclusion about imaging and scaling Imaging, recall, is representation that is effected through resemblance. Our discussion of mathematical statuary ended with the conclusion that we can see that mathematical representation of nature so far always involved some features that, in retrospect, with hindsight, we saw as necessary failures in resemblance. That does not imply that mathematical representations of the sort now available are also thus necessarily deﬁcient, though some humility

, ,  

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in even that respect is appropriate. But in application, the practice of actual construction of models of situations, the idea of ‘perfect’ modeling is so far from realistic that it can certainly not be maintained. The resemblance of even so ‘obvious’ an example of a scale model to its original is, as we have just seen, not nearly as simple as may strike the eye at ﬁrst glance. True, a scale model of a vessel or propeller under study in a naval or aeronautics laboratory will ‘look like’ a real one. But for it to have any use at all for the purpose at hand, there must be a delicately achieved pertinent similarity (to a pertinent degree of approximation) between the situation of these propellers revolving ‘similarly’ when ‘similarly’ immersed. This may well, and typically does, come at the price of dissimilarity: as Galileo already appreciated, the scaling must be different for different parameters. The ‘‘selective’’ in ‘‘selective resemblance’’ is a delicate, highly nuanced, contextually sensitive qualiﬁcation—and this point is general: it pertains to all pictorial representation.

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3 Pictorial Perspective and the Indexical Imagery, as deﬁned above, is pictorial exactly if it bears hallmarks of perspectivity. In the notion of perspective, as so often, we have a cluster concept, with multiple criterial hallmarks. There is no deﬁning common set of characteristics, only family resemblances among the instances. Whether or not something is aptly called perspectival depends on whether some appropriate subset of these hallmarks are present, but what amounts to ‘‘appropriate’’ we cannot delimit precisely either. The hallmarks listed above were occlusion, marginal distortion, texturefading (grain), angle, and with special importance, explicit non-commitment and the ‘‘horizon of alternatives’’. These are all characteristics that relate to the content of the representation. But there is another notion closely connected to perspective which does not appear here. This does not pertain simply to content, but to how we relate to it; it comes to light when we very naturally think that for the painter or photographer, a picture is showing how the pictured scene ‘‘looks from here’’. The painter’s eye is located with respect to the content of the painting in a way that he himself can express with ‘‘this is how it looks to me from here’’. The viewer may naturally say that this is how the scene ‘‘looks from there’’. If I say, for example, that the photo shows the town as seen from the top of the church tower, that indicates something like ‘‘that is how it would look to me if I were on top of the church tower’’. These are indexical statements, with the words ‘‘here’’ and ‘‘there’’ playing the context-sensitive role. For a critic describing a painting this may not be relevant. While s/he may refer to the painter’s view, no special interest may attach either the painter’s subjective situation or what the critic’s own would be. But that

60  :  changes when we turn to representations subject to different norms and use. If we look to the painting or photograph to help us get around in the town, that does require us to locate ourselves with respect to the view presented by that picture. Since scientiﬁc representations are typically produced so as to serve some such practical end, at least in principle, this connection between perspective and the indexical becomes important there. The connection shows up in the sciences, for example, when talk of frames of reference is conducted in terms of observers (whose frames they are, so to speak). We need to look closely into both the character of perspective and the role of indexical judgments (such as self-attributions and self-locations) to see whether that is just an irrelevant heuristic or whether it brings to fore a fundamental connection between perspective, measurement, and theoretical representation.

Pictorial Perspective and the Art of Measuring la pittura e` una specie de natural ﬁlosoﬁa, perch´e l’imita la quantit`a e qualit`a, la forma e virt`u delle cose naturali.1

The histories of perspective in painting, measurement, geometry and technology are thoroughly entangled.2 Geometry is so-called because it began as the art of ‘earth-measuring’, and in Dutch its name is still ‘‘the art of measuring’’ (meetkunde). But as we’ll see, a famous treatise on perspective was called by that name as well. Examining some of this history it will become quite clear that a picture in the modern perspectival style is essentially the outcome of a measuring procedure. Conversely, a measurement outcome is, in paradigm cases at least, a pictorial representation of the object measured. This paradigm example I will extrapolate subsequently: measurement falls squarely under the heading of representation, and measurement outcomes are at a certain stage to be conceived of as trading on selective resemblances in just the way that perspectival picturing does. Astrolabe and triangle How do you measure the width of a river while remaining on the shore? Or the height of a tower while remaining on the ground?

    

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The ﬁrst clue to the answer is of course that the two problems are essentially the same, related by a simple rotation from horizontal to vertical. The second clue is also geometrical: in a right triangle, the ratio of height to base is determined by the ‘‘angle of sight’’ along the hypotenuse. At this point today’s student reaches for trigonometry, but these practical problems were solved long before that was available. By the end of the Roman Empire in the fourth century much of Greek mathematics was lost to the West, not to return there from Arabic sources until almost 800 years later. During these centuries practical surveying, architecture, and the scholarly study of practical geometry did continue, however. The practical techniques of the Roman surveyors survived, were preserved, collated, and taught among both artisans and cleric-scholars.3 A representative text, the practical geometry manual of Hugh of St. Victor, in the century just before Euclid’s Elements became available again, is divided into three parts: altimetry, planimetry, and cosmimetry (measurement relating to the earth, to the sun, and to other aspects of the cosmos). In retrospect we see the methods there presented as justiﬁable within geometry and geometric optics, but what is taught there is simply the practical technology of measurement. The instruments designed for this use—cross staff, quadrant, and even the astrolabe introduced into the West about a century before this manual’s date—consist basically of a ruler with a sighting device (alidade) at the center, and part of a circle on which degrees are marked. These had signiﬁcant use in navigation, but let us here concentrate on land-measurement.4 The surveyor measuring the height of a tower, for example, adjusts the alidade until he can see the top through the two apertures. The angle thus formed determines the ratio of the height of the tower to the distance from the tower. Determines how? Though most of the mathematic theory was lacking, the astrolabe can be manually calibrated on relatively small similar triangles. This presumes understanding of geometric similarity; sufﬁciently much of Euclidean geometry was retained to understand this. The distance from the tower may itself not be measurable directly if it is far away, so Hugh’s manual gives several forms of ‘two station’ methods to use. Suppose that the astrolabe sighting is done at two points P and Q, at an unknown distance from the tower. Measure the distance between P and Q, and a few practical steps, starting with the two alidade readings, will yield the height of the tower:

62  : 

Figure 3.1. Perspective Altimetry

Let h be the height of the tower. The direct measurement by astrolabe at P gives the ﬁrst ratio A = h/PT. The reading at Q gives the second ratio B = h/QT. Only the distance PQ is measured directly. From these three data, the height h can be calculated directly.5 We will soon see this same conﬁguration again. Alberti’s De Pictura When Alberti wrote his monograph on the technique of perspectival drawing in 1435 a great deal of Greek mathematics and geometric optics had been assimilated in the years since Hugh’s manual of practical geometry. But his way of writing was not so different from Hugh’s, because Alberti was also a surveyor applying those practical arts as well as thinking about the theory behind them.6 Since the practical geometry manuals focused on geometric ﬁgures created by physical objects and lines of sight, they were an obvious source for his study of perspective. His great innovation was to think of the visual cone or pyramid cut by the picture plane: [Painters] should understand that, when they draw lines around a surface, and ﬁll the parts they have drawn with colours, their sole object is the representation on this one surface of many different surfaces, just as though this surface which they colour were so transparent and like glass, that the visual pyramid passed right through it. . . . (Alberti 1991: Book I, 48)

    

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Art historian S. Y. Edgerton refers to Alberti’s invention as ‘‘Windows 1435’’. A painter drawing from life is as it were drawing on a window through which he is seeing his subject. In fact of course, the plane on which he draws (the canvas) is not the plane (imaginary window pane) cutting his visual pyramid. But he will succeed in accurately rendering his subject if what he produces is precisely what it would be if he did draw on that ‘window’ plane. There was an elementary exercise for this skill whose sign we see in many paintings of that era: the checkerboard ﬂoor or pavement. Francesco Rosselli’s Supplice de Savonarola (c. 1498), for example, depicts the central square in Florence precisely with such a checkerboard pavement seen in one-point perspective. The following illustrations show respectively how the painter is imagined to see and paint, and what it is that he sees as it will appear on the picture plane:

Figure 3.2. Window and Checkerboard

Notice how the left-hand diagram is really just the one above of the ‘‘two station’’ altimetry, ﬂipped horizontally, but with the picture plane and some other sightlines added. The painter’s eye corresponds in the geometry to the top of the tower of the earlier illustration. So the point of view is the opposite, as it were, but the geometry involved is the same. Alberti’s concern was with technology. He began his development of perspectival drawing techniques by making a box with a small eye-hole in one side.7 In the box there was a checker board laid horizontally on the bottom. Let us call what the checkerboard looked like, if viewed through the peephole, the checkerboard appearance. (That is to say, the box ﬂoor’s appearance in observation or measurement made through the

64  :  peephole.) Studying this set-up in various ways he could make a drawing which, if placed upright in the box at a certain point, presented the same checkerboard appearance to the eye. In fact, viewers could not distinguish between the two when looking through the peephole.8 Looking back at the second illustration you can see that parallel lines orthogonal to the picture plane converge to a point on the horizon, while lines in the other two orthogonal directions remain in place. So the lines between the tiles that are parallel to the picture plane remain parallel to each other; similarly the up-down lines remain vertical. This is one-point linear perspective. In two-point perspective parallelism is preserved in only one of the three directions; but this is hard to ﬁnd in the history of painting till much later. We can think of one-point perspective as the style of representation that captures what is seen in the Alberti experimental situation—the unmoving single eye at the peephole. We can equally think of it as the style in which of three orthogonal directions in space, two are preserved in effect in a grid of parallel lines, while in the third direction all straight lines converge to a point. Masaccio’s Trinity, Botticelli’s Episodes in the Life of Lucretia and Episodes in the Life of Virginia, as well as the philosophers’ favorite, Raphael’s School of Athens, can be mentioned as examples of this style of visual representation. The Art of Measuring: mechanization of perspective Alberti’s technique was applied by contemporary painters, though not nearly as rigorously as either Alberti or Vasari would have it. On the other hand, it was of use also to architecture, technical drawing, and machine design. Alberti’s geometric and practical studies of perspective include ways to ‘mechanize’ the process of perspectival drawing: So attention should be devoted to circumscription; and to do this well, I believe nothing more convenient can be found than the veil . . . whose usage I was the ﬁrst to discover. It is like this: a veil loosely woven of ﬁne thread . . . divided up by thicker threads into as many parallel square sections as you like, and stretched on a frame. I set this up between the eye and the object to be represented, so that the visual pyramid passes through the loose weave of the veil. (Alberti 1991, Book II, 65)

His ‘‘veil’’was precisely the painterly window we have been discussing, but now realized as a practical technological artifact. The veil with its grid

    

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is a measuring instrument, designed to measure not such simple quantities as length or weight but—as I shall discuss further below—cross ratios, projective structure. This is explicitly recognized in Albrecht Dürer’s treatise, where the technique is presented in a part entitled Unterweysung der Messung—‘‘Teaching of Measurement’’, generally translated as ‘‘Art of Measurement’’. The mathematically precise and practical character of this way of rendering the appearances implied its possible mechanization.9 The basis was in effect a very careful and systematic form of measurement, in which certain geometric features are faithfully captured on the picture plane. This way of understanding the episode is supported by a look at the machines that Dürer designed to produce perspectivally correct drawings, which show how far this can go.10 In his most advanced artifact even the human element consists of fully determined mechanical motions. Measurement is precisely what this was, as basis for pictorial representation. The content of such a visual perspective is the content of a complex, technically advanced measurement outcome.

Figure 3.3. Dürer, the Draughtsmen of the Lute

66  :  The heart of an experiment is typically a sort of measurement: the set-up produces or lends itself to a phenomenon that is meant to provide information about the character of some target object, event or process. The artiﬁcially produced or isolated phenomenon is treated as providing data about the target, to provide us with a ‘view’ of it. Dürer’s machines do exactly that, in elementary fashion: they produce drawings that provide us with a view of the object from a given vantage point.

Perspective versus Descartes’s frames of reference Alberti knew that his method worked, in the sense that it produced a realistic view of three dimensional objects. What he didn’t know was why it worked. [Projective geometry] presents the key mathematical concept behind Alberti’s method, the cross ratio.11

To understand perspective mathematically we have to go on to seventeenthcentury geometry. But there another style of representation, more familiar to the scientist, was developed as well: Descartes’s analytic geometry which coordinatizes space with a generalized rectangular frame. We must take care to distinguish these, for perspective and perspectival painting are often drawn on as metaphors for the new ‘spatialized’ or ‘geometrized’ representation of states and motions introduced in the seventeenth century. At ﬁrst blush that ﬁts: description of a ship or its motion in a geometric frame of reference relates that structure to an origin and an orientation in space. But the origin of such a frame is nothing like the position of the eye in perspective, and the directions in space are there only as a conventional orientation, introduced to provide a user-friendly description. So that metaphor, however appealing and pervasive, needs to be kept at arm’s length. Illustrations sufﬁce to show how different Descartes’ representation is from that of a painter, even in structure. Both drawings depict a cube. But in the ﬁrst, in none of the three directions do parallel lines converge. Parallel lines are depicted there as parallel, and equal line segments are depicted as equal. For this representation pertains to Euclidean geometry with its characteristic symmetries. A coordinate system is indeed a speciﬁc

    

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Figure 3.4. Frame of Reference . Perspective

representation of the space in the manifold R3 of triples of real numbers. But coordinate transformations, moving us from one system into another, preserve both distances and parallelism. The move from one visual perspective into another does not! Despite the common features and despite the attraction of the metaphor, depiction in a Cartesian frame of reference is clearly not literally a perspectival representation of anything. My drawings do both utilize projection. When I drew the cube in the Cartesian frame I could not show it to you in its proper three dimensions, but had to project that on a plane surface as well, and in that respect our two ﬁgures are similar. In the ﬁrst drawing, projection is orthogonal. Orthogonal projections occur naturally as well: the sun’s rays are parallel (on our scale, within our modest degree of accuracy) so a shadow on a properly positioned wall is an orthogonal projection. Lines converging to an ever more distant vanishing point approach a set of parallel lines—thus in orthogonal projection we have as it were an eye ‘at inﬁnity’, but of course not at any particular point at inﬁnity. The difference between Cartesian frames of reference and visual perspectives comes out even more clearly if we allow for a ‘motion picture’. Galileo’s diagrams strike us still as faltering ﬁrst few steps. For example, in his study of falling bodies and of projectiles we ﬁnd separate diagrams for the time axis and for different spatial distance axes. Descartes’s geometry combines the spatial dimensions and links the geometric ﬁgures to algebraic equations in a way to us so familiar that we can readily read his treatise on the subject.

68  :  For an illustration, consider two cars rolling along a broad one-way road, one on the left and one on the right. The road consists of equal squares, and the two cars reach the end of each square simultaneously. We can imagine a spatial frame of reference that has the left border of the road as Y axis and the orthogonal border of the ﬁrst square as X-axis. The two cars are moving in parallel, with the same velocity, and are thus depicted in a geometric representation of this situation. Now a painter sets up his easel at beginning of the road, and he has his eye precisely on the Y axis. He sees the two cars ready to start, the time is t = 0. The left and right borders of the road are the lines X = 0 and X = 1. Of course the painter sees these converging in the distance. To picture this, let us modify the diagram showing the checkerboard ﬂoor drawing. The line X = 0 is straight, orthogonal to his easel, but the line X = 1 slants in the Xnegative direction, meeting X = 0 on the horizon. The cars begin to move. He sees them reach each horizontal line simultaneously. But the right-hand car is moving along the hypotenuse of a perceived triangle, so covers a larger perceived distance than the left-hand car in the same time interval.

Figure 3.5. Speed in Perspective

Within the painter’s perspective, the right hand car is moving faster than the car on the left. If he were making a motion picture, or simply taking notes of where in his visual ﬁeld the cars are at t = 1, t = 2, etc. he would be making a measurement of the velocities, but the content of his measurement would be what the motions look like and not what they are. Knowing the geometry of this space and the laws of projection, he can of course draw on registered relations between the two perceived motions to

    

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obtain information about what the motions are really like. These motions are observable processes, they have determinate speeds and directions which are in fact the same, although the appearances (which are the contents of the measurement outcome) are different.12 The ‘view from nowhere’ and the indexical The above should be a caution against talk of ‘‘perspective’’ when discussing coordinates and frames of reference. But why then is such talk so pervasively prevalent? Perspective is a cluster concept, and while careful not to extend its use so broadly as to make it practically vacuous, we should also take care not to narrow our use so much as to lose touch with common discourse. In fact, I was ignoring the connection with the indexical, remarked on at the outset, and we should now bring it back in. First of all, it is true that despite its particularity due to an arbitrary choice of origin and direction, we can think of the Cartesian representation of extension, duration, and motion as embodying a ‘zero-point perspective’: the ‘view from nowhere’ or ‘the point of view of no one in particular,’ to use some famous phrases.13 That is precisely because the chosen frames of reference, the coordinate systems, are inessential to the geometry taken in and by itself. The Cartesian representation is God-like; or so at least Leibniz eventually depicted it: the distinction between the appearance bodies have with respect to us and with respect to God is, in a certain way, like that between a drawing in perspective and a ground plan. For there are different drawings in perspective, depending upon the position of the viewer, while a ground plan or geometrical representation is unique. Indeed, God sees things exactly as they are in accordance with geometrical truth, although he also knows how everything appears to everything else, and so he eminently contains in himself all other appearances.14

But this Leibnizian God has no need of representations either, nor of anything else crucial to us, ﬁnite and situated observers. The coordinate system plays no role for God, and plays no role in geometry conceived in pure abstraction—but what role does it play for us? We can see the shift back and forth between these two ways of ‘seeing’ the world in how theories are presented in physics. In his 1905 paper that introduced the Special Theory of Relativity Einstein wrote:

70  :  Examples . . . lead to the conjecture that . . . the same laws of electrodynamics and optics will be valid for all coordinate systems in which the equations of mechanics hold. . . . We will raise this conjecture (whose content will hereafter be called ‘‘the principle of relativity’’) to the status of a postulate and shall also introduce another postulate, which is only seemingly incompatible with it, namely that light always propagates in empty space with a deﬁnite velocity V that is independent of the state of motion of the emitting body. These two postulates sufﬁce for the attainment of a simple and consistent electrodynamics of moving bodies based on Maxwell’s theory for bodies at rest. (Einstein 1905/2005: 124)

This passage concentrates on the ‘‘objective’’, that is on what is invariant, valid for all frames of reference—which makes reference to frames of reference inessential. But Einstein’s exposition thereafter takes on a different form: If we wish to describe the motion of a material point, we give the values of its coordinates as functions of the time. Now we must bear carefully in mind that a mathematical description of this kind has no physical meaning unless we are quite clear as to what we understand by ‘‘time’’. We have to take into account that all our judgments in which time plays a part are always judgments of simultaneous events. If, for instance, I say, ‘‘That train arrives here at 7 o’clock,’’ I mean something like this: ‘‘The pointing of the small hand of my watch to 7 and the arrival of the train are simultaneous events.’’ (Ibid.)

This shift from the ‘God’s eye view’ to a frame of reference identiﬁed with an observer equipped with clock and measuring stick continues in the well-known thought experiment he then presents for the measurement of a moving rigid rod. We now inquire about the length of the moving rod, which we imagine to be ascertained by the following two operations: (a) The observer moves together with the aforementioned measuring rod and the rigid rod to be measured, and measures the length of the rod by laying out the measuring rod in the same way as if the rod to be measured, the observer, and the measuring rod were all at rest. (b) Using clocks at rest and synchronous in the rest system . . . , the observer determines at which points of the rest system the beginning and end of the rod to be measured are located at some given time t. The distance between these two points, measured with the rod used

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before—but now at rest—is also a length that we can call the ‘‘length of the rod.’’ (Ibid., 128) Not long after Einstein’s creation of this new theory, Minkowski recognized that Einstein had in effect displayed an elegant new mathematical entity, distinct from those utilized in classical models. Light paths are to be represented by curves in this space along which the space-time interval equals zero . . . motions of bodies by paths on which the points have time-like separation . . . and so forth. That is to say, according to the new theory, we are to use this mathematical object in that way to represent the natural phenomena in this domain. At this point observers and frames of reference are left behind. Neither perception nor individual cognition is a salient topic of inquiry in the context of use of Minkowski space for the representation of rigid motion, electric current, magnetic ﬁeld, and transmission of light. Frames of reference can be thought of as attached to any material body or to none; certainly talk of relatively moving observers is de trop. But if we focus on the theoretical models in and by themselves, we are ignoring the use they have. If someone is to use Einstein’s theory to predict the behavior of electrically charged bodies in motion, bodies with which s/he is directly concerned, choice of a coordinate system correlated to a deﬁned physical frame of reference is required. The user must leave the God-like reﬂections on the structure of space-time behind in order to apply the implications of those reﬂections to his or her actual situation. The physical world picture in abstracto is as far removed as possible from this use, it embodies, in Eddington’s words, ‘‘the view of no one in particular’’. But to put this picture to use, something must be done by the user, and this is where choice of reference frame comes in. Hence Weyl’s words are equally apt, when he refers to coordinate systems as ‘‘the unavoidable residuum of the ego’s annihilation’’.15 In sum then, the use of ‘‘perspective’’ and ‘‘perspectival’’ in connection with depictions of events in varying frames of reference cannot be banished completely. There is a close connection between them and the acts of observers locating themselves with respect to the theoretical model—acts of self-location, expressible by the actor in such terms as

72  :  ‘‘I am here, and this is how it looks from here’’. I will come back to this after we have looked at some more general examples of use and self-location. Geometric projection and perspective The Cartesian style of scientiﬁc representation is meant to be of the phenomena as they are, and not as they appear to the observer. But how the phenomena appear in observation or in measurement—their appearances—must also be covered.16 There too, all that is crucial to the actual information captured is in the invariants. What is invariant in the move from one visual perspective to another? The subject of projection and perspectivity belongs now to projective geometry, which became an autonomous subject only in the nineteenth century. But to the extent relevant to our present discussion it was developed far enough within Euclidean geometry in the seventeenth century, notably by Pascal and Desargues.17 We can ask the question ‘‘What is invariant in the move from one visual perspective to another?’’ in a different way, if we take for granted that the invariants are informative about the represented landscape. How, or to what extent, is perspectival drawing mimetic? To what extent, or in what fashion, is the drawing a ‘copy’ of what is drawn? Alberti had already noted that the ‘visual pyramid’ is replete with similar triangles, so that certain proportions are preserved as the objects are imaged on the painterly ‘window’.18 But this pertains to proportions on lines parallel to that ‘window’, and does not get us all that far. There are relations however that are invariant under projection, and will therefore be the same in all perspectives on an object regardless of the position of the eye (or camera). The main projective invariant is the cross ratio, a four-term metric relationship. It is not actually an easy relationship to grasp for the visual imagination! Consider a sequence of four points A, B, C, D on a line, in that order.19 Let us call A and B the major points and C, D the minor points. Then the cross ratio CROSS(A, B; C, D) is the ratio of two distance ratios, namely of the minor points to the major ones: (CA/CB) : (DA/DB).20 Here we have the correct generalization of the checkerboard pattern template that we saw above: the cross ratios of 4-tuples of points on the ‘windows’ and on the ‘ground’, that correspond to each other by the projection from the ‘eye’, are the same. In the checkerboard ﬂoor, the

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Figure 3.6. Cross Ratio Invariance

distances between adjacent points are equal, so the cross ratio is 4/3—a special case that is easily visualized and reproduced by the painter.21 Take any point P and draw lines connecting P with the four points in question. It can be quickly established that the cross ratio is the same for all projections of those four points onto other straight lines.22 In the diagram, where point P now plays the role of ‘eye’, CA/CB : DA/DB = CA /Cb : dA /db, and similarly for other sections of the pencil of projection lines. This picture we will revert to below to show how taking a photograph can be used to make a distance measurement. The invariant content of a perspectival image is the structure ‘common’ to the images produced on any ‘window’ that cuts the pencil of projection lines. That ‘common’ structure is the structure entirely captured in a description of the cross ratios between such sequences of collinear points, in addition to the basic relationships of incidence and order of points and lines. It is this invariant content that carries the information in the perspectival image that is independent of the choice of origin and orientation—except of course that no one ‘window’ can contain an image of more than a ﬁnite part of a selected half of the space ‘on its other side’. Measuring with perspective Alberti’s ‘demonstrations’ as well as the famous display by Brunelleschi of his painting of the Florence Baptistery, had, as Feyerabend pointed out, all the characteristics of a scientiﬁc ‘demonstration experiment’.23 What they demonstrated was that the new techniques in painting correctly and

74  :  accurately ‘latched onto’ certain aspects of geometric structure. The rough and ready way to say this is ‘‘they got the proportions right’’, thinking ﬁrst of all of the proportions in the human body, and secondly in architectural plans. But ‘‘proportion’’ is neither speciﬁc enough nor general enough, as we have just seen. We can’t equate the preservation of cross ratios with preserving proportions—though that is what it is in simple cases. What is true is that the spatial structure is being correctly and accurately ascertained in certain respects—and this sort of ascertaining is precisely what measurement is all about.24 When Dürer called his seminal work on perspective Unterweysung der Messung, that was not idiosyncrasy but insight. To begin its exploitation, let us see precisely how the perspectival drawing is a measurement, revealing quantities in the way any measurement outcome is meant to do. Imagine you have such a drawing—or a pinhole camera photo—of a long straight road, on which you see a person as well as the mile 1 and mile 2 markers.25 How far is the person from the mile markers? We can use the above diagram. Let the line A, B, C, D be the drawing (or the photo, still in place in the camera, if you like), while that person and the markers are at A , B , and C on the line that represents the road. (The line A bCd is not relevant here.) The only qualiﬁcation to be entered concerning the cross ratio concerns inﬁnity. Imagine a fourth point D on our line A B C placed somewhere beyond C and then moved further and further outward. As it moves, the line PD begins to approach PD, the horizontal line that never reaches the ground. In the case of a painter looking through his window, the line PD is the one at eye level, going to the horizon. The mathematical conceit to be introduced at this point is that the two parallel lines ‘‘meet in a point at inﬁnity’’. No need to reify that ﬁction: we can remain with the familiar and much more elementary tactic of taking the limit.26 The degenerate cross ratio CR(A , B , C ) turns out, on this construal, to be just C A /C B . The invariance of the cross ratio under projection still applies: CR(A , B ; C ) so deﬁned is numerically equal to the original CR (A, B, C, D) from which it is this degenerate projection. The cross ratio of those initial four points therefore equals precisely the ratio C A /C B . But the denominator C B is just the distance between the two mile markers, i.e. one mile. The quantities in the cross ratio CR(A, B;

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C, D) on the other hand can be found by measuring them in the drawing or photo itself . Therefore the distance between the person at A and the second mile marker equals a determinable quantity in the produced image.27 In this fashion we shall thus have measured a distance on the road by means of that drawing or pinhole camera operation. The measurement outcome, which ‘indicates’ the value of that quantity is precisely the photo thus produced: The image = the outcome of a measurement, of a certain metric quantity, performed on the depicted situation. Conversely, the outcome of this measurement is the ﬁnal state of the ﬁlm which represents the object thus measured with the value of the measured quantity precisely indicated on its ‘face’. As in any typical measurement, our procedure here does not convert pure raw data into information in an assumption-free or presupposition-less fashion! That the horizon is ‘at eye level’, and that the distance between the two markers is precisely one mile, is assumed in our taking the photo to be representing the situation in the way that we do. In this simple case those assumptions can be independently checked by observation. In other cases they are themselves supplied by a theory, with no such easy access. As Bacon said, experience must be literate; seeing what the measurement outcome reveals requires being able to ‘read’ it. Reading is always reading under the aegis of our entire prior state of opinion plus current suppositions—in that sense, all our reading, even in the general sense of taking notice of measurement results of any sort, is always theory-laden.

Mapping and Perspectival Self-Location The development of the art of Perspectiva into new branches of geometry greatly favored the evolution of cartography. In maps we have scale models of terrain, but projected onto a plane, thus producing occlusion of a sort not inherent to three-dimensional imaging. Maps do not usually have an obvious perspective; but we see perspectivity when, for example, the curvature of the earth makes marginal distortion inevitable as a result of this projection that lowers the dimensionality. A map too is the product of a measuring procedure, but they bring to light a much more important point about ‘point of view’, essentially independent of these limitations in

76  :  cartography. The point extends to all varieties of modeling, but is made salient by the sense in which use enters the concept of ‘map’ from the beginning. A map is not only an object used to represent features of a terrain, it is an object for the use of the industrial designer, the navigator, and most of all the traveler, to plan and direct action. This brings us to an aspect of scientiﬁc representation not touched on so far, though implicit in the discussion of perspective, crucial to its overall understanding: its indexicality.28 Representations have their uses Use enters the very concept of representation, as we have seen: to be a representation is to be something used or taken to represent something. But in addition, representations have their uses. They are typically produced for a certain use, with a certain purpose or goal. An artist may paint, sculpt, or select an object to represent scenes of war for example, for display to the public, either simply for their appreciation or because s/he ‘has something to say’. A cartographer draws a map to represent the structure of the Eurasian railroad system, for use by its management for planning or alternatively for use by the public for getting around the continent. Bohr creates his 1913 model of the atom to explain certain spectrographic data. A map is close kin to a scale model; Rutherford’s atom was perhaps thought of as a scale model (with dimensions dilated rather than contracted); Bohr’s atom registered its resistance to being thusly viewed immediately. While there is great diversity in these examples of the use of representations, many have in common the goal to convey information. The activity of representation is successful in that case only if the recipients are able to receive that information through their ‘viewing’ of the representation. Our discussion of the propeller, for example, displayed a theoretical analysis; how can it guide real propeller construction? In science the original creation of a model may have been a purely theoretical activity, but eventually it provides input for an application, where conditional predictions made on the basis of that model feed into planning and action. What are the conditions of possibility for success in this use? Mapping: the cartographer’s art A map is a graphical representation of geographical or astronomical features, but this may range from a sketch of a subway system, to an interactive,

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zoomable, or animated map on a computer which constantly changes in front of the eyes.29 The familiar Mercator projection maps come theoretically under the same heading as perspectival drawing. A 3-D animation that gives the viewer the sensation of ﬂying over the surface of Venus does too, but at some considerable remove from the primitive form we looked at above. The word map’s application has of course spread by metaphorical and analogical extension. So in addition to the above clear cases there are representations of much more abstract landscapes, like the chromosome map of an animal species, the color map showing variations in hue, intensity, saturation, luminance, shade, and tint, or the site-map of a website to guide one to locations in virtual reality—not to mention the map of sociological theory,30 the human intelligence map,31 and the mediology-theory map.32 The general concept of a map is not so different from that of a model, though the one is extrapolated from a graph with spatial similarity to certain features of a landscape, and the other from a table-top contraption. But there is a signiﬁcant difference in the connotations about the use for which they are designed. A map is designed to help one get around in the landscape it depicts. What are the conditions, pertaining to map and user, of the possibility of such use? A meteorological model The cartographer’s art is as old as philosophy or science itself—but let us choose a contemporary instance that gives us maps with bells and whistles. The Aviation Model (AVN) for numerical weather prediction is not something to display on a table top, it is a computer program and what it produces as speciﬁc individual representations—in effect, weather maps—are displayed on a computer monitor. The AVN is one of the older national models in use by the US National Weather Service, developed mainly to aid in forecasting for aviation. Its resolution is about 100 km, and it tends to perform better than the other models in certain weather situations, such as a strong low pressure area near the East Coast of the USA. It has a forecast product, the Aviation Model Output Statistics, which provides detailed weather information in three hourly increments spanning the 48 hours after the Aviation Model is run. The forecast weather information includes temperature, dew point, wind speed and direction, cloud cover and obstructions to visibility, probability

78  :  of precipitation, precipitation form (rain, sleet, freezing rain, or snow), and probability of severe thunderstorms. Model output statistics are a blend of raw model output (data taken directly from the model grid) and statistical analyses of previous forecasts for the region. A forecast is ﬁrst taken from the AVN weather model, but that is then adjusted on the basis of equations that relate the forecast output of the AVN to what is actually observed at the forecast location. The result is to skew the model’s forecast output to make it more accurate for the location for which a forecast is being produced. The ﬁnal ‘‘skewed’’ product is the AVN MOS product presented as weather forecast. The AVN itself requires input to be run at all, of course: namely initial conditions and lateral boundary conditions obtained from operational weather centers in the relevant area. We can summarize what the model does in the following form: The model presents a space of possible states and their evolution over time—the input locates the weather forecaster in that space, at the outset of the forecasting process. Notice what the forecaster, if s/he is also its user, has to provide in order to create expectations that may be tangibly fulﬁlled: something to the effect of ‘‘I am now here in the space of possible states’’.33 Using maps: Kant’s point34 What we have just seen in the case of the AVN, the entry of an indexical when its product is put to use, is entirely analogous to what someone using any map has to do to make the map useful for ﬁnding his or her way around. What is crucial to any application is a judgment to the effect ‘‘I am now here on the map’’.35 Suppose for example that I am lost, and buy a map. Maps normally do not have an arrow labeled ‘‘You are here’’. But even if the map does have that, the problem is really the same: I have to locate where I am with respect to that arrow. Imagine I found this map lying in the gutter. Imagine instructions about the signiﬁcance of such labeled arrows: ‘‘If you stand in front of a map under condition C, then. . .’’. You still need to supply the indexical premise ‘‘I am in front of this map in condition C’’. The extra information needed by the user to use the map cannot be there, already encoded in that map. When I do have that extra information,

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I can express it by pointing to a spot on the map and saying ‘‘I am there’’—a self-ascription of location on the map. It is not as if there is an object or event that is indescribable, ineffable, beyond the reach of objective or impersonal description. This act of selfdescription too can be described and the information can be included on a bigger map (with the label ‘‘location of vF’s map-reading at time t’’). But then what I need to use this new map is still a self-ascription of location with respect to it. The problem of practical use has not been altered. With this new map, I can go on to self-ascribe a location by the different words ‘‘I am vF and it is now t’’. An attempt to replace or eliminate these self-ascriptions leads to an inﬁnite regress, using an inﬁnite series of maps. But even given the accuracy of the whole inﬁnite series of maps, the regress does not succeed in eliminating the need for self-ascription. For I will still be lost, unless I can locate myself with respect either to at least one of these maps or to the series as a whole, and this I can only do by asserting a self-ascription which is not deducible from the accuracy of those maps. Besides self-location in this narrow sense of ‘placing oneself ’, orientation with respect to direction is equally crucial. An example will immediately help to continue our discussion of measurement from magnitudes (distances) to less metric features. Imagine being in New York with a city map. You go on the subway; at one stop you check the station sign, and it says ‘‘18th Street’’. This allows you to point to a spot on the map, labeled as 18th Street station on the line. You exit and ﬁnd yourself on 18th Street, at the corner of 7th Avenue; but which way is West? Being unfamiliar with the buildings, and too far from the Hudson to see it, you walk one block along the street to see the name of the avenue that crosses it there. Is it 6th or 8th? Now you are oriented with respect to direction too. This involved two measurements: checking the station sign, or the initial street signs as you exited, located you in a spot on the map. Walking one block to take a new sighting was a second measurement which allowed you to orient yourself with respect to direction. Indeed, we can take that as our paradigm situation for how we can draw on science in action or practice. These measurements had as function to locate and orient you with respect to a ‘public’ representation of the object or situation of interest. That means: they were the practical, instrumental means for arriving at those crucial indexical judgments that

80  :  you needed to make use of the map. Although there are both simpler and more complex cases of measurement, this case is paradigmatic in what it reveals about measurement and the use of ‘public’—generally theoretical—representations. It was Kant who placed the inevitable indexicality of application center stage in thinking about how experience, understanding, and reason are related.36 His reﬂections on this subject came at a crucial stage in his philosophical development, in his early essay ‘‘Concerning the ultimate ground of the differentiation of directions in space’’: No matter how well I may know the order of the compass points, I can only determine directions by reference to them, if I am aware of whether this order runs from right to left or from left to right, and the most precise map of the heavens . . . would not enable me [without this orientation] to infer . . . on which side of the horizon I ought to look for the sunrise.37 . . . the most precise map of the heavens . . . would not enable me . . . to infer . . . on which side of the horizon I ought to expect the sun to rise if it did not, in addition to specifying the positions of the stars relative to each other, also specify the direction by reference to the position of the chart relative to my hands. (Ibid.)

There is here a precise and perfect analogy between theory, model, and map. The point about the map, made already by Kant, applies to a model of any sort. The activity of representation is successful only if the recipients are able to receive that information through their ‘viewing’ of the representation. But what are the conditions of possibility for this reception? The recipient must be in some pertinent sense able to relate him or herself, his or her current situation, to the representation. There is one objection to Kant’s point which cannot be simply dismissed. Might each location in the cosmos not have a uniquely deﬁning description in terms that are not in any way indexical or context-dependent? And if so, could that description not be everywhere substituted by a person at that location for the words ‘‘I, here, now’’? Could Kant’s thesis not be evaded in that way, at least in principle? This objection I will take up later, by connecting it with the attempts to develop a purely structuralist view of science by Russell and Carnap, and relate it to arguments such as ‘‘Putnam’s Paradox’’. For now we will look into the implications of Kant’s thesis at a less abstract level.

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GPS, The automated ‘self-locating’ map (?) Does Kant’s analysis still apply, now that we have Global Positioning Systems (GPS)? GPS satellites transmit signals, which the GPS receivers receive passively; the receivers do not transmit data at all—hence no data about their own location. Having such a receiver we obtain talking, self-locating ‘‘maps’’ that don’t need anything from us to help them. When installed in an automobile, the system is designed to show an instinctively recognizable display to drivers allowing them to see where they are in relation to surrounding topography. In advanced versions, the interface includes a bird’s-eye view of adjoining streets and buildings. So how does this work? The auto receives signals from several GPS satellites, with data about the satellite’s location and the time of emission. These signals, sent simultaneously by the satellites, arrive at the receiver at different times due to the difference in distance. From these differences the receiver’s location in the coordinate system deﬁned by the satellites can be determined. The local map coordinates are deﬁned in that coordinate system, and this map also shows the roads and buildings situated in the neighborhood of that location. Well, is anything missing? Yes: the driver has to make the judgment that the moving point on the map display is his own location. If s/he does not know how the system works, that is obvious. But even if the design is understood, a judgment tacitly intervenes; after all, only a slight malfunction in the computerized calculations could result in a signiﬁcant displacement. The case is not so different from a person standing in front of a map which has on it an arrow with the legend ‘‘You are here’’. That person has to make the judgment ‘‘I am now at the location indicated by that arrow’’. This judgment could very easily be false, for example, if the wrong map had been afﬁxed at that place. Suppose the GPS system makes the sound ‘‘Right turn, 500 feet’’. I have to supply the judgment that I am now in the right location with respect to the source of this sound—that it is not, for example, received as a radio signal from the GPS that is currently guiding my cousin in his new VW, sent to impress me. Imagine the automation of ship navigation along similar lines. In the old days the mate took his astrolabe and reported the elevation and compass direction for certain stars; then the captain took those data and the data from the chronometer to the maps on his table. There he plotted the data and arrived at a location on the map; on the basis of this he issued his

82  :  orders for the course to be steered. All of this except the very last step can be automated—the GPS can collect the data that will play the role of what mate and astrolabe provided before. But if the captain is to issue his orders, when he sees a cross appearing on the map depicted on his computer monitor, he has to add ‘‘we are now there, and the ship is currently moving in that direction’’. The situation has not changed in principle, just become less labor intensive. It will perhaps be objected that the captain’s role can also be automated, assigned to an ‘automatic pilot’, integrated with the GPS. But then, who is the user? The person who programs the automatic pilot with a course to steer, and then turns it on—in the full conﬁdence that the map point taken as input is precisely where s/he is at that moment. So again, the situation has not changed in principle for the user.

What is in a map? The short answer is ‘‘Nothing!’’. That is, if we take the physical object by itself, considered entirely without reference to use, to us, then there is nothing in it to determine its semantic content. But we can expand on this point. Even relative to the conventions in force in our community or society, for pictorial representation of terrain, there is some information—taking this in a broad sense—that cannot be in the map itself. So what is in a map? The topic of self-ascription belongs to pragmatics and not to semantics. That is a fancy way to say that what the self-ascription does cannot be equated with, but adds a crucial step to, the content of a map or to the bare impersonal belief that the map ‘‘ﬁts’’ the world. The bearing of this puzzle: we must generalize this to scientiﬁc representation per se. The body of ‘pure’, ‘fundamental’ science, in the sense of the totality of accepted scientiﬁc information, can in principle be written in coordinate free, context-independent form. That is possible for theoretical science, even if it includes the history of the universe or the evolution of biological species on earth. But to draw on this body of science in technology—whether in practical applications or even to test it or use it to explain something, or add to it through research—the scientist or scientiﬁc community must supply something extra, which does not come with that body of science, but serves to locate the user with respect to it.38

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Let me put this again, somewhat differently, in terms of models. ‘‘Model’’ is a metaphor, whose base is the simply constructed table-top model. We use this metaphor when we talk of cosmological models, Hilbert space models, and the like. We could have used the word ‘‘map’’, and made maps the base of our metaphor equally well. Suppose now that science gives us a model which putatively represents the world in full detail. Suppose even we believe that this is so. Suppose we regard ourselves as knowing that it is so. Then still, before we can go on to use that model, to make predictions and build bridges, we must locate ourselves with respect to that model. So apparently we need to have something in addition to what science has given us here. The extra is the self-ascription of location. Have we now landed in a dilemma for our view of science as paradigmatically objective? If we say that the self-ascription is a simple, objective statement of fact, then science is inevitably doomed to be objectively incomplete. If instead we say it is something irreducibly subjective, then we have also admitted a limit to objectivity, we have let subjectivity into science. But the threat we sense here takes its force from equivocation. Something more than what is contained in the printed map, physically constructed model, or computer monitor display is needed. But what is this ‘‘more’’? Not a mysteriously different sort of fact which cannot be encoded on a map! The scientiﬁc story can be complete in the sense of describing all the facts, including that someone does or does not have the ‘‘extra’’ needed for him or her to draw on a particular bit of science. It is just that describing the having of it is no substitute for the having! We will just have to admit a non-pejorative sense of ‘‘subjective’’, if the essential indexical has to be labeled as something subjective. Whatever sense we give it, we may be charged with giving a special role to consciousness in science. But that implication is there only in the premise that there is use of science. To the extent that use implies consciousness and agency, this premise does so too—what more need be said? Self-location in a wider sense We can broaden the concept of self-location. Suppose I see letters on a piece of paper. If I take them to constitute a text in my own language, I am locating myself with respect to what I have before me—to be contrasted with taking them as a text in another language. In very simple cases there

84  :  can easily be such a contrast: suppose an Italian speaker sees the inscription ‘‘burro’’. He may well take it as being in his own language, hence a sign for butter; but he could take it to be Spanish, to refer to a donkey. In a more complex case, Sherlock Holmes sees what is ostensibly a simple innocuous message in English but declares it to be a message in code having a quite different meaning. At ﬁrst blush, such inscriptions are far from maps, and what happens here is far from the use of a map to locate oneself in a landscape. But the mere understanding of the inscription as a text requires relating it to one’s own language—either through taking it to be an expression in one’s own language or through a translation procedure.39 Language may seem too special an example here: it is the seat of meaning if anything is. But to call, classify, something as a map or a model is to locate it in what Wilfrid Sellars called ‘‘the space of reasons’’—at least as this phrase is now broadly understood. By itself this is not yet self-location. It is just to classify the item as having semantic content, and as having a role in reasoned discourse and in practices subject to norms of rationality. We can to some extent separate our understanding of the item, in the sense of grasping its semantic content, from the understanding of our own situation that comes with locating oneself ‘in’ or ‘with respect to’ the item. But the latter comes in train, so to speak. This applies tellingly to any observation report made (as it usually is) in theory-laden language. If I say ‘‘Lo, phlogiston escaping!’’ I am locating my own situation in the logical space provided by phlogiston theory. If instead on the same occasion I say ‘‘Rapid oxidation happening!’’ I am locating my situation in a logical space provided by modern chemistry. If Lavoisier had heard someone shout the former, it would have been helplessly academic of him to reﬂect that he was hearing just another false statement. If you hear me say either of the above, you are well advised to ﬂee the room, if you take yourself to be with me, that is, if you are inclined to echo my self-locating act (even if you will echo it in a different-theory-laden form).

Visual perspective and the metaphor I have resisted the ‘‘perspective’’ terminology in this discussion of selflocation, all the way from the introduction of Kant’s point to Lavoisier’s

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predicament in the crowded theater where someone shouts ‘‘Lo, phlogiston escaping!’’. Perspectivity and indexicality are closely related, but the distinction is of equal importance. The terms ‘‘perspective’’ and ‘‘perspectival’’ are central to how we understand scientiﬁc representation. For this very reason it is all the more important to constrain their use, to keep their meaning from spreading too broadly, though equally important not to narrow them myopically. The most literal use pertains to the modes of visual perspective that became important with the innovations of Renaissance painting. What are its characteristics? First of all, a visual perspective has an origin—the painter’s eye in one-point perspective being the paradigm example. Secondly, it has an orientation: the direction in which this eye or the camera is looking. Thirdly, the content of this visual perspective is expressible in an indexical judgment, ‘‘that is how it is from here, looking in this direction, with that my left, this my right, yonder my above. . .’’. Fourthly, there is the systematic spatial distortion due to projection, so that lines orthogonal to the plane, that together with the eye deﬁnes the orientation, intersect. Fifthly, it is subject to the limitations of occlusion, marginal distortion, and degradation of the grain (coarsening of the threshold of distinction), and we take the content as belonging to a range of alternatives. The prevalent use of ‘‘perspective’’ for frames of reference corresponding to geometric coordinate systems can be misleading. But we must not object too much. This use is a ﬁrst step metaphorical extension only (in one direction, I would like to say, if only to illustrate the metaphor). But the geometric or physical frame of reference thus conceived is a depersonalization of visual perspective, relinquishing all but origin and orientation, which can be arbitrarily chosen. So I think it best to restrain even this extension of the term, or at least emphasize that it needs to be used with much care. Now we have come across another use that does not come directly under the heading of visual perspective. If I stand in front of a map and say ‘‘I am here, looking in this direction’’ (pointing to a spot and indicating a direction in the map) I am placing myself in a perspective within the depicted landscape. . . . To speak this way is to allow ourselves that term when the ﬁrst, second, and third features of visual perspective are there. We have therefore less reason to object to this use of ‘‘perspective’’ than in the previous case. Nevertheless it will be important to distinguish the literal visual perspective of eye or camera from each of them, and they from each other.

86  :  The question will come up a few times below, with either just origin and orientation or indexicality at issue, whether to spread these terms still further in application. For the most part I will resist that; the important relations between representation and perspective will be obscured if we allow much broader use. This despite the already current broad usage in everyday language, which can’t always be resisted without irritating the ear.40 The question is least simple when it comes to verbal description. When is a description perspectival? When it is a description from a certain point of view—but ‘‘point of view’’ is subject to the same ﬂuctuations in use as ‘‘perspective’’. A hallmark we can look for in the case of language is the occurrence of indexical terms or phrases. Among these I include demonstratives such as ‘‘this’’, ‘‘those yonder’’, as well as the more obvious ‘‘I’’, ‘‘you’’, ‘‘here’’, ‘‘now’’. But the indexicality is sometimes hidden, not apparent in the surface structure, so we do not have a perfect criterion here. On the other hand I would resist calling a description perspectival on the sole ground that some reference to the individual or communal users’ experience is indispensable to understanding its terms. That would take in even such ostensibly paradigm ‘impersonal’ examples as ‘‘for any body of gas, pressure will alter with temperature if volume is constant’’. In the actual construction and use of speciﬁc models in practice, we may discern a speciﬁc perspective. But on the other hand, general scientiﬁc theories, in their ‘ofﬁcial’ formulation, are not perspectival descriptions, and their models—if we consider the entire range of models for a given theory—are generally not perspectival representations.41 When it comes to terminology, let’s practice tolerance. Whenever it matters, we should refer to the speciﬁc characteristics themselves, not simply to the perspectivity of which they can serve as hallmarks. Restraint to the extent indicated here is especially preferable in a technical philosophical context, while at the same time it is easy enough to understand the wider usage so commonly found.

Concluding Empiricist Postscript From an empiricist point of view, the task that science sets itself is to represent the observable phenomena. What lessons can we draw now for the aspirant empiricist philosopher of science ?

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The idea of representing phenomena need not, and if practical purposes are kept in mind, must not be restricted to one of copying. But more than that, as we have seen, representation useful for particular purposes will involve selective distortion, and representation is closely involved with useful misrepresentation. Even when likeness is crucial to the purpose, we must look for likeness only in respects that serve the representation’s purpose—and only to the extent that they do so. Given that the aim of science is to provide empirically adequate theories about what the world is like, we should conclude that wherever the representation does trade on likeness, the general rule of selectivity targets the observable phenomena. A model often contains much that does not correspond to any observable feature in the domain. Then, from an empiricist point of view, the model’s structure must be taken to reveal structure in the observable phenomena, while the rest of the model must be serving that purpose indirectly. It may be practically as well as theoretically useful to think of the phenomena as embedded in a larger—and largely unobservable—structure. This we can quite easily illustrate with examples in the visual arts as well. In the sciences, perhaps the longest surviving example is the familiar Zodiac, with the stars arranged in constellations that are memorable as embeddings in such ﬁgures as the Ram, the Scales, the Fish (Aries, Libra, Pisces). In mathematical physics the illustrations are easily found, but much more abstruse, as one sees how useful it is to embed structures to be studied in larger spaces.42 Mnemonics—and perhaps realist philosophies—can then be served by pictorial glosses on those spaces. But there are two more points that spring to the eye here. First of all, a view of science would hardly be empiricist if it ignored measurement, which is science’s main initial access to the phenomena. Spatial measurement is explicitly perspectival, and its deliverances relate to scientiﬁc models precisely in the way that visual perspectives relate to physical space. Second, this turns into a general point: as Hermann Weyl put it graphically, there will be, even in the most theoretical sciences, an ‘‘ineliminable residue of the annihilation of the ego’’ to provide the conditions for relating the theoretical models to speciﬁc empirical situations. All the revolutionary developments in the theories of space and time as well as the upheavals in atomic physics testify to that. The former brought frames of reference to center stage, the latter engendered what is in fact precisely called ‘‘the

88  :  measurement problem’’ as fulcrum for philosophical interpretation. But thereby hangs a tale. . . .43 Secondly, a view of science would hardly be empiricist if it ignored the uses of science, as a resource for praxis. How are theories and models drawn on to communicate information about what things are like, to guide our expectations in practical affairs, to design instruments and technological devices, to ﬁnd our way around in the world ? Here especially enters the indispensability of indexical judgment, even in contexts where the term ‘‘perspective’’ may not be apt. In the end, as we shall see, that entry of the essential indexical will prove to be the crucial clue to how science can depict at all.

PA RT I I

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Windows, Engines, and Measurement Scientiﬁc theories represent how things are, doing so mainly by presenting a range of models as candidate representations of the phenomena. That is right, but it is a view of science, so to speak, ‘from above’; it is a view focusing on what is achieved when it has been achieved. There is a long journey from the initial encounter with nature to the achievement of an even temporarily stable representation. That journey is hugely oversimpliﬁed if described—as it used to be, and still sometimes is1 —as simple collection of data by observation or measurement followed by invention of theories to account for what was found. Following upon both recent and not so recent inquiries into scientiﬁc experimental practice, instrumentation, and measurement we can pursue the question of just how that task of representing the phenomena is achieved, and how it is to be understood. A measurement is at the same time a physical interaction and a meaningful information gathering process. In the examination of measurement in Part I I already announced the thesis that I mean to argue: measurement falls squarely under the heading of representation, and measurement outcomes are at a certain stage to be conceived of as trading on selective resemblances in just the way that perspectival picturing does. The words at a certain stage, perhaps puzzling in their ﬁrst occurrence, refer to the end of that long journey from the initial encounter with nature to the achievement of an even temporarily stable representation. We will look at this process both from within the journey itself and from above, that is, from the point of view achieved at its end. Perspectival drawing provided us with a paradigm example of measurement. The process of drawing produces a representation of the drawn object, which is selectively like that object; the likeness is at once at a rather high level of abstraction and yet springing to the eye. While the information

92   : , ,   about spatial conﬁguration is captured in an invariant relationship that is quite difﬁcult to formulate in words or equations, it is conveyed to us in a user-friendly fashion. The example is paradigmatic also in that it displays so clearly that the representation (the measurement outcome) shows not what the object is like ‘in itself’ but what it ‘looks like’ in that measurement set-up. The user of the utilized measurement instrumentation must express the outcome in a judgment of form ‘‘that is how it is from here’’. And ﬁnally, the coin has another side: it is precisely by a process engendering a judgment of that form—that is to say, by a measurement!—that any model becomes usable at all. But this paradigm example of measurement also hides much, precisely because of its beauty, simplicity, familiarity, and narrow scope. While its development was a great theoretical and technological advance in the history of our culture, it has been assimilated to such an extent that the advance has become all but invisible. For all forms of measurement, and all the roles it plays in relating theory to fact, familiarity breeds obliviousness. To understand what measurement is, we need to examine how systematic inquiry builds continually on what has been previously achieved, both in theory and in technological practice, and thus changing the conditions under which given interactions count as measurements at all, or as measurements of particular quantities. So that journey of inquiry, from initial steps to stable representation, is to be scrutinized. Nevertheless, this inquiry will be unproductive unless guided by the sciences’ categorical imperative, that is, the achievement of theories and models that do adequately, and sufﬁciently accurately, represent the phenomena under study.

4 A Window on the Invisible World (?)1 Constantijn Huygens was ecstatic when he had looked into Cornelis Drebbel’s microscope and spoke of a ‘‘new theater of nature, another world’’—he exhorted painters to ‘‘portray the most minute objects and insects [thus seen] with a ﬁner pencil, and then to compile these drawings into a book to be given the title of the New World’’.2 The microscope is perhaps the best example of an instrument as an aid to the senses. Detection by means of instruments is to be distinguished from observation, in the sense in which I use that term: observation is perception, and perception is something possible for us, if at all, without instruments. This simple and commonplace distinction does not, of course, settle any questions in epistemology. Rather, it raises the further question: how is this use of instruments to be conceived then? As Huygens does, as a window through which we can see into another realm? Does Huygens’s reaction to the optical microscope remain as intuitively tempting when it comes to the advanced technology now involved in experimentation? Just what is it that we do by means of those instruments that are so typically taken to disclose the unobservable? If we simply put some white powder to the tip of the tongue to check whether it is salt or sugar, we are making an observation, conducting a primitive experiment, and in effect performing a measurement without instruments. But we cannot take this simple case as very revealing. In general there is no such simple relation between observation, experiment, and measurement. This is in part because of the complexity of the instrumentation involved. But it is also because measurements occur only as special elements of the experimental procedure by which objects are deliberately placed in unusual, artiﬁcially designed conditions—conditions in which they are made

94  : , ,   to respond to the questions put to them. That intricate construction of a well-designed instrumental set-up for experimentation is what we must inspect ﬁrst, to understand the intricacies of measurement in general. How shall we understand it, if not through Constantijn Huygens’s so charmingly naïve eyes?

Instrumentation’s diversity of roles Instrumentation plays many roles, and we cannot place all its roles in experiment and other laboratory uses under one simple heading. The sorts of measurement that I have brought up so far as examples involve instruments in what Michael Heidelberger classiﬁes as the representative role.3 This role the instruments have in relation to a theoretical context: ‘‘the goal is to represent symbolically in an instrument the relations between natural phenomena’’ (ibid.). The examples he gives are clocks, balances, and measuring rods. An oscilloscope is also illustrative of this role. Relative to a theory much more technologically advanced instruments can play this role as well. The Scanning Tunneling Electron Microscope used in nano-technology provides a good example of such an advanced representative role, which can be understood as such—that is, as playing this role—only relative to theory. But a caution is in order: given the need for theory as intermediary, we must remind ourselves that representation need not be pictorial representation; and when the results are presented in pictorial form there is a question as to how they are to be taken.4 Instrumentation has several more roles. Just which role given instruments are playing in a particular experimental set-up is a non-trivial question, the answer to which is also typically theory-laden. Heidelberger points to an imitative role, played when instrumentation produces phenomena, in controlled artifacts, meant to mimic effects ‘‘as they appear in nature without human intervention’’. As example he mentions an apparatus used to simulate production of an enzyme in an organism. A fountain, so oriented as to produce a ‘‘rainbow’’ when there are no rain clouds, plays that role too. So did Otto von Guerrike’s electrical generator to produce imitation lightning. Galison and Assmus tellingly—and surprisingly—locate Wilson’s early development of the cloud chamber in a tradition of mimetic

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experimentation in nineteenth-century science in geology and metereology (see Galison and Assmus 1989). When a phenomenon created artiﬁcially is taken to imitate a naturally occurring phenomenon, a substantial theoretical claim is involved. The purpose of mimetic experimentation is to learn something about the natural phenomena, such as lava ﬂow, lightning, rainbows, or cloud formation by studying an artifact constructed in tightly controlled circumstances. This presupposes that the two are indeed alike in relevant respects, and these respects may not at all be entailed by visual or other noticeable similarities. More accurately: it is sub specie the eventually accepted theory that they are relevantly the same—the working hypothesis that they are the same is posited in aid of that eventual, hoped for, theoretical achievement. Similar to, but more remarkable than, such purported imitation is the use of instruments ‘‘to produce phenomena that normally do not appear in the realm of human experience’’, called productive by Heidelberger, manufacture by Boon.5 Those may indeed have never occurred before—or occurred in that form—anywhere in nature. Using his electrical generator, Von Guericke found that he could make a sulfur ball glow. While electroluminescence appears in nature under very different circumstances, the strange happening that involved a relationship between luminescence, rotation, friction, and sulphur was a new phenomenon. In such examples it is not unnatural, even if sometimes confusing, to speak of discovery. A new phenomenon is produced, but the important news is that it occurs, and putatively always occurs, under certain general conditions, which may also be realized in nature—if that this is so, then that is a discovery. This is the sense in which we say, for example, that Faraday discovered electro-magnetic induction. But Faraday’s own description gives us a different impression of the work, when he presented these results to the Royal Society in 1831. There Faraday described ﬁrst of all the induction of momentary currents in a wire when he connected or disconnected an adjacent wire to a battery, or when he moved the wire near the battery (‘‘volta-electric induction’’) Then he described the further effects obtained by inserting iron in the helices of wire, and how currents were induced by the movement of magnets brought nearby (‘‘magnetoelectric induction’’).6 Thus his report does not have the ‘discovery’ form; it is a report of the means by which he created the phenomenal effects he reports.

96  : , ,   All three roles, representative, imitative, and productive, are played by instrumentation in experimental, observational, and modeling work in the laboratory.7 Following upon this taxonomy of roles, we must nuance the account of experimentation accordingly. But ﬁrst a strong caveat: these three roles are easily confused, the experiments and instrumental procedures can easily be misconceived, or conceived in plausible but critically dubitable ways. We need to distinguish them carefully and resist easy or familiar categories before we can properly locate measurement in this technological plethora. ‘‘Observation by instruments’’: our bewitching metaphors Strange words simply puzzle us; ordinary words convey what we know already; it is from metaphor that we can get hold of something new and fresh. (Aristotle, Rhetoric 1410b10–13)

Two metaphors guide our thinking about instrumentation in experimental and other scientiﬁc inquiry. The ﬁrst is that of a window on the invisible world: an instrument opens a window for us into a world beyond the directly observable. In the opening quote from Constantijn Huygens we saw him taking the microscope to alter the observable/unobservable boundary. Whether we follow him in this, whether we extend the meanings of words like ‘‘see’’, ‘‘perceive’’, ‘‘observe’’ to ‘‘see through a microscope’’ and the like, or whether we refuse this extended usage (as I do), does not affect our sense of novelty. In either case, there is a signiﬁcant extension, of some sort, to empirical inquiry. If we grant that these instruments function as windows into realms not accessible directly to the senses then we are directly confronted in experience by something that lies beyond our feeble reach. If we do not regard them as playing this ‘window’ role, we must still do justice to their novelty; and here another prevalent metaphor comes into its own. The second metaphor is well conveyed by saying that our instruments are engines of creation. They create new observable phenomena, ones that may never have happened in nature, playing Heidelberger’s productive role, only sometimes to imitate nature but always to teach us more about nature.8 Once a new phenomenon has been created, it takes its place in nature—for we and our efforts are part of nature. Those new phenomena are themselves observable, and become part of our world. So they become part of what science is meant to ‘save’.

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These are valuable metaphors, each aptly guiding some of our thought about instruments. They do not exhaust the subject: somewhere in between the two lies instrumentation conceived of as creating copies, scale models, likenesses, or ‘signiﬁcant similarities’ of natural phenomena on the scale of the directly visible. Peter Dear (1995: 159–60) gives as striking example William Gilbert’s work on the magnet in 1600. His artifact is a spherical magnet made on a lathe from a lodestone. He calls it a ‘‘little earth’’, a terrella, and while we spontaneously read this as a metaphor, Gilbert makes it clear that he means this very literally: the earth is a spherical magnet, and what he has created here is the very same thing on a scale where we can handle it.9 The instrumentation is here conceived of—in terms that echo Dear’s own distinction—as having a mimetic function, to be contrasted with ‘‘the semiotic, focusing on signs and representation’’. This is close to the ‘‘windows’’ metaphor, in that the set-up is claimed to show ‘‘what things are really like’’, but also close to the ‘‘engines’’ metaphor, because hinging on the production of novel effects. So here the two metaphors overlap. We can speak aptly of representation in all three cases, and the boundaries between the three are blurry, in act if not in conception. In the case of optical instruments we can readily see all three illustrated. Firstly, the images produced by lenses are copies, literal likenesses, if the lenses function as windows upon their domain. Secondly, whether or not that is so, the images produced by lenses are themselves (artiﬁcially produced) phenomena, that have seen their uses in ancient magic and ritual as well as in modern science. Thirdly, phenomena instrumentally produced may in some cases be scaled-down or scaled-up versions of naturally occurring phenomena that are too small or too large to be grasped in perception, or be asserted to be ‘signiﬁcantly similar’ to postulated processes in nature. But there is one very important disconnect between the second conception and the other two. Creation of new phenomena is not in general a case of mimesis. It may indeed provide information about ‘natural’ phenomena, but need not be doing so by presenting us with a likeness. Instrumentation in science, both in experimentation and in application, has many roles. One salient role that I want to emphasize here, that is important as different theoretical leanings compete, is the production of new phenomena that all theories in a given domain must account for if they are to compete successfully

98  : , ,   there at all. That role is played by all that is novel in inquiries into nature by means of instrumentation. Paradoxically, that is precisely the one that tends to favor the ‘window’ metaphor, although it is most clearly the role in which the instruments are engines of creation, not channels for passive observation! What better way to challenge a rival theory than to produce a new phenomenon to be accounted for? The new phenomena are not created for nothing or with no consequences: theory must submit itself to their tribunal. There are many famous examples of such creation of new phenomena to play this role. Hertz himself reported on the importance of his experiments in electromagnetism in this way: What we here indicate as having been accomplished by the experiments is accomplished independently of the correctness of particular theories. Nevertheless, there is an obvious connection between the experiments and the theory in connection with which they were undertaken. Since the year 1861 science has been in possession of a theory which Maxwell constructed upon Faraday’s views, and which we therefore call the Faraday–Maxwell theory. This theory afﬁrms the possibility of the class of phenomena here discovered just as positively as the remaining electrical theories are compelled to deny it. (Hertz 1962: 19)

The well-known story of Poisson’s challenge to Augustin Fresnel’s prize essay furnishes a ready example (even if not generally told with perfect historical accuracy).10 Fresnel submitted his essay for a scientiﬁc prize competition on the diffraction of light, basing his analysis on the wave theory. Poisson, a member of the panel which evaluated the essay, pointed out that Fresnel’s analysis had a strange consequence. It implied that the center of the shadow of a circular disc would have a bright spot. This consequence contradicted the Newtonian corpuscular theory of light, and seemed in any case highly unlikely. But Dominique Arago, the panel’s chairman, did the experiment and the bright spot appeared! Despite himself Poisson had designed an experiment that created an observed phenomenon—new in what these scientists on the panel had seen, though not new in nature—which only the wave theory was (at that point) able to explain, and which was entirely upsetting to its great rival.11 The ‘‘window into the invisible world’’ metaphor has dominated modern philosophical thinking about science as much as the ‘‘mirror of nature’’ metaphor dominated modern epistemology and metaphysics. It will serve us better to dislodge or at least weaken its grip on our philosophical

      ⁽  ⁾

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discourse, and to think of experimentation in terms of a literal enlargement of the observable world, by the creation of new observable phenomena, rather than a metaphorical extension of our senses.12 The microscope as window on the invisible world The contention that our epistemic situation changed at a certain point, just because of instruments of observation, seemed to come along naturally with the construction of those instruments, though the idea was then as new as the instruments themselves.13 The technological change started after the Renaissance with the development precisely of optical instruments: the microscope and telescope. Catherine Wilson (1995: 57–65, 218–20) describes how enthusiastic the English scientists of the seventeenth century were about this achievement. One writer on the microscope, Charleton, remarked that of course the ancients, lacking optical instruments, would have greeted the atomic hypothesis with skepticism—but our situation is quite different! Robert Hooke writes in the preface of his Micrographia (1665) that with the aid of the microscope we will discover: the subtilty of the composition of Bodies, the structure of their parts, the various texture of their matter, the instruments and manner of their inward motions, and all the other possible appearances of things . . . . we may perhaps be inabled to discern all the secret workings of Nature, . . . manag’d by Wheels, and Engines, and Springs . . . .

And Hooke’s contemporary Henry Power’s Experimental Philosophy (1664) says in his preface: We might hope, ere long, to see the Magnetic Efﬂuviums of the Lodestone, the Solary Atoms of Light . . . , the springy Particles of Air, the constant and voluntary motion of the Atoms of all ﬂuid Bodies, and those inﬁnite, insensible Corpuscles which daily produce those prodigious (though common) effects among us.14

Notice that he takes the optical microscope to reveal even the composition of light! Besides this radical misunderstanding of what the microscope can do, several other errors both of fact and of principle would soon become evident. If appearances are what appear to us then, by deﬁnition, we never do see beyond the appearances . . . ! This insight, clear enough in Locke and Berkeley in the next century, could be the slogan for our entire discussion.

100  : , ,   When caught unawares we do still spontaneously think about it in the same way as those seventeenth-century enthusiasts, however. We simply tend to add into this family of windows such devices as electron microscopes, spectroscopes, MRI scanners, particle accelerators and so forth. In fact, each of these devices creates new phenomena, truly humanly observable phenomena. Whether or not one subscribes to the idea that these devices extend the range of our vision, it is indisputable that they serve for the systematic creation of new phenomena—new phenomena that must also be saved by our theories, sufﬁce to refute theories to be discarded, and serve to gather empirical information.

Engines of creation: engendering new phenomena Even for the optical microscope I am offering a change in view, by favoring and emphasizing the ‘‘creation’’ metaphor over the ‘‘window’’ metaphor. Though valuable as an heuristic guide, to take the ‘‘window’’ metaphor literally acts as a brake on the possibilities for interpretation. Suspend for a moment the question whether any of these instruments provide access to the unobservable. All those instruments which purport to represent the invisible mimetically, can be thought of, not as windows into the nether world, but as productive experimental arrangements. As always, interpretative options allow for alternatives. Even the clearest examples of instrumentation that play a productive role can, conversely, also be reconceived as imitative. The possible conﬂation does not lie along a one-way street. But the one function we can all agree on for instruments used in science—for it is a function served either way—is that of creating new observable phenomena to be saved. Consider the nineteenth-century ether theory that brought light under the canopy of electromagnetism. Following Faraday’s early experiments, place iron ﬁlings on a sheet of paper above an electromagnet; then turn the magnet on. Suddenly the ﬁlings arrange themselves in a pattern of lines of force. What happened? According to the theory, when you pressed the switch, you affected the state of the very ﬁne all-pervasive medium, the ether. This new state is manifest in the arrangement of the ﬁlings. In that ether theory the phenomena of electricity, magnetism, light, and illumination are assimilated to the same pattern of explanation. Turning

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on the light affects and alters the state of this ethereal medium. This altered state is manifest in the illumination and visibility of the objects in this room. From a realistic point of view you can describe Faraday’s device as a detector for the electromagnetic ﬁeld, as a window allowing us to look upon those mysterious lines of force in the ether. In theory it is now certainly of a piece with optical phenomena. At the phenomenological level it was at the time of a piece with the well-known visible phenomena of magnetic attraction. The behavior of lodestone and iron had been recreated, artiﬁcially produced—in a like yet different form—in the laboratory. But look, we have here also a phenomenon that had never occurred before in history: pressing one object and thereby instantaneously altering a ‘dust’ distribution on another object into such a regular pattern as this. Faraday had created a new phenomenon. But this event had at once great theoretical value—for it was a controlled phenomenon so closely related to known natural phenomena of electricity and magnetism that it was required to be accounted for in any future theory on offer in this area of physics. Faraday had constructed an engine to produce a fascinating variety of new phenomena. The story continues. When representing the distribution of electric charge as analogous to what happens when iron ﬁlings are scattered in the vicinity of a bar magnet, Faraday and Barlow displayed the change in the position of the ﬁlings as a continuous function of their position relative to the poles of a magnet. The examples can be multiplied: brought to light are salient regularities in the new observable phenomena that theory must account for.

The microscope’s public hallucinations I submit that the analogous point is to be made even for the simplest optical microscope. Microscopes too create phenomena, to be accounted for by our theories, they too can be conceived of as engines of creation rather than windows upon the invisible world.15 To say what they create, I must direct attention to a special sort of phenomenon that nature also produces spontaneously. The main examples are certainly—but not only—optical phenomena. They include reﬂections in the water, mirages in the desert, and rainbows.16 The subject is ancient: Plato’s characters in the Sophist

102  : , ,   struggled with the questions of how or in what sense reﬂections in water are real or unreal, and Aristotle began the theory of the rainbow in his Meteorology. The ﬁrst point about these, as about light itself, is that we use nouns. In fact we use count nouns; we talk about all of them as if they were things. The second is that the phenomena themselves show that we are wrong to do so. They refuse to allow us to represent them to ourselves as things, or even as properties of things in any straightforward way. Consider the rainbow. We realize pretty soon that there is no real material shining arch standing above the earth, although at ﬁrst it looks that way. As a second guess we might think that certain parts of the clouds or haze are colored. But that cannot be maintained because if we move, we see the rainbow in a different location on that cloud or haze background.

Figure 4.1. Geometry of the Rainbow

In fact, we realize then that our usual way of speaking involves us in falsehoods. I see a rainbow and you say you see it too. See what too? How can you be seeing the rainbow I see, when yours is located in a different place? Nor are they simply in a different place in our respective visual ﬁelds, in the way the clouds are. For if that were so, we would see the colors ‘attached’ to the same part of the cloud, modulo parallax. If on the other hand I say there are two rainbows, and you agree, we are not even counting the same things. In fact, we are not counting things at all.

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But thirdly, we are not hallucinating. Hallucinations are private, subjective. These rainbow observations are like hallucinations, in that they are not of real things. But they are unlike hallucinations because they are public. Nature creates public hallucinations. So public, in fact, that the camera captures them as well! The observations are scientiﬁcally signiﬁcant in part because they can also be made indirectly, so to speak, with the camera as instrument. Let me put all this a bit more technically. In some sense we do represent shadows, moving spots of light, reﬂections, mirages, and rainbows as things. But the phenomena themselves refuse to let us maintain this representation. The reason is that certain crucial invariances are lacking. If the rainbow were a thing, the various observations and photos would all locate it in the same place in space, at any given time. However, there are signiﬁcant invariants here that dreams and after-images lack. In the case of rainbows this invariance is found in the relations between sun, cloud, and location of the eye or camera. The subtended angle is always 42 degrees, with that location (of eye or camera) between sun and cloud. The larger situation exhibits invariances that allow us to represent it—the situation taken as a whole, not the rainbow!—to ourselves as a structure in nature independent of our subjective experiences. Just to prevent a possible confusion, let us note a difference between two differences. Think of a tree by a pond, reﬂected in the water. There is a difference between a reﬂection and a rainbow that makes the one a ‘picture’ of something real and the other not. That difference is not the difference between a real thing and an image. Different observers also do not see the reﬂection in the water in the same place. For the reﬂection too, the important invariance lies in the relation between light source, water, and eye or camera. But there is a further factor in this invariant set of relationships: the tree. The rainbow and reﬂection are alike in being consistently mis-classiﬁed as things. But they are unlike in that we can classify the latter, and not the former, as ‘copy’ or ‘picture’ of a real object or event that it resembles in a certain respect.17 A catalogue of images Nature creates these public hallucinations. Already in ancient times, concave mirrors and lenses were used to do the same, namely to create (artiﬁcial) public hallucinations. Even a concave mirror and a lighted candle can

104  : , ,   produce a ‘stand alone’ image that is under some viewing conditions as ‘real’ as a hologram.18 It took more than a thousand years before this art of imitating nature provided the resources for a systematic exploration of nature. That is how I suggest we should understand the microscope as well. Van Leeuwenhoek did with his lenses essentially the very same sort of thing as Newton did with his prisms—namely, imitate the ability of nature to create public hallucinations. There are objections to this, especially for the simplest cases, such as van Leeuwenhoek’s magnifying glass or the optical microscopes we remember from high school biology classes. I will take those up below; let’s try for a systematic overview ﬁrst. For images I want to display a division into several kinds and sub-kinds. For mnemonic ease I’ll supply a diagram, with an elaboration on its labels. Graven Images

Public Hallucinations

Private Images

‘‘COPY’’NOT QUALIFIED ‘‘COPY’’QUALIFIED

painting photo sculpture

reflection shadow

rainbow mirage fata morgana

after-image dream hallucination

〈microscope image〉

Figure 4.2. Image Categories

(Graven Images) On one side are the images which are in fact things, such as paintings and photos. (Private Images) On the other extreme are the purely subjective ones like after-images, dreams, and private hallucinations. These are personal, not shared, not publicly accessible. Indeed, we are pretty clearly dealing

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there with discourse that reiﬁes certain experiences which are ‘‘as if ’’ one is seeing or hearing. (Public Hallucinations) In between these two are a whole gallery of images which are not things, but are also not purely subjective, because they can be captured on photographs: reﬂections in the water, mirror images, mirages, rainbows. For those I will use the term ‘‘public hallucinations’’. Some of these public hallucinations are actually ‘‘of ’’ real things: e.g. the reﬂection of a tree in the water. When you see the reﬂection of a tree in water you are not seeing a thing; a reﬂection is not nothing, it is something, but it is not a thing, not a material object.19 That is clear enough because of the way it moves when you move, quite differently from, say, a log ﬂoating in the water. Some public hallucinations are not ‘‘of’’ real things; e.g. the rainbow. But of those which are not, some—only some—would still lend themselves to being conceived of or identiﬁed as pictures of real things. If an image would so lend itself I’ll call it ‘‘ ‘copy’-qualiﬁed’’ (following the Sophist’s distinction between making copies and creating appearances). But of any ‘‘copy’’-qualiﬁed image we can still ask: is it really of something real, or is it not? That is always a question of fact transcending the experience itself.

Objections to this view of ‘observation by instruments’ There are two sorts of immediate objection to seeing all such instrumentation used in experimental inquiry as productive rather than mimetic. The ﬁrst is theoretical, and points to the basis in the data obtained for inference to real entities ‘perceived through’ the instrument. The second is phenomenological: looking into a microscope just does not give us the experience of seeing an image, it is phenomenologically just like looking through a window or telescope. Let us take the phenomenological objection ﬁrst; it has been made strongly and tellingly by Paul Teller (2001b): if you have your eye glued to the microscope you do not have the experience of seeing an image. You have the experience, for example, of seeing paramecia on a slide, not of seeing paramecia-images. This supports the conclusion that when we look

106  : , ,   into a microscope we are seeing the things that we experience ourselves as seeing—in fact what is seeing except detecting through visual experience? I agree to the phenomenology, but not to the conclusion. Much instrumentation that offered us the opportunity for such new experiences—so experientially like looking through windows—has tended to disappear from scientiﬁc use. Output from instrumentation tends now generally to be publicly perceivable print-outs or computer displays. We don’t tend to think of this as introducing a loss; on the contrary—so we should not attach too much importance to this change. The microscope’s output can be sent into a scanner which transmits to a computer or projector—then we see the paramecia on the wall or the monitor. We are having a different sort of experience then, for we say after only a little urging that we are seeing an image. Even when the working of the simple optical microscope is explained, the instructor will set up lenses and show how images are optically produced; no need to pay much attention to the eye or its physiology for this explanation to be quite satisfying. So it seems, surely, that nothing about the status of the microscope or its deliverances can follow from concentration on one of these experimental arrangements to the exclusion of the others. But I want to add to this demurral some further points about the phenomenology as well. Just how different is the experience of looking through a microscope from looking at a rainbow? First of all, no more do we have the experience of seeing a rainbow-image when we, as one says, see a rainbow. Secondly, we must be very careful about how we take such expressions as ‘‘have the experience of seeing an image’’. That sort of assertion has the same surface form as ‘‘have the experience of seeing a boat’’—but remember, images are not things, and be wary of drawing a parallel between the two propositions. Don’t we, it may be objected, sometimes have the experience of seeing an image, as distinct from seeing a real thing, and aren’t those kinds of experiences phenomenologically different? Yes, we do, and they are. Recall Macbeth’s ‘‘Is this a dagger which I see before me?’’, reporting on his experience which is like seeing a dagger although there is no dagger there. There was literally nothing—no real thing—that he saw.20 That does not contradict his having had the experience. So the question that we really have to answer is this: Under

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what conditions do we correctly classify someone’s experience phenomenologically as the experience of seeing an image of something? I envision three cases: Assertion: ‘‘S/he is having the experience of seeing an image of X’’. Cases: Graven Images. S/he is seeing a real thing that we classify as a ‘picture’ of an X. (We have a case in point not only in a display of paintings or statues but also if e.g. a microscope is hooked up to a projector or monitor and s/he sees the screen optically transformed into a picture). Illusion-A. S/he judges that s/he is seeing a real X, while we take him/her to be having an illusion or hallucination, whether private or public. Illusion-B. S/he judges that it is as if s/he is seeing a real X while s/he herself takes that to be an illusion. There are certainly phenomenological differences between these three sorts of cases, as well as between them and cases of ‘real’ seeing. And the type Illusion-B could be subdivided further: s/he might still be either right or wrong in her judgment that it is only as if s/he is seeing a real X. We might sometimes have to add: s/he is having the experience of seeing an X-image, but is mistaken: what is actually happening, contrary to what she thinks, is that s/he is seeing a real X. Sometimes one can’t be sure whether something is illusory or not. The question of how to classify it becomes a factual question, and we may or may not be content to remain agnostic about it. Brieﬂy then, ‘‘seeing as an image’’ is a code for a classiﬁcation of experiences that refers both to the spontaneous judgment on the experiencer’s part, and to what is really happening to that person. What then, precisely, is Teller pointing out about looking into a microscope? In the experience he describes he spontaneously judges that he is seeing e.g. real paramecia, and that he has no inclination to correct that judgment as illusory. He is quite rightly contrasting that sort of experience with such experiences as those of seeing rainbows, mirages, or reﬂections when they are not delusory. In other words, he contrasts it with Illusion-B, namely with experiences in which the spontaneous judgment

108  : , ,   includes a classiﬁcation of that very experience as a misrepresentation or (public or private) hallucination. This genuine phenomenological difference pertains, however, in the ﬁrst instance not to what is really happening to him, but to his response to what is happening to him. That is, if we describe experience as having two sides (what is happening to one that one is aware of, and the judgment expressing what one takes that to be) it pertains not to the ﬁrst but to the second side of experience. So questions as to the ﬁrst side—e.g. the question whether the experience of ‘seeing’ in a microscope is or is not a public hallucination—is not settled by his report. For that further question is a theoretical question about what happens in a microscope. Turning then to the theoretical objection: it is certainly the case that we can represent the images produced by the microscope to ourselves as images of real things (with the same structure as those images). In fact, we can represent what we see as indirectly observed real things behind the microscope’s lenses.21 Part of the reason, as Ian Hacking and Wesley Salmon pointed out in different ways, is that there are strong correlations between what the various instruments bring to light (so to speak) when trained on the same microscopic objects. We are invited, in effect, to engage in inference to ‘the best explanation’ or to ‘the common cause’, so as to posit unobservable objects or events to explain those pervasive correlations. I agree with the core of their argument but not its conclusion. Having argued so often against the epistemic status of such patterns of inference, I prefer to leave that part of the issue aside here.22 The similarity I am pointing out to the rainbow (and which I propose as basis for a possible way of thinking about microscopes) is not a similar lack of invariances that could be interpreted as invariances in possible relations to real objects. The success of that family of instruments (microscope, electron microscope, radio telescope) does indeed derive in part from the possibility of representing their products as images of real things existing independently of any relations to those instruments. The similarity I want to point to is just this: their products are images; they are optically produced, publicly inspectable images. It is these images that are like the rainbow (they cannot themselves be represented as independent things). The difference, that we cannot think of the rainbow as an image of a real arch, while we can think of the microscope image as of a similarly structured object, is important but irrelevant for my present argument. The

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point I am making is that the microscope need not be thought of as a window, but is most certainly an engine creating new optical phenomena. It is accurate to say of what we see in the microscope that we are ‘‘seeing an image’’ (like ‘‘seeing a reﬂection’’, ‘‘seeing a rainbow’’), and that the image could be either a copy of a real thing not visible to the naked eye or a mere public hallucination. I suggest that it is moreover accurate and in fact more illuminating to keep neutrality in this respect and just think of the images themselves as a public hallucination. But secondly, what are the practical implications? To keep neutrality in this respect does not prevent us from gathering empirically attestable information by means of the microscope, or to base e.g. medical advice on what the microscope shows us. We should recognize that a false contrast is made if we oppose ‘‘merely’’ producing images to producing something informative about the objects with which the instruments are placed in interaction. Speciﬁcally, the important correlations between the output of different instruments in similar situations, and between the outputs of the same instrument in a situation subject to variation, do not need to be explained by unobservable external causes in order to be intelligible or useful. We are indeed able to change the images produced in predictable ways, in the case of e.g. microscopes, other than by changing our own relation to the instrument. This ability derives from our latching onto signiﬁcant regularities in the phenomena, including now among these the objects and events under study, the instruments, and the relations between the two.23 Hard and soft core science What I have proposed here is contrary to the way microscopy and similar instrumentation are usually presented, and it is easy to imagine further misgivings. When a new physical, observable artifact, a radiograph or cloud chamber track, is produced the rationale is clearer: we have the newly produced artifact, and why do we say that it represents something else, except sub specie a certain theory? But in the case of the simple optical microscope we have no such physical artifact, we only have the images. But images are observable phenomena, even if they are not objects, as their capture on ﬁlm shows. There is some linguistic regimentation or articulation to be done: the statement, ‘‘There is an image’’ stands in for for a long description of a set-up in which certain physical phenomena—such

110  : , ,   as blackenings of photographic ﬁlm—will happen. Without stretching ourselves very far, we can report on our sightings made by means of a microscope in the same way as we report our rainbow-observations.24 Rainbows are not objects, events or processes. Our use of the noun hides from view the more sophisticated understanding that we’ll now immediately display when pressed. But our common way of speaking has not actually changed, and it need not change, it is just ﬁne for all practical purposes—it is just ﬁne FAPP, as they say in philosophy of physics.25 I think we can relate to our experiences with microscopes in the same way. Our common way of speaking, about seeing things through the microscope, need not change, as long as it is properly understood. Confusions of the two need to be kept at bay. Speciﬁcally, in any ordinary way of speaking it is not correct to say that we have the experience of seeing a rainbow-image, nor of a paramecium image. In ordinary language the correct report is that we have the experience of seeing a rainbow and paramecium. As long as ordinary discourse is not ﬁltered through some theory it does not imply that those are objects.26 What about the observable/unobservable distinction then? The main points of our discussion are not much affected by just where precisely the line is drawn. I draw the line this side of things only appearing in optical microscope images, but won’t really mind very much if you take this option only, for example, for the electron microscope. After all, optical microscopes don’t reveal all that much of the cosmos, no matter how veridical or accurate their images are. The empiricist point is not lost if the line is drawn in a somewhat different way from the way I draw it. The point would be lost only if no such line drawing was to be considered relevant to our understanding of science. Personally I see such a line appearing in a number of contexts, not solely in the debate over theoretically postulated entities. Here is a quote from Steven Weinberg, in his 1998 review of Kuhn’s work: It is important to keep straight what does and does not change in scientiﬁc revolutions, a distinction that is not made in [The Structure of Scientiﬁc Revolutions]. There is a ‘‘hard’’ part of modern physical theories (‘‘hard’’ meaning not difﬁcult, but durable, like bones in paleontology or potsherds in archeology) that usually consists of the equations themselves, together with some understandings about what the symbols mean operationally and about the sorts of phenomena to which they apply. Then there is a ‘‘soft’’ part; it is the vision of reality that we use to

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explain to ourselves why the equations work. The soft part does change: we no longer believe in Maxwell’s ether, and we know there is more to nature than Newton’s particles and forces. The changes in the soft part of scientiﬁc theories also produce changes in our understanding of the conditions under which the hard part is a good approximation. But after our theories reach their mature forms, their hard parts represent permanent accomplishments.(Weinberg 1998: 50)

Weinberg is not exactly known for sympathy with philosophy.27 But I can submit a challenge to philosophers, in Weinberg’s terms: if you are going to distinguish between a hard and soft part of science, in some such way, tell us how you draw the line.28 You can’t get out of this by pointing out that there is a continuum on which the line is drawn, or that the line will be drawn differently in different contexts, historical or social. For that is the case for all or almost all distinctions we make, and does not make those distinctions unreal or unimportant for understanding.29

Experimentation’s diversity of roles In distinguishing the various roles of instrumentation in experiment we automatically induce a corollary taxonomy of experiments, distinguishing for example mimetic experimentation from the creation of new phenomena. But there is another way to classify experiments, namely in terms of the roles they play in the development of high level theories. Philosophers of science, unlike philosophers of technology or historians of science, used typically to concentrate on the function of experimentation as hypothesis testing. The experimenter reads over the theoretician’s shoulder, and designs experiments to test whether the theorist has not gone too far and made the theory empirically inadequate. This characterization is simple and appealing. It is regrettably over-simple for many reasons, by now amply exposed in the literature, although sufﬁciently many famous examples illustrate it to explain the traditional philosophical fascination. In contrast to the hypothesis testing role, there is another function of experimentation, generally also described in the language of discovery, but actually an essential ingredient in the joint evolution of experimental

112  : , ,   practice and theory. We may describe it as theory writing by other means. One example is Millikan’s work on the charge of the electron. Popularly presented, he measured the charge. More accurately, he found the charge to be assigned to the electron in the then developing atomic theory, by measurements made on the motion of oil droplets drifting down in the air between two plates, which he could connect and disconnect with a battery. By friction with the air, the droplets could acquire an electrostatic charge; and Millikan observed their drifting behavior, calculating their apparent charges from their motion. After some ‘smoothing’ of the data. naturally, he could point to a number of which all apparent charges were integral multiples, and offer that as the charge of a single electron. ‘‘That number was something he discovered. Remarkably the number of such charges per Faraday equals Avogadro’s number! No one could have predicted that!’’ This way of announcing his achievement certainly sounds as if by carefully designed experiment we can discover facts about the unobservable entities behind the phenomena. But this makes it also sound like a fortuitous discovery, as if Millikan had stumbled on it. Quite to the contrary, Millikan approached the experiment within a developing theoretical context. Instead we should see the experiment as able to ﬁll a blank in the theory then under development. The theory is written, so to say, step by step. At some point, the principles laid down so far imply that the electron has a negative charge. A blank is left for the magnitude of the charge. If the theory is to be continued at this point, there are two ways to go on. The theorist could hypothesize a value, ask the experimentalist to design a test, offer a second guess, test again. But that is not how it goes in such a case: the experimental apparatus writes a number in the blank. What I mean is: in this case the experiment shows that unless a certain number (or a number not outside a certain interval) is written in the blank, the theory will become empirically inadequate. For the experiment has shown by actual example that no other number will do; that is the sense in which it has ﬁlled in the blank. So regarded, experimentation is the continuation of theory construction by other means. Recalling the famous Clausewitz view of war and diplomacy, I call this the ‘‘Clausewitz doctrine of experimentation’’. It makes the language of construction, rather than of discovery, appropriate for experimentation as much as for theorizing. One way in which things may go badly for a theory is if the numbers placed in various blanks by such experimental procedures do not bear the

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relations that the theory says they should. So when Perrin showed that about a dozen experimental ways of ascertaining Avogadro’s number all gave the same result, his work gave atomic theory a stability that it could not have attained if the results had been different. Similarly for Thomson’s multiple ways of measuring the electron’s mass to charge ratio.30 In these two cases too, the experiments were approached within a developing theoretical framework that needed consistent empirical grounding for its theoretical parameters. For instruments we now have a cross classiﬁcation to draw on: a diversity of roles of instrumentation in experiment, and a diversity of roles of experiment in theorizing. In the former, measurement shows up as a single role among others, the representative role. In experiments, however, measurement is crucially involved in all cases. All the examples of production of new phenomena involved set-ups in which multiple measurements were being made as well. Although it may seem as if electron microscopes, spectroscopes, spectrophotometers, x-ray cameras, refractometers, polarimeters, cloud chambers, scintillators, and the like have been relegated to a ‘lesser’ status in my account than the scientiﬁc realist would give them, there is no implication at all that the role of measurement has vanished from the scene. What needs still to be elaborated, however, is an account of measurement itself, as it appears in the process of joint experimental-theoretical development and as it appears through the eyes of the eventually achieved theory.

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5 The Problem of Coordination A scientiﬁc theory is typically presented with laws, principles, or equations that involve terms speciﬁc to that theory: s = vt + (1/2)at2 , PV = rT, F = ma, action = reaction to mention only a few of the classical standbys. Although these terms are, taken literally, symbols for functions with speciﬁed mathematical character, they are often pronounced as if they were nouns already familiar before the theory’s introduction: ‘‘distance’’, ‘‘velocity’’, ‘‘time’’, ‘‘pressure’’, ‘‘temperature’’, and so forth. That nomenclature is introduced more by way of informal commentary than explanation, and certainly does not deﬁne the theoretical terms.1 But the choice of familiar words does signify something: they point to the sort of data to enter, and the sorts of measurements that can help to determine the values of those functions. The theory would remain a piece of pure mathematics, and not an empirical theory at all, if its terms were not linked to measurement procedures. But what is this linkage? That question, which turns out to bring many further questions and complexities in its train, poses what was once generally known as the ‘problem of coordination’. The term had appeared in Mach’s writings on mechanics and thermodynamics, was salient in the discussion of the relation between mathematical and physical geometry that extended from the nineteenth century into the twentieth, and came to special prominence through the writings of Schlick and Reichenbach when logical empiricism was beginning to break with the neo-Kantian tradition. Since then it has not been so much in use, but the problem to understand just how a scientiﬁc theory is more than its mathematical guise reappeared at every juncture in the subsequent history.

116  : , ,   The questions What counts as a measurement of (physical quantity) X? and What is (that physical quantity) X? cannot be answered independently of each other. To echo another such realization, I am not ashamed to admit that this brings us to the famed ‘hermeneutic circle’ (Eco 1992: 64). We shall examine this apparent circularity by focusing on the one hand on its more abstract consideration by Reichenbach, and on the other hand the practical response in history examined by Mach and Poincar´e.

Coordination: a historical context Assigning a value, a location on a scale or in a larger space, is precisely what Ernst Mach called ‘‘coordination’’.2 After a historical survey of how thermometry developed (which we will look at in more detail below), Mach began a critical discussion of how the involved choices had been made. Precisely in the midst of this critique he introduces the term ‘‘coordination’’: That number which, conformably to any chosen principle of coordination, is coordinated with a volume indication of the thermoscope, and consequently uniquely with a state of heat, is called the temperature of that state. (Mach 1986: 52)

How much would already have to have been in place before this could be offered to identify the quantity temperature? It can make sense only at a point when there is no longer any ambiguity about what counts as a thermoscope or thermometer, or any doubt about whether the volume of the thermometric substance is uniquely correlated with a ‘state of heat’, or about what is meant by that. Mach is very clear that this is at best a late stage in the historical development. He sees clearly that it involves both idealization with respect to the stability in the instruments that count as thermoscopes and choices whose status needs to be critically examined. But some leeway for choice is brought into the open precisely in the phrase that introduces the term ‘‘coordination’’. He continues The temperature numbers are dependent on the principle of coordination, t = f(v), where v is the thermoscopic volume, and, consequently, for the same state of heat they will vary greatly according to the principle adopted. (Ibid.)

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As he points out, different such principles of coordination (choices of function f ) were introduced. Galileo made the temperature numbers correspond to increments starting from a chosen initial volume: t = N if and only if the volume equals v0 (1 + Nα) while Dalton’s function had the form t = Nif and only if the volume equals v0 βN There is a simple mathematical transformation that leads us from one to the other (even if not as simple as a transformation from Fahrenheit to Celsius!), so although the two are different they are, one might say, not fundamentally different. When, however, Amontons and Lambert choose the pressure of a mass of gas of constant volume as temperature index, they are choosing a quite different principle of coordination. In the numbers assigned their method of thermometry is not markedly different from those made by Galileo’s scale. But in this case, the relationship between the scales is not a matter of mathematics or logic—it is due to relations in empirical and physical contingent fact. So what is meant by coordination? Just for now let us take thermoscopes and the indicated physical correlations between heating, expansion, and pressure for granted. Then the initial statement by Mach identiﬁes the physical magnitude (parameter, quantity, observable) temperature. Since he identiﬁes its value as ‘‘dependent on the principle of coordination’’, hence relative to a choice among coordinating principles, we must take it that only what is invariant under these choices is signiﬁcant. This might seem to relegate the coordinating principles themselves to insigniﬁcance. But that is not so: in the absence of any such coordinating principle, attributions of temperature would have no empirical content at all, no reference in what is already established as measurable. To put it brieﬂy: on the one hand, coordination requires for its possibility some recognized empirical regularities, while on the other hand it is required for the new theoretical assertions to have any empirical content at all. The term ‘‘coordination’’ was central in Schlick’s Algemeine Erkenntnislehre, where the account of the process of cognition is based on idea of coordination (Zuordnung).3 While I will ignore his generalization as only tangentially relevant to our subject, its historical role underlines the impact of this problem on the philosophy of science that was then developing.

118  : , ,   Here Poincar´e (whose analysis of this problem for time measurement we will look into below) had been a central ﬁgure, and almost every discussion had to orient itself with respect to his conventionalism. Einstein took an active part, and in fact uses the terminology of coordination quite explicitly in his 1921 essay, ‘‘Geometry and Experience’’: It is clear that the system of concepts of axiomatic geometry alone cannot make any assertion as to the behavior of those objects of reality which we designate as practically rigid bodies. To be able to make such assertions, geometry must be stripped of its merely logical-formal character by the coordination of experienceable objects of reality with the empty conceptual schemata of axiomatic geometry.4

But the radical, foundational taking by the horns we ﬁnd in Reichenbach’s 1920 book, The Theory of Relativity and A Priori Knowledge. There we see him landing in serious difﬁculty with the question of how terms in the theory of space, time, and motion—whose primary reference is to mathematical objects—can be linked to items or aspects of the physical. The reason for his difﬁculties lie precisely in the radicalism of his philosophical approach. Coordination and Reichenbach’s problem Mach’s discussion had encountered no difﬁculty in principle because he stated his principles of coordination in a context where certain other physical parameters and their measurement were assumed to be given. The measurement of length or volume and volume change, for example, is simply taken for granted in the passage I quoted. In 1920 Reichenbach, on the other hand, realizing that he is addressing the most fundamental level of physical theory, allows himself no such luxury. He wants to ﬁnd a general coordination of mathematical spaces and their structure with physical relations. In his later Philosophy of Space and Time he gives as ﬁrst example how units of length can receive their coordination: ‘‘a meter is the forty-millionth part of the circumference of the earth’’( Reichenbach 1958: 15). But the recurring example of the concept of a straight line, or more generally a geodesic, as having as physical correlate a light ray, or the path of a freely falling body, is not equally easy to understand. Nor is the idea that congruence relations have as correlate coincidence with transported rigid bodies.5 In these examples it is quite unclear how to identify the physical correlate without using any geometric or kinematic terms. How are we to describe rigidity or free

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fall without using the language of geometry, or of mathematical physics in general? In the example that most preoccupied him about the theory of relativity, non-simultaneity is to be thus related to non-connectability through any signals, whether by light emission and reﬂection or by material transport. Not only the modal character of ‘‘connectable’’ but also the required identity-over-time (genidentity) of the material are as puzzlingly theoretical as any of the terms in physical geometry. And so we ﬁnd him perplexed. He writes in 1920:6 It is characteristic of modern physics to represent all processes in terms of mathematical equations. But the close connection between the two sciences must not blur their essential difference. (Reichenbach 1965: 34) The mathematical object of knowledge is uniquely determined by the axioms and deﬁnitions of mathematics. (Ibid.) The physical object cannot be determined by axioms and deﬁnitions. It is a thing of the real world, not an object of the logical world of mathematics. Offhand it looks as if the method of representing physical events by mathematical equations is the same as that of mathematics. Physics has developed the method of deﬁning one magnitude in terms of others by relating them to more and more general magnitudes and by ultimately arriving at ‘‘axioms’’, that is, the fundamental equations of physics. Yet what is obtained in this fashion is just a system of mathematical relations. What is lacking in such system is a statement regarding the signiﬁcance of physics, the assertion that the system of equations is true for reality. (Ibid.: 36)

So how can empirical signiﬁcance be achieved? The examples he has in mind, as we just saw, are the use of rigid bodies as choice to set the relation of spatial congruence (in effect, to measure length), the choice of a light ray path in vacuo as physical correlate for geodesics, the choice of a certain periodic process as setting the unit of time. Question: how are these physical correlates to be identiﬁed without use of geometric or kinematic terms? Look back to Mach’s examples of Galileo’s and Dalton’s choices for temperature scaling. These choices are expressed in the above equations in which one variable is set equal to a function of another variable. So is the choice in question simply a choice of a function? A function relating what to what? Isn’t a function a mathematical object itself, deﬁned in terms of a relation between mathematical objects? So Reichenbach writes:

120  : , ,   The coo¨ rdination performed in a physical proposition is very peculiar. It differs distinctly from other kinds of co¨ordination. For example, if two sets of points are given, we establish a correspondence between them by coo¨ rdinating to every point of one set a point of the other set. For this purpose, the elements of each set must be deﬁned; that is, for each element there must exist another deﬁnition in addition to that which determines the co¨ordination to the other set. Such deﬁnitions are lacking on one side of the co¨ordination dealing with the cognition of reality. Although the equations, that is, the conceptual side of the co¨ordination, are uniquely deﬁned, the ‘‘real’’ is not. On the contrary, the ‘‘real’’ is deﬁned by co¨ordination to the equations. (Ibid.: 37–8; my italics)7

Here Reichenbach was imagining, and discounting, the following naïve sort of reply: what is called for is simply a function, a mapping, between mathematical objects and physical objects or processes—what is puzzling about that? The reason he discounts it is because to deﬁne a function we need to have the domain and range identiﬁed ﬁrst—and the question at issue was precisely how that can be done without presupposing that we already have a physical-mathematical relation on hand. So what does Reichenbach mean, when he seems to point to a solution with the words ‘‘On the contrary, the ‘‘real’’ is deﬁned by coordination to the equations’’? We are here at the historical point where the split between the neo-Kantian tradition and logical empiricism begins, at least in Reichenbach’s intellectual life.8 In this early book, Reichenbach argues that at least one essential element of the Kantian a priori can still be maintained. So to effect the necessary coordination of abstract mathematical structure to concrete empirical reality he posited a special class of mathematical–physical principles—‘‘coordinating principles’’ or ‘‘axioms of coordination’’—whose role is precisely to insure those conditions of possibility. These principles, Reichenbach argues at that point, are to be taken as given or imposed a priori, and so to be sharply distinguished from mere empirical laws (‘axioms of connection’).9 In this respect then he follows Kant’s original conception, though he adds the innovation that the a priori is historically relative, the coordinating principles must be changed as the sciences develop. As we should guess from Mach’s example, Reichenbach can coordinate a mathematical representation with physical objects, events, and processes only in a context where something is already given that will make that

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possible. Thus we must doubt whether Reichenbach really did, by this reconception of the a priori as historically relative, provide something given-in-context that makes it possible to link mathematical objects to physical magnitudes. How could principles, or having principles, provide the conditions in which we can relate something mathematical to something physical?10 What exactly are the required conditions—isn’t that the question to be answered before we can think about what would make those conditions possible? In fact, it seems that Reichenbach himself was not able to maintain this reaction to his problem of coordination. At a somewhat later point in his writings he offers ‘‘coordinative deﬁnitions’’ instead of axioms of coordination. This was, I think, deﬁnitely not meant to be just a verbal change: he means to trade at that point on the conventionality of deﬁnition, the possibility to bring the deﬁnition—and therefore, supposedly, the coordination—into being by an act of stipulation. At the same time he means to be still actively inquiring into the empirical conditions under which such deﬁnitions can play the requisite role. What he did not do is change the ahistorical setting of his problem: the coordination is still apparently to be conceived of as possible in the absence of any previous such coordination. But is that possible at all? We have to ask more or less the same question again as before: how can such coordinative deﬁnitions be meaningfully introduced except in a historical context where there are some prior coordinations already in place?11 I submit that they cannot.

The problem of coordination reconceived How should we see the problem of coordination now? Not as a problem so posed as to preclude its own solution! Nor does it seem that we get very far by posing it as a problem that is solved by choosing a particular object to be designated as standard of unit length, or light-ray paths as physical correlate for geodesics. Coordination must determine how measurement can establish a value for what is measured. But that it can do so appears to presuppose an understanding both of the measurement procedure and of what is measured—that is, of the terms between which that relationship is established through coordination. So the prior question at issue is precisely:

122  : , ,   What is measured? In the assertion ‘‘the length of A was measured’’, or ‘‘the temperature of B was measured’’ or ‘‘the duration of C was measured’’, what exactly is the referent of the subject term? This question will look different depending on where we take our vantage point. There are actually two distinct vantage points common in the philosophy of science literature—not just about measurement but about experiments, models, theories, . . . But the divide becomes especially salient in the ways measurement is discussed. • We have no problem identifying what is measured by a particular procedure if we look at it ‘from above’, that is, in retrospect, from within a theory that is already stable and established, and deals with that physical magnitude. Looking at the matter from this vantage point we say that what was being measured all along, however well or poorly, in the centuries leading up to that achievement, was precisely what we say it is now. • Nor do we have a problem in our discussion of measurement if we look at it ‘from within’, that is, if we inquire into the introduction of measurement procedures at historical stages where there were already measuring procedures for certain other physical magnitudes taken as given. The examples of Mach on temperature and Poincar´e on time measurement, that we will look at below, illustrate this very well. It is only if we try to understand coordination without locating ourselves either in the ahistorical ‘above’ or the historical ‘within’, but pretend to a ‘view from nowhere’, that we set ourselves an impossible problem—or fall into metaphysical metaphors. If in our retrospective view today we want to describe the historical process, we can of course only speak our own language, the language we have today. And today it is a truism that an item has temperature T if the stable procedure that we now designate as temperature measurement, when applied to this item, has outcome value T. So if today we have to say what the earlier activities we focus on—such as Galileo constructing a sort of thermometer—were all about, and have to answer what was the intended referent, while measurement procedures

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and theories were still evolving, our answer had better be coherent with what we now take that referent to be. So we can only say: all along they were referring to the magnitude identiﬁable as measured by the procedures that would eventually be accepted as measuring it. So yes, it makes sense for us to say that Galileo’s activity was about measuring temperature, although we have to add immediately (speaking in our own voice, using our own words) that his apparatus did not really do that, that there was a mistake involved in his set-up. Notice how this answer echoes Reichenbach’s statement that what is measured, and thus through measurement coordinated with a theoretical concept, is not deﬁned independently of that coordination, but deﬁned by the coordination. At ﬁrst blush this formulation too may sound circular, even viciously circular, if not debilitatingly anthropocentric or egocentric. But it is not. What would make it viciously circular is only Reichenbach’s tacit assumption that what was being done could in principle be done without any historically prior achievements along such lines. But to understand this properly we must follow the process of coordinating both in its general form and in speciﬁc examples, to show how the joint evolution of measurement and theory can happen and can come to a stable resting point. What is measured? Can we identify the parameters that are measured by means of the measuring procedure itself alone? Well, what does ‘‘alone’’ mean here? Does it mean ‘‘with no theoretical background in which to operate’’? But with no theoretical background, very few procedures classify as measurement procedures. For example, as Mach and later Poincar´e emphasized, procedures for measuring mass presuppose something about the items to which they are applied, e.g. that they are subject to the law that action = reaction.12 This is precisely why Mach, and later Cassirer, Schlick, and Reichenbach, in their different ways, concerned themselves so much with the problem of coordination.13 But it is also through measurement results that theories gain empirical support, so there must be some trade-off, or some subtleties to how empirical support can be gained, if we are not to ﬁnd ourselves lost in a circular argument.

124  : , ,   The idea that the parameters can be identiﬁed simply through the measurement procedures was behind Bridgman’s operationalism, and we often see its strong appeal still in scientiﬁc writings. But the speciﬁcation of the measured parameter even in terms of the (ﬁnal, stable) measuring procedure can never amount to a complete identiﬁcation or deﬁnition. For the procedure will classify only when it is actually applied, and even then only establish order among the items to which it is applied, at the times when it is applied.14 To try and stick with an ‘operationalist’ view of what is measured would land us in the absurdity that a process or object has no character at all while it is not being measured. In practice a theory eventually emerges which encompasses the measurement procedure itself as well as the items measured, and provides the coordination. Thus in the case of temperature, the kinetic theory which was developed after thermometry had been developing for two centuries, provided the parameter which then was identiﬁed as precisely what was measured by the thermometer. So a deﬁnite identiﬁcation, a complete deﬁnition, of the measured parameter is possible but only through, at the hands of, and relative to the theory offered and ﬁnally accepted to account for the stability of the measurement procedure.15 We need to see this development in its historical perspective, and to recognize and distinguish two facts: (a) an empirical fact that has been discovered: namely, the very stability in the procedures found in this historical development, and the reliability of the predictions concerning these and their correlation with other measurement procedures derived from the mature theory in which they are now theoretically embedded. (b) a historical fact: namely, that choices in the development of these measuring procedures went hand in hand with the development of the theory in question, so that we cannot identify an aspect of nature that is measured if we refer only to those empirical procedures without using concepts provided by the theory.16 What now counts as simple passive measurement is a hard-won achievement.17 What were the choices involved? What epistemic status can we assign to these choices? We will not gain much clarity if we continue simply in the current abstract vein. We need to look at the actual history to get any grip on this at all.

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Mach on the history of the thermometer18 As Mach emphasized, it was already a matter of actual practice, before modern science turned its attention to the question of a quantitative measure, to evaluate bodies as hotter or colder or of equal temperature on the basis of common experience. But judgments based on perception do not yield an unambiguous ordering—as was clear from many common examples. Iron and wood never feel equally hot or equally cold no matter how long they are left undisturbed in contact, for example. Evaluations of hotter and colder tend to be reversed if the perceptions are reversed in order. As Poincar´e would later say for time measurement, psychologists could ignore this but not the physicist: some grading procedure with results that could ﬁt into a single scale was desirable. To pursue this desideratum is already a choice. Almost from the very beginning, the aim was to establish a numerical scale. The ideas of hot, cold, as hot as, and hotter than were to be transformed into the idea of a quantity with the same general structure as height or volume or speed. The air thermometer The word ‘‘thermometer’’ came into use in the mid-seventeenth century. In the ﬁrst half of that century there had already been widespread use of instruments that responded to temperature changes, based on the expansion and contraction of air. Galileo is sometimes credited with the invention; certainly there were examples in Italy in his time. Retrospectively they are called ‘‘air thermometers’’. A ﬂask with a long neck is vertically inserted into a container of water, with the opening downward. As the ﬂask becomes colder, the water in the neck rises. Similarly, when the apparatus becomes warmer, the water goes down. The differences in water height were at ﬁrst measured with compasses, but then divisions were marked with numbers attached. Records were made of how low and how high the water would be in winter and summer, or during the night and the day. The very set-up cries out for a simple numerical scale: how high is the column of water in the tube? Changes in that height indicate precisely the change in volume of the contained air. The guess is that the air expands with increase in temperature and contracts as it becomes colder. This guess does not presuppose that we already know how to measure how much

126  : , ,   warmer or colder it gets: the rough and ready comparison by the senses will do for that. But if the change in height of the water column, correlated with the volume of enclosed air, can be used as indicating how much hotter or colder the air became, then we will have a numerical scale induced by the most basic, most venerable form of already familiar measurement: length. Is there an empirical question involved in this choice? What cannot be ascertained without the use of a good thermometer is whether the change in volume is quantitatively proportional to the change in temperature. There will certainly be empirical questions to come, but this is not one. Prior to the construction of a thermometer, there is no thermometer to settle that question! So here the scientist confronts a choice, one that may be provisional to start, with hope of vindication of some sort or other later on: to use changes in volume as quantitative measure of how much hotter or colder one thing is than another. There are a number of drawbacks in this arrangement, if we try to think about it in current terms as measuring temperature. The water and the glass are also relevantly affected by temperature changes, even if not to the same degree as the air. The initial temperature of the apparatus will also play a role. But most important from that point of view is the effect on the water level of changes in atmospheric pressure—the instrument is, we might say now, part thermometer and part barometer. This could not even be realized or taken into account until the notion of atmospheric pressure was in play. The history of the barometer began during the same century, and like that of the thermometer was initially beset by theoretical disputes about the possibility of a vacuum. Hence the problem of effects of atmospheric pressure on the air thermometer did not appear till it could appear. The problem that the air thermometer was actually a mixture of thermometer and barometer was in fact pointed out by Blaise Pascal in mid-seventeenth century, after his barometric experiment on Puy-de-Dˆome. The liquid thermometer How was this problem to be avoided; how could one have an instrument to gauge temperature differences that was not ‘‘part’’ barometer? In the second half of the century the use of the liquid thermometer became common: a liquid in a glass tube, ﬁlled completely when at very high temperature and hermetically closed. When it cools to room temperature, the liquid no

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longer ﬁlls the whole tube—arguably, and at the time contentiously, one now witnesses the creation of a vacuum. (Questions on that score were crucial then; I shall ignore them here.) A common form had its length divided into equal parts. Detailed instructions recorded at the Accademia del Cimento in Florence, speciﬁed that in a good construction snow or ice will not make the liquid column descend below the 20 degree mark, nor the hottest day dilate it beyond 80 degrees.19 Since the tube is entirely closed, this arrangement eliminates the effect of atmospheric pressure, though not the effects of temperature changes on the liquid or the glass.20 The assumption or choice of deﬁnition that differences in temperature are proportional to resulting differences in volume continues, though now instantiated to liquids. Two obvious questions: do different liquids agree with each other in this respect? and do they indeed always expand when, by other common criteria, they are becoming warmer? These questions did not receive very satisfactory answers. The early makers of liquid thermometers used quite a variety of different liquids. Water was not a good candidate, since its volume does not keep contracting as temperature drops toward freezing.21 None of the liquids is of any use below its freezing point anyway. But in the regions well above their freezing point it was at ﬁrst thought that they all expanded and contracted in the same proportion when put in the same environment controlled for effects other than temperature variation—an appearance which in turn was refuted by Dalton for the case of high temperatures.22 The results are thus sensitive to the choice of liquid, partly because their response to temperature changes varies at very low and very high temperatures in a certain range—and the liquid state does not persist below or above a certain point. Even in ‘good’ ranges, the expansion in volume is not proportional for different liquids. Dalton wrote, a good century into this evolution, that liquids were found to expand unequally, ‘‘but no two of them alike. Mercury has appeared to have the least variation’’.23 But even for the otherwise very good choice of mercury, the effects on the glass tube are not negligible in comparison to the effects on the liquid. The expansion of mercury is only seven times that of glass. The gas law and kinetic theory These ﬁndings occasioned a return to the gas thermometer, especially in view of the discovery and reﬁnement of what is now often known

128  : , ,   as the ideal gas law (Boyle’s Law, the Boyle-Mariotte Law, the Law of Charles and Gay Lussac, Charles’s Law—a multiplicity of names partly due to nationalist partiality), PV = rT, with r a constant characterizing the gas. If pressure is kept constant, the volume varies proportionally with the temperature, for all gas in the same way—conversely, if the volume is constant, the change in pressure is proportionate to the change in temperature. Obviously the experimental ﬁndings could at best hint at this relationship before thermometry had been stabilized. But of gases, unlike of liquids, it could not be said that they expand unequally ‘‘but no two of them alike’’. Guillaume Amontons pointed out, in effect, that if a law of this form is correct, then the temperature scale can become a ratio scale: it will have an ‘absolute zero’, marked by the theoretical condition of zero pressure.24 Is this law correct? To begin its correctness is supported by experimental ﬁndings, made with thermometers of the previous generation, accepted as a rough guide while recognized as inadequate. Then, as conﬁdence in the kinetic theory increases, theory cuts the Gordian knot with respect to all questions about how the scale is to be ﬁxed. The gas law is supported by its incorporation in the kinetic theory—and reﬁned as well, so that it also becomes itself qualiﬁed there as having been only an empirical approximation to what really happens. But the scale is now anchored in prior scales for mechanical parameters. Once T is identiﬁed with the mean of a mechanical parameter, the postulated mechanical behavior—‘‘the type of motion we call heat’’, in Clausius’s incomparable phrase—is now the referent of what is measured. We can also put it the other way around: the parameter, identiﬁed by the eventually stabilized procedures for its measurement, is now classiﬁed by the theory as one aspect of the logical space that the theory provides for location of items in its domain. The history of thermometry did not stop here, but this much will sufﬁce to make the important points concerning coordination. Choices and conventions The new type of instruments that emerged in this evolution, before the kinetic theory rewrote the story of what has happened, were selected as correct for thermometry in a process that involved three main choices. Mach’s history of the subject emphasized perspicuously the sense and

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extent of these choices involved in how the scientists arrived at their results. The ﬁrst choice, which we have been emphasizing from the beginning, concerns what is to be taken as representing the quantitative change in temperature. The choice of change in volume passes the initial test of rough agreement with the admittedly imprecise deliverances of experience in familiar circumstances. The vindication of such an initial choice in the future is of course hostage to fortune. The second is the choice of thermoscopic substance, to answer ‘‘change of volume of what?’’ And the third is just what above, at the outset of our discussion, we saw Mach calling the ‘‘principle of coordination’’ for assigning numbers, that is, for setting the scale. (op. cit: ch. II, sections 11–12). There are coherence constraints on the choice of thermoscopic substance. We have seen that if it had been decided to stay with liquids, there would have been quite a lot of arbitrariness in this choice, ﬁrst of all because they do not appear to expand at the same rate even in the same range. Secondly, different liquids would have to be chosen for different ranges, as they near their different freezing or boiling temperatures. The gas law saved the day: it seemed that different gases expand at the same rate as each other under the same conditions. Again there are limitations, since real gases too could liquefy and solidify at different temperatures. The last choice, to set the scale, may now seem trivial. But it is not. The signiﬁcant choice involved is best seen in the discussion of whether there is an absolute zero, a minimum to the temperature scale. The main convention, historically, goes back as far as Galileo: to take the melting point of ice and the boiling point of water as selecting ﬁxed points on the scale. We have already seen that the change in volume of water will not be a convenient one overall. Using the gas law, for a particular gas, we can deﬁne the temperature scale either in terms of that for volume at a constant pressure or in terms of that for pressure at a constant volume. So Amontons proposed the pressure p of a mass of gas at constant volume, and chose as reference point the pressure p0 at the melting point of ice. Then the temperature scale can be deﬁned with an arbitrary scaling constant k as t = kp/p0 . The numbers k and the pressure scale can be chosen so that the difference between the temperature of melting ice and that of boiling water is 100, and there is an absolute zero of temperature for p = 0, whether this can be attained in nature or not.

130  : , ,   This is a very natural way to proceed, but if it is thought that the lower bound is a ﬁction, one could choose a different principle of coordination, not bounded below any more than bounded above. Mach points out that Dalton had in effect chosen such a scale, though of course that is not the one that caught on. The choice is settled later by the absorption of the entire subject of inquiry into physical theory. In the kinetic theory the magnitude with which temperature is identiﬁed has an absolute zero point. But it is well to note that this was not part of the empirical facts guiding the evolution of thermometry, though the kinetic theory was developed in a way that was accountable to that evolution of empirical procedures. The very simple operations we now perform with thermometers are an obvious example of ‘passive’ measurement—in our theories we have the interaction as well as the object and apparatus completely classiﬁed, ‘well understood’. This stage is an achievement. The story of thermometry is a success story, and such success was by no means guaranteed a priori. The conditions of possibility for such a successful coordination—here we are at Reichenbach’s question—we have now seen to involve empirical regularities that are contingent and choices subject to conditions of coherence.

Poincar´e’s analysis of time measurement The second half of Poincar´e’s essay ‘‘The measure of time’’ is the more famous because of its connection with special relativity.25 But I will concentrate here on the ﬁrst half, where Poincar´e begins with the problem that we do not and cannot have a direct intuition of the equality of successive time intervals (equality of duration of successive processes).26 This is not a psychological point. Two successive periods of a clock cannot be compared by placing them temporally side by side, that is why direct perception can’t verify whether they lasted equally long. This had been a recurring topic of discussion; Poincar´e mentions Andrade but across the Channel too it was a topic among the British Idealists and for the young Russell.27 Psychologists may ignore this, Poincar´e writes, but the physicist cannot. In the case of two sticks we can check to see whether they are equally long (at a given time) by placing them side by side; that is we can check spatial

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congruence (at that time) by an operation that effects spatial coincidence (at that time). We can check whether two clocks run in synchrony during a certain interval if we place them in spatial coincidence. These procedures do not sufﬁce for checking whether two sticks distant from each other in time or space are of equal length, nor whether distant clocks are running in tandem, nor whether a clock’s rate in one time interval is the same as some clock’s rate in a disjoint time interval. But in physics, criteria for spatial and temporal congruence are needed. Poincar´e is concentrating on this need. What measures duration is a clock, and physics needs a type or class of processes that will play the role of standard clocks. What type or class to choose? One answer might be: the ones that really measure time, that is, mark out equal intervals for processes that really take equally long. While certain philosophers or scientists might count this demand as intelligible, it must be admitted that there could be no experimental test to check on it.28 We cannot compare two successive processes with respect to duration except with a clock; but clocks present successive processes that are meant to be equal in duration. This is similar to Mach’s point about thermometry: whether the melting of ice always happens at the same temperature, or the volume of a substance expands in proportion to temperature increase, can be checked only with something functioning as a thermometer—and thus cannot be ascertained in order to check whether thermometers are ‘mirroring’ temperature. So Poincar´e presents the choice as a decision: to begin, he says, it was decided to ‘‘admit by deﬁnition’’ that successive periods of a pendulum had equal duration. This is not a historical claim about conscious or explicit decisions, let alone about the actual practices of time reckoning before the modern era. The clock mechanism driven by a descending weight was in use before Christiaan Huygens and Galileo independently invented the pendulum clock. Poincar´e is writing a ‘‘just so’’ story, a way to understand how choice not subject to direct empirical check is involved, no matter which mechanism is settled on to begin. Some choice—which may be subject to equally chosen constraints but not to empirical veriﬁcation—is needed to provide an initial starting point. Coherence conditions But it certainly is the sort of choice that is also subject to strong coherence constraints, which are not a matter of choice. The class of processes chosen

132  : , ,   as initial standard for equal duration must be such that if two are in spatial coincidence, they run at the same rate, they run in synchrony. The decision to choose the pendulum, in such bare form, cannot be sustainable in practice, precisely because it does not have this coherence. Different pendulums will disagree due to alteration in temperature, friction, barometric pressure, etc., and possibly beside these due to at that time unknown or uncontrollable factors. Comparing the behavior of different pendulums does not display stable synchrony. In 1672 Richer took a pendulum clock from Paris, latitude 49◦ North, to Cayenne in French Guyana, latitude 5◦ North.29 He found that he had to change the length of the pendulum to make it agree with standard time reckoning, precisely because as Huygens and later Newton were able to explain, the force of gravity is not the same everywhere on earth. So the next development, according to Poincar´e, was to correct mechanical clocks such as pendulums by the sidereal clock, which is based on the passing of a star across the meridian—the unit then being in effect the duration of one rotation of the earth. This too is a decision, and this time it was made explicitly.30 Yet this adoption of the sidereal clock was subject to a similar objection, if in a somewhat more theoretical guise: the movement of the seas will act like a brake, slowing down the rotation of the earth. Far from being a merely academic or skeptical doubt, this point was used to explain an apparent acceleration in the movement of the moon. Poincar´e wishes to reveal by these examples two problems that arise in developing a measurement procedure for duration. The ﬁrst is the initial one, illustrated with the pendulum: we cannot place successive processes side by side so as to check whether their endpoints coincide in time. So there is no independent means for checking whether successive stages of a single process are of equal duration: the question makes sense only after we have accepted one such process as ‘running evenly’.31 As said above for temperature, duration = the magnitude identiﬁable as measured by the clocks that would eventually be accepted as measuring it. This insight requires us to move all our concern to coherence constraints that put bounds on what we can choose to regard as ‘running evenly’. In another context, to illustrate that deﬁnitions are not totally arbitrary but in general have factual presuppositions, Poincar´e gave the example of

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the light-clock. Suppose we design a device in which light is emitted at one end of a meter-long path, reﬂected at the end, and absorbed again at the starting point. The interval between emission and absorption could be taken as the unit of duration. Now we carry this device and many faithful copies all over the place to be used as clocks. Any objection to this? Yes: in nineteenth-century physics it is entailed that these clocks will not give consistent results, because even rotating them with respect to each other will result in violations of synchrony. No: after the Michelson-Morley experiments, or more accurately, after Einstein’s special theory of relativity was accepted, these light-clocks are perfectly acceptable in principle: there is no necessary incoherence. The second problem is brought out by the astronomers’ rejection of the equality of sidereal days. Their rationale for this is an application of Newtonian physics to the earth and its seas. The application of this physics presupposes that the reference for its time parameter has been ﬁxed. Fixing that parameter by taking sidereal days to be equal runs into trouble precisely with the theory’s representation of the very processes that deﬁne the sidereal day. So Poincar´e diagnoses the astronomers’ practice as follows: they deﬁne duration in such a way as to ensure that Newton’s laws come out correct.32 But this decision, that the time parameter should be ﬁxed so that Newton’s laws will come out true, does not sufﬁce by itself! Newton’s laws could be re-expressed in terms of different ways of measuring time; they just would not have the same simple form. Suppose for example that with a certain adopted standard of measurement and its time scale t, these laws are satisﬁed. Now adopt a new scale T, which is a logarithmic function of t. It would be silly to reﬂect that Newton’s laws as originally stated are not satisﬁed if t is replaced by T: the thing to do is to translate Newton’s laws by means of the transformation of t into T. The result will be an empirically adequate theory if the original was—it will just be not nearly as simple or user-friendly. So in rejecting the sidereal day as time unit, the astronomers, according to Poincar´e adopted not just the above criterion but the following more complete one: Time is to be deﬁned in such a way that the equations of mechanics will be as simple as possible.33 Coherent rivalry and choice This quick conclusion we’ll have to look at critically in a moment, for surely Poincar´e is going a bit fast here! Simplicity is perhaps only a code

134  : , ,   here for a cluster of pragmatic desiderata for our theories. But we can add that the issue became concrete later on due to a proposition advanced in cosmology by Arthur Milne in the 1930s.34 By that time the corrected astronomical time reckoning that had become standard had been supplemented at least at the theoretical level with atomic clocks. It was part of the accepted theory that atoms of each chemical element and compound absorb and emit electromagnetic radiation at their own characteristic frequencies. As reckoned by the sidereal standard, these resonances had been taken to be inherently stable over time and space. In Milne’s time there was as yet no practical application—the ﬁrst atomic clock was not built till 1949. Recall now that there are coherence constraints: the class of ‘‘standard’’ clocks must be such that whenever two are in coincidence, they run at the same rate, they run in synchrony. In principle there could be two such classes, disjoint from each other, and not combinable in a way that satisﬁes this constraint. Milne proposed that this is indeed the case, and that astronomical (sidereal) clocks run differently from atomic clocks. The disparity is negligible outside cosmology. The relation between them, he submitted, was an exponential function, so that the two give us entirely different pictures of cosmic history: for the one, the past is ﬁnite and for the other it is inﬁnite. Although a lively topic at one stage in the development of cosmology, the idea did not have any continued impact. What it illustrates and is important, though, is the point of principle already made by Poincar´e. If the class of measurement interactions (of one sort or another) cannot have an ‘‘if and only if ’’ deﬁnition but only be required to satisfy certain coherence conditions, then the possibility of such disparities (and hence leeway for convention or choice) is entailed. Einstein’s critique; an example with length In his largely sympathetic discussion of Poincar´e’s views, Einstein famously added one telling criticism.35 While choice is involved in setting up the principles of coordination, this sort of choice can enter at many different points, and therefore it is not correct to single out any one element as a stipulation, deﬁnition, or convention. This becomes clearer when we consider the development of measurement processes for more than just one observable. Without some element of choice or convention, coordination is inevitably circular. So far I have,

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in all the examples, taken length measurement for granted. But suppose we consider length in the same way as we considered temperature and time measurement, where the principles of coordination presupposed length measurement as given. In order to ensure standards of length comparison that would remain applicable across time, another such history of coordination unfolded. One telling example occurred early on in a proposal by Huygens. After his invention of the pendulum clock, and his appreciation of how its period is related to latitude, he proposed that the unit of length should be one third of the length of a pendulum with a one second period at the latitude of Paris—not the English foot, not the pied du Roi, which was 16/15 of the English foot, but the pes horarius. Here time measurement is presupposed for the coordination of length. While this proposal did not prove practical, we see a similar entanglement of several principles of coordination at the most familiarly known stage: the construction of the standard meter embodied in the platinum bar to be kept in the French State Archives. Iron copies were distributed to other countries, and needed to be sent to France every ten years to be compared with the primus inter pares copy kept in S`evres. So we see two decisions installed as deﬁnitive: that such metal bars remain congruent to themselves under transport, and so provide a standard for congruence with respect to length, and that a particular such bar deﬁnes the unit of length. But what precisely was the convention? It was decided at the Convention of the Meter in 1875, and was precisely this: the meter is deﬁned to be the distance between the midpoints of the ends of the m`etre des archives at the temperature of melting ice. But is the temperature of melting ice always the same? The convention presupposes temperature measurement, just as the conventions for temperature measurement presuppose length measurement. The circle can be broken by entering a choice at one point or another, but there is no specially privileged point for entering a choice. Thus Einstein separated Poincar´e’s rather simplistic pointing to speciﬁc ‘conventions’ or ‘deﬁnitions’ from the real insight into the inevitable holism in the jointly constructed theory and coordination. As Ryckman observes, Einstein’s admission of Poincar´e’s viewpoint ‘‘is not a concession to conventionalism, that is to the freedom to choose any geometry we like, but to the inevitable epistemological holism of a theory in principle capable of explaining its own measurement appliances, and so its ties to observation’’ (Ryckman 2005: 64). But this holism will isolate the theory

136  : , ,   from what it is about unless the coordination is effective. This can be come about in many different ways, with an element of choice appearing at different points—just not without entering any choices at all. Let’s summarize it this way: any way you slice the cake into equal pieces, some choice is involved, but you can’t single out one slice as the unique result of choice and the others as compelled. Poincar´e was simplistic also in his emphasis on simplicity as playing a decisive role in the choices made along the way. There are many desiderata for theories in science. The bottom line is certainly and only empirical adequacy. Pragmatic considerations such as simplicity provide desiderata as well, but they are various and diverse. There is a practical need to align the continuation of physical theory in form with what is in place already. Thus the desire for explanation, or for conformity to certain favored patterns such as common cause models, could outweigh simplicity. None of this qualiﬁes the important point, that the crucial and necessary coherence constraints on what qualiﬁes something to play the role of physical correlate to measurement leave much leeway—so that choice is inevitably involved—and that there is no independent recourse beyond coherence. To be sure, this requirement of coherence is not simply one for logical consistency. Whether a sort of mechanism can be used to deﬁne the family of standard clocks depends on empirical regularities that may or may not obtain. The central coherence condition on the family of standard clocks, recall, was that if two are in coincidence, they run at the same rate, they run in synchrony. That is a matter of empirical fact. But if two sorts of mechanism satisfy this condition, there is no matter of fact as to which runs evenly, and a choice or convention alone can decide on one of them. There is a natural drive once the role of such choices is recognized, to reformulate theory in a format independent of those choices. The most salient example is the ever greater generality with respect to coordinatization, which had begun earlier in mechanics, and the later extreme freedom in this respect aimed for in general relativity and even more in the space-time theories of Weyl and Eddington.36 The other great example is the advent of group-theoretic formulations, in which invariance under as wide a class of transformations as possible is the guiding aim. But eventually these theoretical representations, which abstract from applicable models to that extent, have to be brought back into applicable form, and there choices determining the particularity of that form come back into their own.

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Observables coordinated: two morals There are at least two morals to be drawn here, and the ﬁrst is an antifoundational moral we can draw from Reichenbach’s impasse. The rules or principles of coordination that can be introduced to deﬁne particular sorts of measurement cannot even be formulated except in a context where some forms of measurement are already accepted and in place. In the use of astrolabe and ruler to measure distant lengths the local measurement of lengths and angles was taken for granted. In the construction of thermometry, the measurement of volume and weight, and later pressure, was already in place. In the measurement of duration, measurement determining pendulum periods, and later on astronomical alignments, was presupposed as already possible throughout. There is no presuppositionless starting point for coordination. We are, to adapt Neurath’s metaphor, sailors engaged in a continuing construction, renovation, alteration, and repair of the ways in which we measure our ship while already at sea. Reichenbach, I would say, erred on the side of philosophical prudence, precision, and caution, because he realized the crucial importance of bestowing genuine empirical content on the equations in a theoretical description. A case of philosophical scruples . . . but scruples, as the Church emphasizes, are also a source of error.37 There is a second moral. The opposite error is more tempting, and less excusable. It is the error that Reichenbach described and from which he rightly shied away (even if too far). For, as he saw, the natural temptation in response to the problem of coordination is simply to impose a parallel vocabulary and declare victory. One might say, for example, a point in Minkowski space corresponds to a real or physical spacetime point, which is a compact convex part of the  with zero measure.38 Reichenbach would be quite right to retort that this parlance is meaningful only after the problem of coordination is solved. It cannot be a solution! For this ‘‘solution’’ begins by thinking of the  as a structure, a set of elements with certain relations between those elements, described precisely in the terms we use in geometry. So it takes for granted that we can represent the  mathematically. And what is that, if not to say

138  : , ,   that there is a certain ‘representing’ relationship between the  and some mathematical structure?—thus assuming precisely what was to be illuminated.39 Reichenbach offers another familiar example as illustration. Boyle’s Law is an equation: PV = rT. To give this physical meaning, its terms must be ‘coordinated’ with physical quantities, pressure, volume, and temperature of gases. But what is, for example, the temperature of a gas? It changes with time, and differs from one body of gas to another, so isn’t it a function that maps bodies of gas and times into the set of real numbers? But if I say that, am I not already taking for granted that I have at my disposal a description of bodies of gas and times? Could this be a description that does not use terms relating to spatial boundaries and numerical comparison? I can try to cut the Gordian knot by saying ‘There exist physical quantities P*, V*, and T* which pertain to bodies of gas, and when ‘‘P’’, ‘‘V’’, and ‘‘T’’ refer to these respectively then ‘‘PV = rT’’ is true’. That is to do just what I said above: to create a parallel vocabulary and declare victory. But it is an empty victory.40 What that sort of naïve response ignores is that there is no independent epistemic access to the parameters to be measured—no access independent of measurement, that is. How could one decide, before a detailed theory is in place, whether or not the changing height of a column of mercury mirrors the changing temperature, except by use of a thermometer?41 So, before thermometry has been established, how could one decide that at all? The answer is that the question is absurd.42 A metaphysician may postulate that there are such physical magnitudes P*, V*, and T*, but s/he is building castles in the air if the next step involves some god-like vision or ontological telescope to compare their values with what our instruments show. Instead, as Mach’s and Poincar´e’s analyses show, measurement practice and theory evolve together in a thoroughly entangled way. Somewhat hesitantly one might say that the measured parameter—or at the very least, its concept—is constituted in the course of this historical development.43 Choices are made, and once made may encounter resistance, whether in experiment or in theory-writing or (more usually) in combination of the two—or else vindicated by smooth progress on both fronts. Mach oversimpliﬁed by keeping theory as much in the background as possible, and Poincar´e oversimpliﬁed by suggesting that it is mainly a matter of submitting deﬁnitions, in such a way as to keep theory as simple as possible.

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But their simpliﬁcations, even if going too far, honor the insight that the parameter that is measured is identiﬁed in the historical process by the envisioned eventual stable measuring practice, while it is differently identiﬁed in retrospect by the theory that draws on that history for its credentials.44 Within the vantage point of the accepted theory, once such stability has been achieved, we can speak meaningfully of the accuracy with which a given instrument gauges a given observable. At that late point, when the parameter has found its place in the theory that has emerged now, it is of course a characteristic which is no longer deﬁned by measurement, but by its role in the theory. That is right and legitimate also from an empiricist point of view, for this discourse draws on how objects, phenomena, and instruments are classiﬁed by (or classiﬁed relative to) the accepted theory.45 It does not presuppose an impossible god-like view in which nature and theory and measurement practice are all accessed independently of each other and compared to see how they are related ‘in reality’. The two ways of looking at the matter we must combine in a synoptic vision. The ﬁrst is ‘from within’ the historical process in which measurement procedures and theory are stabilized. The second is ‘from above’ with the theoretical description of the domain including the measurement interactions already in hand, a stage achieved in that historical process but no longer involving any explicit reference to its own history.46 We are not done with measurement, even with this realization. There is still a whole battery of questions about measurement that are not answered by our escape from the simplistic notion of number-assigning. On the one hand, a measurement procedure is a physical interaction. On the other hand, the measurement outcome ‘locates’ the object or process measured in a certain space of possible states. The physical and intentional aspects of measurements are to be looked into separately, before we bring them back into a synoptic view of how theory relates to experiment and observation.

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6 Measurement as Representation: 1. The Physical Correlate We have looked at measurement ‘from within’, and will now look ‘from above’: that is, try for a view that we can have after the measurement procedures, concepts, and theories have stabilized. This resting point in the conceptual and scientiﬁc development is, to be sure, ﬂeeting and momentary only—yet marking a context in which some things can be taken for granted pro tem, where it makes sense to ask about the world-picture of currently accepted physical theory. For the question What is measured? the most direct answer is Physical magnitudes that characterize the objects measured. At the point where the pertinent theory is stably established, what the relevant physical magnitudes are and how they are measured is itself speciﬁed theoretically. We cannot separate the questions What is measured? and What is a measurement?, and both of them have, eventually, speciﬁc answers from within the pertinent theory, which will classify certain interactions as measurements and their ﬁnal stages as outcomes. The interaction in question I will call the physical correlate of the measurement. The criterion for what sorts of interactions can be measurements will be, roughly speaking, that the outcome must represent the target in a certain fashion—, selectively resembling it at a certain level of abstraction, according to the theory—it is a representation criterion.

Physical conditions of possibility for measurement Recall for a moment Nelson Goodman’s example of the Duke of Wellington’s portrait that resembled the Duke’s brother more than the Duke. It shows that what the painting is a painting of is not settled by resemblance.

142  : , ,   Granting that, let’s ask though how we should classify this example, while thinking of how Dürer’s Art of Measuring depicts the perspectival painter’s art. Perhaps the portrait resembled the brother more, because the brother was in fact the artist’s model while painting the portrait? That would not have been so remarkable—perhaps the artist had noticed how much the brothers did resemble each other, and the Duke was not always available for sitting. Not only not remarkable: in no way threatening the status of the painting as the Duke’s portrait. To take a more extreme case, Dante Gabriel Rossetti’s painting The Girlhood of Mary Virgin depicts the Blessed Virgin in the house of her parents. But the model for Mary was Christina Rossetti, while Mrs. Gabriele Rossetti was the model for St. Anne. That does not make the painting a family portrait of the Rossettis. It is without a single doubt a picture of the Virgin Mary’s childhood, although the painted ﬁgures—equally undoubtedly—resembled Christina and Frances Mary Rossetti more closely than either the Blessed Virgin or her mother. What if the artist commissioned by the Duke of Wellington had not only used the Duke’s brother as model, but had used one of Dürer’s machines for drawing, so as to produce the painting? In that case we say that the drawing is a measurement outcome, but the measurement was surely not a measurement of the Duke! So now the ‘‘of’’ in the two contexts diverge: we will still grant the result the status of portrait of the Duke, however it was made, but not that the operation carried out was a measurement of the Duke.1 While the criteria for what a portrait is a portrait of are various, cultural context-dependent, and sensitive to social factors, that is not quite so for measurement, however similar they may be in other respects. In the case of measurement the physical conditions of the object-instrument interaction have more weight, and are in fact the subject for a good deal of foundational research in the sciences themselves. Focusing on the physical correlate The answer What is measured are physical magnitudes that characterize the objects measured can be elaborated, after science has reached the requisite stability, in the form: What is measured is a physical system of a certain sort. Being of a certain sort, this system is characterized by certain parameters, and a

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measurement made upon this system is a measurement of one of those parameters that characterize it. For at that stable point in the joint evolution of theory and practice there is a theoretical classiﬁcation in place, of the objects that can be measured, before those measurements are made. Even the humble substances to be found on my breakfast table today are already so classiﬁed, by currently accepted theory, that it makes sense to ask of any one of them whether it is a sodium compound or whether it contains electrolytes; and there are measurement outcomes that would answer the question. While Reichenbach focused on the physical correlate of such theoretical notions as geodesic, we can now raise questions about the physical correlate of measurement per se. Measurement is an operation by whose means we gather information; but this is of course done by way of a physical interaction between apparatus and object, to play the information-providing role. Measurement always involves a physical interaction between ‘object’ and ‘apparatus’; that interaction we can call the physical correlate of the measurement.2 This distinction, between the measurement and the physical interaction by whose means it is accomplished, comes into force when we ask just what the theory, speaking in its own terms, must provide by way of a representation of the measuring process. If that interaction is in the theory’s domain, the theoretical description will be of this interaction in the same terms as any other physical interaction, and involve no terms that signify anything intensional or intentional. At some point below we shall narrow our focus to the measurement outcome, which is the information provided by the ﬁnal physical state of the apparatus in that interaction. There the same distinction needs to be made: the ﬁnal physical state of the apparatus is the physical correlate of the measurement outcome. That is not a matter of meaning, but concerns a physical precondition for meaning. No information is gathered at all unless the outcome means something; but it can’t mean anything relevant unless it appears at the end of a suitably structured physical interaction, and takes a particular sort of physical shape. Let us concentrate on the case in which physical theory is taken broadly enough so that this interaction is in its proper domain. We take it for granted then that the theory provides a representation of that physical interaction, in physical terms.

144  : , ,   We can put this partly in terms of language. A claim of the form ‘‘This is an X-measurement of quantity M pertaining to S’’ makes sense only in a context where the object measured is already classiﬁed as a system characterized by quantity M. To so describe an object is already to classify by theory. Therefore the claim is theoretical or at least theory-laden, and has to be treated as such.3 It is a claim which will change in content and in truth conditions as our accepted theories change, since our classiﬁcation of physical systems changes along with the theories. Here too we see a difference in the views from within and from above. From the posterior point of view the new theoretical claims are not displacements but reﬁnements of the old—precisely because the old claims are now seen as (usually in some way imperfect) versions of less precise claims formulable in the new theory.4 This context-dependence of the claim may not be visible when we discuss a measurement procedure that was stabilized so long ago that we now take the description of the relevant objects and apparatus for granted. No one today would think of the use of a thermometer as being a meaningful practice only within certain theory-laden contexts. The practical use of these instruments has attained such stability that it is for now at least not at all sensitive to changes in the background theories. But that context-dependence springs to the eye in cases where the relevant theory is relatively new and has some claims to descriptive completeness—such as relativity theory or quantum mechanics.5 In both of these, the behavior of the measurement apparatus is indeed within the domain of the theory itself, and the criteria for qualifying as [physical correlate of] measurement are to be presented in the terms of the theory—with consequent questions of coherence and consistency inevitably forced on us.6 Philosophical reactions to this point have included a good deal of extremism. There has been outright denial, as in the operationalist and early positivist delusion that we could have a hygienic ‘observation language’ in which the measurement operations and their outcomes could be described free from all theoretical content. Though we still sometimes see the logical positivist and logical empiricist circles of Vienna and Berlin identiﬁed with this point of view, it was in fact quickly abandoned there. The critique by Hanson, Toulmin, and Kuhn in the ﬁfties and sixties removed it from the philosophical scene altogether—or should have. The opposite extreme in denial came then, ostensibly in a famous chapter by Kuhn: scientists in different centuries live in different worlds.

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That extreme is still worth some scrutiny even now. Understood in its most blatant literal sense, it involves a denial that there is any such thing as the physical interaction prior to or independent of the theoretical context in which it is classiﬁed as such or such a measurement. Just to say this shows it up for what it is; but for that very reason, shows that this was not what was meant. The real point is that occurrence of the physical interaction is one thing, its playing the role of representation of the value of a certain parameter is another; and today everyone understands that.7 To escape from both extremes, we need to separate quite clearly the historical process, examined in the preceding chapter, in which measurement and theory are jointly stabilized, from the retrospective characterization of measurement in the eventual established physical theory. The question of coherence How is the measurement interaction represented in the theory? The answer will be different depending on which measurement interactions are meant, and which theory. Stepping back from the speciﬁc cases for now, we ask the more reﬂective question: what must the answer be like, if it is to be satisfactory? The main criterion must surely be this: the theoretical characterization of the measurement situations is required to be coherent with the claims about the existence of measurement outcomes, their relation to what is measured, and their function as sources of information. For this coherence it is required that the theoretical characterization of the measurement interaction allows for a coherent story about how its outcomes provide information about what is being measured. These coherence conditions we need to spell out when we characterize the physical correlate of measurement, in a context where it is assumed that both sides of the interaction have already been well understood and classiﬁed. Recall that ‘‘This is an X-measurement of quantity M pertaining to S’’ makes sense only when the object measured is already classiﬁed as a system characterized by quantity M. This will be so if the context of discourse is governed by a theory; and it is in this kind of context that we can sense a challenge to consistency or coherence. What does this theory, or an accepted background theory, imply about the measuring process whence the theory reaps its credentials?

146  : , ,   On the simplest view, the outcome of a measurement of quantity M on system S reveals the value that M has at the time the measurement is made. Even in the most familiar cases, however, this must be qualiﬁed. The process takes time, so is what is revealed: the value of M at the outset, or at the end, or . . . ? The ﬁnal reading on a thermometer placed in a cup of tea we can take to correspond to the tea’s ﬁnal temperature. Why? Because the theoretical description of the interaction implies, via a pertinent conservation principle, that the two end up with the same temperature. The coherence of this reading of the thermometer is thus underwritten by theory. But the question that was most likely being asked was how hot the tea was? In other quite familiar cases, measurements are destructive: the patient does not survive the test [unaltered], the photon is absorbed, the metal sample fractures or is vaporized. Then the measurement does not reveal the ﬁnal state of the object measured, since the object exists no more. But the test result should at least show whether, for example, the patient did have one illness or another. Measurement can interfere or destroy—and perhaps the most direct indication it typically yields is of the state of the object at the end. But such a process will not serve its function as measurement (rather than, say, as preparation) unless some information can be gleaned about what the target was like before the measurement—isn’t it?8 So, granting that the outcome has its signiﬁcance as such only in a theoretical context, let us ask: what information can be derived from the measurement outcome, concerning the initial state of the object, sub specie the theory? In the case of the thermometer the question would be: given the ﬁnal reading, what can be inferred via thermodynamics about the tea’s initial thermal state? It looks like circularity threatens at once: to infer the initial temperature of the tea, we’d have to know the initial temperature of the thermometer, among other things. But the circle is easily broken: thermometry already relies on some identiﬁed ﬁxed point, such as the freezing point of water, and the thermometer can be prepared to be initially in thermal equilibrium with such a ﬁxed reference point. Can the circularity be so easily evaded in general? I think not. The familiarity of these classical examples and the stability long since achieved in their treatment may tempt us to assume that, but in fact, there are puzzles to be faced.

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General theory of measurement The greatest felt tension between the measurement procedures that we know by acquaintance and their description in a theory is found in quantum mechanics. Not surprisingly, therefore, that is where we ﬁnd also the most extensive treatment of the physical side of measurement. Under the heading of ‘‘quantum theory of measurement’’ we do not ﬁnd solutions to philosophical problems.9 Instead we ﬁnd there simpliﬁed and idealized but straightforward accounts of the physical [part of the] process. These accounts have a general form that has considerable claim to universal applicability in the empirical sciences. I shall take that form, depicted non-technically, as my central example of theoretical descriptions of measurement. Initial measurement set-up A proper measurement process must involve, besides the object on which the measurement is to be performed, a separate system to play the role of measurement apparatus. The procedure is meant to address a particular parameter, property, quantity, or observable (essentially interchangeable terms) that pertains to the object. The apparatus is to be coupled to the object in such a way that a process will occur that counts as a measurement.

Figure 6.1. Measurement Schema

What are the criteria? If that process is a measurement of property A, say, then there must be a corresponding property B, of the apparatus,

148  : , ,   whose ﬁnal value will be the outcome. That process has to be governed by an equation which has to guarantee something about properties A and B and that ﬁnal value—but classical intuitions beware! In addition, the system denoted as apparatus has to be such that it will play this role quite independently of what state the object is in—give it any object in any state and it will play its role.10 What must the governing equation guarantee, in order that this process really be the measurement of some property pertaining to the object? In a classical context we might say something like: if the object is thus or so then the end-state of the apparatus has to indicate precisely that.11 This assumes that the parameter measured already has a deﬁnite value before the measurement, a value that can be revealed by the measurement outcome. But this ‘‘revelation’’ assumption is at best controversial and certainly generally rejected in foundational studies of quantum mechanics. Measurement does not ‘reveal’ (?) Complacency about measurement was sorely tried as the quantum theory developed. There are now many arguments in the literature to show how that theory requires a revision in our demands for what constitutes measurement. Instead of repeating these here, I will illustrate such reasoning with a simpliﬁed example that displays the sort of cautions we must entertain.12 Consider a strange situation characterized by three physical parameters, A, B, and C. The data are that, for each, measurement yields values 0 and 1 with equal frequency. We are able to measure them two at a time, and always ﬁnd opposite values. Here is a simple theory that generalizes on these data: For X, Y = A, B, C: the probability that X has value 1 equals 0.5, as does the probability that it has value 0. But the probability that X and Y both have value 1 equals 0 if X = Y But this simple theory is self-contradictory. For it assigns probability 1 to both of the following: No two of A, B, C have the same value Each of A, B, C has value 0 or 1

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It is impossible for both to be true! What went wrong? The data listed were about measurement outcomes, while the extrapolated probabilities were absolute and unconditional. The proper generalization must give due attention to the fact that what was found was how things appear in the measurement set-up: For X, Y = A, B, C: the probability that X has value 1, conditional on its being measured equals 0.5, as does the probability that it has value 0. But the probability that X and Y both have value 1, on the condition that the two are measured together, equals 0 if X = Y Now there is no contradiction, but rather the consequence that A, B, C cannot all be measured together. So that is our solution. How good a solution is it? We want to ask, surely, what value C has while A and B have values 1 and 0 respectively. But how this question is to be understood depends on what we take for granted here. Can we take for granted that when not measured, A, B, C still can only have values 1 or 0? Or could they have other values when not measured? Or perhaps have no value at all? And secondly, can we take for granted that if a measurement of A shows value x, then at that moment, A has value x? Or that it had value x at the beginning of the measurement? Could we perhaps reason like this? Suppose A is measured and the outcome is 1. Now we can predict that a measurement of one of the others, B or C, will have outcome 0, with certainty. On this basis, can we assert that B and C already have value 0 at this moment? If we do, we will have to add that joint measurements of B and C that are actually carried out are systematically deceptive, for they never show them having the same value. Before seeing an example like this, or at least before having any inclination to take it seriously, various assumptions involved in such reasoning would likely have been taken for granted when thinking about measurement. Classical intuitions (if such beings exist) suggest two postulates: Value Deﬁniteness: Each physical parameter always has some value, namely one of the values which may be found by measurement. Veracity in Measurement: Measurement of a parameter faithfully reveals the value it really has.13

150  : , ,   These two postulates can be consistently added to our above story, but then they imply that either measurement is subject to or involves a systematic alteration, or else some sort of conspiracy in nature constrains when measurements are made: when A and B both have value 1, we are lucky or clever enough not to measure the two of them! Perhaps we should put it this way. The conjunction of these postulates would be an attempt to say that the world is basically the same, whether things are being measured or not. But given the above story, the two postulates are both true only if things are not basically the same whether things are being measured or not! So the attempt fails: some difference between measured and unmeasured world will have to be admitted as a possibility here. Paul Feyerabend’s (1958) name for the postulate of Value Deﬁniteness was Classical Principle C. This principle must be rejected, he argued. From Bohr to Feynman, physicists have expressed similar opinions: an observable (measurable parameter) might not have a speciﬁc value outside the context of measurement. However, the second postulate—Veracity in Measurement —has also been much looked upon as a candidate for rejection or revision. To keep the ﬁrst postulate and reject the second, by means of an explanation through uncontrollable disturbance by measurement, would not be a happy option. It would imply some sort of conspiracy again: if A and B do sometimes both have value 1, how does the ‘‘uncontrollable’’ disturbance in measurement carefully and systematically hide that fact? Finally, rejecting Value Deﬁniteness would by itself already imply a weakening of Veracity in Measurement. For if, at a certain time, parameter A has no value, and is measured, then this measurement yields a value as outcome, but clearly does not reveal a value. When we specify what counts as a measurement of A, we describe a physical arrangement which must have one of two outcomes (indicator values), in this case 0 or 1. For this to be a measurement, hence to play a role in information gathering, it must surely do something that is revealing about what is measured? But what sort of information it does yield, and how much, we shall have to consider very carefully. General theory form What the apparatus is like at the end must reﬂect some pertinent feature of what the object was like initially. There must be a transfer of some character

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of the object’s initial state to the apparatus’ ﬁnal state.14 Otherwise there would be no way to use this process to gather information about the object on which the measurement is performed. How this ‘transfer’ requirement is to be made precise will of course be different in different theories. For the general form we must allow for the case in which the relation between physical state and measurement outcome is only characterizable in terms of probabilities. A deterministic theory can be thought of as the special case in which all probabilities are zero or one. The measurement situation modeled as theory prescribes, when the apparatus is itself in the theory’s domain of application, must include a speciﬁcation of the following factors:15 a family M of observables (physical magnitudes) each with a range of possible values; a set S of states—physical states of both the system measured and of the measuring system; a stochastic response function P s m for each m in M and s in S, which is a probability measure on the range of m; with P s m to be interpreted as the probability that a measurement of m will yield a value in E, if performed when the state is s. Suppose now that one sort of process represented in the theory is that of the interaction that qualiﬁes as measurement of an observable. The situation depicted then involves two systems, the object measured S and the apparatus R by which it is measured. Together S and R constitute a larger system, a ‘two-body’ system, S + R. The family of observables must then include some that pertain just to that object S, some that pertain just to the apparatus R, and some that pertain to both at once, that is, to system S + R—and similarly for the states. But there must also be a constraint on how this situation evolves, as the two objects are coupled and interact. Classifying R as an apparatus, for measuring observable A for example (an observable pertaining to the object to be measured) entails that this interaction will take a certain form, which qualiﬁes the designation as ‘‘apparatus for measuring A’’. In fact, this interaction must be such that something pertinent about the initial state of the object is reﬂected in the ﬁnal state of the apparatus. So imagine the apparatus as having a dial with a pointer—as a system it is characterized in part by an observable B, the pointer observable,

152  : , ,   whose possible values are precisely the numbers which the pointer can indicate on that dial. Now the criterial condition, in its strictest form, must be this: Criterion for the Physical Correlate of Measurement: PB ﬁn (E) = PA init (E) where ﬁn is the ﬁnal state of the apparatus, and init is the initial state of the object on which A was being measured. That is, the ﬁnal state of the apparatus must reﬂect, in its probabilities pertaining to pointer observable B, the probabilities pertaining to measured observable A in the initial state of the measured object.16 This includes as a special case that the pointer observable B on the apparatus would most certainly show e.g. value 17 if inspected at the end, on the supposition that the measured object was initially in a state in which observable A ‘most deﬁnitely’ had value 17. To this extent then the old criterion of Veracity (or revelation) is being honored still. The coherence constraint But notice the qualifying clause ‘‘if inspected’’ and the qualifying quotation marks that we found it necessary to insert in the preceding paragraph. For the probabilities PB ﬁn (E) and PA init (E) are understood as conditional probabilities—conditional on measurement! At this point one begins to have the sense that there may be ways into the enchanted wood but no ways out. What does the Criterion for the Physical Correlate of Measurement come to when we look at it in this light? Firstly, the theory will have implications for results of other measurements (of observables A , A , etc. ) on systems with the same sort of preceding history, depending on the results from the A-measurements. This is one way in which inconsistencies could arise with the theory when different measurements are made, so we have here an empirical coherence condition. This condition depends so much, however, on the speciﬁcs of the theory in question that it may not be possible to say more about it in general. Secondly, the theory must imply that if two other apparatus were coupled to these systems, to the object at the beginning and to the apparatus at the end, the probabilities of the corresponding outcomes would be the same.

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Figure 6.2. Coherence of Measurement

That is, suppose that we had two further apparatus, R1 and R2, the former one also used to measure A on system S and the latter to measure B on apparatus R. Then the Criterion’s being satisﬁed mutatis mutandis for all measurement interactions will guarantee the coherence of the ‘exterior’ measurements with the ‘interior measurements’. For a comparison of the probabilities pertaining to R1 and R2 should show an accord that ‘reveals’ that the accord between S and R demanded by our Criterion was satisﬁed. The theory must predict that if a second independent measurement was made to secure the premise that the object had been prepared in a certain initial state, and a third measurement were made of observable B on the apparatus end-state, then the demanded accord would be found with the requisite probabilities. This is therefore a coherence constraint on the theory: it must ﬁrst of all have this internal harmony in what it predicts, but secondly, comparisons of results in the two kinds of set-ups must be empirically vindicated. So here we have a coherence condition that is partly internal consistency and partly empirical. The theory is to satisfy that coherence constraint—and that is the most we can ask of it. Put in terms of the sort of description the theory provides for physical systems in general, this ‘physical theory of measurement’ is not plagued by any sort of circularity.17 At the same time, of course, it does not pretend to aim at a deﬁnition of measurement in terms that have nothing to do with measurement! In practice, the level at which a theory confronts experience is not that of raw data taken from individual measurement outcomes, but of the ‘data

154  : , ,   models’ constructed on their basis, and the further smoothing of the data models in which for example sequences or discrete variables are replaced by continuous functions.18 But the conceptual problems—such as the ‘measurement problem’ of quantum mechanics—refers to the individual outcomes of measurement interactions, modeled in the above fashion. I will return to this distinction between data and data models below. The point is that, however we conceive of this, the above coherence constraint will have to apply to how the data from measurement are to be accommodated by the theory’s theoretical models. Veracity reconsidered The criterion imposed on the physical correlate of measurement is as strong as possible, given the general form of a physical theory that was under consideration. That form, in turn, was kept very permissive, so as to allow for the form that quantum mechanics took when it was formulated deﬁnitively c. 1925. As a result, the conditions on measurement had to allow violations of the two ‘classical’ principles that we had noted: Value Deﬁniteness and Veracity in Measurement. As to Value Deﬁniteness, nothing in the Criterion for the physical correlate of measurement precludes the observable A to have no value (or only an ‘unsharp’ or ‘fuzzy’ value) at the outset. In fact it is not even implied that the pointer observable B will have a speciﬁc one of its possible values in the ﬁnal apparatus state. All that is implied is that if measurements be made, to measure those observables, the possible values will appear as outcomes with certain probabilities. Value deﬁniteness is not implied in any sense, way, or form.19 Veracity is implied only in the very much weakened form of accord among conditional probabilities. But if the theory speciﬁes nothing beyond those conditional probabilities then no stronger criterion can even be formulated for the physical correlate of measurement, to the extent that the theory can cover that. If we now return to the empirical assertions, we are not bound to stay within the theoretical description, and we can refer freely to the actual outcomes of measurements, as typically summarized in data models. These data models are constructed from the raw data that are actually gathered, so we are dealing here with actual frequencies, and probabilities that have a good ﬁt to those frequencies. Suppose now that such a ‘summary’ is pretty well a picture of a state in a theoretical model, to the extent of

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displaying probabilities derivable from that state. Then the theory may, and generally does, have implications for how that state evolves in time.20 Thus predictions can be made about what will be found if new measurements are made in that situation at a later time. As an example, we can take the Stern–Gerlach apparatus, named after Otto Stern and Walther Gerlach’s famous 1920 experiment. There is a classical picture behind the idea of the experiment: imagine a ‘beam’ of particles being emitted, with a particle being like a classical dipole with two halves of charge spinning quickly. In a magnetic ﬁeld, such a particle will begin to precess. This way the particle’s position becomes perfectly correlated with its spin value. That is also the result, mutatis mutandis, with the situation described in quantum mechanics. When the spin in a given direction can have only two values, the beam is split into two separate beams, ‘upper’ and ‘lower’. The early attempts to realize the experiment encountered many difﬁculties (cf. Bretislav and Herschbach 2003). In the ﬁrst realization a beam of silver atoms (produced in an oven, at temperature 1000 ◦ C) was collimated by two narrow slits (0.03 mm wide) and traversed a deﬂecting magnet 3.5 cm long with ﬁeld strength about 0.1 tesla and gradient 10 tesla/cm. The splitting achieved was only 0.2 mm, and there were doubts as to the data obtained. (Of course the set-up is now described in rather more ideal terms when it is used to illustrate quantum properties.) The apparatus can be rotated, so as to measure spin in any direction. Thus data on different spin observables can be collected on some samples produced in the oven, and on the basis of the frequencies in those samples, it is possible to infer—via the theoretical description and classiﬁcation of this process—just what state is prepared by that source.21 Then the proportions of the output in the two channels in later measurements can be predicted on that basis. It would be illegitimate to conclude that the silver atoms exiting in the upper channel were prepared in the oven in a state of spin-up. Rather, the oven prepares a beam of atoms in a state which is such that the probabilities of a Stern–Gerlach measurement having outcome up or outcome down are deﬁnite, with the result that the relative proportions in the two channels are deﬁnite. But still we can see now that Veracity is honored at some appropriate level. The outcome does not reveal a prior state for an individual silver atom, but the frequencies in the outcome do give information about the prior state in which the source prepares what it

156  : , ,   sends out. If it were not so, the role of measurement would not be played at all, since the outcomes would not serve to provide information about what the measurement is performed on. In practical terms it is precisely the source on which the measurement, taken as a whole, is performed.

What is not measurement In discussing the physical correlate of measurement we are displaying physical preconditions, conditions for the possibility of measurement. These do not deﬁne what measurement is, and in fact there will always be many interactions conceivable in nature that satisfy those conditions but are not measurements. For measurement is information gathering, a measurement outcome is something that has meaning, is in fact a representation of what is measured, and that point does not reduce to a physical condition. In the foundational literature on physics, this distinction is mostly honored by neglect—reasonably so, since the focus there is on what is within the intended domain of application of physics. In the sense of ‘‘measurement’’, as the term is typically used there, there are (according to the pertinent theory) measurements going on all the time in the stratosphere and at the sub-atomic level, far away from what we can do or use. So in this usage of the term, there is no reference to practice or to our information gathering. On the macroscopic level too we can think of processes that connect two situations separated in time or space. These could be so correlated that it would be possible in principle to get information about the one by inspecting the other—provided of course we knew of that correlation! But that something could be done does not mean that it is done. That something could be assigned a representational role does not mean that it has one. There are contexts where the distinction between a measurement and its physical correlate can be neglected, but philosophical reﬂection is not one.

7 Measurement as Representation: 2. Information A measurement outcome is something physical: an event, the end-state of the apparatus, or an object (photo, graph, list of numbers) produced by the measurement process. On the other hand, measurement is informationgathering, so a measurement outcome has a meaning (i.e. information content). The information provided is ‘‘read’’ off the measurement outcome, which is a physical state or event, but one with meaning to the literate.1 The term ‘‘measurement’’ is an endorsing term. If we call something a measurement, we imply that there is something correct or valuable in the way it yields a representation (even if on this particular occasion it went wrong). Measuring is an operation by which we can produce or gather information; and here ‘‘information’’ too has an endorsing sense. The communication engineer’s neutral usage of ‘‘information’’ has begun to modify common usage to some extent. But even now it would still be puzzling or provocative to hear ‘‘We get information from observation, measurement, ﬁctions, lies, and popular astrology.’’ If we insist that measurement is information gathering, we mean in part that ﬁctionalizing or speculating or guessing is not measurement. Measurement is an operation, using something that functions as an instrument, to gather information. The instrument is being used, in Heidelberger’s terminology noted above, in a representative role, that is, the operation has as outcome a representation of the object or situation operated on.2 Since many sorts of operations can have such outcomes, this notion needs to be narrowed. Attention to more complex measurement situations, that need to be assimilated via a series of steps in model building, leads—I shall argue—to the need for a broad concept in which measurement is itself a speciﬁc form of [self-]location.

158  : , ,  

What is measurement—number-assigning ? We do ﬁnd simplistic answers sometimes even in places where it matters.3 Lord Kelvin, with a well-deserved reputation as scourge of purported sciences outside the ken of physics, wrote: I often say that when you can measure what you are speaking about and can express it in numbers you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind [and] you have scarcely . . . advanced to the stage of science . . . . (Thomson, Lord Kelvin, 1891: 80-1)

This sort of view was developed more systematically by Norman Campbell (Campbell 1920, 1928, 1943) in his inﬂuential writings on measurement. More historically important to the subject is S. S. Stevens’s seminal paper for the behavioral sciences in Science 1946, which appears at ﬁrst sight to express Kelvin’s view, for he writes [W]e may say that measurement, in the broadest sense, is deﬁned as the assignment of numerals to objects or events according to rules. (Stevens 1946: 677) [W]e may venture to suggest by way of conclusion that the most liberal and useful deﬁnition of measurement is . . . ‘the assignment of numerals to things so as to represent facts and conventions about them’. (Ibid., 680)

Stevens’s article was responding to a commission which had attempted to arrive at a general, comprehensive account of measurement. The focus in that historically important paper was quite narrow: the mathematical foundations of measurement, in the sense of a study of conditions under which an algebraic structure is embeddable in the real number continuum.4 Almost all of his discussion pertains solely to measurement scales and the sort of invariance that is required for meaningful scaling, and is only superﬁcially linked to either the process or the role of measuring in scientiﬁc practice. The algebraic structures in question are described informally in terms of physical addition operations, but this is kept skimpy and abstract. Stevens is able to maintain those slogan formulations only by, on the one hand, extending the concept of number beyond its normal use, and on the other hand, by restricting himself to the abstraction involved in a mathematical point of view.

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In fact the idea of an essential link between measurement and numberassigning comes from taking a myopic view of mathematics and an equally myopic view of measurement. Mathematics is not the study of numbers. Numbers are merely a particular if useful and user-friendly instance of mathematical structure. Granted, the numbers’ salience is not the only reason to think of measurement mainly in terms of numerical outcomes. Some paradigmatic examples of measurement are indeed numerical. But as Dürer’s Treatise on Measuring already brought out, we will have to look beyond those. Kinds of scales The guiding idea for the study of measurement scales is that the grading must be thought of as reﬂecting characteristics of operations on something physical—operations that can plausibly be called measurements in some ‘ordinary’ sense. Presumably Stevens was referring to this with ‘‘according to rules’’, though what sort of rules for operating on real physical things count as measurement recipes was left aside. The constraints on the possible outcomes that he describes are designed so as to guarantee that what can be measured is precisely what can be represented as graded on a real-number scale. Mathematically speaking there are many structures other than the real number continuum. Even in the ﬁeld called ‘‘foundations of measurement theory’’, the conception is already so generalized that the standard numerical version can only be seen as a particular case. The authors in that ﬁeld are aware of the limits of numerical representation, and Stevens himself introduced the now commonly found distinctions: nominal measurement is the assignment of (numerical) labels without implying any algebraic structure; ordinal measurement assigns a rank ordering; interval measurement is ranking on a scale where only the intervals between elements are numerically comparable; ratio measurement is ranking on such a scale where there is also a minimum, and the ranking can be represented by non-negative numbers with the ratios between these numbers reﬂecting a physical relationship as well.5

160  : , ,   Each of these categories has its examples. The Mohs hardness scale for minerals is the typical example of an ordinal scale.6 If we ignore absolute temperature then our thermometers, whether Fahrenheit or Celsius, provide an interval scale—the Kelvin scale for temperature with its absolute zero is a ratio scale.7 But Stevens’s taxonomy is not exactly a table of categories supported by a transcendental deduction! It looks nicely hierarchical: we can suppose that ordinal measurement must be a special case of nominal measurement with the labels reﬂecting the ordering. Then each category besides the nominal presumes an ordering which is linear: two assigned labels x and y must either be the same or have one greater than the other. As was much emphasized in nineteenth-century discussions, notably by Mach and Poincar´e even this ordering requires a contingent empirical regularity for its coherence.8 To be able to order at all, even in its most minimal logical sense, one needs at least a criterion of equality. Suppose this be speciﬁed, and suppose that by this criterion A and B are both equal to C. Does it follow that by the same criterion A will be equal to B? No, it does not, at least not logically, if the criterion refers to a performable physical test. But the relationship of equality, that is of having the same position in the ordering, must be transitive or the ordering falls apart. (Similarly of course for the relationship of greater than or after if applicable). What criterion is proposed is up to the proposer, and if there is a a variety of plausible candidates for the criterion then this may be a matter of choice or even convention. But whether a proposed criterion can be adopted will depend in part on contingent empirical regularities, on pain of incoherence in practice. Returning now to ordinal measurement: the numerical relationships between the assigned numbers are not all of them signiﬁcant. If we rankorder some items by assigning the numbers 1 to 10, merely on the basis of some ‘greater than’ relationship, there is no signiﬁcance implied by the fact that 7 is as much greater than 5 as 5 is than 3, or that 6 is 3 times as great as 2. What is signiﬁcant is what is invariant under ‘re-scaling’. This is not an explanation; it is a remark that connects various notions connected with each other. Take the example of a nominal scale: the members of a soccer team are numbered 1 through 11. It is important only that different players receive different numbers, the ordering does not matter. That is equivalent

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to saying that all permutations of this assignment result in equally admissible numberings: each such transformation is an admissible re-scaling. But what decides what is and is not important, what are precisely the relevant features to be preserved in all and only the admissible transformations of the scale, is not decided by mathematical characteristics. It is decided by two factors: the measurer’s purpose and the relationships that can be accessed instrumentally or practically on which to base the assignment. These factors are outside the domain of the mathematical theory of measurement. Signiﬁcance and invariance There is an important difference between what we might call ‘‘scale invariance’’ and ‘‘scaling invariance’’. In the preceding Part, in the discussion of scale models, we saw that a perfectly proportional decrease or increase in geometric shape may be quite the wrong thing for scale modeling, depending on what is meant to be studied or predicted. This is a point about ‘‘active’’ transformations, in which e.g. something literally of great size is replaced by something of small size for the purpose of some research. But scale transformations such as the permutation of a nominal assignment, or change from centimeters to inches in length measurement, are just changes in bookkeeping. The books must be kept straight—what is important must be properly represented; but this is a purpose-dependent norm for representation, not something telling about nature. Signiﬁcance is to be equated with invariance. That is, the procedure by which the numbers are assigned may favor only one large class of ways to assign them, as against other assignments, and only what is common to the favored class matters. This was an important insight exploited by Stevens: the types of scales are to be distinguished in terms of the families of transformations under which they are invariant.9 Interval measurement illustrates this well. An admissible transformation is in this case a combination of a dilation (in a sense that includes contraction) and a translation. As familiar example take the Fahrenheit (F) and Celsius (C) ‘‘centigrade’’ scales for temperature. Suppose that the high today is 41 ◦ F and yesterday’s was 32 ◦ F. Does that mean that the temperature today is more than 20 per cent higher than it was yesterday? The scale transformation from F to C is this: F = 32+(9/5)C. So today the high was 5 ◦ C, and yesterday it was 0 ◦ C. The ratio of today’s temperature to yesterday has no invariant sense. But in ratios of intervals we ﬁnd an

162  : , ,   invariant. So suppose that on the third day it is 50 degrees Fahrenheit, so we say that the increase was the same each day. Then in Celsius, carefully calculated, we ﬁnd the same: the increase from 0 to 5 was followed by an increase from 5 to 10. The rules for measuring temperature allow a large class of number assignments. In fact, the only constraint on this a class of numerical assignments is that it is closed under operations of translation and dilation. That means: if Y is an admissible scale then aY+b is too. So what signiﬁcance can a thermometer reading have? The answer, as we also mentioned, was that interval ratios are invariant: (aY2 + b - aY1 - b) / (aY4 + b - aY3 - b) = (Y2 - Y1)/ (Y4 - Y3) There is a way to express the signiﬁcant part of what is found in a temperature measurement with single numbers. That is to choose one interval that can be stably ﬁxed—e.g. the freezing and boiling point of water—as reference, and pay attention only to the ratio of the intervals in which that is divided by the temperature reading. If the numbers assigned to those ﬁxed points are 0 and 100, then the reading will be numerically equal to the percentage selected of the referent. Thus the Celsius scale is the most convenient: a reading of 15 ◦ C indicates saliently the proportion (the interval from 0 to the reading point)/(the reference interval) as 15 per cent. The choice of ﬁxed reference points is conventional, subject only to the possibility of stabilizing the measuring procedure to that extent—and thereby hangs a tale. The connection between invariance and signiﬁcance (often referred to, somewhat confusingly, as ‘‘meaningfulness’’) was studied at length in the literature on measurement. There is an important connection between this subject and the technique of dimensional analysis which we encountered in the discussion of scale models. The seminal article by Duncan Luce (1978) begins with the question why the laws in physics are required to be dimensionally invariant, that is, invariant under the admissible scale transformations. As focus he takes the case of measurement scales: It is pretty well agreed that the problem of meaningfulness within a single measured attribute is closely tied to knowing how two different, but equally acceptable, numerical representations of the qualitative structure are related. Meaningful

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numerical statements are those that remain invariant under permissible changes in the representation. That, however, is very similar to saying, in the more complex case of physical laws, that the representation must be dimensionally invariant. So, on the face of it, there appears to be a close relationship between meaningful statements in a single attribute and dimensionally invariant laws stated in terms of several attributes. (Luce 1978: p. 3)

The result he proves is in effect that features of a relational structure are invariant under the symmetries of that structure precisely if they are representable by a dimensionally invariant numerical function.10 Approximative measurement We need not go far aﬁeld to ﬁnd that linear ordering in a scale is too restrictive, in general. After all, a measurement outcome is not inﬁnitely precise: the length of the table is registered e.g. as 100 plus or minus 1 cm. So here the real outcome is not a number but an interval. These intervals are ordered by inclusion, which is a partial rather than linear ordering. We can introduce a notion of ‘strictly greater’, e.g. by the deﬁnition that one interval is greater than another if all its elements are greater than all of the other’s elements. But of course it does not follow then that the ranks are the same if neither is strictly greater than the other—so the ordering is not linear. If this practical point is granted, we have already left behind the idea that measurement outcomes can always be represented as points on a linear scale. For the class of regions—however delimited—that can be indicated as found locations in this way is not a linearly ordered class. It is a class partially ordered by set inclusion, or by set inclusion modulo differences of measure zero, or some such relation. We should not call this ‘‘locating on an interval scale’’—that term already has an established use, as we saw above. Rather, in this case, the object is located in the space of intervals, or in the larger space of ‘Borel sets’ generated by countable meet and join operations. This is the range most typically encountered where measurement results are not assumed to be ‘point-like sharp’. There an elementary form of statement, which can be either that of a theoretical assertion or of a measurement outcome, relates a physical parameter to a set of its possible values, which is itself linked to a deﬁned region in a much larger state space speciﬁed by a theory.11

164  : , ,  

The scale as logical space Consider a simple theory of gases which characterizes them in terms of three parameters, P(ressure), V(olume), and T(emperature). This theory provides a three-dimensional mathematical space, and measurement will locate given bodies of gas in regions of this space. The ﬁrst point to notice is that this use of measurement makes sense only in the context of the theory in question, which already provides the general framework for classifying the items to be located in that space. Secondly, this is an example in which the classiﬁcation is very simple and particular. In general, measurement of an item classiﬁed as being in the domain of a particular theory will locate that item rather indeﬁnitely as somewhere in a space common to a whole family of models provided by that theory. The farthest but still illuminating generalization of this point was made by Wittgenstein, inspired by his study of thermodynamics and mechanics: 1.13 The facts in logical space are the world. 2.013 Every thing is, as it were, in a space of possible atomic facts. I can think of this space as empty, but not of the thing without the space. 2.0131 A spatial object must lie in inﬁnite space. (A point in space is an argument place.) A speck in a visual ﬁeld need not be red, but it must have a colour; it has, so to speak, a colour space round it. A tone must have a pitch, the object of the sense of touch a hardness, etc. 2.202 The picture represents a possible state of affairs in logical space.

The HSB color space, with dimensions hue, brightness, and saturation is a good example of a logical space, but so is the PVT space in elementary gas theory, phase space in classical mechanics, Hilbert space in quantum mechanics; space and time themselves also serve as examples.12 I submit the following generalization as the proper concept of a measurement operation: measurement is an operation that locates an item (already classiﬁed as in the domain of a given theory) in a logical space (provided by the theory to represent a range of possible states or characteristics of such items).

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The act of measurement is an act—performed in accordance with certain operational rules—of locating an item in a logical space. While the terms vary, descriptions of actual measurement procedures typically bear this out. This has been better appreciated in philosophy of technology than elsewhere. Thus Davis Baird: Measurement presupposes representation, for measuring something locates it in an ordered space of possible measurement outcomes. (Baird 2004: 12).

When discussing the thermometer and, in more detail, the Indicator Diagram for a steam engine, Baird uses the phrase ‘‘ﬁeld of possibilities’’: ‘‘we have on the thermometer’s glass tube a scale that displays the ﬁeld of possibilities that we embrace with the thermometer’’. (Ibid., 2003: 50) In the steam engine case, the indicator is an instrument that produces a simultaneous trace of the pressure and volume inside the working cylinder as the engine runs through its cycle: The instrument harnesses an instance of material agency, the behavior of pressure and volume in a steam engine cylinder as it goes through its cycle. [. . .] At the same time, the indicator presents information. A ﬁeld of possibilities is constructed in terms of the pressure-volume graph on which the indicator ‘‘writes’’. (Ibid.: 52)

As a special case, the logical space can be a scale, which may indicate the location as being in a certain region of a larger space. Thus a pressure measurement locates a gas in a region of the larger PVT space, a momentum measurement locates a body in a region of its phase space. So the locating is typically not in an exact point, but in a region. We already saw this in connection with imprecise measurement, which assigns not numbers but intervals, hence takes its assignments from a partial rather than linear ordering.13 But this holds more generally, since what is measured is usually only some aspect of that ‘ﬁeld of possibilities’. Thus measurement is an act of locating an item in a logical space. The converse does not hold: you can locate me in the logical space of astrology simply by asserting that my Sun sign is Aries. Above I added ‘‘performed in accordance with certain operational rules’’, but by itself that only points to another question. The astrologer’s or soothsayers’s or visionary’s operational rules may not count as yielding genuine measurement. What precisely is needed? That is precisely the heart of the ‘problem of coordination’, hence requires looking into the joint evolution of theory and measurement. But

166  : , ,   we can add that once a stable theory has been achieved, the distinction between what is and is not genuine measurement will be answered relative to that theory. Here is a good example: as Henry Margenau and Adolf Grunbaum discussed, there are certainly procedures that look like simultaneous position-velocity measurements of particles at any scale. But quantum theory classiﬁes them as having no such signiﬁcance—for no operational outcome can reveal characteristics that, according to the theory, the system cannot have (cf. Grunbaum 1957: 713–15).

Data models and surface models It is typical of discussions of measurement in foundations of physics to retain the simplifying picture of a measurement as a single operation, a single interaction with the object to be measured. In actual practice, as pointed out above, that is quite unrealistic. There are measurements that can be done in one shot: when the nurse takes the patient’s blood pressure she does not make a series of measurements and then calculate the mean and standard deviation. But that is in part because no very great accuracy is required here, and in repeated results there is unlikely to be a spread of more than a few points. In the general case this tolerance of inaccuracy cannot be maintained, and a measurement will take the more complex form. The result obtained in one operation we may call a datum, the results obtained on one such occasion the data; these data are subjected to a statistical analysis to reach the ofﬁcial measurement outcome.14 Blood pressure, weight and mass are still very simple examples, where the outcome may be reported as a single number, or pair of numbers (e.g. mean and standard deviation). Other, still quite familiar, cases display greater complexity. On the weather forecast website I consult I can ﬁnd a graph depicting yesterday’s temperature plotted against time. This was constructed from data gathered at various stations in the region, at various times during the day—this graph is a smoothed-out summary of the information that emerged from all these data, it is a data model. The question about the daytime temperatures in this region of one day ago is answered with a measurement outcome, certainly—but that is the graph in question, which is a data model constructed from an analysis of the raw data.15

  : 

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That this is how the results of measurement, and the complexity of their relation to theory, must be conceived was an early and continuing theme in Patrick Suppes’ work: exact analysis of the relation between empirical theories and relevant data calls for a hierarchy of models of different logical type. Generally speaking, in pure mathematics the comparison of models involves comparison of two models of the same logical type, as in the assertion of representation theorems. A radically different situation often obtains in the comparison of theory and experiment. Theoretical notions are used in the theory which have no direct observable analogue in the experimental data. (Suppes 1962: 253)

Our words have a sometimes disturbing plasticity. At one of the local small stations a particular reading is made at a particular time: that is a measurement. But the total process of all those single data collections plus the statistical analysis plus the systematic ‘summarizing’ that ﬁnally yields the graph—that also is a measurement. It is the latter that takes a more scientiﬁcally signiﬁcant form. What is important is that in both cases, the primitive single data collection and the complex complete measurement with a stable result, the outcome must be regarded this way: this is what the object looks like in this measurement set-up. And in both cases, the object measured is thereby located in a logical space, characteristically associated with the type of operation involved. Suppes continued with ‘‘In addition, it is common for models of a theory to contain continuous functions or inﬁnite sequences although the conﬁrming data are highly discrete and ﬁnitistic in character.’’ The graph or other summary of the data found will be abstracted into a mathematically idealized form before it reaches the theoretician’s desk. Let’s call that processed artifact a surface model. Although usage does not restrict the term ‘‘data model’’ to summary representations of frequencies, let us take that as our example. I distinguish data model and surface model in this way: • the data model summarizes the relative frequencies found • the surface model ‘smoothes’—in fact ‘idealizes’—this summary still further so as to replace the relative frequency counts by measures with a continuous range of values. The term ‘‘data model’’ is often used in the more general sense that does not distinguish summarized relative frequencies from probability measures, and

168  : , ,   I will not insist everywhere pedantically on this distinction. Probabilities relate to relative frequencies—a subject all by itself—but the frequencies alone are what can be found in nature.16 Any procedure for replacing a ﬁnite set of data by a graph furnishes an example, and such procedures are common throughout the sciences, facilitated by easily available software. But the replacement is a step in a long process from hands-on or mechanical data gathering to the eventual mathematical structures that can confront the proffered theoretical models.17 The complexity of the journey from raw data to graphical representation is also well illustrated by Ronald Giere’s examination of astronomical observation. The data gathered by optical, radio, and gamma ray detection in today’s astronomical observatories are combined and transformed in a process that yields visual representations color of comets, stars, nebulae, and galaxies. Giere examines how astronomical data, gathered in the form of black and white photos, are processed to yield images in color of a nebula, through a process originally invented by James Clerk Maxwell. Each individual photo captures just one aspect, but attention to the ﬁltering by which it is obtained allows it to contribute to the reproduction of color in the images that are the ﬁnal product. As Giere emphasizes, the images presented . . . are conclusions. These images present a picture that is continuous, or at least very ﬁne-grained. The actual data cannot be that ﬁnegrained. The data are made up of individual events recorded in various detectors at different times and processed by various physical and computational means. The images are constructed using those data, but go beyond the data. (Giere 2006: 48)

Recall that just now we are looking at measurement ‘from above’, that is, at this construction guided by accepted theory, rather than ‘from within’. Surface models and their embeddings A theory provides, in essence, a set of models. The ‘‘in essence’’ signals much that must be delicately expanded and qualiﬁed; I will leave this aside for the moment.18 These models—the theoretical models—are provided in the ﬁrst instance to ﬁt observed and observable phenomena. Since the description of these phenomena is in practice already by means of models—the ‘data models’ or ‘surface models’, we can put the requirement as follows: the data or surface models must ideally be isomorphically embeddable in theoretical models.19

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Since the advent of quantum theory much thought has been given to the form that any possible surface model must take. Consider an experimental situation of a quite simple structure, involving several alternative measuring arrangements, a classiﬁcation of possible outcomes, and some probabilities extrapolated from (imagined) observed frequencies. Then a surface model can be thought of as speciﬁed by three factors: (i) two sets of observable conditions: (a) a set of realizable measurement choices—call it PRC, and (b) a set of possible outcomes—call it PRS; (ii) the surface state P; this is a function which assigns probabilities of outcomes in PRS, conditional on measurements in PRC. So P is deﬁned on at least part of PRC × PRS and its values are real numbers in the interval [0, 1]. This structure is subject to certain minimal conditions which must guarantee that P is mathematically extendible to a classical probability function.20 The numbers assigned by the surface state we can call surface probabilities. What about theoretical models? We already looked at this brieﬂy above. These need to be conceived without prejudice in favor of determinism or causal modeling. The theoretical model could specify, in general, a family M of observables (physical magnitudes) each with a range of possible values; a set S of states; and a stochastic response function P s m for each m in M and s in S, which is a probability measure on the range of m. The number P s m is to be interpreted as the model’s speciﬁcation of the probability that a measurement of m will yield a value in E, if performed when the state is s. From this we can at once see more or less what it shall mean for such a theoretical model to ﬁt a surface model. But not quite yet: it only tells us the probabilities of surface phenomena, on the supposition of a measurement and of a state. The latter is again something theoretical, behind the phenomena. A stringent notion of ‘ﬁtting’ could go as follows: A theoretical model MT ﬁts an experimental model ME just in case MT has some state s such that the function P s m contains the surface

170  : , ,   state of ME, relative to the given identiﬁcation of the measurement setups as measurement of the physical magnitudes m. The situation is thus, so to speak, represented ‘‘from below’’ and ‘‘from above’’. In the surface model we have a representation that can be prepared in the laboratory or observatory, without recourse to the theory and its theoretical models.21 But the theoretician has a look at the same situation ‘from above’, by specifying how it can be represented by a theoretical model. A way to evaluate what s/he provides along this line is presented by the above notion of ﬁt. Let us look at a concrete illustration. The surface model of an EPR experiment As example I want to take a much discussed and described experiment that was inspired by the Einstein–Podolski–Rosen (‘‘EPR’’) paradox, proposed by David Bohm, and eventually carried out by Alain Aspect. There is no better or more generally accessible presentation of this sort of experiment than the article David Mermin wrote for The Journal of Philosophy (1981). This article makes very clear that there are two, quite independent descriptions of the experiment, one ‘from above’ (in quantum theory terms) and one ‘from below’ that can be understood without knowledge of the theory. As Mermin writes, it is not beyond present technology to mass produce the experimental set-up in a form to be sold in local drugstores and operated by anyone.22 There are three unconnected devices, a transmitter and two detectors. The latter we may call L (for left) and R. Each has a dial with (say) three settings or orientations. When the detectors are turned on and the transmitter button is pushed, each detector ﬂashes a light, which is either red or green. For ease of presentation, let’s introduce some notation: L1: the proposition that L has been given the ﬁrst setting. Lx: the proposition that L has been given the xth setting: x = 1, 2, 3. The experiments have each two distinct possible outcomes, the red and green light ﬂashes, which we may represent by the numbers zero and one: L30: the proposition that L has the third setting and outcome zero. Lx0: the proposition that L has the xth setting (x = 1, 2, 3) and outcome zero.

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Lxa: the proposition that L has the xth setting and outcome a (a = 1, 0). After a particular run on this apparatus, some of these propositions can receive a score of T (true) or F (false). Propositions about settings other than the actual one receive no score—they are irrelevant. Suppose for example that for the ﬁrst run, each apparatus was placed in the ﬁrst setting; L had outcome 1 and R had outcome 0. An experimental report looks, in part, as follows: Proposition L1 L2 R1 L10 L20 R10

Score T F T F No score T

L20 received no score. Since it is the proposition that L was placed in setting 2 and had outcome 0, it could have been given F, simply on the basis that L2 was false. But L20 is useless information for it records an outcome for an arrangement that was not actualized, and so does not appear in the experimental report. This single report is in any case not likely to come to the theoretician’s desk. What reaches him rather is like: (S) With initial preparation X, the probability of outcome Lxa, given setting Lx, equals p. (L) For all initial preparations, the probability of (Lxa & Rxa), given settings Lx and Rx, equals zero. which is an extrapolation of a summary of many reports of the above sort, and of the frequency counts made on their basis. There is a gulf not only between the ﬁnite set of data points and the extrapolation to relative frequencies, but also between the latter and probabilities. Accordingly there is a distinction between two intellectual artefacts that lie between the collection of speciﬁc, concrete results on the one hand and what the theorist takes as his or her input on the other. The reported relative frequencies, in their own summarized form constitute

172  : , ,   the data model obtained from repeated applications of a single measurement procedure. What the theory confronts is abstracted from many data models. The abstracting is an idealizing, an extrapolation to a form that could not be reached in actual practice. So the reports (S) and (L) and their cognates, describe an idealized but not yet theoretical structure—the surface model. Patrick Suppes pointed out that through construction of data models the experimentalist is in general bringing the theoretician small relational structures, constructed carefully from selected data. In the speciﬁc examples that Suppes mentions the little structures are algebras; hence he calls them empirical algebras.23 Literature on the foundations of quantum mechanics typically points to such small structures that represent data, but they are not always algebras. They are more generally partial algebras, or just partially ordered sets (‘‘posets’’) with some relations and/or operations. In the Appendix I will illustrate this, but the above already sufﬁces to show how we need to think of the deliverances of experimentation and measurement as taking a much more general form than simple number assignments.

The over-arching concept for measurement The bounds we set to the form that surface models can have, how we conceive that form, has changed historically. The bounds to the structure a logical space, supplied by a theory to delineate its models, can have, that has changed historically as well. All the examples we have examined of measuring procedures are certainly cases of grading, in a generalized sense: they serve to classify items as in a certain respect greater, less, or equal. But as we also saw, this does not establish that the scale must be the real number continuum, nor even that the order is linear. The range may be an algebra, a lattice, or even more rudimentary, a poset. Generalizing on such examples, we can still conclude what I announced at the outset: By measuring we assign the item a location in a logical space. So locating something in a logical space is the over-arching concept under which all actions of measurement can be arrayed. This is the only

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stable stopping point we have found in the successive generalization of the notion of number-assigning. Let’s emphasize again that we cannot equate the two, however. Not every act even of locating something correctly in a logical space is the expression of an actual measurement outcome.24 To say for example that our solar system is a Newtonian mechanical ten-body system, that is to locate our earth, the visible sun, and our sister planets in a logical space: the space of Newtonian models, which is the logical space of Newtonian physics. The bare assertion does not state a measurement result. But it is meaningful in part because it is not unconnected with what measurement can show. Can the assertion be taken as expressing a possible measurement outcome? In this case, certainly. A very large family of measurements normally so-called can be viewed as a ‘battery’, as a single complex measurement, and it can be so viewed within a certain theoretical context. In this complex outcome, our sun and planets ‘look like’ a Newtonian system. There are, however, more abstruse assertions within the language of contemporary physics that are much harder to see as thus connected with measurement, even sub specie the pertinent theory. There is certainly a general concern in the empirical sciences that the conditions of application of their concepts be closely connected with experimentally realizable conditions, but there is no rigid limit to how loose this ‘‘closely’’ can be. Perspectival effects within logical space While I do not want to stretch the term ‘‘perspective’’ very far beyond its narrowly literal meaning, there are places in this discussion of measurement where it seems apt enough. The aptness depends on context, and speciﬁcally on what theories count as accepted in that context. If a thermometer is used by one person to locate the air in his room on the Fahrenheit scale, and by another to locate it in the space of possible mean kinetic energies of its molecules, the two are locating the same thing, by means of the same instrument, in two different logical spaces. That we should not at once call a change of perspective, though it certainly marks a change in ways of thinking about the air. What if a theory then equates temperature and mean kinetic energy? In that case we should say that relative to the theory it is appropriate to call this a change of perspective. This qualiﬁcation is important, though it may be left tacit in a context where the theory has been entirely accepted.

174  : , ,   Quite a different case is posed by two different measurement procedures that are such as to assign always (error and vagueness aside) the same locations in the same space. As an example consider several methods for measuring velocity of a ﬂuid, by pressure probe (use of pressure drop across a nozzle—Bernoulli effect), hot wire anemometer (heat transfer from a heated wire, due to convection heat transfer), and laser Doppler velocimetry (use of light scattering from small seed particles, Doppler shift). These can all be calibrated so as to give the same reading for the ﬂuid velocity; there is no difference even in scaling as there is with the Fahrenheit and Celsius scales. It does not seem right to say here ‘‘there are differences in how things look from here’’, but rather ‘‘things look the same from here as from there’’. The moment we introduce this ‘‘from here and from there’’ distinction, however, we also have reason to warrant the perspective appellation. What if two persons use different but similar thermometers to locate the same room’s air on the Fahrenheit scale and Celsius scale respectively? We can’t say that they use different methods of measurement, nor that they are measuring different parameters. Abstractly speaking we still have a choice: should we speak of two logical spaces, or of different coordinate systems in one space? But the latter choice is here so natural, that it almost seems forced on us. It would certainly not be outr´e to speak of a difference in perspective. The two thermometers assign locations on two different scales, and these can in principle be taken as two logical spaces. But there is a transformation that systematically connects the readings on the one with the readings on the other. Given knowledge of that transformation, we immediately have a concept of a single logical space in which the two scales are, in effect, two coordinate systems. The transformation is linear; we immediately conceive of all the admissible temperature scales as connected to the Fahrenheit scale by linear transformations. With this way of looking at the matter, we are inclined to think of the use of Fahrenheit and Celsius thermometers providing different perspectives on the same magnitude, whose logical space is a more abstract structure. We can see a similarity to the use of radar by two inertially moving trafﬁc policemen, in motion relative to each other, to determine the speed of one and the same car. Their readings are different, for what they obtain is the car’s speed relative to their own car—but whether they know it or not, there is a transformation that translates their different readings into one

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another. The two policemen we say, surely, have before them the contents of two perspectives within the same space—literally in this case. At the same time we should recognize that relative to the pertinent theory there is no real signiﬁcance to a choice between a family of spaces related by a group of transformations and a family of coordinatizations of a single space. Little but bookkeeping ease is involved in the choice between speaking of different spaces, transformable one into another, and different coordinate systems imposed on a single space. Yet if we think in the former terms it seems less apt to speak of differences of perspective. Perhaps the resistance is the more reasonable, the more we recognize that there is high theory in play; and the one important thing is to pay attention to what theories are playing the background role. Reconciling two views: perspective and invariance For measurement the distinction is essential between the ‘giving’ of an object through individual exhibition on the one side, in conceptual ways on the other. The latter is only possible relative to objects that must be immediately exhibited. That is why a theory of relativity is perforce always involved in measurement.25

This is Herman Weyl’s rather cryptic elaboration on how measurement necessitates reference to something playing the role of a coordinate system associated with the perspective embodied in the measurement set-up. But there is a good deal of measurement in which this attributed relativity or perspectivity does not readily spring to the eye. What are we to make of measurements that yield outcomes ostensibly independent of differentiating features of the measurement arrangement? Imagine a container with a movable piston, a pressure gauge and thermometer attached, and a scale on the outside that indicates the volume for the various positions of the piston. Three measurements are being carried out on this object, and together they locate its content in the threedimensional Pressure-Volume-Temperature manifold—itself represented perhaps by the space of triples of non-negative real numbers. The outcome registered at some particular time, locates the container contents in that space. This illustrates well the conclusion, that measurement is an action of locating the measured object in some logical space. But what has happened now to the much emphasized insight that a measurement outcome is after

176  : , ,   all only a representation of the target, and in general does not show what that is like but only what it ‘looks like’ in that measurement set-up? Let us honor these two views of what measurement does with the names Measuring is Locating and Measurement is Perspectival. Are they in tension with each other at all? One small point may help: what is perspectival is not the action of measuring but the contents of the measurement outcome, and locating is an action, not a content. Action and outcome are two different kinds of things. But this distinction does not go all that far: the measurement outcome does after all represent the target as located in a certain logical space. If we understand Measuring is Locating as meaning just that, we are back with two takes on the same thing, on the outcome. If we are to call measurement perspectival, we need to qualify and elaborate if we are to arrive at an accurately made point. Let’s begin with a classic passage in which Poincar´e insists on the outcome’s relativity to set-up: Lorentz could have accounted for the facts by supposing that the velocity of light is greater in the direction of the earth’s motion than in the perpendicular direction. He preferred to admit that the velocity is the same in the two directions, but that bodies are smaller in the former than in the latter. If the surfaces of the waves of light had undergone the same deformations as material bodies, we should never have perceived the Lorentz–Fitzgerald deformation. In the one case as in the other, there can be no question of absolute magnitude, but of the measurement of that magnitude by means of some instrument. This instrument may be a yard-measure or the path traversed by light. It is only the relation of the magnitude to the instrument that we measure, and if this relation is altered, we have no means of knowing whether it is the magnitude or the instrument that has changed.26

‘‘It is only the relation of the magnitude to the instrument that we measure’’—that is either an overstatement or misleadingly put, though it emphasizes something important. The outcome of the measurement operation is a representation of the target, but it represents the object as it appears in that measurement set-up. However, it does not follow that the appearance there is different from the appearance of the same thing in other such set-ups. For after all there are invariants too! In the case Poincar´e is discussing, the parameters measured are not invariant, they have different values in differently moving inertial frames. But there are parameters, even

  : 

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ones deﬁnable from the results of local distance and time measurement outcomes, that have the same value in different frames.27 Note well though that the invariance we are now discussing is not the invariance that was cited as required for signiﬁcance above. There we were concerned with the transformations that connect all the members of an admissible family of scales. Now we are discussing parameters for which the value registered in a measurement outcome is the same under admissible variations in the measurement set-up. As an example imagine again the speed of a car measured by radar from a moving police car. This speed can be registered on a scale of miles per hour or kilometers per minute, and so forth. Leaving aside data about the police car’s own speed relative to the road surface, what is registered is the speed of the target relative to the radar source. If the policeman drives at a different speed himself, that relative speed of the target will be different: it is not an invariant. If on the other hand the radar is used to measure the target’s acceleration, the result will be the same for any speed the police car may have. More precisely: where Newtonian mechanics applies, the acceleration is the same in all inertial frames, for all inertially moving measurement set-ups, while the speed is not. There are therefore measurement outcomes that have no relativity left. Generally, these are after instrumental outputs have been processed with paper and pencil operations, with ﬁnal outcome deduced relative to a theory. This is an important point: a measurement and its outcome can be complex, and include calculations and input from a model or theory. Such a procedure still ﬁts the general idea of an operation performed so as to create a representation of the object; one that locates it in a certain logical space, with a location that it does not have a priori. To see how the activity signiﬁed by the slogan Measuring as Locating intertwines with the fact that Measurement is Perspectival, let us take a look at some examples of how ‘simple’ measurement outcomes are combined in such way to yield an outcome of a thus created ‘complex’ measurement. Think once more of celestial navigation in the days of sailing ships. Navigating consists in locating oneself and guiding oneself from one location to another. The ﬁrst part is the measurement whose outcome will govern that self-guidance. What is one locating oneself in, in this case? In the grid, apparently ﬁrst proposed by Hipparcus: basically our system of latitude and longitude which was also Ptolemy’s. In Ptolemy’s coordinate system as in ours, the

178  : , ,   equator is at 0 and the North Pole at 90 degrees; similarly 360 degrees of longitude cover the earth, though 0 was not at Greenwich. For the ship’s longitude to be determined, it would be sufﬁcient to know what time it was at the same moment—e.g. local high noon—at 0 longitude, since the Earth rotates 15 degrees per hour. Precise time reckoning was a problem—let’s leave this aside here. To determine the latitude by means of local measurements was possible, however. The astrolabe that I mentioned before was succeeded in the eighteenth century by the sextant, to measure the elevation (angular altitude above the horizon) of celestial bodies. The elevation of the sun at noon, is a straightforward function of latitude. In the Northern hemisphere, the elevation of the North Star above the horizon is equal to the latitude. So imagine a sailing ship in the eighteenth century, with a sailor using a sextant to measure the elevation of the North Star. This places the North Star in a coordinatized Euclidean plane. Of course we do not read this as an ‘‘absolute’’ feature, it is its location relative to the observation set-up. The number found by the sailor is reported to the captain, who puts this to use to locate the ship on the earth. That is, he uses the reported number to locate the ship on the map, at a particular latitude—which together with his chronometer reading and his calculations of longitude yield a single point on that map. Note the difference: at this point what for the sailor was the operation of ﬁnding the elevation there of the North Star has become for the captain a position measurement performed on the boat. The outcome of the complex measurement completed by the captain’s manipulation and combination of data being the result of a deduction from that elevation value, relative to premises that the captain already had at his disposal, including some that are very theoretical. This measurement locates the boat in the geographic grid; it can be recorded without adding a ‘‘from here’’. The longitude, latitude, and time can now be entered in the log without any such ‘relative’ marker. As a second example think of three cameras ﬁxed on a grid, taking photos of an ancient coin. In each photo the object has an elliptical shape. In the simple set-up deﬁned by one of the cameras, the shape measurement outcome is ‘‘elliptical, with axes . . .’’. But in the three-camera set-up, complemented with a little computer to process these three individual outcomes, there is a big measurement going on, with an outcome such as e.g. ‘‘circular’’. This too is a shape measurement. But here the outcome seems not to be relative to the cameras.

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Yet this example brings out how context-dependent these judgments of relative/non-relative are. We might have to see something relative where we did not notice it before. In this example I probably tricked you into imagining this as an inertial frame in which objects and cameras are at rest with respect to each other. Shape is not a relativistic invariant, and if you think about it again in the new context that I have now introduced into our discourse, what was classiﬁed as not relative is reclassiﬁed as relative to something after all. But all the shape measurements located the object in the same logical space of shapes. There is no contradiction when it is noted that the location of the object in that logical space is relative to the measurement set-up—and that adding in some other data and bits of theory will yield more complex measurement procedures in which invariants are measured. Finally, single simple measurements may be combined into complex ones over time. Imagine a Stern–Gerlach apparatus with the usual two channels and screens at their ends. When the ﬁrst black spot appears, and it is on the upper screen, we can say ‘‘looks like a particle in a deﬁnite spin-up state was prepared’’. After, say, a hundred spots one might say ‘‘looks like this source is preparing particles in a state in which 1/3 register as spin-up and 2/3 as spin-down’’. There is no falsehood here, nor any contradiction, for the two sorts of states mentioned can look the same in a single measurement. After a long run, the ‘‘looks like’’ can disappear: the outcome is that the source prepares particles in a superposition or mixture of spin-up and spin-down eigenstates . . .’’ etc. This can be the output of a program on a computer coupled to the apparatus; obviously certain assumptions or forms of inference are built into the algorithm that deﬁnes the program. The outcome in this case locates the source output with respect to a certain Hilbert space.

What is a measurement outcome? We cannot distinguish what measurement is by attending solely to the physical correlate of measurement. The representational role is exhibited only if we attend to how the measurement outcome is a representation of what is measured. A measurement is a physical interaction, set up by agents, in a way that allows them to gather information.28 The outcome of a measurement

180  : , ,   provides a representation of the entity (object, event, process) measured, selectively, by displaying values of some physical parameters that— according to the theory governing this context—characterize that object. It is a speciﬁc kind of representation, with some of its features inherited from the nature of representation in general and some others peculiar to it. The various strands in the way we think of measurement are to be disentangled—though without implying for a moment that the distinctions drawn are more than distinctions of reason, or that the different aspects we can distinguish could exist all on their own. To disentangle some of the strands in the way we think of measurement, we can begin by emphasizing again the dual character, common to all that properly belongs under the heading of technology: that we describe the processes and objects involved both in physical and in functional terms. What have we found so far? First of all there is the relationality in the concept of measurement outcome. An outcome is an outcome of something, of a process, which must be of a sort that satisﬁes various stringent conditions. This is already correct on the purely physical side. But however smoothly the story goes, of how certain implied correlations qualify a physical process as being usable for gathering information, there is still more to it. First of all, more is in play than a comparison of amounts of information transmitted, and more than a correlation between physical states of object and apparatus. The measurement was designed to answer speciﬁc questions, and the information provided by the outcome is relevant to their answer. Notions such as relevance and reference, as well as the questions that deﬁne the context, are at best assumed as given when a technical measure of the amount of information is applied. The information borne by a measurement outcome refers to a speciﬁc object or process (what was measured) and is selective with respect to characteristics of that referent. Such very simple examples as, the information provided by a reading of a thermometer (concerning the ambient temperature) or a pressure gauge (concerning tire pressure), may suggest that information transmission is something that can be explicated in purely physical terms.29 But if a physical item is classiﬁed as the outcome of a measurement (of a speciﬁc kind) then it is classiﬁed as a representation, and therefore in effect classiﬁed under an intensional concept. The outcome provides a representation of the measured item, but also represents it as thus or so.

  : 

181

We can see this as soon as the phrase ‘‘carries the information that . . .’’ is employed. To adapt an old example from Frege: to carry the information that the Evening Star is within 15 degrees of the sun is not to carry the information that the Morning Star is thus, although these are the same object, and although therefore correlation with the position of the Evening Star is automatically correlation with the position of the Morning Star. The intensionality of the concept of a measurement outcome consists in the fact that it is something that has meaning. In reporting the outcome one says, for example, that the pressure was [found to be] 17 psi; that report is a sentence expressing a proposition. Even if, given the background knowledge or opinion about the whole set-up, the pressure was necessarily 17 psi if and only if 17 = rT/V, the outcome was not that the value of rT/V was 17—not even in the context deﬁned by that background knowledge or opinion.30 That the concept is intensional is not to be confused with its being intentional. Literally, ‘‘intentional’’ refers to intention, but we take it broadly to include purpose, goal, role, and function. To classify something as a measurement outcome is to classify it as playing a certain role, namely as the outcome of a process with a deﬁnite function. This is entirely in line with the reﬂection that the activity of measurement belongs to technology, and technological concepts have this dual character, of referring to physical entities but partly (and essentially so) in terms of function.31 Fourthly, we must insist on the indexicality of the measurement outcome content in general. That is easy enough to spot when the measurement outcome is the indexical proposition that the iceberg is located 17 leagues to the North-East. But it is an especially signiﬁcant feature when the context is more, rather than less, theory-laden. Suppose for a moment that I take a pressure reading on the tire of my car. The outcome can be reported simply and precisely as attributing a feature to that tire. Where is the indexicality in that? But think of how different a role this report plays from the assertion, written in the very same words, in a historical account of what someone or other did somewhere. For this outcome to be something useful for me, I must appreciate that by means of this measurement operation I have also located my own situation—which involves this tire—as having a place in the theoretical region of pressure-graded objects.

182  : , ,   Although this sort of representation is indexical in the same sense that ‘ﬁrst person’ discourse is indexical (cf. Perry 1993), it is not ‘subjective’ in contrast to ‘public’ or ‘inter-subjective’. Measurement outcomes are public and they are intersubjectively accessible; this is crucial to the methodological requirement of reproducibility in scientiﬁc experimentation. The measurement outcome content typically involves indexical reference to a particular vantage point, but this vantage point is a publicly ascertainable feature of the measurement set-up. This fourth characteristic, the indexicality, is one not shared by representation in general. It derives from the speciﬁc function which this particular method of representation has. There are further features that relate the measurement outcome to speciﬁc kinds of representation, notably to imaging and picturing. Recall that by imaging I mean representation that trades on selective resemblances, and that the special case of picturing was introduced above as follows: Overlapping these categories of representation that trade on selective resemblances lies still a further salient case, which shares some crucial features found in visual perspective, a development which in art we associate speciﬁcally with Renaissance painting. Perspective involves . . . such features as occlusion, marginal distortion, texture-fading. For cases of imagery in which such features of perspective are present I’ll use the terms picture and picturing —these can include cases of kinematic and visual but perhaps also still further forms of imagery.

The most easily recognized cases of picturing are ones in which the resemblance is not at a very high or abstract level: it is just a sharing of selected properties. The subway map shares the topological structure of the subway system, say; it is a picture of that system. But resemblance, we recall, can be higher order: the spatial structure of a set of letters on a page may be the same as the temporal structure of a set of events named by those letters. The use of visual or kinematic imagery to depict things that are not visual or kinematic is rife, and not excluded by our notion of imagery. So a measurement outcome may well purport to give us information by means of a selective resemblance to what is measured, although the pointer-observable may be of a very different character than the measured observable. Indeed, the Criterion for the Physical Correlate of Measurement entails that resemblance at such a structural level is required to be implied by the

  : 

183

theoretical classiﬁcation of something as a measurement. So this, the trading on selective resemblance at some level, is our ﬁfth characteristic. And ﬁnally, there is the sense in which measurement outcomes are perspectival. This has been explored extensively by Ronald Giere (2004; 2006), whose work has placed this topic at center stage. There is historic precedent. Poincar´e (1897: ch. 1) emphasized strongly the ‘perspectival’ (‘relational’ in the sense of relative rather than derived) character of the outcome. Poincar´e overstated it perhaps (‘‘It is only the relation of the magnitude to the instrument that we measure’’), but the point is clear. What the outcome reveals is not directly what the measured object is like, but what it ‘looks like’ in that measurement set-up. The latter point was exploited saliently by Einstein in his 1905 STR paper, where measurements of time and of length are taken to yield only data explicitly relative to the state of motion of the measurement set-up.32 To assess this question of perspectivity properly we must take stock of its hallmarks, such as occlusion, marginal distortion, and texture-fading. Literally, marginal distortion is distortion in proportions toward the outer edges of a perspectival drawing or painting. In a more general sense, marginal distortion is distortion that is the result of the limits inherent in a given mode of representation. Such would seem to be a feature of almost any sort of measurement at all, given that instruments have a limited range, and become inevitably less reliable and less exact near the limits of that range. A liquid thermometer is not to be used near the freezing and boiling points of that liquid, for example. Perhaps a simple detection of presence or absence, with a yes–no answer, would be a counterexample, but then only if the means of detection could not possibly give uncertain results—an unlikely case. In the context of quantum mechanics we have a more extreme sort of example. That an object’s location may, as far as its state is concerned, be probabilistically smeared out over the whole of space, while we can only get our measuring apparatus to ask whether or not it is in this room (or similarly small ﬁnite region), means that information gatherable beyond a narrow range is maximally indeﬁnite—that is marginal distortion. Occlusion and texture-fading are easy to spot in the sort of measurement that is paradigmatic for our concept, namely the sort of spatial measurement I discussed with reference to Hugh of St. Victor, Alberti, Brunelleschi, and Dürer.

184  : , ,   It is tempting to see these same concepts as applying in quantum mechanics due to the impossibility of simultaneous measurement of noncommuting observables. If one observable is measured, the extent of information that can be gathered with respect to incompatible observables is drastically reined in: that remind us of occlusion. That the observable really measured, when a measurement is ofﬁcially designed to measure a sharp observable, will necessarily be a fuzzy version of that observable—that sounds like texture-fading. But to so use the terms is at best an analogical extension of their literal meaning. Occlusion means hiding —changing something so that it can’t be measured is therefore not occlusion.33 In the case of quantum mechanics it is certainly not the case, for example, that the measured value of momentum implies a real but hidden value of position. On the other hand, if a system is in an eigenstate of an observable, and an incompatible observable is measured, then that state has changed. So we have two reasons not to call this occlusion. As to texture-fading: the fuzzying discussed for quantum measurement does not have anything to do with distance, which was what concerns texture-fading in pictures. But though the analogy is far from complete, it is not far-fetched to analogically extend the notions of occlusion and fading in this way.

Relating the views ‘from above’ and ‘from within’ As long as we consider what happens in measurement purely from the theoretical point of view, the only criteria are theory-internal. The criteria of adequacy cannot go beyond coherence. That is why the theoretical point of view remains empty unless the problem of coordination is also faced and taken into account. ‘‘Concepts without percepts are empty’’—this Kantian motto transposes to the empiricist point that theory without coordination is empty. To understand how theoretical parameters become coordinated, we have to look into the historical process in which measurement procedures and theory evolve together, in a thoroughly entangled way. Nor can this evolution start in anything but prior meaningful discourse relating to what eventually will be delineated as the theory’s domain. The scientiﬁc realist will interpret this process, in which practice and theory jointly stabilize, as establishing that the theory latches on to the blueprint for the universe.34 But the one empirical fact is precisely this:

  : 

185

that the practice, both experimental and theoretical, stabilizes, and that ‘nature cooperates’ to the extent that, perhaps temporarily, no or less resistance is experienced in this practice.35 For the time being, at least, the expectations engendered by empirical predictions are satisﬁed, the retrospective evaluation says that thus guided empirical judgment has been well calibrated.

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PA RT I I I

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Structure and Perspective In the preceding parts we have uncovered some conditions for the possibility of scientiﬁc representation. Viewed in one particular way, all of these conditions can be brought under one heading: the crucial role of use and practice. Although description in language is at best one mode of representation, this crucial link to use and praxis points us toward the study of pragmatics: the study of language in which word-thing relations are seen as abstractions from word-thing-user relationships.1 The asymmetry of representation and the possibility of misrepresentation, for example, we saw to derive from use rather than from independently speciﬁable relations between representor and represented. Nothing is a representation unless it has a certain kind of role in use and practice. In addition, besides their status as representation deriving from use, some representors have a use, which they can have only in a context in which indexicals and self-reference are available. While I gave maps as paradigmatic example, this use is central to all the practical sciences, where scientiﬁc representations are drawn on so as to apply scientiﬁc knowledge in practice. This is well illustrated also by the problem of coordination, which was seen to be unsolvable except in a context where some coordinations are already achieved and present. Coordination, which assimilates theoretical terms to the language in use, is not to be understood as a completely explicit or conscious historical process. We cannot think of theoretical or other newly introduced terms as made subject to principles of coordination, except in a context where it is already possible to rely on other terms, ‘old’ terms, as ‘already coordinated’, as meaningful.2 Meaning and use must indeed be bestowed on newly introduced terms, but this makes sense only if we think of them as introduced into an already extant language, into our own language in use.3 The distinction between what is newly introduced theory, and what is language in which the instruments and measurements are already described, is historically conditioned. The phrase ‘‘we already knew how to describe’’ signals reliance on our own language at that historical moment. How could it be otherwise? There is no moment outside history, and at each moment

190   :    in history we not only can but must rely on the language in which we conduct our business, the language we live in. This is Neurath’s insight about mariners at sea, but extended to the conditions of possibility of scientiﬁc representation in every respect. The epistemic situation as here described has seen responses not just in philosophy of science but in metaphysics, and not just in past centuries but in our own time. We will follow the resulting probl´ematique as it developed through the twentieth century and into our own, starting with the Bildtheorie of Hertz and Boltzmann as precursor to structuralism in the philosophy of science. Structuralism about science is, roughly speaking, the contention that scientiﬁc representation is of structure only. The obvious question, what is structure and what is not?, is the ﬁrst that any advocate of structuralism must answer. The answers have tended to dissolve into vacuity or inconsistency when pressed to precision—as we shall see, going through Russell’s, Carnap’s, and Putnam’s arguments as structuralist conceptions emerged, faltered, and took hold. Structuralism unfortunately involved, during most of its problem-beset history, another attempt to achieve the simultaneous interpretation of all language and theory without relying on our prior language-in-use. Seen in this way, reminiscent of Reichenbach’s attempt to conceive of and solve the problem of coordination in an extreme form, structuralism too pursued an impossible ideal.4 But just as we can see a real and viable process of coordination behind Reichenbach’s reach for the ‘unconditioned’, so it seems to me that we can see a genuine and viable sense in which structuralist views of science are right, at heart.5 I shall advocate a version, an empiricist version, of structuralism. Once again, the redeeming clues are to be found in pragmatics. The empiricist view I propose will, I hope, do justice to the strong structuralist trend found in philosophy of science without subordinating it to any form of metaphysical realism, and without giving in to the attendant illusions of reason.

8 From the Bildtheorie of Science to Paradox The arguments by scientists and philosophers in the decades just before and after 1900 concerned not only the directions that science should take, but also the very idea of what science was or was to be. As I shall present it, the advocacy of the Bildtheorie, the view that what science gives us is representations, important as it is in its own right, is also integral to the origins of what is now known as structuralism in the philosophy of science. Structuralism about science—the thesis that a science represents only structure in its domain—became increasingly and recurrently a salient theme in twentieth-century philosophy of science. This development had two motivating philosophical controversies in its past. One was in the philosophy of mathematics, and speciﬁcally of geometry, where the conception of a theory of space was gradually replaced by the conception of geometry as a branch of abstract mathematics. The other, which we will consider ﬁrst, occurred in the debates among physicists about the status and use of models, where we see a clear foreshadowing of later debates over scientiﬁc and structural realism.

The Bildtheorie controversy One word of caution, however. At times the debates about how to conceive of science, the scientiﬁc practice, its products, and its criteria of adequacy, became entangled with those about the scientiﬁc acceptability of atomic theory. It is important not to confuse or conﬂate the two. The protagonists of the Bildtheorie1 in the time of structuralism’s infancy can be found on both sides of the atoms debate. Mach, antagonist to atomic theory, plays

192  :    an important role. So did some of Hertz’s misgivings about atomistic representation of the phenomena, but Hertz’s work was instrumental to the development of atomic theory. Boltzmann, a major protagonist of the Bildtheorie, was an enthusiastic advocate and developer of the kinetic theory and statistical mechanics. There is undeniably a distinctive anti-scientiﬁcrealist tenor to the Bildtheorie conception. But the anti-realism takes a sophisticated form, not to be confused with philosophical prejudice against theories that postulate unobservables.2 Planck against the heretics On December 9, 1908 Max Planck addressed the Student Corps of the Faculty of Natural Sciences of the University of Leiden. His announced topic was The Unity of the Physical World-Picture, but his intent included a polemic against those scientists who had turned against realism in the past ﬁfty years. His target was mainly Ernst Mach, but he says that this heresy ‘‘enjoys great popularity, particularly in circles of natural scientists’’ (ibid.: 129). In Planck’s eyes those so misguided had forsaken the faith of their fathers. He speaks against them with passion: When the great masters of exact research contributed their ideas to science: when Nicolaus Copernicus tore the earth from the center of the universe, when Johannes Kepler formulated the laws named after him, when Isaac Newton discovered general gravitation, when your great countryman Christian Huygens put forward the wave theory of light, and when Michael Faraday created the foundations of electrodynamics . . . [Mach’s] economical point of view was surely the very last thing which steeled the resolve of these men in their battle against traditional views and towering authorities. Nein! . . . it was their rock-solid belief in the reality of their world picture. (Planck 1992: 131)3

Planck considered Mach’s heresy to be a mistaken if understandable ‘‘philosophical manifestation of unavoidable disenchantment’’ when the mechanical world view began to disintegrate. But there is some irony in this episode, as we shall see. Perhaps unwittingly, very likely without any conscious polemical purpose, Planck casts his opposition to Mach’s view of science in a way that leaves the issue entirely entangled with the issue of the reality of atoms. His more general discussion of the aim and structure of physical theory is in effect interrupted by the long diatribe, in the last section, against Mach’s views on atomism.4

  BILDTHEORIE    

193

What complicates the disentanglement further is that Planck himself uses the terms of the Bildtheorie of science, which are more adapted to less realist views than his own. Not just in the title of his lecture but throughout, Planck speaks of physical theories as pictures and of the product of science as a whole as a world-picture. This is the language of the Bildtheorie of science, it honors the view of science as representation that was also the common coin of his opponents. While the term Weltbild was already used in much philosophical literature, the introduction of this term into a view of how we are to understand the scientiﬁc enterprise is generally attributed to Hertz.5 But it is possible to see the real opposition between Planck’s and Mach’s views on science, as well as, and properly distinguished from, the opposition between their views on the reality of atoms—by carefully selecting ﬁrst of all the question that Planck himself calls the most fundamental: What do we really mean when we speak of a physical world-picture? Is it merely a convenient but basically arbitrary intellectual concept, or should we take the opposite view, that it reﬂects actual natural processes quite independent of us? (Planck 1970: 4)

The phrase ‘‘basically arbitrary’’ is certainly not innocent of polemics; that is certainly not how his opponents would speak. We ﬁnd Planck’s view of science, its aim and structure, spelled out more fully in the last section of the lecture. He begins by spelling out how he sees the history of modern physics as a story of progress: the old system of physics [sometime before 1900] was not like a single picture, but more like a whole picture gallery; since every class of natural phenomena had its own picture. And these different pictures did not all hang together; one could take any one of them away, without affecting the others. In the future world-picture, this will not be possible. Each one is an indispensable component of the whole and, as such, has a speciﬁc meaning for observed nature; while, conversely, every observable physical phenomenon must ﬁnd its precisely appropriate place in the picture. (Ibid.: 21–2)

The crucial point follows this immediately: ‘‘In this respect, it differs essentially from ordinary pictures, which certainly need to correspond to the original in some particulars, but not in all—a distinction to which, in my opinion, physicists have not hitherto paid enough attention’’ (ibid.: 22).

194  :    On the contrary, that is precisely what the rivals in this view of science explicitly contradict, when they speak in terms of ‘‘pictures’’ along the lines made prevalent by e.g. Hertz and Boltzmann. Indeed, Planck is here rejecting, in effect, the very core of the Bildtheorie while keeping its picture terminology. For there is no point in emphasizing that science presents us with representations of natural phenomena, if not to convey that success in science will consist in constructing an image of nature that is adequate in certain respects and trades on resemblance at best in part, as opposed to constructing a true and accurate copy. That this passage is not meant simply as a bit of futurology, a vision of the best conceivable future, but an expression of what Planck takes to be the deﬁning aim of science, is then made clear: A constant, uniﬁed world-picture is, as I have tried to show, the ﬁxed goal which true natural science, in all its forms, is perpetually approaching; and in physics we may justly claim that our present world-picture, although it shimmers with the most varied colors imparted by the individuality of the researcher, nevertheless contains certain features which can never be effaced by any revolution, either in nature or in the human mind. This constant element, independent of every human (and indeed of every intellectual) individuality, is what we call ‘‘the Real’’. (Ibid.: 25)

The passage I quoted at the outset follows now, and just after that there comes, at least to our eyes, a curious ending to his polemics. He wishes simultaneously to withdraw from the ‘representation’ or ‘picture’ view and to embrace it on a higher (deeper?) level: Those great men did not speak about their ‘‘world-picture’’; they spoke about ‘‘the world’’ or about ‘‘Nature’’ itself. Now, is there any recognizable difference between their ‘‘world’’ and our ‘‘world-picture of the future’’? Surely not! For the fact that no method exists for proving such a difference was made the common property of all thinkers by Immanuel Kant. (Ibid.)

With this uncertain—faltering?—clarion call on behalf of scientiﬁc realism still in our ears let us proceed to the more thorough-going views of science as representation. Maxwell, Hertz, Boltzmann, Mach Planck directs himself primarily against Mach when he insists that the scientiﬁc world-picture must ‘‘[differ] essentially from ordinary pictures, which certainly need to correspond to the original in some particulars, but

  BILDTHEORIE    

195

not in all’’. But it is in Boltzmann’s writings that we see the contrary view most clearly.6 Boltzmann presents his own point of view as deriving mainly from Maxwell and Hertz, two of the heroes of the then recent achievements in electromagnetism.7 Maxwell’s writings are not exactly unambiguous. In fact he is often taken as postulating the reality of the ether and of the electromagnetic waves in the ether, while sometimes despairing of any purely mechanical theory of their character. However, as Boltzmann emphasizes, Maxwell speaks of the envisaged mechanisms as merely analogies, partial analogies, that allow us to get an imaginative grasp on the equations. The equations must on the one hand ﬁt the observed magnetic, electrical, and optical phenomena, and on the other hand allow of some understanding of the theory as a description of a physical process. But as far as description goes, we receive mainly analogies with other forms of material propagation, diffusion, and interaction—with gases, ﬂuids, and heat. Maxwell himself cautions us against thinking of this as a true description of reality behind the phenomena: By a judicious use of this analogy [between Fourier’s equations of heat conduction and the equations of the electrostatic ﬁeld] . . . the progress of physics has been greatly assisted. In order to avoid the dangers of crude hypotheses we must study the true nature of analogies of this kind. We must not conclude from the partial similarity of some of the relations of the phenomena of heat and electricity that there is any real similarity between the causes of these phenomena. The similarity is a similarity between relations, not a similarity between things related. (Maxwell 1881: 51–2)

We have noted before that imagery may trade on ‘higher-order’ resemblance: not a sharing of properties, but of relational structure. That the models provided by a science may have that sort of less direct relationship to the phenomena becomes a guiding theme for structuralisms in the next century.8 Then, as Boltzmann sees it, Hertz makes a virtue of necessity and asserts this as a way to understand the scientiﬁc enterprise as a whole. Thus Hertz writes (and Boltzmann cites this passage): If we wish to lend more color to the theory, there is nothing to prevent us from supplementing all this and aiding our powers of imagination by concrete representations of the various conceptions as to the nature of electric polarization,

196  :    the electric current, etc. But scientiﬁc accuracy requires of us that we should in no wise confuse the simple and homely ﬁgure, as it is presented to us by nature, with the gay garment which we use to clothe it. Of our own free will we can make no change whatever in the form of the one, but the cut and color of the other we can choose as we please. (Hertz 1962: 28)

Indeed, with Hertz we begin to have such an emphasis on the representations and their adequacy to the experimental facts as sole anchor, that we can quite understand Planck’s sense that the represented world is mostly counted as well-lost for love of theory: We form for ourselves inner pictures or symbols of external objects; and the form which we give them is such that the necessary consequences of the pictures in thought are always the pictures of the necessary consequences in nature of the things pictured . . . . The pictures which we here speak of are our conceptions of things. With the things themselves they are in conformity in one important respect, namely in satisfying the above requirement. For our purpose it is not necessary that they should be in conformity with the things in any other respect whatever.9

In the ﬁrst part of this passage we see the relationship pattern characteristic of how a symmetry requirement is to be satisﬁed:10 picture 1

event 1

picture 2

event 2

Figure 8.1. Adequacy as Symmetry

The evolution at the top is a logical deduction relative to assumed conditions; the evolution at the bottom is the regularity in nature in those conditions, and the downward arrows stand for the pertinent ‘picturing’ relation. We can illustrate the idea with our earlier example of the Aviation Model (AVN) for numerical weather prediction. The data are entered concerning current and past conditions, and the program generates a model of the current meteorological state of the region, and calculates its forecast

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evolution. This calculation provides models of the region for the next ﬁve or so days, and Hertz’s constraint requires (naturally!) that those accurately represent the conditions on those days—with the models’ success gauged by some measure of accuracy. In this sort of example, there are few if any hidden parameters characterizing unobservable entities, but the pattern Hertz displays is general. It can be cited equally well as a constraint on hortatory astrology (in which natal charts are progressed to forecast the native’s life history) as on quantum electrodynamics, both of which are replete with parameters characterizing unseen inﬂuences. Not to say, of course, that this properly imposed constraint is satisﬁed in all cases. Hertz’s constraint is a crucial condition for the objectivity of scientiﬁc representation. Far from mere exercises of the imagination, merely adding levels of fantasy to the known empirical realities, scientiﬁc representations must allow us to go reliably from what we know to what we will or can encounter further on. This is the empirical constraint on scientiﬁc theorizing, here phrased in a form easily seen to be appropriate to a structuralist (as opposed to a naïve realist) conception of ‘picturing’ by means of models. Alongside of this positive contribution to an understanding of how an abstract theoretical science can provide representations of the empirical phenomena, there was a good deal of polemics in the air. Boltzmann, though always on the side of those advocating the kinetic and atomic theories nevertheless, lecturing in 1899, expressed the heretics’ philosophical point of view most trenchantly: We know how . . . to obtain a useful picture of the world of appearance. What the real cause for the fact that the world of appearance runs its course in just this way may be; what may be hidden behind the world of appearance, propelling it, as it were—such investigations we do not consider to be of the task of natural science. (Boltzmann 1905a: 252)

Finally, we may note Mach’s reaction to Planck’s criticisms of this train of thought. Just as Boltzmann does in this last passage, so Mach attributes those realist misgivings to metaphysical dreams by which philosophers have infected physicists from time to time: In any case, physicists have nothing to seek ‘beyond the appearances’. Whether philosophers will always ﬁnd it necessary to afﬁrm something real . . . whose relations may only be recognized in the wholly abstract form of equations, may be left

198  :    entirely for the philosophers to decide. [. . .] Hopefully, physicists of the 20th century will not let their investigations be disturbed by such meddling! (Mach [1910/1992], 124–5)

So each side depicts the other as having strayed from the true concerns of natural science into mistaken philosophical conceptions of their common enterprise. The ‘‘picture’’ and ‘‘image’’ imagery became pervasive in philosophically reﬂective writing on physics by physicists, increasingly so during the controversies over the interpretation of quantum mechanics in the late 1920s and 1930s. Bohr’s insistence on the use of the complementary wave and particle pictures—neither of which can be regarded as faithfully mimetic representations, precisely because they are mutually exclusive—is too well known to bear repetition. But Erwin Schro¨ dinger, who rejected wave-particle duality, wrote in the same terms. Heisenberg’s uncertainty relations, he wrote, ‘‘changed our conception [of . . . and] even what is to be understood by a physical world image.’’11 This usage continues throughout his writings: we do give a complete description, continuous in space and time without leaving any gaps, conforming to the classical ideal—a description of something. But we do not claim that this ‘something’ is the observed or observable facts; and still less do we claim that we thus describe what nature (matter, radiation, etc.) really is. In fact we use this picture (the so-called wave picture) in full knowledge that it is neither.’’12

We can certainly see major strands of anti-realism in all these writings, but notice that it is not simplistic anti-realism—in fact, most of these writers were actively involved in developing the new physics, including atomic theory and quantum theory. The Bildtheorie view of science takes a general form that is compatible, for example, with what we know later under the name of structural realism. If what science gives us by way of theories and models is to be conceived of as pictures, as representations, then the question is opened as to just what the relevant and appropriate criteria of adequacy are for them. An extreme view would be that a good representation of that sort is one that corresponds in every respect to what it is representing. That is not what even a quite extreme scientiﬁc realist would say, since there is always a good deal of mathematical artiﬁce present. But it is approached, later in the twentieth

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century by various forms of scientiﬁc realism (Salmon, Boyd, or Leplin for example, or to a lesser extent ‘‘entity realism’’), and by the criterion for corresponding ‘‘elements of reality’’ in the paper on the completeness of quantum mechanics by Einstein, Podolsky, and Rosen. But a different criterion is found in the varieties of structuralism in the philosophy of science that we shall go on to examine—criteria to the effect that scientiﬁc models trade for their success on resemblance with respect to structure alone. And there is also of course a still more liberal empiricist conception according to which the base line criterion is just empirical adequacy. Each of these is an answer to the sort of question which, in our initial discussion, we saw proper to the identiﬁcation of adequate representation, which needs to take into account what is the purpose or aim and speciﬁcally what is at stake in the representational practice. Representational Options and Realism: Descartes, Hertz, Poincar´e, Duhem What options were coming into play at this point? The equations come with a narrative about unobservable entities, but this narrative is classiﬁed as providing us with a useful representation, that only needs to ﬁt the regularities in what is observed. This view, though newly salient at the time, is not without historical precedent; witness Descartes’s stand at the end of the Fourth Book of his Principles of Philosophy: Prop. 204. That, touching the things which our senses do not perceive, it is sufﬁcient to explain how they can be . . . ( Descartes 1959: 210)

This is his reply to the objection that his theories may ﬁt the phenomena without being true, something he concedes quite readily: . . . just as the same artisan can make two clocks, which, though they both equally well indicate the time, and are not different in outward appearance, have nevertheless nothing resembling in the composition of their wheels; so doubtless the Supreme Maker of things has an inﬁnity of diverse means at his disposal, by each of which he could have made all the things of this world to appear as we see them, without it being possible for the human mind to know which of all these means he chose to employ. (Ibid.)

But the aim of science is misunderstood if this is taken as an objection: I believe that I have done all that was required, if the causes I have assigned are such that their effects accurately correspond to all the phenomena of nature,

200  :    without determining whether it is by these or by others that they are actually produced. And it will be sufﬁcient for the use of life to know the causes thus imagined, for medicine, mechanics, and in general all the arts to which the knowledge of physics is of service, have for their end only those effects that are sensible, and that are accordingly to be reckoned among the phenomena of nature. (Ibid.)

In support, Descartes cites Aristotle’s On Meteors: ‘‘We consider a satisfactory explanation of phenomena inaccessible to observation to have been given when our account of them is free from impossibilities.’’13 How should we think of scientiﬁc representation if conceived of in this manner? The guiding idea is something like a painting of real terrestrial events surrounded by indications to show not just what is happening but what is really going on. Demons and angels perhaps, or imagined natural but invisible mechanisms made visible by the artist’s prerogative. The corresponding view of its aim or criterion of success would then be that the painting is ‘‘right’’ if it is historically accurate and ﬁts the terrestrial events into a possible intelligible scenario that accounts for them. The criteria of intelligibility are our own—requiring some sort of causal pattern perhaps, satisfying e.g. mechanistic views of interaction. Since intelligibility does not imply knowledge of the truth, those criteria can be satisﬁed without demanding that the scenario is actually true overall.14 The implied agnosticism, or even skepticism, is mostly downplayed in Boltzmann’s writings. While seemingly insisting that not so much is needed for science, his texts leave it open that one could also believe the entire ‘picture’ to be accurate in all respects.15 The skepticism is clearer in Hertz’s verdict, later repeated by Poincar´e, on classical electromagnetism. Hertz’s early book Die Constitution der Materie already introduced his emphasis on representation as the task and method of the exact sciences.16 How can there be an atomic theory, to cover the macroscopic phenomena of e.g. heat or elasticity? Any properties we can ascribe to the atoms will have to be among the only ones we know, hence properties characterizing observable macroscopic objects. But no conception of atoms as just small versions of familiar kinds will be adequate for atomic theory; in fact, he calls any such conception a logical error. Thus, at least, Hertz sees the issue—expressing a concern which may not have seemed pressing to his contemporaries still in the thrall of mechanical explanation.

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We can at the same time see that the entanglement with the question of the scientiﬁc acceptability of atomic theory is only contingently involved with this, as a currently salient example of scientiﬁc representation. Not only was Hertz’s work crucial to the establishment of that theory, but his sentiment concerning the theory’s representations is echoed almost literally half a century or so later by Heisenberg: The atom of modern physics can be symbolized only through a partial differential equation in an abstract space of many dimensions . . . . All its qualities are inferential; no material properties can be directly attributed to it. That is to say, any picture of the atom that our imagination is able to invent is for that reason defective. An understanding of the atomic world in that primary sensuous fashion . . . is impossible.17

Symbolized, hence represented, yes; but not mimetically represented, not picturable in the familiar parameters of macroscopic observation. From Hertz to Heisenberg, the complexities of modeling the unobservable, once even Newton’s ‘‘instant distant correlation’’ version of mechanics gives out, drives the liberalization of the sense of representation in which science is taken to represent the investigated phenomena. Whether or not Hertz’s argument here was telling, the issue led him to the view that all we can have for atomic physics is a theory in which the magnitudes (theoretical parameters) are connected to each other and to macroscopic phenomena by mathematical equations. It does not have seemed necessary to him, as far as success in science is concerned, that some further or deeper meaning needs to be ascribed to those parameters. But we must not understand this to mean that the theoretical parameters are allowed to be empirically ungrounded! They need to be connected to measurement, relative to the theory, in a way that allows for determination of their values in principle. This remained a problem for another few decades, but its appreciation goes hand in hand with Hertz’s conviction that what is essential in the mathematical equations in which the theory is formulated is that they provide a representation of relations between macroscopically observable magnitudes. In the Introduction to his book Electric Waves, Hertz described his own presentation as but one possible representation of Maxwell’s electromagnetic theory, simplifying it ‘‘as far as possible by eliminating . . . those portions which could be dispensed with, inasmuch as they could not

202  :    affect any possible phenomena’’. This presentation is to be contrasted with Maxwell’s presentation of the theory and its representation as a limiting case in Helmholtz’s theory, but Hertz insists that they all have ‘‘substantially the same inner signiﬁcance’’. He continues, in a familiar passage: This common signiﬁcance of the different modes of representation (and others can certainly be found) appears to me to be the undying part of Maxwell’s work. This, and not Maxwell’s peculiar conceptions or methods, would I designate as ‘‘Maxwell’s theory.’’ To the question, ‘‘What is Maxwell’s theory?’’ I know of no shorter or more deﬁnite answer than the following:—Maxwell’s theory is Maxwell’s system of equations. Every theory which leads to the same system of equations, and therefore comprises the same possible phenomena, I would consider as being a form or special case of Maxwell’s theory . . . .18

and this affects how he envisages scientiﬁc methodology: It is true we cannot a priori demand from nature simplicity, nor can we judge what in her opinion is simple. But with regard to images of our own creation we can lay down requirements. We are justiﬁed in deciding that if our images are well adapted to the things, the actual relations of the things must be represented by simple relations between the images. (Ibid.: 23)

Maxwell himself had attempted to prove the existence of models of his equations in mechanics. The theory of the ether was a sustained attempt to provide them with a concrete mechanical underpinning (cf. Stein 1989: 61–2). When Maxwell had his theory fully worked out, he discarded the earlier rather primitive ether models and tried to subsume his theory under the generalized dynamics of Lagrange, which deals with mechanical systems whose internal constitution is not fully speciﬁed.19 In Hertz’s, and later Poincar´e’s, verdict we recognize a deﬁnitive goodbye to the interrelation of matter and ether as a live topic in physics.20 Hertz’s last book, The Principles of Mechanics Presented in a New Form represents physical processes by embedding them in a larger structure consisting of ‘hidden masses’, but no forces. When the kinetic energy of a macroscopic system apparently changes, in a way normally accounted for by postulating forces, the effect is instead accounted for within mechanics by reference to the motions of the hidden masses. But these hidden masses consist of mass-points, entities with zero extension, and so the theory does not appear to afford a literal realistic construal. The Principles of Mechanics is divided into two books, and the basic theory developed in the ﬁrst is

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applied to what could be observable processes through something like what Reichenbach would later have called coordination. Is there anything, we might ask, that goes beyond coordination? This is precisely where Hertz writes what I quoted above: ‘‘We form for ourselves inner pictures or symbols of external objects; and the form which we give them is such that the necessary consequences of the pictures in thought are always the pictures of the necessary consequences in nature of the things pictured . . . . The pictures which we here speak of are our conceptions of things. With the things themselves they are in conformity in one important respect, namely in satisfying the above requirement. For our purpose it is not necessary that they should be in conformity with the things in any other respect whatever.’’21 A decade or so later, close to the date of Planck’s scorching attack on the heretics of his generation, Poincar´e and Duhem were publishing views in harmony with Hertz’s and Boltzmann’s Bildtheorie of science. Duhem made the concept of representation the cornerstone of his view of science. After a thorough critique of more metaphysical and realist conceptions in the ﬁrst chapter of his The Aim and Structure of Physical Theory, he presents his own: A physical theory is not an explanation. It is a system of mathematical propositions, deduced from a small number of principles, which aim to represent as simply, as completely, and as exactly as possible a set of experimental laws. (Duhem 1962: 19)

What could be the use of theory thus conceived? Success with respect to simplicity, completeness, and exactitude in a representation would speak for itself, one might say. But something more is needed to show how we can conceive of familiar ways of drawing on theory in prediction, application, and practice. Duhem responds by elaborating on this conception as allowing us to see theory as providing a taxonomy in which to locate the regularities, objects, and processes that are of practical concern. To begin with the regularities: ‘‘Theory is not solely an economical representation of experimental laws; it is also a classiﬁcation of these laws’’ (ibid., 23), thus bringing order and internal connections to what would otherwise be a jumble of empirical generalizations. Secondly, as illustrated by his discussion of taxonomy in zoology, the theoretical descriptions of the materials at hand are to be understood as classiﬁcation. This is a radio: anyone can see that. If

204  :    we add that it is a device that transforms electromagnetic waves into sound-waves, we have not—on this view—expressed a belief in the reality of electromagnetic waves. Instead, we have pronounced the object’s classiﬁcation sub specie physical theory, we have located it in the space of theoretical kinds. Poincar´e’s views on science were generally tending toward what Planck considers the great heresy, though stated with caution and diplomacy. In the spirit of Hertz’s view of science as non-mimetic representation, Poincar´e speaks in Science and Hypothesis of ‘‘images we substituted for the real objects which Nature will hide for ever from our eyes. The true relations between these real objects are the only reality we can attain . . . .’’22 The same point occurs already in the preface, p.xxiv: ‘‘the aim of science is not things themselves, as the dogmatists in their simplicity imagine, but the relations between things; outside those relations there is no reality knowable’’. Of the principle of conservation of energy, for example, he writes that if we try to enunciate it in full generality, ‘‘we see it vanish, so to speak, and nothing is left but this—there is something which remains constant. . . .’’ (Ibid.: 132). This begins to sound quite extreme with respect to what we can possibly know, but the denial is leavened with an insistence on a hard core of increasing empirical knowledge: No theory seemed more solid than Fresnel’s . . . . Nevertheless, we now prefer Maxwell’s. Does that mean that Fresnel’s work was in vain? No, because Fresnel’s aim was not to know whether there really is an ether, whether it consists of atoms, whether these atoms really move in one sense or another; it was to predict optical phenomena.23

Here Poincar´e’s view is quite in harmony with Duhem’s, that the aim of physical theory is to systematize experimental laws, by whatever theoretical means lend themselves to that, and without any implication of reality for the theoretical parameters.

Representation: the problem for structuralism There is nevertheless something paradoxical in all this. If science describes nature, Maxwell’s equations must form a theory about what something

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is like. Mustn’t the theory also say what that something is? Here is the dilemma: • if Maxwell’s equations are statements, what do they say? • if they are not statements, how can they amount to a theory at all? If we leave aside the more instrumentalist (non-statement) options, we can discern here two not very well-distinguished alternatives. The ﬁrst is that no, there is no ether, no mechanical medium subject to wave disturbances, but yes, there is something, it is the electromagnetic ﬁeld itself, which is a thing, and it is not the shape or form of something else. Today that is an often expressed view, perhaps not always clearly distinguished from simple rejection of the classical ether: Fields in empty space have physical reality; the medium that supports them does not (Mermin 1998: 753). On this option, there is no puzzle, just a new ontology, some new and previously inconceivable furniture for the world. The second alternative is more agnostic, and so presumably closer to how Poincar´e shies away from claims to knowledge of the unobservable. It could be expressed like this:24 • The equations only describe a form or structure—if that is the form or structure of something, then that something is an unknown entity. • The ﬁeld is ﬁrst of all an abstract entity (mathematical: e.g. a function assigning values to points in space), though we can of course also give the name ‘‘ﬁeld’’ to whatever it is—if anything—that bears this structure. • That unknown bearer might well have other properties, just as ordinary things have properties beside their shape. But the theory does not describe those. • Science abstracts, it presents us with the structural skeleton of nature only. To begin even this sounds rather reactionary, just when we have discarded the ether and its frustratingly elusive qualities, there is reference again to something, whatever it is, that ‘‘bears’’ the ﬁeld after all. But in retrospect we can understand it as the beginning of sustained attempts to develop a structuralist view of science, and we shall follow this attempt through several successive stages.25

206  :    Unfortunately, these attempts ran into heavy weather. The problems to come show their ﬁrst signs at this time; they will erupt into full-blown paradox within two decades, and not disappear before our own day, if then. Duhem’s problem for Poincar´e On Duhem’s view, as we saw, a physical theory is ‘‘a system of mathematical propositions . . . which aim to represent a set of experimental laws’’ . But how can mathematical propositions represent experimental laws? The latter have terms applying to physical objects and processes, the former do not. Here we recognize what a decade or so later Reichenbach presents as the problem of coordination. In the examples Duhem takes up, the theoretical propositions have terms which—though standing for e.g. real number valued functions—look like names of properties: ‘‘mass’’, ‘‘charge’’, ‘‘wavelength’’, ‘‘kinetic energy’’. In atomic theory, for example, such terms appear to be applied to entities to which no experimental law refers, at least as Duhem sees it. But the problem is not peculiar to that case. Indeed, as Duhem forcefully points out, even the experimental laws involve terminology that is meaningful only because of its links with certain theoretical roles, and not simply because of their extension among the experimentally manipulated or inspected objects. The language in which this classiﬁcation is pronounced is a theory-laden language: The role of the scientist is not limited to creating a clear and precise language in which to express concrete facts; rather it is the case that the creation of this language presupposes the creation of a physical theory. (Ibid: 151).

and it is in some ways amazing that what Duhem taught there needed to be understood all over again a half century later. We are some distance here from a set of pure mathematical propositions, and indeed, the sort of taxonomy that a scientiﬁc theory provides for its domain does not seem to be something that could be presented in the language of pure mathematics. So Duhem attempts an account of the scientist’s theoretical language. It is distinguished by its special jargon, foreign to the discourse of every day, and so we may be tempted to assimilate it to the technical language found in a practical art or craft. ‘‘That would be a mistake,’’ he writes, and continues:

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I am on a sailing ship. I hear the ofﬁcer on watch shout out the order: ‘‘All hands, tackle the halyard and bowlines everywhere!’’ A stranger to things of the sea, I do not understand these words, but I see the men on ship run to posts assigned in advance, grab hold of speciﬁc ropes, and pull on them in regular order. The words uttered by the ofﬁcer indicate to them very speciﬁc and concrete objects, arousing in their mind the idea of a known manipulation to be performed. Such, for the initiated, is the effect of technical language. Quite different is the language of the physicist. Suppose the following sentence is pronounced to a physicist: ‘‘If we increase the pressure by so many atmospheres, we increase the electromotive force of a battery by so many volts.’’ It is indeed true that the initiated person who knows the theories of physics can translate this statement into facts and can do the experiment whose result is thus expressed, but the noteworthy point is that he can do it in an inﬁnity of different ways. ( Duhem 1962: 148–9)

On this matter, Duhem criticized Poincar´e, who expressed too simplistic a view of scientiﬁc description. Poincar´e wrote that a scientiﬁc fact ‘‘is nothing but a brute fact stated in a convenient language’’, and likens the relation of that convenient language to more familiar discourse to the relation between French and German. Duhem sees a much weaker relation between theoretical description and observable phenomenon: Between an abstract symbol and a concrete fact there may be a correspondence, but there cannot be complete parity; the abstract symbol cannot be the adequate representation of the concrete fact, the concrete fact cannot be the exact realization of the abstract symbol; the abstract and symbolic formula by which a physicist expresses the concrete facts he has observed in the course of an experiment cannot be the exact equivalent or the faithful story of these observations. (Ibid.: 151)

So now we can see the theory of scientiﬁc representation, of theoretical description and mathematical picturing, getting into deep water, as Duhem begins to press this chasm between representation and what is represented, which is not bridged by a simple reference relation.26 While we cannot say that he has as yet perceived the deep perplexities which this forebodes, we can’t help but sense the foreboding. Weyl on isomorphism So we have seen that while Duhem starts by regarding a physical theory as a set of mathematical propositions, this quickly turns out to be not a set of propositions in pure mathematics but in a scientiﬁc language involving

208  :    mathematical symbolism. The character of that language is left more as a challenge than as a solved problem. Poincar´e, though unguarded in the passages Duhem criticizes, was much more engaged in the foundations of mathematics and of physical geometry. For him theoretical physics has a pure part, which is pure mathematics, connected with scientiﬁc practice through what he calls conventions—e.g. to ﬁx the extension of ‘‘congruent’’ in nature. These conventions are what Poincar´e offers for the task of coordination, as we saw before. It is precisely because he takes the problem of coordination to be solved, by this means, that Hertz’s dictum that Maxwell’s theory is just Maxwell’s equations could be one we can hear on Poincar´e’s lips. It would be anachronistic to attribute either to Duhem or to Poincar´e a clear sense of just how much of a problem this skates over, how deep a problem is appearing here for the structuralist program. Scientiﬁc theories are, or provide, pictures, representations; they do so by drawing on the resources of mathematics; but just how do those ‘pictures’, those mathematical constructions, represent what they represent?27 This is the problem of coordination not in its practical form, in which it is solved again and again in scientiﬁc practice, but in the more abstract form to be faced when structuralism is developed into a general view of science—and the threat it poses is that it is possibly not solvable at all. This problem appeared explicitly when Bertrand Russell and Rudolf Carnap developed their aspirant-structuralist views in the late 1920s, which we will examine in some detail below. Before delving into the more philosophical contexts in which those writers operate, we can ﬁnd the problem displayed graphically in a lecture by another philosopher–physicist. Hermann Weyl expressed the fundamental insight as follows in 1934:28 A science can never determine its subject-matter except up to isomorphic representation. The idea of isomorphism indicates the self-understood, insurmountable barrier of knowledge. [. . .T]oward the ‘‘nature’’ of its objects science maintains complete indifference. (Weyl 1934: 19)

The initial assertion is clearly based on two basic convictions: • that scientiﬁc representation is mathematical, and • that in mathematics no distinction cuts across structural sameness.

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The former is just the structuralist conviction we have witnessed so clearly in the mathematical physicists of the late nineteenth and early twentieth century. The latter is a take on what mathematics is and does, expressed by one of its most philosophically profound masters. Care is needed with Weyl’s point. If two structures are isomorphic, but are parts of a third structure, then they may be distinguishable by their relations to the whole. The Euclidean plane contains many mutually congruent triangles. In addition, ‘‘isomorphic’’ often has a restricted sense in context: ‘‘isomorphic groups’’ can mean ‘‘structures related by a group isomorphism’’, but though the same as far as the group operations are concerned they may be quite different in other respects. If however we consider two structures, taken by themselves, and an isomorphism which preserves all the deﬁnable structural aspects, then indeed they are not discernible in any mathematically representable way. But surely we can point to differences between domains with the same structure? Weyl illustrates this with the example of a color space and an isomorphic geometric object. Just a few years before his writing the CIE 1931 color space was created by the International Commission on Illumination, based on experiments on human color vision performed in the 1920s. This color space is a region in the projective plane. If we can nevertheless distinguish the one from the other, or from other attribute spaces with that structure, doesn’t that mean that we can know more than what science, so conceived, can deliver? Weyl accompanies his point about this limitation with an immediate characterization of the ‘‘something else’’ which is then left un-represented. This—for example what distinguishes the colors from the points of the projective plane—one can only know in immediate alive intuition. (Ibid.)

When two objects are represented by isomorphic structures—hence, when their representation presents no difference between them—we may still know that they are distinct and in un-represented ways different objects.29 In practice there would always be further differences that could be represented as well (by enlarging the mathematical model), but Weyl clearly wants us to think of the limiting case where there is nothing more to do in the modeling. Even then it seems we must grant that we could know that the two represented objects are distinct. But Weyl’s solution to how this is possible, appealing to our immediately alive intuition, may not be entailed.

210  :    What could the additional knowledge be? Weyl’s assertion sounds paradoxical, it sounds diminishing with respect to scientiﬁc knowledge. We seem to be left with four equally unpalatable alternatives: • that either the point about isomorphism and mathematics is mistaken, or • that scientiﬁc representation is not at bottom mathematical representation alone, or • that science is necessarily incomplete in a way we can know it to be incomplete, or • that those apparent differences to us, cutting across isomorphism, are illusory. In his comment about immediate alive intuition, Weyl appears to opt for the second, or perhaps the third, alternative. But on either of these, we face a perplexing epistemological question: Is there something that I could know to be the case, and which is not expressed by a proposition that could be part of some scientiﬁc theory? The problem with Mary Weyl’s paradox is quite abstract, in comparison with the discussions of the Bildtheorie physicists. Perhaps his brief illustration of the isomorphism of the color space to the projective plane is abstruse as well. But this example, to which he offers his ‘‘intuitionist’’ philosophical response, • what distinguishes the colors from the points of the projective plane—one can only know in immediate alive intuition has a narrative version that was proposed in philosophy of mind for a quite different purpose. It is known, after its creator, as Frank Jackson’s Mary Problem; and here is its statement in David Lodge’s novel Thinks: That’s Frank Jackson’s Mary, the colour scientist. The idea is that she’s been born and raised and educated in a totally monochrome environment. She knows absolutely everything there is to know about colour in scientiﬁc terms—for example, the various wavelengths that stimulate the eye in colour recognition—but she has never actually seen any colours. Notice there are no mirrors in her room, so she can’t see the pigmentation of her own face, eyes, or hair, and the rest of her body is covered. Then one day she’s allowed out of the room, and the ﬁrst thing she sees is, say, a red rose. (Lodge 2001: 53)

  BILDTHEORIE    

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Having all the knowledge about color that theoretical physics and physiology can provide, will Mary be able to classify the rose as red—and if so, how? Weyl would point out that the projective plane has many symmetries, so that there are in effect many isomorphisms between the color space and that mathematical structure. Transform an isomorphism appropriately and you generate another isomorphism. Like Locke’s ‘‘inverted spectrum’’ problem, only worse! So if what Mary knows is just this structure, that is certainly not enough to identify which color is which. If her knowledge includes differentiating connections of the colors with things outside the color space—e.g. of what is ﬁltered by a ruby—then the same problem reappears in principle. If all she knows is the structure of the entire domain, colors and rubies and whatever else may be included, that will not be enough to single out speciﬁc features. Even if they happen e.g. to have a unique location in the spatial conﬁguration, that may not be one identiﬁable by Mary in her own (though only partially accessible) frame of reference. Can’t Mary do better than a disembodied mind endowed with a complete mathematical representation of nature? She can indeed. Somewhat later in the novel we read an imagined sequel, about the day when Mary will be allowed to come out: She glanced down at her own hands . . . sheathed in serviceable pigskin which she was permitted to remove only at night, in total darkness, with the assistance of the blind maidservant Lucy, thus preventing any inadvertent glimpse of the pearly pinkness that—so she understood—tinted the translucent plates covering the dorsal surfaces of her ﬁnger-ends. Well, she would soon be able to see her ﬁngernails along with many other things . . . . (Ibid.: 154–5)

When the gloves are removed in broad daylight, she will see her ﬁngernails, and know where they are located in the color space, because she knew already how to identify her nails and had already the knowledge that they are pink.30 Suddenly that theoretical object, the color space, will be subject to coordination (to that extent), she will be able to say ‘‘this is pink’’, and if she can gather some more such clues, soon know her practical way around the colored world. The change is that, at the point of success, she will be able to locate any pink object she sees in the same region of the color

212  :    space—be able to say ‘‘this, in front of me, is there in the color space’’, and the many symmetries of the projective plane will no longer respect what she can discriminate. This sounds like a way to success, but note how crucially it relies on the indexical: what Mary must be able to say at this point is ‘‘these are my ﬁngernails; my nails are pink’’. So let us repeat the question that we posed for the conclusion of Weyl’s reasoning, which was: Is there something that I could know to be the case, and which is not expressed by a proposition that could be part of some scientiﬁc theory? There is some sense, undoubtedly, in which Mary can answer this question afﬁrmatively for herself—but in what sense?31 In the solution here imagined, the crux is the self-attribution of a location in color space. Is that the only way? This question we will pursue throughout Part III, and we will return to it explicitly at the end.

9 The Longest Journey: Bertrand Russell The ﬁrst sustained, rigorous development of a structuralist view of science appeared at the hands of Bertrand Russell, in whose writings the philosophical motivation precedes a precise formulation drawing on mathematical logic. The appeal lay perhaps mainly in the motivation, though in retrospect his arguments were couched in a by then pass´e ‘mirror of nature’ view of mind. The rigor came from his founding of theoretical physics in a mathematics constructed along logicist lines, which is also precisely what proved his undoing.

Prolegomena to Russell’s conversion to structuralism Bertrand Russell provides an excellent example of a philosopher who initially resisted structuralist leanings, insisting speciﬁcally on a quite contrary view. This connected directly with philosophy of mathematics, because on this early view, he took geometry to be not simply abstract mathematics but the theory of physical space. Lobachevsky and Helmholtz When non-Euclidean geometries were ﬁrst created, the question arose whether there could be an empirical test to decide whether space is Euclidean. In principle it seemed easy enough. For example, in Euclidean space the interior angles of a triangle add up to 180 degrees. Gauss’s earlier studies of surfaces with negative curvature could be applied here. In hyperbolic geometry, in a three-sided ﬁgure whose sides are arcs of minimal length, the interior angles sum to less than 180 degrees, the defect increasing

214  :    with the area. But the extent of this defect depends on the curvature of the surface, which may be very small in which case the differences from a surface of zero curvature (Euclidean plane) would show up signiﬁcantly only in very large areas. So Lobachevsky looked into astronomical data. He suggested that one might ‘‘investigate a stellar triangle for an experimental resolution of the question.’’ The ‘‘stellar triangle’’ he proposed was the star Sirius and two different positions of the Earth at six-month intervals. But if there was such a defect in the sum of the interior angles, it was still within the limits of measurement error. Indeed, for any defect you can name and any size, there would be a curvature constant small enough to guarantee that the defect would still be within those limits. So, if there are also limits on the size of the regions we can inspect in this way, in the course of human history, the difference between Euclidean and hyperbolic space may be beyond what such measurements can reveal. But there is a deeper problem. The measurement results will also depend on what we take as measurement standard, and what presuppositions are in place with that choice. This point can be graphically (if imperfectly) illustrated by von Helmholtz’s examples of mirror-universes. Helmholtz imagined that we are making measurements to determine whether our space is Euclidean. Do the interior angles of a triangle add up to 180 degrees? Suppose they do. At the same time he imagines that we are reﬂected in a huge concave mirror. In that mirror we see little people moving around with rulers and ‘doing the same thing’ as we do. Of course, they get the same results, and announce that they live in a Euclidean space. We want to disagree; they are moving around on a concave surface, and their measuring rods change length as they work. All geometrical measurements . . . with regularly varying images of real instruments would yield exactly the same results as in the outer world. . . . I do not see how men in the mirror are to discover [the in-]correctness of Euclid’s axioms. (Helmholtz 1956: 661)

The reﬂection in the mirror is a perfect image of what is reﬂected. If we do not look at any relation of these two sides to ourselves (or any third object or background in which they are located) then they have precisely the

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same structure. Thus this example illustrates an isomorphism. Two objects which are isomorphic are mathematically no different from each other. We my overlook this if we, very naturally, think of the two as parts of a larger structure (or as related to ourselves as onlookers). In that larger context such parts may indeed be distinguishable, namely by their relations to other parts or to the whole. But if we are dealing with the world as a whole and a model of the world—as in a theory of physical space—then we have no such recourse. Mathematicians’ reaction One response to the birth of non-Euclidean geometry had thus been that the question might be subject to empirical test. This had not fared well, in view of this relativity to the measurement standard, and was by no means the only or even the most acceptable idea about this new subject at the time. Another main reaction was that to understand non-Euclidean geometries they had to be interpreted within Euclidean geometry.1 To be intelligible they should be readable as strangely worded descriptions of parts or aspects of Euclidean space. That this is possible was shown initially for the hyperbolic plane by Beltrami in his ‘‘Saggio di Interpretatione della Geometria Non-Euclidia’’ (1868).2 Beltrami himself expressed his goal as maintaining Euclidean geometry as the one true theory of space. Felix Klein perceived the limits of this effort and ﬁrst attempted to interpret both Euclidean and non-Euclidean geometries within projective geometry. But very soon through Klein’s work, and even more radically through Riemann’s, there came into being such a cornucopia of geometries that any true theory of space underlying all of them could be very little more than pure logic. The mathematicians’ response was that none were privileged, all geometries were on a par and certain to remain parts of mathematics. But how do they relate to the physical world which we investigate empirically? With the focus shifted to interpretation, through such puzzles as Helmholtz’s mirror worlds, that problem has now taken on a very different shape. It is no longer a straightforward question that could be settled by measurement. Klein’s idea that a single projective space could be the domain of either a Euclidean or non-Euclidean geometry introduced a new way of viewing the matter. Thus with Klein, and also Lie, we get a different feeling: they display the Euclidean and non-Euclidean geometries as pertaining

216  :    to a single space, the space of projective geometry, but with different metrics—or if you like, different congruence relations. Different congruence relations, corresponding in practice to different measurement standards, imposed on the same space give it a different geometry. We can see an initial locus here for the problem of coordination. The mathematicians’ attitude tied in well with Helmholtz’s illustration: with different measurement techniques, we may be mapping out different congruence relations. To ﬁnd out whether two different objects are congruent we would presumably do something like this: transport a meterstick from one to the other. But that we have designated this instrument as a meter-stick, hence as establishing congruence, is a choice, and may have alternatives. Noting then how this ties in with Helmholtz’s mirror imaging, and the latter with Weyl’s point, we can now see how pervasively Weyl’s paradox shows up. Within the realm of mathematical structure, we cannot make the distinctions clearly used in empirical scientiﬁc practice. The physicist—or more likely, the realist philosopher of science—‘‘wishes to ﬁnd out what things are like’’. But the mathematician says ‘‘that makes sense only while taking for granted your choices when you accepted or created measurement standards and kept in place a whole host of presuppositions—apart from that, there are no real differences to be found’’. Bertrand Russell: There must be ‘‘a fact of the matter’’! When Bertrand Russell wrote his Essay on Geometry he followed Klein’s lead in this reconstruction of the different geometries, which ranged Euclidean and non-Euclidean geometries under projective geometry. He even gave what purported to be an a priori, or perhaps transcendental, argument that physical space must be of this sort—in modern terms, this means that physical space must have constant curvature.3 But Russell added that there must be a unique real relation of congruence. Given that, it follows that there is a fact of the matter, whether space is really Euclidean or not. Geometry, at least the geometry used in physics, must be according to Russell a non-vacuous theory of real spatial relations. But how are those real relations identiﬁed? Russell maintains that we are directly acquainted with them, through intuition. Having been pressed to elaborate on this view by Poincar´e’s review in 1899, Russell dismissed the question.4 That is, he said, like asking me to spell the letter ‘‘A’’.5

  :  

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The view that Russell was resisting may aptly be called an early form of structuralism, though it is anachronistic to name it thus.6 Russell insisted to the contrary that there was something more to be known, and in fact captured in scientiﬁc theories representing nature, than mathematics affords—something more than mathematical structure. (At times he took this to an almost absurd extreme, asking even ‘‘wherein . . . lies the plausibility of the notion that all points are alike?’’7 ) Certainly, for the mathematician, all congruence relations, and all possible denotata of their primitive terms, have equal status—but for the empirical sciences there must be more, there must be a fact of the matter that goes beyond what mathematicians can describe.

Russell’s structuralist turn When physics describes the world, how much does it describe? Perhaps there are real spatial relations in contrast to gerrymandered artiﬁcial metrics, as Russell had been contending. Which ones are denoted by the geometric language used in physics seems not to matter, in principle anyway. But then, better to say perhaps that science does not describe nature down to that level of detail. Better to say, perhaps, that science describes only structure without content. We see Russell taking this line in his Problems of Philosophy (1912). Indeed, he elaborates there a structuralist position that is uniform with respect to the entire content of the physical sciences. Let us begin with what he says about space and geometry, and then note how this is extended to the ether, electromagnetic waves, atoms, and so forth. Russell insists that to understand a proposition (so that it may be capable of being judged true or false) we must be acquainted with all its ultimate constituents. This includes certain concrete individual entities but also the properties and relations expressed by its predicates. However, we are acquainted only with those things which are part of our direct experience.8 Physics speaks of things well beyond the ken of direct encounter in experience, things in the External World. Unfortunately, it is exactly that External World which science purports to describe. How is this possible? Russell could have said that science describes nature simply by saying that there exist entities, with which we are not acquainted, but which have

218  :    the same properties and stand in the same relations, as enter our direct experience. Beginning with the theory of space, he appears to say precisely that at ﬁrst: If, as science and common sense assume, there is one public all-embracing physical space in which physical objects are, the relative positions of physical objects in physical space must more or less correspond to the relative positions of sense-date in our private spaces. There is no difﬁculty in supposing this to be the case. [. . .] thus we may assume that there is a physical space in which physical objects have spatial relations corresponding to those which the corresponding sense-data have in our private spaces. It is this physical space which is dealt with in geometry and assumed in physics and astronomy. (Russell 1912/1997: 30–1)

The reason he has given for this is that the objects are postulated in the ﬁrst place as causes to explain our sensations, and causation presupposes spatial contiguity. But in that motivating discussion he was moving back and forth between his rather strange ontology and the common sense picture. Attempting to be more conscientious we see the gloss fading rapidly: Assuming that there is physical space, and that it does thus correspond to private spaces, what can we know about it? We can know only what is required in order to secure the correspondence. That is to say, we can know nothing of what it is like in itself, but we can know the sort of arrangement of physical objects which results from their spatial relations. We can know, for example that the earth and moon and sun are in one straight line during an eclipse, though we cannot know what a physical straight line is in itself. . . . Thus we come to know much more about the relations of distances in physical space than about the distances themselves. . . . We can know the properties of the relations required to preserve the correspondence with sense-data. . . . (Russell 1912/1997: 31–2)

Note well that now we are asserted to know, not the distances or other spatial relations between bodies, but only the properties of those relations. That is, we know what the abstract structure is. If we ‘‘cannot know what a physical straight line is’’ we certainly cannot know such other relations in physical space as congruence. All we know about that, presumably, is what the axioms of a geometry can say about congruence. Has the ‘‘real property’’ realism adopted in response to von Helmholtz, Klein, and Poincar´e been given up? That is not clear. What has been given up certainly is any pretence to knowledge of those real properties themselves.

  :  

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Every analogy with familiar things, like waves in water, planets, and billiard balls was already heavy with disanalogies. So Russell says: we can only infer the properties of the properties, and the properties of the relations—the type of structure: We can know the properties of the relations required to preserve the correspondence with sense data, but we cannot know the nature of the terms between which the relations hold . . . . [A]lthough the relations of physical objects have all sorts of knowable properties, . . . the physical objects themselves remain unknown in their intrinsic nature . . . .’’ (Russell 1912: 32, 34)

The Analysis of Matter (1927) makes this precise: this structure is exactly, no more and no less, what can be described in terms of mathematical logic. The logic in question is strong, and today we would see it as higher order logic or set theory. But still, how little this is! Science is now interpreted as saying that the entities stand in relations which have such properties as transitivity, reﬂexivity, etc. but as giving no further clue as to what those relations are. . . . whenever we infer from perceptions it is only structure that we can validly infer; and structure is what can be expressed by mathematical logic. The only legitimate attitude about the physical world seems to be one of complete agnosticism as regards all but its mathematical properties. (Russell 1927: 254, 270)

Newman’s objection to Russell The mathematician M. H. A. Newman made the crucial critical point in a review article concerning The Analysis of Matter:9 . . . it is meaningless to speak of the structure of a mere collection of things . . . . Further, no important information about the aggregate A, except its cardinal number, is contained in the statement that there exists a system of relations, with A as ﬁeld, whose structure is an assigned one. For given any aggregate A, a system of relations between its members can be found having any assigned structure compatible with the cardinal number of A. (Newman p. 140)

We can illustrate Newman’s point quite easily. Suppose I have seven neighbors, and I insist that they instantiate a model M I have of a rigidly hierarchical social structure. This model M we can envisage as follows: it has seven elements and a ‘directly under’ relation. To visualize it, its elements are the general,

220  :    directly under him two colonels, and directly under each colonel there are two majors. How can I say that my neighbors instantiate this hierarchical structure? To justify that I have to deﬁne a relation which has the same properties as the relation directly under has in my model. So I arbitrarily label my seven neighbors as follows: 1, 10, 11, 100, 101, 110 111. Then I deﬁne the relation as follows: neighbor 1 does not bear to anything; any neighbor with a label of form Y1 or Y0 bears to the neighbor labeled Y; and that is all. Of course there is no sense to the idea that I have discovered a hierarchical structure in my neighborhood by carrying out this ‘‘pencil and paper’’ operation. But if I were to say that ‘‘my set of neighbors is hierarchically structured in the fashion of model M’’ means only that there is a relation on that set which has the same properties as the directly under relation in M, then the trivializing result follows. For there is indeed such a relation on that set of neighbors provided only there are seven of them. In general, equality of size between two sets means just that there is a one-to-one correspondence between them, and that correspondence can be used to single out a copy in the one set of any relational structure there may be displayed in the other.10 You can easily see that this is von Helmholtz’s move with concave mirrors, repeated at a more abstract level. The very reasons that drove Russell originally to ‘‘real property’’ realism, and then later drove him away from that epistemologically uncomfortable position, have returned to plague him again. Russell capitulates Russell capitulated. In a letter to Newman, he reverted to the ‘‘real property’’ realism of his early days. The only difference is an update from real spatial relations to real spatio-temporal relations: Dear Newman, [. . .] It was quite clear to me, as I read your article, that I had not really intended to say what in fact I did say, that nothing is known about the physical world except its structure. I had always assumed spatio-temporal continuity with the world of percepts, that is to say, I had assumed that there might be co-punctuality between percepts and non-percepts. . . . And co-punctuality I regarded as a relation which might exist among percepts and is itself perceptible. (Russell 1968: 176)

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He had always assumed that? On the contrary, that is precisely what he shied away from in The Problems of Philosophy and precisely why he tried for an extreme structuralism in The Analysis of Matter. However that may be, Russell’s response clearly re-introduces his earlier epistemically charged ‘‘real property’’ realism, slightly updated, with the claim that we are directly acquainted with certain crucial spatio-temporal relations. Science does not merely assert that there are relations with the requisite formal character needed to give sufﬁcient complexity to the causes it postulates. On the contrary, it asserts that (to put it succinctly) those entities postulated, and known at best by description, bear certain relations which we know by direct acquaintance (both to each other and to entities that we know by direct acquaintance). Russell does not tell Newman very much, and mentions only one such spatio-temporal relation—contiguity, ‘co-punctuality’—that is instantiated by items of direct acquaintance. But he probably had in mind the construction of space-time from relations among events in which he followed Whitehead’s approach, so may have intended all signiﬁcant space-time relations or at least all those that can be instantiated in experience. Disentangling Russell’s response We need ﬁrst to isolate within Russell’s response the part that is essential and sufﬁcient for countering Newman’s objection. Newman himself had pointed out an obvious minimalist way to deal with his point. He suggested that Russell could repair his position by distinguishing between ‘important’ and ‘unimportant’ relations. It would not matter just what distinction is meant by this, as long as there is one. Then the repair would go as follows: Relations fall into two classes, important and unimportant. The important relations in nature instantiate the following structure. . . . Newman’s argument is then blocked. Given a one-to-one mapping from graph G to set S, any given relations on G will of course have images in S. But there is no guarantee that their images will be important relations. Hence the repaired assertion is contingent, possibly true and possibly false, and hence informative. But this is just a logical maneuver. Without any information about that supposedly (postulatedly?) factual distinction between the important

222  :    and unimportant, we have no grip at all on what the assertion says about the world.11 Newman in fact derided his own suggestion as a counsel of despair. One way to take Russell is that he accepted Newman’s suggested way out but added something so as to regain informativeness. He added that the ‘important’ relations are exactly those with which we have direct acquaintance. This ties in with the epistemology in his earlier Problems of Philosophy: to understand a proposition requires acquaintance with each of its constituents. Since we do understand some general propositions it follows from this that we are directly acquainted with certain Universals—all the ones that appear as constituents in the propositions we understand. But are we acquainted with all the arbitrary relations Newman recognizes? That does not follow. For example, we understand the proposition that every set has a well-ordering. That entails acquaintance with the (higher order) Universal being a well-ordering, but not acquaintance with e.g. a particular well-ordering of the real numbers. So within Russell’s epistemology there was indeed a natural division into important and unimportant relations to which he could appeal. To complete the repair then, Russell has to say that a scientiﬁc theory tells us something about the structure of certain relations with which we are acquainted, instantiated by those postulated entities in nature with which we are not acquainted, and about whose qualitative properties we have no idea.12 But now we can also see that the repair is still not ﬁnished. There is the danger that it does not go far enough, and may still leave us with very uninformative scientiﬁc theories, so two tasks remain. Task One: those ‘‘certain relations with which we are acquainted’’ must be speciﬁed to some extent. Russell clearly saw this, for here we arrive at Russell’s speciﬁc addition to Newman’s way out: ‘‘ spatio-temporal continuity [of the physical world] with the world of percepts’’. At the very least the terms ostensibly spatio-temporal in the theory denote the very spatiotemporal relations with which we are directly acquainted, and they relate the postulated entities both to each other and to the observed entities of our direct acquaintance. But this is a postulate: Russell is telling us that the ostensibly spatio-temporal terms are univocal, that they denote the same relations when they appear in physical theory as when we use them ordinarily.

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Task Two: to show that this sort of construal of theories, with that elaboration added, now has the resources to be sufﬁciently informative. Suppose that the space-time geometry of the physical world is indeed as science says it is, but when it comes to any other relations and properties attributed to those physical things out there, science can only specify the formal structure. Then won’t Newman’s argument apply again?13 The assertion that there are certain relations with a given formal structure on those things will be automatically true, provided only it is logically compatible with the space-time geometry in question, plus some assumptions about cardinalities. To put it the other way around, any additional physics beyond the space-time geometry might then give information only about cardinalities. That is equally disastrous unless the entirety of physics reduces to providing space-time models. Such was the dream of W. K. Clifton’s ‘‘space theory of matter’’, revived eventually in ‘‘geometrodynamics’’, and may still be alive in some programmes in the foundations of physics. But it is certainly not something that we would be ready to accept as a priori certain, I would think, nor something to which we would want to indissolubly connect a philosophical position.

Conclusion So Russell’s extreme structuralism collapsed and his repair, through a reversion to ‘‘real property’’ realism, does not obviously save him. In effect, it takes him back to the point where he had hoped that some admixture of structuralism with realism might make his peace with von Helmholtz’s lights and mirrors. But there is something striking in his repair, that alters the view he presents of science quite radically. In The Analysis of Matter scientiﬁc theories are presented as being completely formulable, without loss, in the language of pure mathematics. Such a formulation would involve no direct reference to what we encounter in experience, let alone indexicality, self-reference, or self-location. Reference to actual individuals would be achieved entirely by objective description. This aspect of his structuralist view of science is lost in the repair, when our direct acquaintance with certain entities separates what science is about from what logical gerrymandering concocts.

224  :    This story has its sequel in writings by Rudolf Carnap, Hilary Putnam, and David Lewis, where we will recognize the themes from Helmholtz, Russell, and Newman in a new setting. The problems are transposed there to a context governed by concerns in philosophy of language and analytic metaphysics. They become clearer there and so, if anything, more devastating. But when the problems become clearer, so do the possibilities of solution. Is there after all a viable form of structuralism about science? I shall argue for a speciﬁcally empiricist structuralism, which escapes trivialization by recourse to resources that we have been encountering all along the way: the role of indexical and ostensive reference.

10 Carnap’s Lost World and Putnam’s Paradox Rudolf Carnap’s most famous work purveys a structuralist philosophy of science, and runs into essentially the same problem as Russell faced—the problem that Newman pointed out. But unlike Russell, Carnap exhibited a ﬂuctuating awareness of the difﬁculty besetting his programme, and of the limited options his epistemology allows for escape.1 The option he ﬁnally chooses is in the pattern that Newman suggested, somewhat ironically, to Russell, but Carnap attempts simultaneously to refer to experience and to claim that the notions he needs are experience-independent. The basic problem returns after some decades when Hilary Putnam puts it to good use in his seminal critique of metaphysical realism. In both cases I shall argue that the solution—or rather, dissolution—of the problem is possible with the introduction of indexical reference, and I shall explore the role of the indexical further in the next chapter where I will discuss structuralism independently of this history.

Carnap: Der Logische Aufbau der Welt Carnap published his Logische Aufbau two years after Russell’s Analysis of Matter, and begins Part Two with the announcement we shall maintain and seek to establish the thesis that science deals only with the description of structural properties of objects.

And he immediately speciﬁes what this will amount to: A property description indicates the properties . . . , while a relation description . . . does not make any assertion about the objects as individuals.’’ (section 10, p. 19)

226  :    There is a certain type of relation description which we shall call structure description. Unlike relation descriptions, these not only leave the properties of the individual elements of the range unmentioned, they do not even specify the relations themselves which hold between these elements. In a structure description, only the structure of the relation is indicated, i.e. the totality of its formal properties. (section 11, p. 21)

The crucial problem The problem that we saw so clearly spelled out by Hermann Weyl, then appears in the next section: Thus, our thesis, namely that scientiﬁc statements relate only to structural properties, amounts to the assertion that scientiﬁc statements speak only of forms without stating what the elements and the relations of these forms are. Superﬁcially, this seems to be a paradoxical assertion . . . in empirical science, one ought to know whether one speaks of persons or villages. This is the decisive point: empirical science must be in a position to distinguish these various entities. . . . (section 12, p. 23)

How does Carnap deal with this problem? He admits that it looks as if the use of deﬁnite descriptions will be successful only if eventually it relies on some ostensive description—that is, some recourse to pointing or other indexical or demonstrative form of reference. He takes that look to be deceptive: However, we shall presently see that, within any object domain, a unique system of deﬁnite descriptions is in principle possible, even without the aid of ostensive description. (pp. 24–5)

But his elaboration immediately lets us doubt this irenic assurance: It is of especial importance to consider the possibility of such a system for the totality of all objects of knowledge. Even in this case it is not possible to make an a priori decision. But we shall see later that any intersubjective, rational science presupposes this possibility. (p. 25)

What we see here is a vacillation between several possibilities. (A) Is nature so structured that everything can be uniquely identiﬁed by means of a description that captures only that structure? He has an illustration: the stations in the Eurasian railroad system can be each uniquely identiﬁed by listing just the railroad connections between them. But if this system had some global symmetries, then the identiﬁcations would not be unique.

 ’     ’ 

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What if we bring in more structural features, besides the railroad connection relation? Can we be sure that if we bring in enough—when we are at the level of ‘‘a system for the totality of all objects of knowledge’’ we will not be plagued with such non-uniqueness? Here we come to (B), ‘‘any intersubjective, rational science presupposes this possibility’’. If that is so we have a transcendental justiﬁcation for the assertion that knowledge of structure includes all the knowledge there is to be had. Disturbingly, Carnap throws out yet a third option in this Part of the Aufbau: that (C) we should adopt an ontology in which isomorphism implies identity. Quite a move, for a father of logical positivism and the author of Pseudoproblems in Philosophy! Return to the problem In Part Four of the Aufbau Carnap returns to this problem.2 He has been envisaging reconstructions of the body of science, or other parts of knowledge, and it seems that he views the structuralist programme announced in Part One of the Aufbau in the same way as Russell at this point. A purely structural statement must contain only logical symbols . . . . Thus . . . the problem arises whether it is possible to complete this formalization by eliminating from the statements of science these basic relations . . . . That this elimination is possible becomes obvious . . . . (section 153)

The elimination must replace relation descriptions by structure descriptions, as we saw above, that is to say, by assertions that there are relations having certain properties, and instantiated in certain ways. But now (section 154) he notices a difﬁculty—in effect the very same difﬁculty that M. H. A. Newman pointed out to Russell just a little later. If a statement to the effect that there are relations having certain properties and are instantiated in certain ways is framed entirely in logico-mathematical terms then it can be satisﬁed in any set of sufﬁcient size. For the relations in question can be taken to be suitably chosen sets of ordered pairs, triples, and the like, and the suitable choice is possible provided enough elements are available. Thus Carnap writes in section 154, about the argument that he had given to support his claim that the elimination is possible, that he had assumed that:

228  :    . . . after a replacement of one set of basic relations by another, the constructional formulas of the system would not remain applicable . . . . However, our assumption is justiﬁed only if the new relation extensions are not arbitrary, unconnected pair lists, but if we require of them that they correspond to some experienceable, ‘‘natural’’ relations (to give a preliminary, vague expression). If no such requirement is made, then there are certainly other relation extensions for which all constructional formulas can be produced.

In other words, he admits that if the ‘suitable choice’ of sets of ordered pairs is unconstrained, if they can be ‘unconnected pair lists’, then a theory so framed will be true provided only there are enough elements to choose the pairs from. Practically all empirical content would be lost, in such a structuralist reconstruction of a scientiﬁc theory! In contrast to relations of this sort [unconnected lists of pairs], we wish to call relation extensions which correspond to experienceable, ‘‘natural’’ relations founded relation extensions.

So one of the properties that must be attributed to the relations mentioned in a structure description is this ‘foundedness’. A scientiﬁc theory will be reconstructed as an assertion to the effect that there are founded relations instantiated in a certain pattern.3 As Newman pointed out, any distinction of this logical form blocks the trivialing argument. In that sense, Carnap has solved his problem, formally speaking. But does this not give up entirely on the structuralist programme he had announced? That a relation is ‘experienceable’, ‘natural’ is not something that can be formulated using only logical vocabulary. Carnap sees this very well, and makes what seems in retrospect a desperate move. He submits, though guardedly, that this ‘‘founded’’ is itself a new logical term: ‘‘It is perhaps permissible, because of this generality, to envisage foundedness as a basic concept of logic.’’ But this concept was introduced in an analysis of how one might add non-logical content or reference to a structure description!4 The two sides of Weyl’s coin As noted above, we see two problems for the concept of science as a representation by means of mathematical models. First of all as Weyl says explicitly, there are distinct but isomorphic structures discernable in nature,

 ’     ’ 

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but mathematical models will capture only the structure that is common to them. Thus to have speciﬁc knowledge of one of such a pair, we must know more than that it is adequately represented by some such mathematical structure. Secondly, as Klein and Helmholtz had pointed out, given any signiﬁcant limitation on what is observable or detectable by us, there will be many non-isomorphic structures that ﬁt what we do observe or detect. I said above that these are two sides of one coin. In what sense is that so? Russell reacted to the latter problem by insisting that we have knowledge that goes beyond what can be mathematically represented: we are acquainted with the physical congruence relation, whose mathematical representation is just one of many congruence relations that can be imposed on a projective space. One of these relations has a privileged epistemic status, it is the one that ‘enters’ our experience. Now we see Carnap opting for a precisely similar solution to the former problem. He assigns a privileged epistemic status to certain relations, and connects that status with what can be experienced. We hear an echo here of the ‘problem of coordination’ which due to Schlick’s and Reichenbach’s discussions must have been salient in Carnap’s world. The problem Carnap has encountered pertains to the connection between any structural representation and what it represents; preoccupied with coordination, it is seen as relating speciﬁcally to the theory-experience relation. We see the essential problem in much more general form when Hilary Putnam takes it up some four decades later.

Putnam’s Paradox At the American Philosophical Association in 1976, Putnam produced his most famous argument against metaphysical realism. He called it his modeltheoretic argument. David Lewis called it Putnam’s Paradox, because, he claimed, this argument would show that almost any theory at all is true. For Lewis, therefore, the argument can be cited in support of a stronger realism than the realism that Putnam targeted, namely a stronger realism which can resist the argument. I will begin with what I shall call the core of the argument.5

230  :    The argument Suppositions: 1.   has inﬁnitely many pieces 2. Theory T is consistent 3. Theory T says that there are inﬁnitely many things Exactly one meta-mathematical result will be required in the proof; after that it will proceed precisely like Newman’s argument against Russell. That result is the Loewenheim–Skolem–Tarski–Vaught theorem, which shows that if a theory has an inﬁnite model then it has models of every inﬁnite cardinality.6 1. Since T is consistent and has (only) inﬁnite models, it has a model of the same cardinality as  . 2. Let be a one-to-one correspondence between the elements of that model and the pieces of the world. (This correspondence exists by deﬁnition, given that the two sets have the same cardinality.) 3. Each term of the theory will have an extension in the model; DO: assign to each term precisely the image under of its extension in the model. Take that image to be its extension in  . 4. Since all the theorems of T are true in the model (i.e. given the terms’ extensions in the model) all of them will also be true in   when given those images as extensions in  . 5. So T is a true theory about  .

Figure 10.1. Putnam’s Paradox

An example, an objection, and a ‘way out’ Let’s think about an example of a simple theory and see how Putnam’s argument applies. I will call this the Water Theory.

 ’     ’ 

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A1) There are inﬁnitely many distinct bodies of water A2) There are inﬁnitely many things that are not water Assume that   is or can be regarded as having inﬁnitely many things in it. Then there is an extension that can be assigned to the phrase ‘‘body of water’’ such that both A1 and A2 come out true. That is not very surprising! But as Putnam suggests, the theory can contain an extra axiom for every assertion we want to make—or if you like, every one that is ever in the fullness of time observed to be true—of the form ‘‘X is a body of water’’ or ‘‘Y is not a body of water’’. Then there will still exist an assignment of referents to the terms ‘‘X’’, ‘‘Y’’ and the like, and an extension to ‘‘body of water’’ which will make A1, A2, and all the extra axioms true. And so forth: for the argument is very general, and does not rely on any speciﬁc assumptions about the content of the theory. Whatever constraints we want to introduce Putnam challenges us to make explicit; then he will add them to the theory under consideration . . . all those constraints are after all ‘‘just more theory’’. David Lewis then offered the following way out. When Putnam writes ‘‘Map the individuals of M one-to-one into the pieces of  , and use the mapping to deﬁne relations of M directly in  ’’ Lewis insists that we must add ‘‘in such a way that the deﬁned relations in   are natural relations’’. The ‘‘it’s just more theory’’ objection is evaded, because the distinction between natural and unnatural relations does not enter into the formulation of the theory. Rather, it is a distinction in nature which must be respected. Must be respected how? Respected in our view of just what it is that is asserted by a scientiﬁc theory, that is, in our spelling out of the truth conditions for that theory. We can understand the theory as a non-trivial assertion about what things are like only on an interpretation of the terms as referring to natural relations. The terminological link with the passage I quoted from Carnap about foundedness is not accidental: we have here essentially the same form of solution to essentially the same problem faced by the two offered conceptions of science. What is that postulated distinction in nature? Plato is said to have touched on it with his famous phrase ‘‘carving nature at the joints’’. One sometimes hears an item of faith: the relations that will ultimately be described in successful science are in fact precisely the natural ones. But this seems

232  :    almost empty! The argument seems to be that to understand what a theory says we must interpret it as being about the natural relations, to assert that it is true is to say that it is instantiated in natural relations, and so, it follows at once that if science being ultimately entirely successful means being true, then the relations it will describe will be natural. For this argument to be logically compelling, ‘‘natural’’ cannot be independently meaningful; but if it is not independently meaningful, the conclusion has no bite at all. In any case, what cannot be meant here is that the relations that will ultimately be described in structural terms are in fact precisely the natural ones—unless one were to count ‘‘natural’’ as structural, more or less as Carnap wanted to count ‘‘founded’’ as logical, and with no better reason.

Staying with Putnam: the Paradox dissolved Putnam himself did not present the argument as a paradox, but as an argument against what he called metaphysical realism. As I read him, Putnam clearly signaled how the apparent paradox dissolves upon scrutiny. The puzzle only begins with the reﬂection that Putnam established his conclusion for theories in a very large class of languages, perhaps even all languages (if that makes sense). The crucial next step is to add that our own language—the very language in which we state his argument and develop our scientiﬁc theories—is one of those. Infer then that his conclusion is true of our own language. There you have it! Trying to block this universal instantiation looks absurd, doesn’t it? But the conclusion, that our ideal theories cannot be false, looks equally absurd. So what can we say in retort? To our dismay it seems we have no recourse but to say, with the metaphysicians: Putnam’s argument pertains only to languages lacking the semantic glue to stick their words ﬁrmly to their referents—so, our language must be different, our words must be somehow guaranteed to refer to natural relations only. We seem to have no other way out. But that is not so. We should make a sharp distinction between how Putnam’s argument applies to an arbitrary language under study, and how we could apply it ourselves to our own language. An arbitrary language under study is in need of an interpretation. If we can keep the pragmatic dimension as ﬁxed or given for the time being, that interpretation must

 ’     ’ 

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certainly have at least as core an assignment of extensions to its singular and general terms. But what is an assignment? The term ‘‘assignment’’ carries the connotation of ‘‘something having been assigned by someone’’, but Putnam’s argument actually just concerns functions. The text of Putnam’s argument needs to be read very carefully. It proceeds by means of instructions to us to do something, and includes as implicature that we can do them. This may look like an inessential feature of the argument, but actually shows precisely what we can resist. We are apparently being led through a chain of reasoning destined to the conclusion that we can regard the  as having a certain structure. There exists a model of the ’s cardinality, there exists a function that will map the one onto the other . . . Select one of each! Under what conditions can we do that? ‘‘We can regard the  as having that structure.’’ That is an assertion about us; and what is its basis? The basis is the argument that there exists a certain function which maps the model and  one-to-one onto each other. Fine, that function exists—but what does that have to do with us? We have an interpretation for the given language only if we can deﬁne or identify such a function. To do that we must be able to describe both the function’s domain and its range, hence both the syntax of that language and  , as well as the way in which the former gets mapped on to the latter. To illustrate the illegitimate slippage from ‘‘there exists’’ to ‘‘we have’, let us look at two examples, one quite practical and the other abstract. Here I have an apparently blank sheet of paper. It is the best sort of paper, heavy bond, paper with character. Looking carefully at it, I see striations and marks, hardly discernible but deﬁnitely there, and upon consideration realize that they are as complex and manifold as the nighttime sky. There exists a function that maps the main landmarks and streets of Paris more or less continuously into these marks and striations, in a sufﬁciently detailed way to rival even the best commercially available maps.7 So, this sheet is in fact an exquisitely detailed map of the city of Paris: how much will you pay me for it? I assert the existence of this function. Its existence follows from some simple but plausible assumptions about the small imperfections that all paper made from wood must have.8 These assumptions are not nearly as abstruse as those going into the proof of the Loewenheim–Skolem–Tarski–Vaught

234  :    theorem. Yet you balk . . . You have a good point; you would not call this sheet a map because in fact, although the function exists, it is not true that you can regard this sheet as a map of Paris. To illustrate the same problem and at the same time the ordinary, acceptable use in which the slippage from ‘‘there exists’’ to ‘‘we have’’ causes no problems, take this simple problem in geometry: The Euclidean sphere of radius 1 can be coordinatized as the set of points (x,y,z) in R3 satisfying x2 + y2 + z2 = 1. Determine the distance along great circles between two arbitrary points on the sphere as a function of the coordinates. What is asserted in the opening sentence? That there exists a suitable mapping of the sphere into the set of triples of real numbers. The ‘‘can’’ in ‘‘can be coordinatized’’ has no literal signiﬁcance, for how could we do it? How could we select a point to be assigned (0,0,1) for example? In fact, the sphere as a mathematical entity has perfect rotational symmetry, so any description of any point on it applies equally to all points—unless it is a description that relates some points to things outside the sphere, in which case we are not considering the sphere by itself.9 Going back now to the context of Putnam’s argument we must similarly conclude the following. As long as we are not given an independent description of both the domain and range of an interpretation, we do not have any such interpretation, nor any way to identify one. Given an independent description of the interpretation’s domain and range, however, whether the theory is true under that interpretation depends entirely on how the mapping is deﬁned, using those descriptions. Remember here the Water Theory I gave above: if we do not have our own resources to describe both this theory (its language and axioms) and the world, then we can’t have an interpretation of it. But if we have those resources, then we can insist that all admissible interpretations must assign water as referent to the word ‘‘water’’—and upon that interpretation, the theory’s truth or falsity is a contingent matter. On my reading, Putnam spells this out quite clearly. In ‘‘Realism and Reason’’, where the argument was ﬁrst introduced, he defuses one point with the laconic ‘‘because the world is not describable independently of our description’’ (p. 496). And in ‘‘Two philosophical perspectives’’ he is very explicit: ‘‘This simply states in mathematical language the intuitive

 ’     ’ 

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fact that to single out a correspondence between two domains one needs some independent access to both domains’’ (Putnam 1981: 74). It is of no earthly relevance to know that there exists some such function such that it, if only it could be described, would furnish an interpretation under which the theory is true. The only question of interest for us, if we do really have such an interpretation, is whether the theory is true under that interpretation. What happens if we want to apply Putnam’s argument to our own theories formulated in our own language? We shall be able to grasp such a theory if we can grasp an interpretation of the language in which it is formulated (i.e. our language!). Well, as noted, we can grasp an interpretation—i.e. function linking words to parts of  —only if we can identify and describe that function. But we cannot do that unless we can independently describe  . So Putnam’s model theoretic argument, if applied to our own language, meets up with a dilemma: (A) if we cannot describe the relevant elements of  , neither can we describe/deﬁne/identify any function that assigns extensions to our predicates in  ; (B) if we can describe those elements of   then we can also distinguish between right and wrong assignments of extensions to our predicates in  . For example, on alternative (B), we would insist that a right assignment of an extension to ‘‘water’’ is water, and that all other assignments are wrong. That the theory is true on some other such assignment is simply irrelevant to whether it is true. I take it that this is also how Putnam meant the apparent paradox to be resolved: by noting the dilemma posed by (A) and (B). The response is Wittgensteinian, in that it focuses on us, on our use of theories and representations, and brings to light the impasses we reach when we abstract obliviously from use to use-independent concepts. In terms of our prior discussion, we must emphasize the crucial role of the indexical here. A theory says nothing to us unless we can locate ourselves, in our own language, with respect to its content.

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11 An Empiricist Structuralism The mechanization of the world-picture culminating in the seventeenth century had as equally dramatic sequel the mathematization of the worldpicture that culminated in the twentieth.1 The former led quite reasonably to a new conception of science itself, contrasting sharply with its conception in the Aristotelian tradition. The latter transition has a rightful claim upon us to develop views of science so as to do justice to its revolutionary impact on the sciences themselves. It is this claim that I see honored in the century long attempts to develop structuralist views of science, the recurrent defeats of such views notwithstanding.2 Accordingly I shall argue here that the defeats are not inevitable but that structuralism ﬁnds its proper articulation only in an empiricist setting.

What could be an empiricist structuralism? When structuralist views are expressed intuitively they come in such slogans as ‘‘all that science describes is structure’’ or even ‘‘all that we can know is structure’’. Such slogans are provocative, and meet immediately with resistance. Doesn’t science tell us also what water really is, namely H2 O, that when water is cooled it will ﬁrst contract in volume and then at a certain point begin to expand, that it will refract light and dissolve sugar . . . ? None of this sounds at ﬁrst blush like a description of the structure of water but rather of what it is and what it is like. Such examples are easily multiplied. By choosing water, I chose an example that is as telling for the empiricist as for the scientiﬁc realist, though the former may discount that phrase ‘‘what water really is’’. The latter, on the other hand, will insist that similar examples of the unobservable abound: science tells us what electric currents really are, what atoms are really like, how the mass-energy distribution curves space, and so forth.

238  :    At ﬁrst sight, then, structuralist sentiments are greeted with an incredulous stare. But in an empiricist setting, scientiﬁc theories are viewed precisely as theories, not as our sole wherewithal for getting around in the world. On the contrary, theories are artifacts, constructed to aid us in planning and understanding—not to usurp those functions. Accordingly, the stories about nature, about what things are like, which spell out a way the world could possibly be like for such a theory to be true, take on a lesser role. They allow us to move around in the theory, to exercise the imagination, even to get to the point intellectually where we can draw qualitative consequences via the theory without actual calculation.3 The two poles of scientiﬁc understanding, for the empiricist, are the observable phenomena on the one hand and the theoretical models on the other. The former are the target of scientiﬁc representation and the latter its vehicle. But those theoretical models are abstract structures, even in the case of the practical sciences such as materials science, geology, and biology—let alone in the advanced forms of physics. All abstract structures are mathematical structures, in the contemporary sense of ‘‘mathematical’’, which is not restricted to the traditional numberoriented forms. And mathematical structures, as Weyl so emphatically pointed out, are not distinguished beyond isomorphism—to know the structure of a mathematical object is to know all there is to know. As we have seen, this poses a problem for any structuralist view of scientiﬁc knowledge. Essential to an empiricist structuralism is the following core construal of the slogan that all we know is structure: I. Science represents the empirical phenomena as embeddable in certain abstract structures (theoretical models). II. Those abstract structures are describable only up to structural isomorphism. In the empiricist version, the structuralist slogan is clearly and substantially qualiﬁed. It does not take reliance on theory for us to know many things about water—about observable phenomena in general—and so the slogan must be read as meaning, at best, that all we know through science is structure. Nor does the formulation as I. and II. imply that there is in nature, or in the phenomena, a form/content or structure/quality distinction to be drawn. The structuralism in ‘‘empiricist structuralism’’ refers solely to the

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thesis that all scientiﬁc representation is at heart mathematical. Empiricist structuralism is a view not of what nature is like but of what science is. This modesty leaves it squarely placed, however, in the range targeted by Weyl’s paradox, Putnam’s paradox, Newman’s argument against Russell, and Carnap’s puzzles over how science (as he conceived it) could be about the phenomena at all. But its modesty does remove some of the problems. I will now attempt to isolate the fundamental problem that was at the heart of all these conundrums.

The fundamental remaining problem for a structuralist view of science Bertrand Russell and Rudolf Carnap are only two of the philosophers who attempted to develop a structuralist conception of science. Some later developments are explicitly so named (viz. Stegmueller’s ‘structuralist view of theories’) while others did not adopt the name, but are clearly of the same ilk (e.g. the ‘semantic view’ of theories). Starting with John Worrall’s work in the mid-Eighties, Structural Realism made great inroads into philosophy of science at many levels.4 Each of these is implicitly (and sometimes explicitly) challenged through the puzzles and paradoxes that we saw for the earlier advocates.5 As will have been apparent, I see the correct response to those challenges as emerging with proper clarity only in the discussion surrounding Putnam’s ‘model-theoretic argument’. The form of this response is not linked differently to the different forms of structuralism. Bertrand Russell’s response to Newman was not totally clear, and could be understood as simply accepting Newman’s invitation to postulate a distinction that separates relations into important and unimportant. But Russell, like Carnap, insisted on linking that separation to (one’s own) experience, which is—as I see it—a deﬁnite step toward the crucial clue of self-reference. Only in the discussion of Putnam’s Paradox did this role of indexicality become entirely clear. Attention to the indexical, which we found to be indispensable already in the use of maps, models, and theories at an early point in reﬂection on science and practice, is crucial to defuse the challenge from Newman’s argument and its variants. What is needed to complete the task is to grasp a fundamental underlying problem about representation. It is not one to be stated in just a few words

240  :    if we are to feel its proper impact, but we can bring it out properly in successive stages. To do so, I will begin by revisiting Reichenbach’s problem of coordination in a more abstract form. How can an abstract entity represent? The empiricist structuralism formulated in theses I. and II. above looks pretty well like precisely the view we saw Weyl as expressing, and for him it led at once to what I called ‘‘Weyl’s paradox’’. The basic perplexity emerges in its purest form when we ask just how Principle I. is to be understood. What does it mean, to embed the phenomena in an abstract structure? Or to represent them by doing so?6 We can understand how one mathematical structure can be embedded in another—the word ‘‘embedding’’ refers here to a certain kind of function. So Principle I. can mean this: that a scientiﬁc theory will ﬁrst of all represent the phenomena by means of mathematical structures, and then show how those structures ﬁt into larger ones, the theoretical models. So far so good—that relates well to the discussion we had earlier of data models, surface models, and theoretical models. But it presupposes that we understand the ﬁrst part of this two-part process: representation of the phenomena, which are concrete, by mathematical structures, which are abstract. Hence the most fundamental question is this: How can an abstract entity, such as a mathematical structure, represent something that is not abstract, something in nature? The perplexities appeared clearly in the passages, that we already discussed in Part Two, from Reichenbach’s 1920 discussion of coordination. Recall how perplexed Reichenbach sounded: The mathematical object of knowledge is uniquely determined by the axioms and deﬁnitions of mathematics. (Reichenbach 1920/1965: 34) The physical object cannot be determined by axioms and deﬁnitions. It is a thing of the real world, not an object of the logical world of mathematics. Offhand it looks as if the method of representing physical events by mathematical equations is the same as that of mathematics. Physics has developed the method of deﬁning one magnitude in terms of others by relating them to more and more general magnitudes and by ultimately arriving at ‘‘axioms’’, that is, the fundamental equations of physics. Yet what is obtained in this fashion is just a system of

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mathematical relations. What is lacking in such system is a statement regarding the signiﬁcance of physics, the assertion that the system of equations is true for reality. (Ibid.: 36)

Indeed, a theory in mathematical physics is, if just taken in and by itself, a mathematically formulated theory, in just the way that e.g. Euclidean and hyperbolic geometry are. So taken, it is only a theory about mathematical objects. How could it have more content, to make it something different from pure mathematics? No use, it would seem, to just add an extra axiom such as ‘‘the above axioms are true of reality’’. At least at ﬁrst sight, that would be ‘‘just more theory’’, as Putnam would say, and the question Why is this more than just mathematics? applies—or would seem to apply—equally to the newly extended theory. If words are all we have . . . how can we answer this at all? The ‘offhand’ realist response What did Reichenbach mean with his above remark ‘‘Offhand it looks as if the method of representing physical events by mathematical equations is the same as that of mathematics’’? He imagines in effect the following naïve reply: what is called for is simply a function, a mapping, between mathematical objects and physical objects or processes—so what is puzzling about that? True, we have no difﬁculty with mappings between two sets of mathematical objects. But notice: to deﬁne such a mapping we must identify its domain and its range, plus the relation that effects the coordination between them. For example, if two sets of points are given, we establish a correspondence between them by co¨ordinating to every point of one set a point of the other set. For this purpose, the elements of each set must be deﬁned; that is, for each element there must exist another deﬁnition in addition to that which determines the coo¨ rdination to the other set. Such deﬁnitions are lacking on one side of the co¨ordination dealing with the cognition of reality. Although the equations, that is, the conceptual side of the co¨ordination, are uniquely deﬁned, the ‘‘real’’ is not. (Ibid.: 37)7

So here is the problem baldly stated. If the target is not a mathematical object then we do not have a well-deﬁned range for the function, so how can we speak of an embedding or isomorphism or homomorphism or whatever between that target and some mathematical object?8

242  :    Vacuity of the ‘offhand’ realist response The natural temptation is simply to impose a parallel vocabulary and declare victory. To illustrate this earlier on, recall, I took Reichenbach’s own example of Boyle’s Law, PV = rT. To give physical meaning to this equation its terms must be co-ordinated with physical quantities. But what is, for example, the temperature of a gas? It changes with time, and differs from one body of gas to another, so isn’t it a function that maps times and bodies of gas into the set of real numbers? This suggests that we could answer the question of how ‘‘PV = rT’’ is more than just a mathematical theory by asserting that it ‘‘mirrors reality’’. Suppose we just answer with ‘‘There exist physical quantities P*, V*, and T* which pertain to bodies of gas, and when ‘P’, ‘V’, and ‘T’ refer to these respectively then ‘PV = rT’ is true’’. If this is added as a postulate to the theory, we can only be mystiﬁed as to what wheels it turns at all! Although verbally apt, all this does is to create a parallel vocabulary and declare victory—but it is an empty victory. The metaphysical realist’s response depicts nature as itself a relational structure in precisely the same way that a mathematical object is a structure. On this view, if the mathematical model represents reality, it does so in the sense that it is a picture or copy—selective at best, but accurate within its selectivity—of the structure that is there. As one might say then, there is no problem, because this depicted physical system is a set of parts connected by a speciﬁc family of relations—so of course there are functions deﬁned on this set of parts with range in the model and vice versa. A little uncharitably we could summarize this as: a physical system is an abstract entity with concrete parts or elements—uncharitably, since a physical system is meant to be something physical, not something abstract.9 But a function that relates A and B must have a set as its domain. If A is e.g. a thunderstorm or a cloud chamber—a physical process, event, or object—then A is not a set. Fine, the realist can answer, but A has parts, and the function’s domain is the set of these parts. No reason why the elements of the set need be mathematical entities after all. Moreover there are speciﬁc relations between these parts; and these relations have as their extensions sets of sequences in that domain. The function provides a proper

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matching provided the images of these relations are relevant relations in the model. (*) the function relates two structured sets, call them S(A) and B: S(A) = SA1, SA2. where: SA1 = the set of parts of A, SA2 = the family of sets which are extensions of relations on these parts B is a mathematical object, representable correspondingly in the same general form Set, Relations. This follows the most general format of standard mathematical modeling: the format of relational systems, the subject for what Whitehead called Universal Algebra.10 What is taken for granted here is the relation between A, the physical entity, and S(A), a relevant mathematical representation of A. But why is the relation between A and S(A) any different from the one we asked the question about, namely the relation between A and B? Isn’t S(A) an abstract entity used to represent A? We cannot very well answer ‘‘how can an abstract mathematical structure represent a concrete physical entity?’’ by saying ‘‘this is possible if we assume the latter is represented by some other mathematical object’’. Let us not be hasty. First of all, the realist can reply, there is no question but that the sets S(A), SA1, SA2 are real if A is real. When we say that B is an adequate representation of A we mean simply that S(A) is isomorphic to or embeddable in or homomorphically mappable into B. So far, so good—we can agree to all that (modulo whatever view of mathematics we have), this use of an elementary part of set theory must be legitimate. But the realist’s next problem is that this sounds like there is more uniqueness than there is—that there is less selectivity in what S(A) can be than there must in fact be.11 The respect in which B represents A may be one thing or another: that depends precisely on how we ‘divide up’ this entity A and which aspects of its relational structure we select—i.e. what we choose for SA1 and SA2. What indicates here the relationship between A and S(A) is so far no more than our description or denotation of SA1 as ‘‘the set of parts of A’’, etc. The word ‘‘the’’ is however not justiﬁed here, it is a misnomer, given that A can be divided up in various ways.

244  :    And just what is this dividing up? It is nothing more nor less than the act of representing A as having SA1 as its set of parts—i.e. of A as consisting of the members of SA1. Would all be well for the realist response (which thinks of the coordination as a simple two-place relation between the two items) if there were only a single, unique way of representing A in this manner? The question is moot. For, as a matter of fact, and as we have seen before, as long as A can be represented as having a set of parts of the same size as that of B, some such relationship as (*) above will certainly exist between A and B! That is precisely the point that (by now, famously) trivialized Russell’s structuralism and drives Putnam’s model theoretic argument. The realist has one ﬁnal gambit, namely the one that we already saw in David Lewis’s introduction of natural classes. S/he insists that there is an essentially unique privileged way of representing: ‘‘carving nature at the joints’’. There is an objective distinction ‘in nature’ between on the one hand arbitrary or gerrymandered and on the other hand natural divisions of A into a set of parts, and similarly between arbitrary and natural relations between those parts. So A is already a highly structured entity, and the abstract relational system we need is a precise copy thereof. We could now ask about this ‘copy’ relation; but there seems no need to press that. For what precisely is this gambit, this insistent assertion? It is a postulate (if meaningful at all). What can it possibly mean? The word ‘‘natural’’, its crucial term, derives its meaning solely from the role this postulate plays in completing the realist response. Honi soit qui mal y pense they’ll tell us.12 Unless we subscribe to this sort of metaphysics, and something like a substantial correspondence theory of truth and representation, the realist maintains, we cannot make sense of our own practice.13 But their point will be moot, and the postulate superﬂuous, if the problem dissolves upon scrutiny, as I shall now argue.

The two main dangers for an empiricist Let us look at the problem afresh, without reference to how a scientiﬁc or metaphysical realist might approach it. The problem of coordination, taken in its most abstract form, is the problem of understanding what makes for

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a representation, what the relation can be between a representation and what it represents. There is an ambiguity to be avoided, which we looked at brieﬂy in the Part I. A painting of an angel has representational content: it depicts an angel. Whether it also has a referent, whether there is something real that it depicts, is another question: that depends in part on whether angels exist. What is at issue in the problem of coordination is the relation between a representation and its referent. Under this heading we can address how such a speciﬁc item as a geodesic in a mathematical space relates to something physical—but that makes sense only in a context already replete with theory and previously established language or other form of representation. The philosophical impulse to consider a problem in its purest possible form must be respected too, and so we see the problem successively generalized—but this can go too far, possibly to the point of losing sense in the end. There is clearly a practice of representing by means of abstract, and not just concrete, artifacts. This practice we can examine, to see just what mappings, embeddings, or structural comparisons are involved in scientiﬁc representation. If these, however, remain on an abstract or mathematical level, then we still have to face the problem of how this practice can relate to the phenomena. For familiar natural phenomena such as daybreak and starlight, electric light or electromagnetic effects, science provides us with models. Often the initially startling event is that a model, of a sort we have been trusting, does not save them. The question how an abstract structure can represent something, transposed to such a context, is just this: how, or in what sense, can such an abstract entity as a model ‘‘save’’ or fail to ‘‘save’’ this concrete phenomenon? What is the pertinent relation that holds or does not hold between the mathematical structure described by our equations and that natural or artiﬁcially produced process? There are really two cases here, hidden by my easy use of ‘‘us’’ and ‘‘our’’. Consider two scientists, one who offers a model for how the frog population density varied in the Netherlands in the twentieth century, and another who offers a quite similar model for the dinosaur population density in a larger area at a much earlier time. For their models to be adequate—hence, for their theories to be true—those observable phenomena must be embeddable in their models. But the cases are different, for the ﬁrst scientist can draw on actual measurement results of the relevant

246  :    population density, and the second cannot. So for the ﬁrst we face the question of what the adequacy of a model—the conditions under which it accurately represents its domain—amounts to in a concrete practical setting. For the second scientist we have to understand how a phenomenon somewhere and somewhen, which is not encountered in human experience or targeted in actual measurements or observations, can be said to ‘‘ﬁt’’ a theoretical model. There are of course many examples of the latter sort, some much more recondite or abstruse than dinosaurs—think for example of extra-galactic phenomena, even ones that a theory classiﬁes as beyond our ‘event-horizon’ so that we are precluded from ever gathering even very indirect evidence about them. Metaphysical postulates of the sort we just discussed above, concerning ‘structure in nature’, seem to me to be specially inspired by the latter case, of the phenomena that are not encountered in our practice. On the one hand, theories have many models that are never actually constructed by any one—in the sense that a mathematician can say that a certain equation has many solutions that are never actually written down by anyone. On the other hand, those phenomena are never actually represented, in any concrete way by anyone. With human participation entirely foreign to the context, what can constitute adequacy or truth except a direct theory-model to nature relation? How can we assert that those unknown phenomena ﬁt models of our theories, except in a sense that implies the metaphysical realist’s postulation of ‘structure in nature’, consisting of universals or the like? That rhetorical question present the ﬁrst of the main dangers for the empiricist that I want to take up here. The second danger to the empiricist, implicit in discussion of how theories or theoretical models can represent phenomena when those phenomena have actually been subject to measurement, is subtler. There the pertinent ‘matching’ seems so obvious and clear: the theoretical model and the data model are both abstract structures, and it is not difﬁcult to understand how the latter may be embeddable in the former. But a theoretician at a desk, matching his theories’ equations to graphs or density functions or the like, is not someone directly in touch with nature. So doesn’t a reﬂection that focuses on the data models for assessing empirical adequacy, lose contact with reality altogether?

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Phenomena far outside experience Theories represent the phenomena just in case their models, in some sense, ‘‘share the same structure’’ with those phenomena—that, in slogan form, is what is called the semantic view of theories. My own variant upon this theme is that the phenomena are, from a theoretical point of view, small, arbitrary, and chaotic—even nasty, brutish, and short, one might say—but can be understood as embeddable in beautifully simple but much larger mathematical models. Embedding, that means displaying an isomorphism to selected parts of those models. Here is the argument to present the ﬁrst challenge. For a phenomenon to be embeddable in a model, that means that it is isomorphic to a part of that model. So the two, the phenomenon and the relevant model part must have the same structure. Therefore, the phenomenon must have a structure, and this shared structure is obviously not itself a physical, concrete individual—so what is implied here is something of the order of realism about universals. What are we to make of this argument? As I see it, it is closely related to similar arguments that convict us of realism about universals or of postulating ‘truth-makers’ of some sort, as soon as we talk of truth. The plausibility of the argument comes from our having to speak in a general way when abstracting from particular cases. But the argument loses its plausibility if we just look carefully at a particular case, and then reﬂect that what we say about the general case is simply about all the particular cases in a summary fashion. Let’s begin with a simple example: during a period of optimal conditions a certain bacteria colony grows exponentially, doubling in size in equal intervals of time. The equation for exponential growth is N(t) = N(0)ekt , with k a constant reﬂecting the doubling time. It is easy to calculate the doubling time τ: 2 = ekτ , so τ = ln(2)/k. The solutions for this equation are speciﬁc functions deﬁned on some interval (t, t ) for speciﬁc values of N(0) and k, with range in the real numbers. This is an example of a very simple theory (the equation) and its models (the solutions of that equation). But the growth is discontinuous: even bacteria do not reproduce continuously in the mathematical sense. So the ‘empirical substructures’ of these models are the sequences N(0), N(τ), N(2τ), . . . . .

248  :    Now we consider a real phenomenon: the actual number of bacteria in a certain colony located in Antarctica in a certain interval of 12 hours several million years before humanity emerged on earth. Its growth during that interval: does it ﬁt one of these models? What does that mean? At the beginning of those 12 hours there was a certain number N (0) of bacteria in the colony and at each time t = τ, 2τ, . . . within those hours a number N (τ). Is the function N actually one of the empirical substructures of one of those models? Yes or no? If yes, then the phenomena ﬁt that model, and hence the theory, in the sense required. In this discussion there was at no time an implied assertion that there exists a structure, in the sense of a universal or similar abstract entity, that the phenomenon instantiates, in addition to the phenomenon itself. Or let us put it this way: there was no such implication, unless all predicative statements have such implications. We are here at a point of contact with our discussion above of the notion of truth. If all predicative statements had such implications, that would mean, for example, that ‘‘Snow is white’’ is true only if in addition to snow there exists also a universal, whiteness, to which snow bears a certain relation. I take it that is not the case: ‘‘Snow is white’’ implies the existence of nothing at all other than snow, and similarly, the statement that there were at time t 500 bacteria in this colony implies the existence of nothing at all beyond the bacteria and the colony.14 Rejection of the correspondence theory of truth may be familiar from much other philosophical literature, but a few—even cursory—remarks may still help to underline this point. ‘‘Snow is white’’ is true. Does this imply that besides snow, there is also whiteness? The case for Yes! begins with the remark that snow cannot by itself make the statement true, because the existence of snow does not imply that snow is white . . . snow could have existed and have had some other color. So (?) there must be something in addition to snow, whose ‘cooperation’ with the also needed snow will make the statement true. I am carrying coals to Newcastle by responding to this (Quine’s ‘‘On what there is’’ appeared long ago): all that is needed for ‘‘Snow is white’’ to be true is that snow be white, and that does not imply the existence of anything but snow. But then, the objection goes, what about sentences that are not particular but general, like ‘‘All statements made by Francis Bacon just after eating oatmeal were true’’? While in the above particular example, the use of

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‘‘true’’ succeeds only in echoing the statement to which it is attributed, so that it is dispensable, that is not the case with such a general form. Correct; but nothing more is needed in nature for the general statement to be true except all that is needed for each of the particular statements, made by Bacon just after eating oatmeal, to be true. And none of those needed metaphysical underpinning, unless they were metaphysical themselves. (We needn’t deny that metaphysical statements need metaphysical grounds, but that is not the fault of truth talk.) There is a lot more said in the discussions about truth, but I’ll leave it to the literature; the main point sufﬁces here.15 The same point applies to isomorphism. Let’s take a particular case of a concrete physical, observable object: this table top is metrically isomorphic to a Euclidean square. That is true, but simply because this table top is square—c’est tout! It is true because the top’s sides are of equal length and the angles between them are right angles. It could be paraphrased as ‘‘the table top instantiates the Euclidean square Form’’, but the cash value of the assertion carries no metaphysical commitment: it is just that the table top is square. When we want to discuss something of greater generality, such as a theory’s empirical adequacy, we must speak of all phenomena of a certain sort and how they relate to mathematical models. But (just as for the example of Francis Bacon’s statements) nothing more is needed for the general statement than whatever is needed for the particular cases falling under it. And that does not include ‘structure in nature’ in a sense that goes beyond the simple point that some table tops are square, some birds are fast, some battles go to the strong, and so forth. Another question remains, though, about representation. There is no such thing as representation apart from or independent of our practice. So how can we say something like: this theory accurately represents that bacterial growth phenomenon (in those 12 hours long, long ago, and far, far away) although the relevant model was never constructed and the bacterial colony was certainly not encountered in human practice?16 The structural relationship between the model in question and the phenomenon, that we just described, does not sufﬁce to make the model a representation of the phenomenon. This was amply argued in our initial discussion of representation. If a model were offered to represent the phenomenon, that structural relation would determine whether the

250  :    model was adequate with respect to its purpose, for example, whether the representation was accurate with respect to the rate of growth in volume or in numbers, as measured by this sort of clock or that, . . . . By hypothesis, no model for this particular phenomenon was ever offered, so the point is moot. Yet we would like to say that if the equation does have such a solution—equivalently, if the theory has such a model—then that (equation, theory) does correctly represent that phenomenon. It seems to me that the only points to be made in response to this are verbal. If the theory is offered, that amounts to the offer of a range of structures—the structures we call models of the theory—as candidates for the representation of the phenomena in its domain. If this range contains a candidate that would satisfy the structural constraint—if the phenomenon is embeddable in it, understood in the innocuous sense explained by the above examples—then the theory is empirically adequate (and indeed true, if the domain contains only observable phenomena). Perhaps it would have been better if the word ‘‘model’’ had not been adopted by logicians to apply to structures never offered in practice. For undoubtedly, in many contexts, something is called a model only if it is a representation, and the sense in which any solution of an equation is a model of the theory expressed by that equation certainly does not have that meaning. But it is too late to regiment our language so as to correct that, and we will just need to be sensitive to usage in different contexts. The problem in concrete setting We turn now to the second case: the phenomena are encountered in practice, measurements are made, data models constructed, and so forth. How exactly are theories confronted with the phenomena given by measurement and experimental results in ﬁeld work or laboratory practice? Ronald Giere and Paul Teller have both discussed this with reference to detailed accounts of empirical research.17 I will also take a speciﬁc example, to discuss here in terms we set out before. The observable phenomenon makes its appearance to us ﬁrst of all in the outcome of a speciﬁc measurement, or large set of such measurements—or at slight remove, in a data model constructed from these individual outcomes, or at a slightly further remove yet, in the surface model constructed by extrapolating the patterns in the data model to something ﬁner than our

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instruments can register. In such a scientiﬁc inquiry, from the gathering of raw data to the achievement of something that can properly confront the pertinent theory, we see moves toward increasingly abstract representation. Well before the theorist displays his or her wares, experimentalists are already constructing models of the data to convey the outcomes of their inquiries. Describing the situation in this way, three ‘stages’ are displayed: the observable phenomena subjected to measurement, the appearances they present in progressively more abstract ‘outcomes’, and ﬁnally the theory whose theoretical models are accountable to those—already quite abstract—deliverances of scientiﬁc inquiry. Young’s 1802 double slit experiments constituted, together with Fresnel’s ‘‘light spot’’ in a shadow, the crucial phenomena that Newton’s particle models for light could not ‘save’. That was their signiﬁcance at the time; now we see much more in them, for there are more sensitive versions of Young’s experimental design. Speciﬁcally, somewhat more than a hundred years later, once the photo electric effect was discovered and its quantum character appreciated, such diffraction experiments needed to be revisited and reinterpreted. The reinterpretation focuses on a variation, not performable or even contemplated in Young’s time, in which the source is so slow that only a single black spot appears on the screen during non-negligibly large time intervals. (Experiments in which individual photons were isolated could actually not be performed till the last quarter of the twentieth century.) If carried out in this form, the interference pattern will still appear, but only gradually. Similarly of course for each of the two separate single slit experiments resulting with either of the slits closed. So let us imagine what the data could be. Think of the screen divided into N small areas, indexed as x = 1, 2, 3, . . . and consider for each the proportion of hits in that area after the ﬁrst n hits on the screen. Then the data gathered are recorded in three relative frequency distributions rel(n,x), relA (n,x), relB (n,x). For example, relA (50,17) denotes the proportion of hits, among the ﬁrst 50 hits, that occurred in area 17 when only slit A was open. This summary is the outcome of the speciﬁcally carried out experiment, though ‘policed’ since the recording may have been subject to noise or random errors or inaccuracies; it is the data model. But it is certainly not yet what any theory about the phenomenon must confront. A ‘‘curve

252  :    smoothing’’ program replaces the three relative frequency functions with three probability functions p, pA, pB, the result being what I prefer to call a surface model. (This involves an empirical hypothesis—future repetitions of the experiment could conceivably yield a bad ﬁt with these functions.) The surface model is p, pA, pB Theoretical model T1: p is a mixture of pA and pB. Empirically refuted. Theoretical model T2. p, pA, pB are determined by a geometric probability model, in the way of quantum mechanics. This works. Our diagnosis of this procedure: the theory to phenomena relation displayed here is an embedding of one mathematical structure in another one. For the data model—or, more accurately, the surface model—which represents the appearances, is itself a mathematical structure. So there is indeed a ‘matching’ of structures involved; but is a ‘matching’ of two mathematical structures, namely the theoretical model and the data model. At this point, we can be sure, the metaphysical realist will ask for the next step: so then what is the relation between data model and the phenomena it models? But the dialectical situation is different now that we are in a concrete setting, different from the bacteria colony in the Antarctica example: here the measurements are really made and the modeling is explicitly a case of representation. The form of the metaphysician’s question insinuates a substantive presupposition: that there is a relation between data model and phenomena, which determines whether the data model represents the phenomena, and which has nothing to do with anything but the two of them. Once again we ﬁnd ourselves with an idea akin to, of a piece with, the correspondence theory of truth, the idea that there is a user-independent relationship between words and things that determines whether a sentence is true or false. Such an idea cannot be carried through without postulating a good deal of ontological ﬂora and fauna beyond concrete individuals. But we have discussed this issue sufﬁciently above, we don’t need to repeat the argument against such presuppositions. What we need to ask ourselves now, though, is precisely how can we phrase the question we had originally, about how an abstract entity can represent something physical, so as to make sense of it in this concrete context.

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The problem in concrete setting revisited and dissolved How can we answer the question of how a theory or model relates to the phenomena by pointing to a relation between theoretical and data models, both of them abstract entities? The answer has to be that the data model represent the phenomena; but why does that not just push the problem one step back? The short answer is this: construction of a data model is precisely the selective relevant depiction of the phenomena by the user of the theory required for the possibility of representation of the phenomenon. But this answer needs to be ﬁlled out As we have just seen, when a theoretical model is said to represent certain phenomena, there is indeed reference to a matching, namely between parts of the theoretical models and the relevant data models—both of them abstract entities. Note now, the crucial word in this sentence: the punch comes in the word ‘‘relevant’’. There is nothing in an abstract structure itself that can determine that it is the relevant data model, to be matched by the theory. A particular data model is relevant because it was constructed on the basis of results gathered in a certain way, selected by speciﬁc criteria of relevance, on certain occasions, in a practical experimental or observational setting, designed for that purpose.18 Looking into the model by itself we will ﬁnd nothing to imply that it is or is not a representation of e.g. a bacterial colony or tides in a sea inlet. (In the same way there is nothing in a printed map taken in itself that can make it a map of Austin, TX instead of, say, a kabalistic charm.)19 Rather, the relation to be identiﬁed is not a 2-place relation between data model and phenomenon but a 3-place relation that involves the user. Representation is a relation between the abstract structure and the phenomena constituted by the user. Nothing represents anything except in the sense of being used or taken to do that job or play that role for us. We can also reach this conclusion by a different route, thinking ﬁrst of the phenomena. Psillos writes quite correctly: Ergo, the structure of a domain is a relative notion. It depends, and varies with, the properties and relations that characterise the domain. Differently put, a domain has no inherent structure, unless some properties and relations are imposed on it. Or,

254  :    two classes A and B may be structured by relations R and R respectively in such a way that they are isomorphic, but they may be also structured by relations Q and Q in such a way that they are not isomorphic. (Psillos 2006: 562–3)

That is, the phenomenon, what it is like taken by itself, does not determine which structures are data models for it—that depends on our selective attention to the phenomenon, and our decisions in attending to certain aspects, to represent them in certain ways and to a certain extent.20,21 Have we now lost or sidestepped Reichenbach’s problem, or have we rather landed in it ourselves? Have we in effect succumbed to a post modern il n’y a pas de ‘hors-texte’? No, but the way to see that we have not ‘lost the world’ will explain why the sea-change I propose is of the sort that tends to be described as a Wittgensteinian move. An example: the metaphysician meets practice To understand how the above maneuver does not simply push Reichenbach’s question one step back, we have to think of the assertions made in context, by the scientists involved. Something will come to light if we listen to the assertions that such a scientist would initially make in the ﬁrst person. Let us take a very simple example that puts the discussion into ﬁrst-person discourse, by imagining ourselves in the scientist’s role. The example I’ll choose will accordingly not require the agent’s being very advanced in a particular science. Best to have the theory in question deliver models no more complicated than the data models engendered by an empirical inquiry. Imagine a township council meeting where I have represented the deer population growth in Princeton by means of a graph. We have been discussing a theory, call it T, about the Princeton environment for deer, including luscious gardens as well as the council’s culling instinct, its tendency to experiment with birth-control measures for the local fauna, and the like. I point out that theory T provides models that ﬁt very well with the structure displayed in the graphical representation—call it S—that I have just presented. Now someone notices that I am pointing to a match between two abstract mathematical structures (a model of T and a graph), and expresses doubts about whether we are staying in touch with reality: : Yes, T ﬁts well with this graph, your representation S, but does T ﬁt the actual deer population growth in Princeton?

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Well, this is Princeton. Still, to begin at least I do not realize that the questioner is a metaphysician. So to begin I take this as a legitimate request for more information about how I arrived at the graph. I describe the measurements that sampled values of various parameters over time, how we had set up stations for deer observation, the procedures followed and the precautions taken against sampling bias. The challenger responds: Yes, I understand, I can see that you carried out those procedures diligently and responsibly, and that the outcomes are summarized properly in your graph. But the question remains: theory T ﬁts the summarized outcomes of your measurement procedures, but does T ﬁt the actual deer population growth in Princeton? Now I am beginning to see that the question relates to a rather more foundational worry, and I think that perhaps I can guess what it is. So I explain that I did take guidance from the theory, that the parameters measured were precisely those that T treats as relevant to population growth in this sort of case. I add that although the measurement procedures were mostly standard in this scientiﬁc area of inquiry, it was also in light of T that I took for granted in some cases that values obtained for some parameters gave information about the values of certain parameters less accessible to direct measurement. But I add that none of this biased the inquiry so as to guarantee that the results would be in accordance with T. The measurement outcomes could have been quite different, while the actual outcomes were, as I just showed, in accordance with T. To the challenger this signals a complete misunderstanding of his concern. I took his worry to be about a problem in epistemology! But his interest is in what is really the case, and how we can understand how things really are, independently of our knowledge and of any procedures followed to gain knowledge. So he tries to convey that to me: I understand that in this case your claim to knowledge about the deer population growth in Princeton is warranted. But there is still the real deer population growth, which is something in the world, distinct from anything in your graph, distinct from anything in the content of your warranted knowledge claim, distinct from the object of knowledge that you have constituted in your practice (put it how

256  :    you will)—and that, the real deer population growth, is what we want theory T to match! Although I can see the logical leeway on which he trades there is no leeway for me in this context, short of withdrawing my graph altogether. Since this is my representation of the deer population growth, there is for me no difference between the question whether T ﬁts the graph and the question whether T ﬁts the deer population growth. If I were to opt for a denial or even a doubt, though without withdrawing my graph, I would in effect be offering a reply of form: The deer population growth in Princeton is thus or so, but the sentence ‘‘The deer population growth in Princeton is thus or so’’ is not true, for all I know or believe. The ﬁrst conjunct would be the content of the graph which I am presenting as representation of the deer population growth, and the second conjunct would express my doubt that this graph does correctly represent the deer population growth. Such a response would be as paradoxical as any of Moore’s Paradox forms, like ‘‘It isn’t so, but I believe that it is’’ or ‘‘It is so, but I do not believe that it is’’. In fact, I will have become incoherent if I let this challenge lead me into any such concession. Notice here that, unlike me, the challenger would not be (and is not) inconsistent or incoherent. He may be doubting that my data model is a good one, that it correctly and accurately represents the deer population growth. For him that is logically independent of whether the graph was correctly constructed from results to be taken seriously as veridical. But that my data model does have that status is precisely the content of my assertion when I present the graph with the words ‘‘here you see the deer population growth in Princeton’’. His challenge does not offer any evidence to the contrary. But what if I, disturbed by his question, take it as an injunction to doubt my own memory or competence, and I take that seriously? I will not have identiﬁed his metaphysical concern, but heard him as speaking in this scientiﬁc context, where the task of coordination is past. (The theoretical terms are already closely connected with the practical procedures that deﬁne the relevant

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kinds of measurement.) If I do, I will ﬁrst of all be bracketing, taking back, putting on hold, my presentation of the graph. What I cannot do is to both present the graph as representing something and say that perhaps it doesn’t represent that at all. This example brings out the crucial point: in the chain [theory]–[data model]–[reality] the last link is one that is expressed in indexical judgments. The assertion that a given graph represents the phenomenon I have been observing and measuring is on a par with the assertion that a certain spot on a map is the point where I am. The conditions of correctness in both cases pertain to the use to which the structure—whether abstract or physical—is being put, and how. When I presented the graph to the audience interested in deer population growth in their borough, I was saying ‘‘we are here’’ in a logical space of possible growth patterns. And as Kant explained—as we discussed in Part I—the ability to self-attribute a position with respect to the representation is the condition of possibility of use of that representation. The ‘link’ to reality When Newton claimed that his theory of gravitation ﬁt the phenomena, he meant in part that his equations entailed (under certain simplifying assumptions) Kepler’s laws of planetary motion.22 The latter he took as descriptions of those phenomena, the planetary motions. What Kepler’s laws gave him was in effect what I have called a surface model, a structure constructed from data painstakingly amassed by astronomical observations. The matching Newton demonstrated was therefore between mathematical structures: between a substructure of his model of the solar system and that Keplerian structure. As long as we restrict our attention to this process, we are leaving aside the ﬁnal question: but how does the last structure taken into account relate to what is meant to be the target of all this representing? The answer, brought to light already in our initial discussion of maps and models in use, is that at this last step we come to what can only be expressed in indexical judgments. We can use a map to get around the region in which we ﬁnd ourselves if and only if that map is the thing we

258  :    use and locate ourselves in to represent the pertinent features of that region. This point transposes to the use of any model, and pertains speciﬁcally to the assessment of whether a theory succeeds in correctly representing what it is meant to represent. Take any such assertion as The exponential function represents in smoothed summary form the growth of the investigated bacterial colony. Despite appearances, that is an indexical statement: the phrase ‘‘the investigated bacterial colony’’ receives its reference from the context in which it is used. The assertion is not made true by anything that can be formally described within semantics, understood as limited to word-thing relations, ignoring the role of the user and context of use.23 Nor does the phenomenon, what it is like, taken by itself, determine which structures are data models for it. That depends on our selective attention to the phenomenon, and our decisions in attending to certain aspects, to represent them in certain ways and to a certain extent.24 So once again we arrive at the conclusion: there is nothing useful to be found in 2-place structure-phenomenon relations alone when we try to understand representation. Anything we see by way of such relations is something abstracted from the 3-place relation of use of something by someone to represent something as thus or so. The ‘Loss of Reality’ objection dissolved Perhaps, by now, the second danger for the empiricist in this context has been entirely and successfully evaded. I hope so. But at the risk of sounding completely de trop let’s see the challenge in its most strident guise: Oh, so you say that the only ‘matching’ is between data models and theoretical models. Hence the theory does not confront the observable phenomena, those things, events, and processes out there, but only certain representations of them. Empirical adequacy is not adequacy to the phenomena pure and simple, but to the phenomena as described! An empiricist account of what the sciences are all about must absolutely answer this objection.25 Let us therefore honor it with a special name: the Loss of Reality Objection.

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The empiricist reply must be, in effect, the step that leaves the entire game of metaphysics behind, and frees us forever from its illusionary charm and glamour. But just because it is the step out of that so insidiously enchanted forest into realistic common sense, it will have to be a very simple one. On the one hand, we must immediately admit that: the claim that the theory is adequate to the phenomena is not the same as the claim that it is adequate to the phenomena as represented by someone (nor as represented by everyone, or anyone). After all the phenomena are actual objects, events, and processes, while the representations that we or the scientiﬁc community construct of them are the products of our independent intellectual activity. Yet on the other hand it is clear that if we try to check a claim of adequacy, then we will compare one representation or description with another —namely, the theoretical model and the data model. What are we to make of these two points, taken together? For us the claims (A) that the theory is adequate to the phenomena and the claim (B) that it is adequate to the phenomena as represented, i.e. as represented by us, are indeed the same! That (A) and (B) are the same for us is a pragmatic tautology. That it holds depends crucially on who the indexical word ‘‘us’’ functions to denote in an assertion. Appreciating that the equivalence for us is a pragmatic tautology removes the basis for the Loss of Reality objection.26 A pragmatic tautology is a statement which is logically contingent, but undeniable nevertheless. Similarly, a pragmatic contradiction is a statement that is logically contingent, but cannot be asserted. That is possible, because the logical contingency pertains to its content, while deniability or assertability is a concept pertaining to use. Assertion, denial, calling into question, and the like are actions by a language user. The semantic status (e.g. expresses something contingent) and the pragmatic status

260  :    (e.g. is deniable) diverge quite obviously when the statement has some indexical or context-dependent element or feature. The classic example is Moore’s Paradox, which I already cited above. But in the cases we are concerned with, there is often no explicitly indexical word or phrase to be seen. That does not settle the matter. Consider for example ‘‘There are no statements’’. This is logically contingent since there are statements in the actual world but there are also logically possible worlds where there are none. Yet it is not assertable, since its assertion would be a statement—someone asserting it would stand convicted of falsity by that very act. In any context of use, the statement is false in the world of which that context is a part. Yet the statement does not have in it such an indexical as ‘‘I’’ which plays the crucial role in the usual examples, as in Moore’s Paradox.27 Similar remarks apply to Tarski’s famous schema ‘‘ The sentence ‘—’ is true if and only if—’’. Suppose we ﬁll it with ‘‘Grows is’’ or with ‘‘Grnunkj’’. Then we do not obtain a truth, in the ﬁrst case because we do not have ‘‘Grows is’’ as a sentence of our own language, and in the second because we do not acknowledge the word ‘‘Grnunkj’’ as a word of our own language at all. The distinction expressed with the indexical ‘‘our own’’ is crucial, but is not shown by the words in the sentences under discussion. We can make the distinction in a general way by asking about sentences both whether they could be true and whether they can be asserted by us. If the world had been such that our language had developed so that the word ‘‘snow’’ had been a word for grass, then ‘‘Snow is white’’ would not have been a true sentence, though snow would still have been white. So the Tarskian equivalence is not a necessarily true statement, but its contraries are not assertable by us for sentences in our own actual language. How does this apply here? We can sum up the relevant point quite simply: in a context in which a given model is someone’s representation of a phenomenon, there is for that person no difference between the question whether a theory ﬁts that representation and the question whether that theory ﬁts the phenomenon.28 Think of what would happen to Moore’s Paradox if we treated pragmatic inconsistencies like logical ones. We would appeal to the logical principle that if (A & B) is inconsistent then A implies the denial of B and conversely. So from the incoherence of ‘‘I believe that it is raining in Peking, but it isn’t’’ we would infer ‘‘If I believe that it is raining in Peking

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then it is’’. From the inconsistency in Moore’s Paradox we would proceed to to the general certainty of It I believe that something is so, then it is, crediting ourselves with clairvoyance. To postulate metaphysical structure in re, or in nature, to save structuralism would be like ‘saving’ Moore’s Paradox by postulating clairvoyance.

Return to our epistemological question After introducing Weyl’s text and examining the four options that it appeared to offer us, I said that we face here a perplexing epistemological question: Is there something that I could know to be the case, and which is not expressed by a proposition that could be part of some scientiﬁc theory? The answer is YES: something expressed only by an indexical proposition. For to use a theory or model, to base predictions on it, we have to locate ourselves with respect to it. This applies speciﬁcally if a theory is assessed by recourse to a data model or surface model. In this assessment the mathematical structure is matched, but the relevance of the matching consists precisely in the user’s relation to that structure. When using a model to ﬁnd our way around in the world we have to be able to say, for example, that the phenomenon we are presently witnessing is classiﬁed in the theory as oxidizing, or as phlogiston escape, or the like. We have to locate our situation in the theory’s logical space, in a way that is similar to our ‘‘We are here’’ with respect to a map. This implies no relevant incompleteness in the theory or model itself. To make this point about science is not to deny that science is (in principle?) so complete that every fact, every phenomenon, is completely represented (representable?) within science. A New York subway map or Paris Metro map is not incomplete because it comes without a ‘‘you are here’’ tag. It is selective, it neither does nor purports to represent more than certain aspects of the topological structure of the system—but that it does completely.

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PA RT I V

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Appearance and Reality To begin this last part, let’s have a moment to take stock. A science presents us with representations of the phenomena through artifacts, both abstract, such as theories and mathematical models, and concrete such as graphs, tables, charts, and ‘table-top’ models. These representations do not form a haphazard compilation though any unity, in the range of representations made available, is visible mainly at the more abstract levels. The insistence on this unity as a hallmark of science has been with us in philosophy since the beginning. In Aristotle we see a remarkable parallel in his views on drama and on physics.1 The Physics presents us with a view of the structure of nature and natural processes, and also, in conjunction with the Posterior Analytics, of the structure of the science that deals with nature. The Poetics presents the structure of the human condition and of human action as they are depicted in tragedies, but also of the structure of those tragedies, that dramatize human existence, themselves. This easily noticed parallel actually extends from this form of the books to how they depict their subjects. A tragedy, writes Aristotle, is a representation of an action . . . . In a well-constructed tragic plot, if some part were transposed or removed the whole would be disrupted and disturbed. The action is to form a tightly structured causal process, ﬂowing jointly from the character of the protagonists and the conditions in which they ﬁnd themselves. The element of chance is to be shunned and if needed at all, kept off stage. The events ‘‘should result from the actual structure of the plot, so it happens that they arise either by necessity or by probability as a result of the preceding events. It makes a great difference whether these [events] happen because of those or [only] after those.’’ (Poetics 52a17–22) Human action and fate must thus be represented as subject to a causal structure that sufﬁces to make them intelligible to us. This kind of intelligibility is precisely what Aristotle asks for in a scientiﬁc theory. The natural path of inquiry ‘‘leads from what is familiar or evident to us to what is by nature clear or conclusive’’. Starting with what is naturally obscure but apparent to us we must seek the reasons why, and thereby

266   :    advance to what is intelligible in itself. ‘‘All inquiry aims at knowledge; but we cannot claim to know a subject matter until we have grasped the ‘why’ of it, that is, its fundamental explanation.’’2 Aristotle himself seems to see the parallelism very well. When in the Physics he comes to what he considers a bad theory (the theory of evolution by natural selection and chance variation, as it happens!) he makes fun of it. It does not meet his standard for scientiﬁc knowledge, for it does not ‘‘deal adequately with the ‘why’. . . . in terms of each type of explanatory factor’’. And he emphasizes that again in the Metaphysics: ‘‘But the phenomena show that nature is not a series of episodes, like a bad tragedy’’.3 This conception of science has not only a venerable history, it has exercised its grip on the philosophical imagination down to the present day. A well-constructed scientiﬁc theory will tell a story, a narrative in which the why is as clearly explained as the what, and we come to understand not only ‘what happens’ but ‘what is really going on’. The question at issue for us today, in philosophy of science, is precisely to what extent this conception can still be a guide to the understanding of the sciences we now have. Theoretical scientiﬁc representation is in general very far from a simple picturing. Creations of almost unimaginable beauty enter theoretical modeling, and their exploration is understandably irresistible to philosopher and mathematician alike.4 But—and this may not be just an empiricist sentiment—these explorations need to be supplemented with inquiry into the modeling relation itself. At ﬁrst blush, a model typically represents the phenomena as embedded in a larger nature or reality. Whether we take this at face value or bracket the question of reality, we need to investigate the relations of this theoretically postulated reality to the phenomena, and to the appearances of those phenomena in measurement. Scientiﬁc knowledge is objective, in a sense that implies maximal intersubjectivity. This intuitive conviction, which empiricists, transcendentalists, and scientiﬁc realists all share, rightly remains regardless of the admitted perspectivity in observation and measurement. But ‘‘objectivity’’ is a term with many antonyms, and therefore diverse meanings, so the very formulation of this thought invites equivocation. I will begin with the ideal of objectivity as it appears in putative completeness criteria that philosophy has purported to ﬁnd in physical theorizing. Some prominent such criteria, once honored, bit the dust in recent history. One, much loved and honored

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in modern times, still sings its siren song to today’s philosopher; I shall call it the Appearance from Reality Criterion. But I shall argue that it must follow the others into the dust, if we are to appreciate the new key in which the sciences are now composed. Once rejected (as, I shall argue, it has effectively been rejected already in scientiﬁc practice) we have the freedom to follow the contemporary abstract structural forms now prevalent in the advanced sciences without the unbearable constraint to satisfy a ‘‘realist’’ imagination.

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12 Appearance vs. Reality in the Sciences If science offers a representation of nature, what precisely does it represent? Paintings and photos depict things as they appear to us in perception. In contrast a scientiﬁc theory may be said to depict things as they are. The differences between how things appear to us and how they are depicted by a scientiﬁc theory can certainly be striking. To urge the intellectual understanding’s superiority over the senses, those differences became fodder for the skeptical arguments Descartes adapted in the Meditations. His skepticism was not directed at inferences blithely made to go beyond the senses, but at trust in the senses themselves. Yet in the end we can only evaluate the accuracy of any representation by attending to how the represented object appears to us, whether in direct observation or in measurement. So even if a scientiﬁc theory or model represents how things really are, the scientiﬁc account is not ﬁnished unless it entails correctly what their appearance will be like under realizable conditions. The phrase ‘‘not ﬁnished unless’’ points to a notion or ideal of completeness. The ideal of completeness here in play—even if we merely see completeness as a regulative ideal, perhaps not humanly achievable—challenges a science to represent the appearances as well as the theoretically postulated reality. This challenge, so very clear to the seventeenth century scientistphilosophers, is perennial. It was posed dramatically for our time by Einstein and his colleagues in ‘‘Can quantum-mechanical description of physical reality be considered complete?’’. This is a challenge that can precipitate radical, revolutionary change in the sciences. As I will argue here, these radical changes radiate outward from the content of the new theories to the very methodology of science itself and to the conception of what science is to be.

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Appearance and Reality: the real and unreal problem Even within philosophy the appearance/reality contrast has had somewhat of an undeservedly bad press. When F. H. Bradley wrote his famous Appearance and Reality he was not aiming to install that contrast as a dichotomy, but to break out of it. He was in fact criticizing what he saw as all the forms of realist metaphysics engendered in the seventeenth and eighteenth centuries. In chapter XII he concludes: We have found, so far, that we have not been able to arrive at reality. The various ways, in which things have been taken up, have all failed to give more than mere appearance. Whatever we have tried [to conceive as beyond the appearances] has turned out to be something which, on investigation, has been proved to contradict itself. (Bradley 1930: 110)

Where there is no contrast to be found there is no concept to apply. Bradley is referring here speciﬁcally to the so-called Problem of the External World that so bedeviled modern philosophy. What he concluded was that it is a problem with no solution—hence a problem to be escaped rather than solved. He was right. The problem of appearance and reality, as posed in modern philosophy, was unsolvable. Accordingly, the problem itself must have been based on a mistake, a mistaken presupposition hidden in its origin or formulation. But as so often happens in philosophy, this pseudo-problem was arrived at in response to a real problem—in fact a real problem that faced the physicist-philosophers of the early modern era. How was the philosophical problem created, and how did it displace the real problem?1

Appearance versus reality at the birth of modern science Early in the seventeenth century the newly emerging modern sciences faced a challenge on many fronts. How is the scientiﬁc characterization of reality, whether in astronomy, or in dynamics, about the constitution of material objects and processes, to be reconciled with the very different way in which things appear to us?

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There were striking successes and glaring gaps. Copernicus’s achievement was a paradigmatic success: the apparent planetary motions could be explained, by means of geometric transformations, on the basis of his suncentered model. But what about the appearance that the earth does not move? It was not part of Copernicus’s achievement, to have shown to his contemporaries that this appearance was compatible with his hypothesis. For in the physics of his time the motion described was taken to entail all sorts of force effects that were not observed. Why aren’t the clouds left behind? Why do cannon balls travel equally far, regardless of the direction in which they are ﬁred? While Copernicus was not in a position to address these questions, Galileo takes the disparity as an indictment of Aristotelian physics and begins to develop a dynamics compatible with Copernicus’s cosmology. By the time of Descartes it is widely accepted that such a dynamics can indeed be developed, and the appearance of rest for loose objects on a moving earth is no longer puzzling—a striking success achieved in the course of a single century. Below I will return to Copernicus’s success, which began this historic process, but I want to do so as illustration for just what Galileo and Descartes saw as a real problem for the emerging natural science. In The Assayer Galileo addressed directly the problem of what later came to be called ‘‘secondary qualities’’, qualities that he did not want to have any fundamental role in the new scientiﬁc image.2 Galileo insisted on a strict discipline to be observed in the new sciences, to eschew the multiplication of entities or features in service of explanation. This strict discipline, in his insistence, implied the demand to couch theories in terms of notions that could be mathematically represented. The qualities that this allowed were measurable quantities—the ‘‘primary’’ qualities, in our later terminology—and they were few.3 Even so, by some later standards, Galileo admitted too many. Descartes reduced the list to just attributes deﬁnable in terms of spatial and temporal extension; others wanted additions, such as impenetrability, but ascetic scarcity was maintained. The world as it appears to us is colorful, noisy, smelly, tasty . . . the world described purely in terms of those primary qualities is none of that. Galileo’s Assayer Galileo famously promised that the colors, smells, and sounds in the experienced world would be fully explained by a physics among whose

272  :    descriptive parameters those qualities were not allowed. With typical rhetorical ﬂair he makes this promise as if it is already certain to be satisﬁed. Giving as sole reason that he cannot imagine the contrary (!) he submits that the perceived qualities fall into those that really belong to the objects and those that are ‘mere names’: . . . whenever I conceive any material or corporeal substance, I immediately feel the need to think of it as bounded, and as having this or that shape; as being large or small in relation to other things, and in some speciﬁc place at any given time; as being in motion or at rest; as touching or not touching some other body; and as being one in number, or few, or many. From these conditions I cannot separate such a substance by any stretch of my imagination. But that it must be white or red, bitter or sweet, noisy or silent, and of sweet or foul odor, my mind does not feel compelled to bring in as necessary accompaniments [. . . .] Hence I think that tastes, odors, colors, and so on are no more than mere names so far as the object in which we place them is concerned, and that they reside only in the consciousness. Hence if the living creature were removed, all these qualities would be wiped away and annihilated. But since we have imposed upon them special names, distinct from those of the other and real qualities mentioned previously, we wish to believe that they really exist as actually different from those.4

He continues with some examples of how interaction with something material can induce different sensations in different parts of the body, and he offers possible explanations of those sensations in terms of tiny particles and motions of the air affecting the sense organs. The colorful, tasty, smelly, and noisy appearances are thus promised to be shown produced as interactional events, in which the relata are in principle completely characterized in terms of primary qualities. Nor is he shy to claim success: ‘‘Having shown that many sensations which are supposed to be qualities residing in external objects have no real existence save in us, and outside ourselves are mere names. . . .’’ (Ibid.: 277). While there is no implication here that the occurrence of secondary qualities is something beyond the sciences’ domain of application, the insistence on their ‘‘residence in the sensitive body’’ suggests an irrelevance for basic physics. This distinction between primary and secondary qualities coincided with one familiar to the Aristotelian tradition. In the latter, the peculiar sensibles are those—such as color and smell—perceptible through a single sense, while the common sensibles—such as shape and volume—are those detectable through several senses. We can determine a shape either

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by looking or by feeling, for example. It can hardly be a coincidence that the primary qualities of Galileo are more or less the common sensibles. And there is a plausible rationale for that: something detectible only through a single sense has something too speciﬁc, too speciﬁcally sense-dependent, to lend itself to being given a sense-independent status as well. Moreover, as is always remarked in this connection, there were already quantitative measures available at least for the paradigm examples of primary qualities—attributes of shape, volume, number—that were important to the experimental methods now being developed. Galileo’s famous injunction that the Book of Nature is written in geometry, and Bacon’s that it needs mathematical literacy to be read, is crucial to this moment.5 The rhetoric focused on what was to be counted as real, what as mere appearance. But there was—more importantly in my view—a methodological reason for Galileo’s insistence on the restriction of descriptions of nature to a small number of basic terms. Scientiﬁc discipline was to preclude the sort of multiplication of concepts that might so easily provide merely ‘‘verbal’’ rather than real solutions to the problems posed by natural phenomena. Really, all the above considerations are broadly methodological. The question for us, in retrospect, is whether to applaud or decry the metaphysical claims offered to buttress the methodological reasons. Galileo’s promises, that the appearances couched in secondary qualities would be entirely explained, were not empty. There were solid achievements behind them already, even if still rather modest in our eyes. Those achievements accumulated into awesome riches by the late nineteenth century. To mention but one salient example, beloved of philosophers: combustion of sodium samples is an observable process (phenomenon) that can be exhaustively described in the terms of basic physics, but this description can also be utilized to explain how that process produces a yellow appearance to the human eye, and of course, to a camera or spectrophotometer. We can say that, indeed; but how much does it mean? How much it means depends crucially on what ‘‘explain’’ means, and on the criteria for what counts as an explanation. Descartes’s radical transposition Both Galileo and Descartes were inclined to wave their hands toward a reduction of all qualities to the primary ones. But Descartes, going

274  :    beyond Galileo’s relegation of secondary qualities to ‘‘residence in the sensitive body’’, initiated a strategy that bedeviled both philosophy and science for some centuries to come. Using well known, in fact ancient, skeptical arguments and techniques he made a radical move that would, if successful, take the problem off the shoulders of physics altogether and place it squarely on metaphysics. In his metaphysical world-picture we are acquainted directly only with our ideas, our subjectively formed representations, but have no logical warrant for thinking that what there is outside the mind is correctly mirrored within.6 When he sought some assurance for the belief that there is such a mirroring, this was limited to mirroring of the primary qualities alone. What of the rest? In Descartes’s posthumous work The World, or Treatise on Light he purports to lay the foundation of a world-picture entirely transparent to the human understanding. Here (unlike in Galileo) we do see criteria for explanation, or understanding. They are the criteria of the mechanistic philosophy which insisted that all interaction must be by mechanical actionby-contact. The set of primary qualities is reﬁned to just the attributes that play a role in mechanical explanation: attributes of extension (spatial and temporal). The primary qualities are therefore quantities, that is, numerically representable magnitudes, entirely and adequately represented in geometry (speciﬁcally in kinematics, that is, the geometry of space, time, and motion) But in the Discourse and Meditations, besides this positive effort to construct a complete physics, the programme is defended by the use of traditional skeptical arguments directed especially against any claims to reality of the secondary qualities. Perception by means of any single sense is subject to many distortions and illusions. The analysis of warmth and material falsity in Meditations III provides a good example of this: As belonging to the class of things that are clearly apprehended, I recognize the following, viz, magnitude or extension in length, breadth, and depth; ﬁgure, which results from the termination of extension; situation, which bodies of diverse ﬁgures preserve with reference to each other; and motion or the change of situation; to which may be added substance, duration, and number. But with regard to light, colors, sounds, odors, tastes, heat, cold, and the other tactile qualities, they are thought with so much obscurity and confusion, that I cannot determine even whether they are true or false; in other words, whether or not the ideas I have of these qualities are in truth the ideas of real objects. For although I before remarked that it is only in judgments that formal falsity, or falsity properly so called, can

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be met with, there may nevertheless be found in ideas a certain material falsity, which arises when they represent what is nothing as if it were something. Thus, for example, the ideas I have of cold and heat are so far from being clear and distinct, that I am unable from them to discover whether cold is only the privation of heat, or heat the privation of cold; or whether they are or are not real qualities: and since, ideas being as it were images there can be none that does not seem to us to represent some object, the idea which represents cold as something real and positive will not improperly be called false, if it be correct to say that cold is nothing but a privation of heat; and so in other cases.7

A more famous passage in Meditations II (sections 11–13) does too. The warmed bit of wax, which loses its secondary qualities and undergoes simultaneous changes in its primary qualities, is nevertheless known to be the same—giving the lie to the impression ‘‘that the wax is known by the act of sight’’ and establishing that some facts ostensibly gained through experience are actually not given through the senses at all. The problem of reconciling the scientiﬁc image with the appearances to us is thus subsumed under the general problem of skepticism, with those appearances located in the individual mind. The problem initially faced in the sciences was thus transposed into one pertaining to mind and matter. The way this problem is ‘‘solved’’ involves large concessions to skepticism with respect to perception and judgment. Descartes ﬁnds a guarantee for our adequate representation with respect to primary qualities but the rest are merely subjective dressing. The gain? This certainly freed the emerging new physics, but it would leave the emerging modern philosophy holding the bag. The cost? It isn’t just the secondary qualities that end up solely in the mind, on Descartes’s construal. All that is known directly turns out to be precisely what is in the mind. And what is there, separate from the putatively known material entities, can at best claim to be a mirroring—with so far no guarantee to be veridical—of whatever has those primary qualities. Neither the primary qualities nor their bearers are directly accessible. A proof of God’s existence and of His main attributes is needed to bridge the gap. The problem of reconciling the appearances with the new sciences’ ascetic world-picture, in which material has only primary qualities, has been putatively solved, the task no longer weighs heavily on the shoulders of natural philosophy. But this came at the cost of introducing the much more difﬁcult—in fact, in the end, unsolvable—‘‘Problem of the External

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Three putative completeness criteria To understand the sciences, to achieve a synoptic vision in which the manifest and scientiﬁc images both receive their due, we must take the opposite tack from all of modern philosophy: ignore Descartes’s transposition of the problem from nature/science to mind/philosophy, and face squarely the accountability of the physical sciences to the public appearances. ‘‘Reality’’ and ‘‘Appearance’’ are philosophically loaded terms. In our context they are to be understood quite prosaically. The theoretically postulated reality of science may be a world consisting of atoms, while the appearances that the theories are meant to save are the contents of measurement outcomes. Recall that it is too limited to think of these outcomes as simply numbers or sentences! The yellow color of the ﬂames, the spatial conﬁguration on a monitor display or in a projection made by a ‘drawing machine’ or camera obscura, the smell of a gas sample, can all count as outcomes to be reported. The smelly, colorful, noisy things around us, such as apples and horses, sunsets and storms, are all to be accommodated. ‘‘Appearance’’ does not refer here to subjectively experienced impressions. All those colorful, noisy things are public, and so are the appearances: the dictate of repeatability ensures that scientiﬁcally admissible experimental results are public. Neither does ‘‘reality’’ refer to Kantian things-inthemselves or to a Cartesian external world: all those smelly, colorful, noisy things—and the colorful, noisy, smelly measurement outcomes—are most certainly real. The theoretically postulated entities of a scientiﬁc theory may be real as well; but in our context of discussion their theoretical status is to be kept in mind. What has science traditionally been held accountable for? If science means to provide a representation of nature, criteria for its success must be related to that task, and must appear concretely in scientiﬁc theory

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choice and evaluation. Such criteria have often been explicitly formulated. Often enough they were not just debated among philosophers, but loomed centrally in famous episodes in the sciences themselves. Most striking, though, is how time and again science has refused submission to ideals imposed from outside, or even from its own past. Scientiﬁc progress may and does involve rejection of previously proclaimed criteria. We shall examine some here, related to necessity, determinism, and causality. The historical pattern While ﬂaunting previously upheld criteria is remarkable, so is the typical aftermath of such ﬂauntings: sustained reactionary philosophical efforts at restoration. The pattern seems to be this. A certain criterion of completeness is held up as an ideal for science. Perhaps it is even said to be satisﬁed already, details apart; and the scientiﬁcally minded glory in its rule. Embattled by new empirical ﬁndings, scientists violate and reject the criterion, and gain a startling success by their own (new) lights. The success is hailed as a triumph over a now discredited philosophical ideal. But then the reaction sets in as well. There is a sense that in ignoring past demands, science has betrayed its trust—now it ‘‘only describes and does not explain’’, now it ‘‘only puts in mathematical form, but leaves nature unintelligible’’. These were the Aristotelian complaints about Cartesian mechanics and cosmology, the Cartesians’ complaints about Newton’s physics as mere mathematization, and Huygens’s complaint that Newton’s theory was not ‘mechanical’, that is, did not explain gravitational phenomena.8 Nearer our own time there were the complaints about Einstein’s discarding of the ether which was argued to leave optical phenomena inexplicable, and lately the Bohmians’ complaints about the Copenhagen school in quantum theory.9 When empirical successes are scored at the cost of violating respected and revered criteria of theoretical success, many philosophers and reﬂective scientists strive mightily to reinstate the rejected criteria, or show them to be ‘essentially’ or ‘really’ satisﬁable after all. Necessity, determinism, causality As ﬁrst example consider the Aristotelian ideal that science must explain how things happen, by demonstrating that they must happen in the way they do. That

278  :    ideal requires that regularities in the phenomena derive from universal necessary principles. Galileo, Gassendi, Boyle, Descartes, and Newton consciously and explicitly refuse to take on this Aristotelian task for science, or to accept it as criterion for scientiﬁc success. Indeed, they claim that the modern era’s scientiﬁc success derives largely from their rejection of that tradition. But already in other passages at the hands of these very same writers, we see the ideal, and that criterion, advocated in an even stronger form! There is talk of laws of nature, not all that clearly distinguished from those lambasted and ridiculed constraints of Aristotelian physics. Even more extreme sentiments take hold: the regularities must derive from not just natural but logical necessity.10 This sentiment is sometimes encountered still, not so much among philosophers but in physicists’ dreams of a ﬁnal theory so logically airtight as to admit of no conceivable alternative, one that would be grasped as true when understood at all.11 More salient in the development of modern natural science, however, is the turn to mechanism as a guiding philosophy, thoroughly entangled with a conviction of determinism in nature. Descartes and Boyle were main advocates of the view of nature as run entirely by means of mechanical action by contact, and their advocacy was nothing short of evangelistic. Thus Boyle describes the world as a ‘‘self-moving engine’’, a ‘‘vast machine’’ running ‘‘by the meer contrivance of brute matter managed by certain laws of local motion’’.12 When that vision was thoroughly diluted—or more accurately, left behind—by Newton’s introduction of gravitational action at a distance, it was the insistence on determinism that remained. Thus, shortly before the revolutions that would shake physics, Maxwell wrote, with an echo of the mechanical philosophy’s inspiration still: When any phenomenon can be described as an example of some general principle which is applicable to other phenomena, that phenomenon is said to be explained . . . . On the other hand, when a physical phenomenon can be completely described as a change in the conﬁguration and motion of a material system, the dynamical explanation of that phenomenon is said to be complete. (Maxwell 1890: vol. ii: 418)

The conviction that a scientiﬁc account is complete only if it is deterministic was thereafter strongly supported within the Kantian tradition. For as Kant

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saw it, the very coherence of experience requires that it takes a form of experiencing ourselves as living in a spatio-temporally deﬁnite causal order. The context in which physics was changing around 1900 included thus a strong conviction, inherited from classical physics and modern philosophy, that all phenomena in nature derive from an underlying deterministic physics. Determinism had become a criterion of completeness: any apparent gap in determinism so far is ﬁlled by statistical laws, but the statistical probabilities can only be a measure of ignorance. Poincar´e begins one chapter in his Science and Hypothesis of 1905 with ‘‘No doubt the reader will be astonished to ﬁnd reﬂections on the calculus of probabilities in such a volume as this. What has that calculus to do with physical science?’’ But the question of the possibility of indeterminism was already salient and in the air before it was related in any way to the quantum, due to different views of statistical thermodynamics (cf. St¨oltzner 1999).13 Max Planck discusses that too in his famous 1908 Leiden lecture. There he praised Boltzmann for ‘‘the emancipation of the concept of entropy from the human art of experimentation’’ (p. 14), but notes that the price seemed to be to relegate the second law of thermodynamics to a merely statistical regularity that admits exceptions. Boltzmann has drawn therefrom the conclusion that such strange events contradicting the second law of thermodynamics could well occur in nature, and he accordingly left some room for them in his physical world view. To my mind, this is, however, a matter in which one does not have to comply with him. For, a nature in which such events happen . . . would no longer be our nature. . . . (Ibid.: 15)

This is a strong profession of faith in determinism, but the mere fact that it seemed appropriate to express this shows how the question could not be ignored. Planck had an explicit antagonist in Franz Seraﬁn Exner, who defended an empiricism along Mach’s lines, rejecting as meaningless any speculation as to whether there exist some unobservable deterministic laws behind the phenomena (cf. St¨oltzner 2002). So Exner emphasized that if some domain appears to be subject to deterministic laws, these could be the macroscopic limit of indeterministic basic laws governing nature at a microscopic level. This theme was later taken up by Hans Reichenbach, who coupled it with his view of how an indeterministic world could still be ‘lawlike’ through his principle of the common cause.

280  :    The challenge to determinism was the ﬁrst, foremost, and most visible philosophical confrontation to arrive for quantum mechanics. The probabilistic resources of classical statistical mechanics had been newly adapted in such a way that, as it seemed then, no grounding in an underlying deterministic mechanics was possible. A vocal part of the physics community was averse to seeking out the logical possibilities here—some rejected explicitly the task of ﬁnding or displaying such hidden mechanisms.14 Nature is indeterministic, or at least it can be or may be—and if that is so, determinism is a mistaken completeness criterion for theory. Now Reichenbach, who did much to provide a rationale for this rejection of determinism, introduced an apparently weaker but still substantive new completeness criterion: the Common Cause Principle.15 This third principle is satisﬁed by the causal models of general use in the social sciences, and for many purposes in the natural sciences as well. They are models in which all pervasive correlations derive from common causes (in a technical, probabilistically deﬁnable sense). But the demonstration in the 1960s and later that quantum mechanics violates Bell’s inequalities shows that the new physics was riding rough-shod even over this third criterion.16,17 However that may be, I’ll now turn to a fourth completeness criterion that seems compatible with these rejections, and appears to be quite generally accepted at least among philosophers and by the general public. But it too was one clearly, emphatically, and explicitly rejected by the Copenhagen physicists.18

Appearance vs. Reality: A deeper Criterion As we saw, physics in the modern era depicts nature as being quite different from how it appears. The theoretically postulated reality is different from the appearances. The disparity became truly salient when Galileo and Gassendi embraced the atomism revived in the Renaissance. Those atoms have only primary properties such as shape, volume, and number; the appearances are colorful, noisy, smelly, and tasty. The disparity is certainly not diminished by Descartes’s further restriction of the real attributes of matter to extension in space and time. Newton, rejecting this ontological asceticism, introduced forces and mass, but these seemed occult additions

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to many of his contemporaries; they certainly did not return closer to the familiar world of experientia. But at the same time, as we saw illustrated above, Galileo accepted a commensurate criterion of completeness, that physics must explain how those appearances are produced in reality. This amounts to a stringent demand: that this noisy, colorful, smelly but tasty world of appearance be fully explained in terms of the attributes that science explicitly counts among its signiﬁcant parameters. Science is understood to be incomplete until and unless it meets that demand: I call this the Appearance from Reality Criterion. Does this Criterion govern and guide the scientiﬁc enterprise as a whole? Examples abound to make us immediately sympathetic to that idea. We credit science with adequate and satisfactory explanations of how many familiar phenomena are produced: how ash is produced when we burn a cigarette or some logs, how methane is naturally produced in a swamp. In the example of how a ﬂame is turned yellow when a sodium sample is inserted, an aspect of visual appearance is explained. But we must look carefully into just what is demanded, how far it can be pushed, and what resistances it may have encountered since those heady early days. What the Appearance from Reality Criterion requires In the examples I gave, scientiﬁc representation of nature is shown to include the appearances in a very speciﬁc, particular way. Their ‘derivation’ does not just amount to showing that they ﬁt into the theoretical representation. That would be a minimal requirement; just the one that Cardinal Bellarmini suggested to Galileo as solely relevant to the Copernican theory. In contrast, the ‘derivation’ must be a demonstration that, and how, these appearances are produced as a proper part of the depicted reality. This stringency in the demand, for such a derivation/explanation, has remained a continuing theme in scientiﬁc realist writing. Jared Leplin writes: A theory is not simply an empirical law or generalization to the effect that certain observable phenomena occur, but an explanation of their occurrence that provides some mechanism to produce them, or some deeper principles to which their production is reducible. (Leplin 1997: 15)19

282  :    The telling phrase is ‘‘provides some mechanism to produce them’’. The appearances are to be explained as produced in the world depicted by fundamental physics. Just what is this requirement? Below we will take up a paradigm example, in Copernicus’s explanation of the astronomical appearances. The mechanism to produce them is described completely in terms of kinematics and geometric optics. The appearances there—like all the examples I listed above—are public, intersubjective, not belonging peculiarly to individual experience but displayed in publicly accessible form. Recall the familiar example of a yellow color in the ﬂame. What is produced, when a sodium sample is inserted in a ﬂame, is light of a wavelength, measurable and recordable mechanically as well as humanly visible, in the ‘yellow’ part of the color space.20 The requirement of production does not preclude indeterministic or stochastic accounts.21 Imagine that the explanation of how a cigarette burns by recourse to statistical thermodynamics were qualiﬁed by the hypothesis that there are random small ﬂuctuations which modify the underlying mechanics. That would still leave us with a derivation (in the appropriate sense) of the burning process.22 A means of production subject to small random perturbations is not an inexplicable miracle, and a stochastic dynamics is not a mere ‘‘black-box’’ predictor; it provides a derivation of the phenomenon through the details of action and interaction. The Appearance from Reality Criterion applies tellingly, however, whenever there is a felt gap in the story: the appearances must clearly derive from what is really going on. The gap is not ﬁlled if one simply adds a postulate to the effect that the Appearances are produced in some speciﬁc way, without displaying that way.23 For it is easy enough, but entirely uninformative, to wave a hand at some supposed relation between the theorizing and measuring agent and those aspects of nature that are measured and represented, and then to claim that this accounts for how the Appearances differ from what science ostensibly describes. If left as a mere claim that does not sufﬁce, and certainly does not satisfy the Criterion. A look back to the three earlier completeness criteria show how they all hinge on involving modality into our understanding of science. According

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to Aristotle science must show why the phenomena and their regularities must be the way they are, deriving from principles that are universal and necessary. The seventeenth-century conception of laws of nature, only just emerging from their theological gestation, delineate what is necessity in nature. The Appearance from Reality Criterion is similar in this respect: it too is a demand for explanation which is satisﬁable only by connections deeper than brute or factual regularity. Therefore, when it is claimed that to be complete physical science needs to derive the appearances from that reality, the term ‘‘derive’’ cannot here just mean ‘‘deduce’’ or ‘‘predict’’. Required is a connection of the order of explanation through necessity and/or causal mechanisms to be displayed, which produce the appearances. When is this demand not met? It is not met if science should simply issue successful predictions of measurement outcomes. Prediction does not sufﬁce by itself even if the prediction is by means of systematic rules of calculation, from the state of nature theoretically described. For calculational and predictive success does not ipso facto imply an explanation of why and how the appearances are produced.

Phenomena versus appearances That empirical science must ‘‘save the phenomena’’ is an ancient creed. The criterion that I call ‘‘Appearance from Reality’’ is however a stronger dictate. The phrase ‘‘to save the phenomena’’ is often rendered colloquially as ‘‘to save the appearances’’ and ‘‘appearances’’ is often used synonymous with ‘‘phenomena’’. As emphasized from the outset, I use these terms so as to mark a crucial distinction. Phenomena are observable entities (objects, events, processes, . . .) of any sort, appearances are the contents of measurement outcomes.24 Both are terms with a past. On philosophical lips they are often loaded, unfortunately, with connotations linking them to mind and thought which are not in any way relevant to my usage. The verbal distinction can for example be found in Kant’s Critique of Pure Reason. Still less must phenomenon and appearance be held to be identical. For truth or illusory appearance does not reside in the object, in so far as it is intuited, but in the judgement upon the object, in so far as it is thought.25

284  :    Both the way in which each of these notions is understood here, and the way the distinction is marked, have to do with perception and perceptual illusions or mere impressions. Nothing in my usage of these two terms refers to these factors, unless very indirectly. How an observable object or process (phenomenon) appears in the outcomes of the measurements is itself an objective fact, a public, intersubjectively accessible fact. But there is a similarity nevertheless to Kant’s insistence on—to coin a phrase—the greater objectivity of the phenomenon. That is, we do have to insist on the distinction: the appearance is determined jointly by the measurement set-up (involving both apparatus and the system to which it is applied), the experimental practice, and the theoretical conceptual framework in which the target and measurement procedure are classiﬁed, characterized, and understood. While usage is diverse, I do not think that my usage is egregious with respect to the historical scientiﬁc literature. For example Hertz, commenting on his research on the effect of ultra-violet light, says in the Introduction to his Electrical Waves ‘‘Inasmuch as a certain acquaintance with the phenomenon is required in investigating the oscillations, I have reprinted the communication relating to it [here]’’. The reprinted paper to which he refers begins as follows (and there we can see what the phenomenon in question is): In a series of experiments on the effects of resonance between very rapid electric oscillations . . . two electric sparks were produced by the same discharge of an induction coil, and therefore simultaneously. One of these, the spark A, was the discharge-spark of the induction-coil, and served to excite the primary oscillation. The second, spark B, belonged to the induced or secondary oscillation. The latter was not very luminous . . . I occasionally enclosed the spark B in a dark case so as more easily to make the observations; and in so doing I observed that the maximum spark-length became decidedly smaller inside the case than it was before. On removing in succession the various parts of the case, it was seen that the only portion of it which exercised this prejudicial effect was that which screened the spark B from the spark A. [. . .] A phenomenon so remarkable called for closer investigation. (Hertz 1962: 63)

‘‘Appearance’’ I reserve strictly for the contents of (possible) measurement outcomes. Phenomena are observable, but their appearance, that is to say, what they look like in given measurement or observation set-ups, is to be

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distinguished from them as much as any person’s appearance is to be distinguished from that person.26 As an especially good example for our present concerns I will take the distinction between planetary orbits and their appearances to the terrestrial observer.27 The Renaissance astronomers’ data detailed what they saw night after night, hence the appearances, the contents of their measurement outcomes. But Ptolemaic and Copernican astronomy, accountable to those same appearances, described the phenomena differently. Am I right to say that the planetary orbits are observable processes, while what the astronomers cited were their appearances? I do not think there is a difference in principle between this case and the way in which we must distinguish what Mount Everest is from how Mount Everest looks from the North, the South, and various other compass directions. That mountain is undoubtedly an observable object—only weird philosophical jargon could decide differently—though of course we infer its shape from what its appearances are in telemetry data and photographs taken from various positions. That data and inferences are needed to arrive at an accurate description of the thing does not mean that it is not something observable, and only a strange departure from common sense would lead us to the idea that e.g. mountains are only theoretically postulated entities.28 The planetary orbits differ from the mountain only in that measurements over time are needed, while in the case of a mountain a number of simultaneous measurements would do as well. There were two developments in techniques of representation before Galileo that we can see as feeding into the kinematic representation developed in his century. The ﬁrst was that of linear one-point perspective in painting, and the second Copernicus’s and Tycho’s mastery of transforming geometric models in astronomy so as to shift the center taken as ‘at rest’. Both concentrated on how a description of the visual appearance from particular vantage points can be derived from a reality admitting of many different vantage points. Both grew from the subject of Perspectiva, a m´elange of geometry, optics, and practical drafting techniques, and both were steps on the way to projective and descriptive geometry. But the sorts of representation that perspectival drawing and geometric technique provided were more than superﬁcially different. For, if I may put this anachronistically, the second dealt not with perspective but with transformations of frames of reference in Euclidean

286  :    space and its consonant kinematics. This technique was mastered in practice by the time of Copernicus, though formalized only by the end of the seventeenth century. In contrast, the study and perfection of perspectival drawing gave rise to the very different subject of projective geometry. That too saw its ﬁrst rigorous development in the seventeenth century, but was then neglected, until coming into its own (with a uniﬁed treatment of Euclidean and non-Euclidean geometries) in the nineteenth century. This is not an incidental historical point. Both forms of representation tend to be called ‘‘perspectival’’ and both tend to be thought of as depicting the appearances. But they accomplish very different tasks. I honor the difference between them with distinct terminologies. The geometric representation of e.g. planets and planetary motion depicts the phenomena. The content of a perspectival drawing of the same events is their appearance, equivalently, the drawing depicts how they appear from certain vantage point.29 At the risk of annoying by repetition: the phenomena, in the sense of the observable parts of the world, whether objects, events, or processes, the sciences must save (in the ancient phrase). These admit also of objective and indeed purely theoretical description, which does not link their reality to contexts of observation or to acts of measurement. What the latter, the measurements, provide are how those observable parts of the world appear to us in the corresponding measurement set-ups. That, the content of a measurement outcome, is an appearance, and the sciences are accountable to that as well. How Copernicus saved the appearances In the ﬁrst book I set forth the entire distribution of the spheres together with the motions which I attribute to the earth, so that this book contains, as it were, the general structure of the universe. Then in the remaining books I correlate the motions of the other planets and of all the spheres with the movement of the earth so that I may thereby determine to what extent the motions and appearances of the other planets and spheres can be saved if they are correlated with the earth’s motions. (Copernicus, De Revolutionibus, Preface.)

Here we have the paradigm example of this distinction, playing a crucial role at the very beginning of the modern era.

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Copernicus described the planetary motions. These motions are observable; they can be observed and also be registered on ﬁlm, whether from Earth or from a satellite or from outer space. Copernicus depicted them clearly in a geometric model. They certainly cannot be identiﬁed with the appearance recorded by any such speciﬁc means as views through a telescope, successive photos over a period of many days, ﬁlm, or video recording. The recording is always from a speciﬁc vantage point, and that point is arbitrary, it has no privileged status either in nature or in Copernicus’s model. What is the content of such a recording then? It is the appearance of the planetary motions, for a photo, ﬁlm, painting, or drawing displays how the recorded object, event, or process ‘looks’ from the chosen vantage point. Speciﬁcally, the planetary ‘retrograde’ motions are, according to Copernicus, (mere) appearances. Mercury’s retrograde motion is a good example of an appearance. Literally speaking this is something that does not happen—it does not exist! Mercury never turns back to reverse its orbital direction. But that is how it appears to the observer or the ﬁlm camera, viewing from Earth. Mercury’s motion is an observable phenomenon, but Mercury’s retrograde motion is an appearance. The Ptolemaic system, concretely depicted by an armillary on a table top, represents the motions of the stars and planets in the frame of reference of the earth. Copernicus’s system of the world represents the motions of the stars and planets in the frame of reference of ﬁxed stars, with the ‘mean sun’ as center. But because Copernicus devised his system by a sort of ‘transcription’ of Ptolemy’s, he can point out that observations from the earth in his system will deliver the same data.30 Consider what is utilized in this ‘pointing out’. First of all, there is what the system represents directly, the postulated ‘‘general structure of the universe’’, as he wrote in the Preface. Secondly, there is the geometric optics, based on the postulate that unobstructed light travels in straight lines with inﬁnite speed. Thirdly—and here we make contact with the visual arts—the appearances to be saved are identiﬁed with the projections through a point on the earth of the celestial motions by those straight light-lines. The appearances, thus conceived, change with time. There were no motion pictures, but of course one could construct a series of stills, and one could furthermore combine these into a single picture of a motion over

288  :    time. This is the birth of modern kinematics together with kinematic, as opposed to static geometric, representation. The most striking illustration to Copernicus’s contemporaries as to us, is the new explanation of retrograde motion of the planets. The following composite diagram shows both the postulated kinematics and the light-lines that project the motions onto an earthbound window, so as to derive the appearances from the real motions.

Figure 12.1. Copernicus’s Model of Retrograde Motion

The above diagram (from Cohen 1960: 39, ﬁgure 10) graphically presents the Copernican explanation of the apparent retrograde motion of an inferior planet such as Mercury or Venus, by depicting how its motion would look from a slower moving Earth also orbiting the sun. Copernicus’s model represents the observable phenomena, that is, certain processes in space and time. What the Copernican does in order to credential his representation is, in effect, to explain by means of geometric optics and projective geometry how the visual appearances (content of outcomes of measurements made by astronomers) are produced from reality. Copernicus can demonstrate that his theory ‘saves’ certain phenomena by showing how the visual appearances derive (via kinematics and optics) from what he postulates concerning them.

Three-faceted representation There was another discipline in Copernicus’s time, located between astronomy and physics. It carried the melodious name of Theorica.31 The aim of

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this enterprise was to construct physical, mechanical models of the heavens that would explain the celestial phenomena. The crystalline spheres are of course well known from the long uneasy history of attempts to reconcile ancient and medieval physics and astronomy. It was very difﬁcult to construct a physically possible cosmos along these lines that would save the phenomena described by the astronomers, but arguably not impossible. The attempt ended effectively with Tycho Brahe’s observations on comets apparently moving unobstructed through the spheres. Although the theories offered in Theorica along these lines failed, they were worthy forerunners of later physical cosmologies like Newton’s System of the World with its universal gravitational force. The models of Theorica, as well as Newton’s, go well beyond Ptolemy’s or Copernicus’s description of the observable phenomena. They depict a postulated ‘‘Reality’’, a nature structured in unobservable ways and populated by largely unobservable entities of which the phenomena are the observable parts. The physical sciences give us representations of nature, and scientiﬁc representation is in general three-faceted. From a purely foundational point of view, the theoretical models that depict the ‘underlying reality’ are the main thing. But some elements or substructures of those models are meant to represent the observable phenomena—the empirical substructures. Finally, it is a requirement for the theory to at least predict, and if possible to ‘‘derive’’ the appearances, that is to say, the contents of measurement outcomes. This division corresponds to three ostensibly different domains: [1] Theoretically postulated reality — Micro structure, forces, ﬁelds, global space-time structures [2] The observable phenomena — Macro objects, motions, tangible and visible bodies, . . . [3] The appearances — Measurement outcomes, ‘‘how things look’’ in observational context The phenomena can be measured and observed in many different ways. How they appear in the measurement outcomes will vary from one way to another—and from one occasion to another—speciﬁcally because those outcomes provide perspectives on the phenomena. So to say that the theory must save the phenomena is not the same as saying that the theory must be in accord with the experimental and observational results. There is certainly a close connection! If the measurement is well designed for its purpose,

290  :    what is to be found at its conclusion—such as pointer positions on gauges, print-outs, numbers, computer monitor displays, or what have you—will be especially telling when different theories try to achieve their designated ends. But we need to keep remembering: the measurement outcome shows not how the phenomena are but how they look. In the modern era, the era of what we now call classical physics, each level has a certain completeness. When frames of reference come into their own, we have eventually a three-level representation: there is the world [1] as described in co-ordinate independent terms, then the world [2] as described in a given frame of reference (co-ordinatization), and ﬁnally [3] as it looks from a given vantage point with speciﬁc orientation. The ﬁrst form of representation admits of many of the second sort, and the second of many of the third sort. The higher level is uniquely determined, in the classical world picture, by the collection of those at the next level plus the transformations that connect them.32 There is the important difference, however, that a single representation on the ﬁrst or second level contains everything in a way that the third most deﬁnitely cannot. The depiction on that third level, that is, of the visual appearances—the true measurement outcomes—is always limited, just as the observer cannot see what is behind his back, so any measurement set-up at all displays only what is inside its range. With the end of the modern period in physics, the advent of relativity and quantum theory, the conception of a physical theory and of the connections between postulated reality, phenomena, and appearances underwent radical change. It is at this point that more drastic challenges appear for the Appearance from Reality Criterion.

13 Rejecting the Appearance from Reality Criterion What should be our take on the Appearance from Reality Criterion today? We need ﬁrst of all to explore the conceptual possibilities, the ways in which a science could feasibly come to violate the Criterion, and then ask how these relate to practice in recent science. Although my main concern is with the natural sciences, it is instructive to see how the issue is broached in analytic philosophy of mind and its take on cognitive science. The question of how the psychological phenomena relate to the physical processes in brain, body, or body plus environment—our new version of the mind–body problem—has seen a series of answers more or less culminating in the supervenience thesis. This move points us readily to logical and philosophical moves available when outright reduction seems out of reach. In fact, that thesis echoes a move familiar from another historical episode: Leibniz’s reconciliation of contingency with the principle of sufﬁcient reason, and both episodes can serve us as guide to other such perplexities. For recent scientiﬁc practice I shall argue that the Copenhagen development of quantum theory exempliﬁes a clear rejection of the Criterion. The famous Measurement Problem in the philosophy of quantum mechanics is not a problem from an empiricist point of view.1 What it marks, I shall argue, is the methodological rejection of the Appearance from Reality Criterion in this new science.2 The rejection may not be unique in the history of science, but is brought home to us inescapably by the advent of the new quantum theory. Even if that theory is superseded (or if fundamental physics develops in a accordance with a new interpretation under which the Criterion can be satisﬁed) our view of science must be forever modiﬁed in the light of this historical episode. The Appearance from Reality Criterion was

292  :    rejected in practice, in an episode that can only be acknowledged as one of genuine scientiﬁc advance, in the face of theoretical and experimental problems. In this respect it is in the same boat as determinism: even if in the future a completely adequate deterministic physics emerges, the history of twentieth physics will have shown that determinism is not a completeness requirement for science as such.3 Since these criteria were violated in actual practice it is possible—and in fact incumbent on us—to conceive of science as not so constrained.

The supervenience of mind challenge Cognition and philosophy of mind are far from my proper concern here. But they offer a remarkable illustration of how the Appearance from Reality Criterion can be challenged. Not only that: the challenge comes in this case within a self-proclaimed naturalist, physicalist, ‘scientiﬁc’ approach to the so-called mind–body problem in philosophy. Are psychology or cognitive science in principle autonomous or must they be reducible to fundamental physics? That is hardly a practical question for a working scientist. There is no such reduction even for current materials science or chemistry, let alone physiology—not to speak of any part of psychology. But we can ask about reducibility in principle, and that has been a central question in the story of physicalism in twentieth-century analytic philosophy of mind. In the 1950s, U.T. Place offered the hypothesis that certain events and processes traditionally classiﬁed as mental (for example, sensation) are identical with events and processes in the brain (Place 1956). He called this the materialist hypothesis. As he pointed out, it is in principle falsiﬁable, namely if the described mental events and processes have a greater complexity than brain events and processes.4 Not only this position, but every claim concerning reduction of the psychological to the physical, bit the dust in the course of the ensuing philosophical debates. The eventual claim, introduced to save materialism or physicalism in principle, became the much weaker one that, though irreducible, mental phenomena supervene on physical reality.5 What this means, roughly presented, is that the actual psychological phenomena could not be different without a difference in the physical state

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(of the organism, or possibly the organism plus its natural environment, or possibly of the entire universe). Leaving aside the more metaphysical notions that are often involved in the idea of supervenience, we can draw one clear consequence from it that will reinforce this reading (though in slightly more technical terms). Suppose that P and M are two languages, one truly physicalist and the other mentalist—whatever that means, whatever might sufﬁce for the one to describe everything in terms aptly used in physics and for the other to describe psychological phenomena (as well) in terms of psychology or cognitive science. What is implied here by the thesis that the psychological supervenes on the physical? (a) To say that we have no possible reduction but do have supervenience implies that M cannot be translated into P sentence by sentence, nor paragraph by paragraph, nor even that the deﬁnable sets of sentences of M are translatable into deﬁnable sets of sentences of P. That is the ‘‘no reduction’’ claim. I will explain the restriction to what is deﬁnable below. Taking that for granted for now, what does the claim to supervenience entail? To understand that, let’s think of all the possible situations that can be described (however partially) in both languages. We can depict them this way if we assume no special relation between the two languages:

Figure 13.1. Failure of Supervenience

294  :    Here the vertical bars depict the situations that satisfy a given maximally complete description P(1), P(2), . . . in the physicalist language P. The slanted bars depict those that satisfy given maximally complete descriptions M(1), M(2), . . . available in the language M. The letters ‘‘X’’ and ‘‘Y’’ name two speciﬁc situations: Both X and Y satisfy P(1), while X satisﬁes M(1) but Y satisﬁes M(2). Now you can see immediately that if there are such situations, then M does not supervene on P! For suppose we are in situation X. Then if we would say ‘‘if the mental phenomena were different then something physical would also be different’’, the possibility of situation Y would contradict our claim. So if we want to redraw the picture, so as to present the case in which the mental is supervenient on the physical, then the lines that divide complete descriptions in M would have to coincide with certain lines that divide complete descriptions in P. In other words, roughly put: a complete description in M has to be in effect a disjunction of complete descriptions in P. Note well: if we rule out reducibility, then those disjunctions cannot be ﬁnite or recursively speciﬁable etc. That is why I wrote ‘‘roughly put’’—I’ve stretched the word ‘‘disjunction’’ beyond its normal use. The point is clear though: at the level of indeﬁnable sets of possibilities, a possibility in principle describable in M can be equivalent to a family of possibilities in principle describable in P. But in principle describable does not, in that case, refer to what is humanly or mechanically deﬁnable or computable or recursively speciﬁable, and so does not imply reducibility. We can sum up this conclusion, taking a little liberty with the notion of disjunction, in this way: (b) The syntactically complete descriptions of everything (to the extent possible in M) correspond disjunctively to sets of sentences in P, as far as truth conditions are concerned. Putting the ﬁrst and second point together shows perspicuously that the correspondence just claimed must be between sets of sentences that are not deﬁnable.6 Obviously no deduction could mirror this correspondence. It would be beside the point to stop here to inquire into the plausibility of this claim. What is important for us, once again, is a meta-issue. On this view, that the mental or psychological only supervenes on the physical, what becomes of the science of cognitive psychology? By classifying psychological

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phenomena—the subject of the science of cognitive psychology—as irreducible, this position implies a pertinent autonomy for that science, its independence from fundamental physics. For the ‘‘supervenience without reduction’’ claim explicitly entails that no mechanism can be displayed even in principle for the production of the (mental) Appearances from the (supposedly physical) Reality. Those appearances depend on the physical in the minimal sense that the appearances could not be otherwise without the physical state being different. But they are not derivable in the relevant sense. The claim of supervenience without reducibility implies that science is not, will not, and cannot be complete in the sense of deriving the psychological phenomena from the postulated physical reality. On the basis that ought implies can we must then also conclude that science as a whole is not required to be complete in that sense. We have here a conception of the sciences that speciﬁcally exempts them from satisfying the Appearance from Reality Criterion. For this conception envisages the development of a cognitive science and psychology in the absence of any physicalist account of how the pertinent phenomena are ‘derived’ or ‘produced’ in physical terms, while maintaining that the physical description is in principle complete. Just to prevent any misunderstanding: in displaying this conception I am not objecting here to the claim that the psychological phenomena supervene on the physical. Nor am I even maintaining that this claim fails to adequately explain why those phenomena are or even must be what they are.7 The point is rather that in the philosophy of mind it became popular, though perhaps without explicitly noticing this, to reject the Appearance from Reality Criterion as a completeness criterion for science as a whole.8 Once we see this logical leeway, we can ask how the supervenience thesis could be maintained elsewhere. The possibility is then no longer outr´e, and can be taken equally seriously, for example, for the various ‘levels’ of inorganic nature. It may be objected ﬁnally that the supervenience claim is, by its very character, neither refutable nor conﬁrmable by empirical evidence, and that therefore it is an empty addition to scientiﬁc psychology or cognitive science. That may be so. But it is actually possible to have grounds to reject a supervenience claim, given (besides any empirical evidence) certain other completeness claims that could be made for a scientiﬁc theory. So the claim of supervenience cannot be part of every sort of interpretative view of science. This we will see concretely illustrated in the case of quantum mechanics.

296  :   

The Great Leibnizian Escape move The ‘‘supervenience without reducibility’’ claim asserts a connection in nature which cannot be displayed by means of a theoretical deduction. Deduction is after all an operation to be carried out with certain resources—linguistic, logical, mathematical—and these resources have their limits. While these limits are the proper domain of meta-mathematics, developed only in the last century, they were vaguely perceived already in the seventeenth century. Here Leibniz stands out as aware of possible limits to science. Like Descartes he seems to have initially harbored the dream of a complete theory of everything whose principles can be known a priori. But after a certain point his vision changes. Then he begins to distinguish between necessary and contingent propositions. As he characterizes them, the former can be proved in a ﬁnite number of steps by reducing them through analysis of the involved concepts to identical propositions or primary principles, while the analysis of contingent propositions goes on ad inﬁnitum. This is why we cannot know the truth of contingent propositions a priori. God alone can know this, not because he can complete the required inﬁnite analysis, but rather because he intuits the whole analysis with one glance. The criterion for distinguishing necessary from contingent truths emerges from the following feature, which only those who have in them a tincture of mathematics will easily understand: in the case of necessary truths an identical equation will be reached by carrying the analysis sufﬁciently far, which amounts to demonstrating the truth with geometrical rigor; whereas in the case of contingent truths the analysis proceeds to inﬁnity, with reasons given for reasons, in such a way that there is never a complete demonstration although the underlying reason for the truth is always there, perfectly understood only by God, who, with one stroke of thought, goes through the whole inﬁnite series.9

Ignoring the overreaching rationalist ambition, we can transpose this view to the relationship—as Leibniz envisages it—between nature as described in fundamental physics and the appearances (whether physical measurement outcomes or psychological phenomena). The assertion here is that given the information that the physical state is thus or so, there is a strict entailment of what the appearances must be. However, as we know from meta-mathematics, not all entailments are capturable by deﬁnable

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consequence operations. So taken, this is precisely the ‘‘supervenience without reduction’’ view, reading deduction for reduction. As I emphasized early on, mere deducibility would also not ipso facto satisfy the Appearance from Reality Criterion. But it is certainly a necessary condition for success by this criterion. Remember after all that we are not discussing criteria for God’s creation, nor for the structure of reality! Our concern is with completeness criteria for the sciences in practice, which evolve within the resources humanly available. So analytic philosophy of mind is in effect offering cognitive science the Great Leibnizian Escape move: there is a logical reduction of the phenomena to the real, but it is not graspable by a ﬁnite mind (read: ‘‘not deﬁnable by ﬁnitary or even recursive means . . .’’). We will now examine the case of quantum mechanics in some detail. The Leibnizian Escape move becomes pertinent only if we are already willing to reject the demand that the Appearance from Reality Criterion poses for science as a whole. But once we are so willing, we must not shy away from it in the natural sciences either. If we are already willing to grant that the appearances may be relatively autonomous, constrained by but not derivable from the postulated theoretical reality, then we can next ask whether, within a given theory, even something like a supervenience thesis is tenable.

The quantum mechanics challenge In quantum mechanics the physical systems are characterized by states that change over time and physical quantities (‘‘observables’’) which are represented by operators on the state space. The theory was developed quite far before there was clarity on what is its vehicle for prediction. The requisite rule was ﬁrst introduced for the special case of scattering by Max Born, and so is still called the Born Rule. We can state it in several ways, but each takes the form of a prediction of outcomes, conditional on the performance of a measurement. In one form it allows calculation of an expectation value, which pertains to a weighted average in a large number of measurements. If observable A is measured on a system in quantum state ψ, the expectation value of the outcome is ψ, Aψ

298  :    In a simpler form, it speciﬁes the probability of any given possible outcome in a single measurement: If observable A is measured on a system in quantum state ψ, and ψr is the eigenstate of A corresponding to its possible value r, then the probability of outcome r is (ψr . ψ)2 The details of calculation, and the simpliﬁcation involved in these formulations, have seen much discussion, but for our purposes we can leave these aside for the most part.10 The dialectic that creates the problem This Rule includes two terms that do not obviously belong to the vocabulary of quantum mechanical—or even physical—description: ‘‘measured’’ and ‘‘outcome’’. As we have discussed at some length, measurement has a physical correlate, and we must assume that this, rather than measurement as such, is what the Rule is about. But then of course the question arises: just what is that physical referent? The puzzle begins in all seriousness when we ask what possible answers there could be within the quantum theory itself. 1. Suppose ﬁrst of all that ‘‘outcome’’ and/or ‘‘measurement’’ do not refer to quantum states and/or their evolution at all.11 Then the theory is certainly incomplete with respect to what happens in nature, no matter how complete within its own domain. 2. Suppose secondly that ‘‘outcome’’ and ‘‘measurement’’ refer respectively to a ﬁnal quantum state of the apparatus, and to the evolution of the quantum state of apparatus + object during the interaction. This supposition allows just three possible cases: • the dynamics is incomplete, or • it is complete and then the Born Rule is either i. superﬂuous or ii. inconsistent with it.12 The reason that it allows only these three is that quantum dynamics is deterministic: the quantum state of an isolated system (whether Apparatus +

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Object, or Apparatus + Object + Environment) evolves deterministically.13 So if the dynamics is complete and correct, then the Born Rule must be superﬂuous if ‘outcome’ and ‘measurement’ are supposed to be in the domain of this dynamics. The Born Rule cannot be deduced from the dynamics, so it is not superﬂuous.14 We are thus left with a classic dilemma between incompleteness and inconsistency; that is (one way to present) the famous Measurement Problem. States, appearances, and phenomena Let us see for a moment how we can describe the situation in the terms of the previous chapter. What matters is that measurement outcomes—in the sense of the measurement outcome contents, rather than the outcomes as observable physical events—are a prime examples of what we are to classify as appearances. The quantum states are then the theoretically described reality. The state of the system, among the possible states postulated by the theory, provides the basis for prediction. What about the middle term, the phenomena? These would be the observable objects on which measurements are made, or any observable events and processes targeted for quantum-mechanical explanation. This includes the instruments involved in measurement, the physical events that are the ﬁnal states of such apparatus (which are meaningful as measurement outcomes), and the entire set-up which is characterized as a measurement.15 The Born Rule is one of conditional prediction. What it predicts is what the appearances will be—with speciﬁed probabilities—under certain conditions. The Copenhagen physicists astonished not only traditional philosophers but also their colleagues by not recognizing, indeed refusing to acknowledge, any need to close the apparent gaps in explanation. For by itself the Born Rule certainly does not give any information about how those appearances are produced. Can we, by looking into quantum theory, ﬁnd an answer to the question of how the measurement outcome comes about? Does this scientiﬁc theory specify, explicitly or implicitly, a process, whether deterministic or stochastic, by which this appearance is produced? This is of course the question whether the Appearance from Reality Criterion is satisﬁed. To answer that, the ﬁrst task is to identify

300  :    measurements and outcomes among the processes and events describable in quantum mechanical terms. Having done that, along the lines displayed more abstractly in the general theory of measurement, we can ask how the Born Rule relates to the dynamical laws of that theory. It is then, precisely, that we run headlong into the (in)famous Measurement Problem of quantum mechanics. Can the measurement outcome be ‘derived’ in the way we have been discussing? What that means is this: can we describe the process in which observable A is measured on a system in quantum state ψ by a suitable apparatus, starting in its ‘ready state’ ϕ and interacting with that system through a certain interval of time, so as to show how the outcome is produced? Even if we amend the last word ‘‘produced’’ to the phrase ‘‘produced with probability . . .’’ there turns out to be a problem, as we’ll see. The reason is, to announce it brieﬂy, that the theoretical description of this interaction, in quantum theoretical terms just does not seem to provide a place for the speciﬁc outcome in question.16

Exploring the case of quantum mechanics In quantum theory we have, it is often said, the most successful theory in the history of science. As far as prediction goes, the riches gained have been beyond the dreams of avarice. . . . But what about the explanation or ‘derivation’ of the predicted appearances? Measurement in quantum mechanics17 Let’s begin by recalling the general format of a physical theory, that we used in discussion of the general theory of measurement: • a family M of observables (physical magnitudes) each with a range of possible values; • a set S of states; • and a stochastic response function Psm for each m in M and s in S, which is a probability measure on the range of m.

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The number Psm (E) is to be interpreted as the probability that a measurement of m will yield a value in E, if performed when the state is s. In quantum mechanics we must ﬁrst of all respect the distinction between pure and mixed states: If s and u are states of a system X, and 0 < c < 1, then there is a state v of S such that for all observables A pertaining to X, and all intervals E, PAv (E) = cPAs (E) + (1-c)PAu (E). We call v a mixture of s and u in proportions (‘weights’) c and 1-c. If there are distinct states of which v is a mixture, then it is a mixed state, and if not then it is pure. The Greek letters ψ, ϕ, . . . are customarily used for pure states. Usually mixtures are introduced, to begin at least, to represent ignorance—and they are suited to doing so even in quantum mechanics. But in that theory that is not their only role, and in general we cannot equate ‘‘X is in a mixture of ψ and ϕ’’ with ‘‘X is really either in ψ or ϕ’’. Secondly, we need to pay attention to how systems can be parts of other systems; for example, if X and Y are systems, so is their composite X + Y. The states of a composite are related to those of its parts, but are not determined by it (‘‘quantum holism’’). To explain that we have to distinguish mixtures from another sort of combination of states: pure state ϕ of X is a superposition of pure states {ϕ(1), . . ., ϕ(N)} of X if and only if for all intervals E and all observables A pertaining to X: if PAϕ(1) (E) = 1 and . . . and PAϕ(N) (E) = 1 then PAϕ (E) = 1. Note that mixtures of pure states are not pure, while superpositions are. And ﬁnally we need to introduce a way in which composite pure states can be constructed: If systems X and Y in states ψ, ϕ, respectively have not interacted, then the state of X + Y is the tensor product ψ ⊗ ϕ But in general, and speciﬁcally if they have been interacting, the state of X + Y is some superposition of states of that form (We speak then

302  :    of ‘‘entanglement’’, ‘‘entangled state’’.) We’ll look at this again when we come to the end-state of a measurement interaction. Now we can describe the standard form of a measurement process in quantum mechanics. The Apparatus to measure observable A starts in its ready state ψ, say, and the Object on which A is being measured starts in its initial state init. At this moment the system Apparatus + Object is in state ψ ⊗ init. Now the two are coupled, and we recall that the criterion for the physical correlate of measurement must be satisﬁed: Criterion for the Physical Correlate of Measurement: PBﬁn (E) = PAinit (E) where ﬁn is the ﬁnal state of the apparatus and B the ‘pointer observable’. The form of interaction, or in other words, the evolution of the system Apparatus + Object during the appropriate interval, must guarantee that this will be so. It must guarantee that, regardless of what the Object’s initial state init was (within the range within which the Apparatus can operate). There is a special case, when the outcome of a measurement is certain, for which we can now introduce some shorthand notation (in the style of Dirac): When PA s ({r}) = 1, then s is called an eigenstate of A, corresponding to eigenvalue r, and is denoted |A, r. To satisfy the Criterion for the physical correlate of measurement, we need at least the following to be the case: If init is an eigenstate of A, |A, r then ﬁn must be a corresponding eigenstate |B, r of B. It is also required, in the standard form of measurement, that an eigenstate of A is not disturbed or changed by measurement of A (although other states can be so changed). So this means that the evolution of Apparatus + Object must be of such a character that |ψ ⊗ |A, r evolves into |B, r ⊗ |A, r, for any eigenvalue r of A And now the peculiarities of quantum theory take over. The fact that evolution of system is, in the absence of external disturbances, unitary guarantees the following:

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If |ϕ is a superposition of {|A, r(1), . . ., |A, r(N)} then |ψ ⊗ |ϕ evolves into a superposition of {|B, r(1) ⊗ |A, r(1), . . ., |B, r(n) ⊗ |A, r(N)} We are almost home; there is a relevant theorem that derives states for parts from states of the whole. At this point one can deduce the character of the Apparatus’s ﬁnal state ﬁn: ﬁn is a mixture of {|B, r(l), . . . , |B, r(N)} This may look very nice! Couldn’t we say that this means that ﬁn really is one of the pure states {|B, r(1), . . ., |B, r(N)}? Couldn’t we add that the measurement outcome is that A was measured to have value r(J) just precisely if ﬁn was |B, r(J)? The adder in the grass appears precisely here. The answer is NO, we cannot say that. The mixed state in which the Apparatus ends up is not identical with any pure state.18 If we knew that an object was really in some pure state, but did not know which, then it would be ﬁne to attribute a mixed state to that object, and use that as a basis for prediction. But the converse is not the case; that a system ends up in a mixed state cannot be equated in general with its really ending up in some pure state. Mixed states are sui generis. And what is more, it is precisely states produced by interactions such as happen in measurement that are demonstrably not ‘‘ignorance cases’’. This point can be richly illustrated with recombination experiments.19 The idea for such experiments was originally suggested by Eugene Wigner.20 In such an experiment, the reconstituted beam could be once more in a coherent state. So a physical measurement interaction need not, in itself, be such that the coherence is destroyed by the separation and recombination. Therefore no information of entanglement was lost, as would happen in a collapse. So now the perplexity can be stated as follows: There is nothing we can see in the quantum mechanical description of the end of the interaction that could be offered as equivalent to the statement ‘‘the outcome of the measurement of A was the Jth eigenvalue of A’’. Yet it would seem that any single measurement would have one such deﬁnite outcome.

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Supervenience? Could the distinction between supervenience and reduction introduce some leeway here? Suppose that the quantum mechanical representation of nature, including dynamics, is complete, and that our statement ‘‘A is measured on X during interval ’’ is true if and only if the pertinent quantum states and their evolution during belong to some set depending on A, X and ; call it S(A, X, ). So far that supposition is just a stipulation, harmless under this supposition. But one could add: we humans are quite good at recognizing whether or not the real situation is thus, but S(A, X, ) is not deﬁnable within the language of quantum mechanics. There are in general many such indescribable, undeﬁnable sets for any theoretical language no matter how rich we make it. Can we tenably add that at the same time the relevant sets are describable perfectly well in the ordinary language used also by laboratory assistants fairly ignorant of the theory? Yes, we can, at least logically speaking, for there is the option that the discourse to which ‘‘A is measured on X during interval ’’ belongs may be precisely supervenient on, but irreducible to, that theoretical language. But in fact, this simple move does not by itself bring us out of the woods. While the logician may explore this loophole in abstract terms, the physicist has a serious obstacle in store. For it certainly seems, both in practice and in theory, that the physical conditions of measurement and the physical correlate of the measurement outcome (the ﬁnal state of the apparatus) are perfectly well formulable in quantum mechanical terms. So the obvious question is whether that physical correlate could be the same even if the measurement outcome contents (the appearance) were different. And that, unfortunately, seems to be obviously so: all the cases in which a measurement of A has outcome value j, for j = 1, 2, 3, . . ., on a particular occasion when none of them were certain, are cases in which the ﬁnal state of the apparatus is the same mixture of its possible pointer states. So the appearances do not supervene on the physical state.21

An empiricist view The view that I will now advocate will not be surprising given the way I stated the perplexity in the preceding paragraph. But it does require

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appreciating an attitude toward theory that has been mainly absent from foundational studies of quantum mechanics. For unqualiﬁed adequacy of the theory, what is required is that the surface models of phenomena ﬁt properly with or into the theoretical models. The surface models will provide probability functions for events that are classiﬁed as outcomes in situations classiﬁed as measurements of given observables. Those probability functions need to be parts of the theoretically speciﬁed Born probabilities for the same situation as theoretically represented in terms of possible states and evolutions.22 The matching required is between two families of probability functions, not between the individual events summarized in the surface model and the states represented in the theoretical model. But is the identiﬁcation of actual events in measurement situations—so classiﬁed in practice—and states in the theoretical models for those situations not presupposed by the question of whether the surface models ﬁt with the theoretical models? There is a good question hidden in this challenge, but also a presupposition of its own. We can view the situation ‘‘from within’’ and point out that in the laboratory practice, as it has evolved in fact, a given process is classiﬁed as a measurement of a certain sort. Then, viewing the matter ‘‘from above’’, the theory must be able to classify it as a physical correlate of that sort of measurement. That will restrict the class of pertinent theoretical models. The observable said to be measured is to be represented by an operator of a particular kind—Hermitean, or at least bounded, etc.—on a given state space, and the apparatus as capable of a certain range of states represented by elements of such a space, and so forth. But none of this entails that what happens in the actual situation must be displayed as entirely identiﬁable in the theoretical model.23 The most stringent demand that can be made here is that the relative frequencies of certain events in this sort of situation must have a good ﬁt to probability functions, extrapolated from them in surface models, which are identiﬁable as parts of corresponding probability functions in the theoretical models. When this demand is met—whether strictly or to some approximation—the theory is borne out by the experimental results, and can be used to make predictions. Take for example the Stern–Gerlach experiment that we brieﬂy looked at above. Suppose that with a given source, the frequency counts in the upper and lower channel show a proportion 2:1.

306  :    Starting with the assumption that the usual sort of theoretical model ﬁts this situation, the range of states possibly prepared by the sources is restricted by the condition PAs ({‘up’}) : PAs ({‘down’}) = 2:1 where A is the pertinent observable (spin in vertical direction, say). This means that if the initial state is pure, it is a superposition of two eigenstates with coefﬁcients whose squares are 2/3 and 1/3. Alternatively the state could be 2:1 mixture of those eigenstates. A further measurement (for spin along a different direction) can decide between these two possibilities. With the information about the state prepared by the source, we can go to the theoretician for a prediction of outcomes if the magnet array is rotated with respect to the source, or if a new such array replaces the counter at the end of the upper channel, and so forth. The prediction will be that the frequency counts ﬁt the relevant new Born probabilities in a theoretical model that satisﬁes the above condition. And that is all. This point was summarized brilliantly if cryptically by Hertz in the famous passage, that I already mentioned in connection with the Bildtheorie of science, where he discusses modeling, in his Prinzipien der Mechanik. The representation provided by the theory must be such that the ‘‘intellectually necessary consequences’’ of the representation must in turn represent the ‘‘naturally necessary consequences’’ of the system. Now, with our eyes on quantum theory, we can read this in a new light: We form for ourselves inner pictures or symbols of external objects; and the form which we give them is such that the necessary consequences of the pictures in thought are always the pictures of the necessary consequences in nature of the things pictured . . . . The pictures which we here speak of are our conceptions of things. With the things themselves they are in conformity in one important respect, namely in satisfying the above requirement. For our purpose it is not necessary that they should be in conformity with the things in any other respect whatever. (Hertz 1894/1956: 1–2; see also p. 177)

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Anti-empiricist confusion There is a charge sometimes made against quantum theory itself, as typically presented in physics textbooks, but also against any empiricist view of physics. That is that the theory is made out to be about measurements rather than about nature, about what people [will] actually ﬁnd rather than about what happens. But that was not what physics was meant to be! After all, it is said, there are events in the stratosphere, where no measuring instruments are present, and physics applies to those events as well. Certainly it does! But the objection trades on a stunted impression of what is meant, either by such a textbook presentation of the theory, or by an empiricist reading of it. First of all, the observable phenomena do not include just actual measurements and their outcomes. They include all the observable entities—objects, events, processes—that there are, have been, or will be, whether observed or measured or not. Secondly, the theory provides a representation of the whole of nature, which includes both observable and unobservable parts. An empiricist will emphasize the term ‘‘representation’’ here, and recall all the ways in which a representation may only partially, and perhaps distortedly, mirror anything real. So the empiricist construal does not contradict the assertion that the theory is about all that can happen, and not just about measurements. Thirdly, the predictions to whose accuracy the theory is directly accountable are predictions of what the appearances will be when something appears, i.e. is measured. But it would be an oxymoron (or anthropocentric metaphor) to speak of accountability to what cannot be checked in practice, to what cannot be accounted. In the case of quantum theory it is sometimes objected that we can check, by actual measurement as well as with our own eyes, that the pointer on a given dial coincides with a particular number, whereas any state we can ascribe to that object will not be localized in a ﬁnite region of space for more than an instant. The conclusion drawn is that observable things are after all not accommodated in the theory: the correct representation of the pointer is as an object wholly in a small spatial region for an extended period of time and this we ﬁnd nowhere in the quantum theoretical representation. But this objection confuses the three levels: Theorica, phenomena, and appearances. What is subjected to a position measurement, may well be an observable object, yes. But the way it appears when it is photographed

308  :    or measured in some other way, is its representation by a measurement outcome. It is not incumbent on the theory to represent it in the same way. (In other words, the Appearance from Reality Criterion is not an inherent constraint on physical theory, but one that is at best imposed on it.) It is incumbent on the theory only to predict what its appearances will be like, and that it can do via the Born Rule to the extent needed in empirical applications: by providing probabilities and expectation values. This will still be very puzzling if you think of ‘Theorica’ as telling it like it is, if you identify reality with the theoretically postulated reality. Think of this, however, as an empiricist must: the theoretical representation, in which no object state is localized in a ﬁnite region of space for more than an instant, is only theory. As is all of our theoretical science. The reality to which it is accountable is only the observable part of the world, and that implies for us that what it is in practice directly accountable to are the appearances—the outcomes of the measurements and observations that are actually made. Advocates of the Appearance from Reality Criterion will not be satisﬁed with quantum mechanics. Some of the resistance to that theory may be explainable in this way. Even more so the attempts to provide the theory with an interpretation that restores obedience to the Criterion at some ‘deeper’ level. Conversely, irenic acceptance of the theory—such as that which we have seen prevalent in the physics community, throughout the last century—would seem to signal an attitude content without any sustained attempt to satisfy that Criterion. This, it seems to me, should allow us to draw the right moral about what are and what are not norms that govern scientiﬁc practice.

APPENDIX TO CHAPTER 1

Models and theories as representations Throughout this book, when I discuss theories and models I remain within what is known as the semantic approach to (or, semantic view of) science. That approach has lately been contested by various writers (Suarez 2003, Frigg 2006, and Morrison 2007, for example) on the basis that the concept of model it involves is not in harmony with what models are. The tension displayed in these critiques, I submit, is not actually due so much to a different concept of models as to different interests in the same subject. On the semantic view, a theory offers us a large range of models—in fact, while a theory may have many different formulations, its set of models is what is important.1 If a theory is advocated then the claim made is that these models can be used to represent the phenomena, and to represent them accurately. A model can (be used to) represent a given phenomenon accurately only if it has a substructure isomorphic to that phenomenon.2 (That structural relationship to the phenomenon is of course not what makes it a representation, but what makes it accurate: it is its role in use that bestows the representational role.) A theory may therefore be taken to represent its domain as thus or so in the sense that the models it makes available for the representation of phenomena in that domain are thus or so. One topic that was never entirely settled in this literature is just what a theory is. Patrick Suppes, one of the fathers of the approach, identiﬁed Newtonian particle mechanics with the set, which he deﬁned, of Newtonian particle systems—in effect, with the theory’s set of models. This was inspired by the form of contemporary mathematics: Euclidean geometry is simply the study of the set of Euclidean spaces, and presenting that geometry consists in nothing more than a deﬁnition of that set (cf. Blumenthal 1980). But of course, Euclidean geometry is a mathematical theory, while Newtonian mechanics is a physical theory, so it seems that something is left out. This was quickly enough appreciated, and various additions were made. Thus I added (van Fraassen 1970) a semi-interpreted language of ‘‘elementary statements’’; Ronald Giere (1985, 1988) identiﬁed a theory with a combination of a theoretical deﬁnition (of the set of models) and theoretical hypotheses (relating the models to the domain of application). In a longer discussion (van Fraassen 1991: ch. 1) I discussed Giere’s scheme in the light of some criticisms by Nancy Cartwright. While not settling on any ofﬁcial deﬁnition of what a theory is, I emphasized that

310  it must be the sort of thing that can be believed, disbelieved, doubted, and so forth. Thompson 2007 is the latest comprehensive account of the development of the semantic approach, with special emphasis on its uses (in Thompson 1989 and Lloyd 1984, 1986, 1987, 1994) to theories in biology, and contrasts various conceptions of theories and models. Lately Margaret Morrison (2007) has emphasized that despite the centrality of models for the understanding of science, important roles remain for theories, though also without offering an ‘‘ofﬁcial’’ codiﬁcation of just what a theory is. Nancy Cartwright, Towﬁc Shomar, and Mauricio Suarez (1995) developed a more instrumental view of theories, as providing tools for the construction of models. At ﬁrst blush it seems that our conceptions of theory and model must be quite different, if on their view a model is something constructed and a theory a tool. But in fact I think that a closer look reveals that our conceptions are entirely compatible; what we concentrate on signals differences in interest and focus. The sense in which a theory offers or presents us with a family of models—the theoretical models—is just the sense in which a set of equations presents us with the set of its own solutions. In many cases, no solutions to a given equation are historically found or constructed for a very long time . . . though mathematically speaking, they exist all along. When the equations formulate a scientiﬁc theory, their solutions are the models of that theory. In this sense, Newtonian mechanics had in its range of models already solutions to the three-body problem, though the scientists following Newton, not being logically omniscient, could not see what it was.3 As Suarez and Cartwright (forthcoming) discuss, the process of actually and historically arriving at a working model can take several forms. Only rarely is it a matter of simply instantiating the theory to some speciﬁc values of some parameters. In some cases, emphasized especially in Ronald Giere’s elaboration of the semantic view, the process is one of ‘‘de-idealization’’. That is, the scientist wishing to construct a model of an actual situation or phenomenon begins with a very idealized model, and then adds more factors characterizing the real situation until a good enough representation of the phenomena is achieved. An example would be to model a real pendulum starting with a model of one in which friction is ignored, thus an ideal case, and then add force terms that correspond to the friction in the actual situation. But there are also processes of model construction in which we see imaginative leaps that are not like that—de-idealization is not at all the only tactic to be found in model construction. For example, the construction of a model of superconductivity by the London brothers—a much discussed example, and central to those critical discussions of the semantic view—involved a change from an analogy with ferromagnets to an analogy with diamagnets. Nevertheless, what is arrived at is a model of the background theory. If the ideal pendulum is a Newtonian model, and it is de-idealized, then the result is still a Newtonian model. For if it is not, then it is a counter-example to Newton’s theory,

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and hence not a de-idealization. Historically this could happen as well—with the resulting model not being a de-idealization and not a solution to the problem as originally deﬁned—and sometimes famously does. For example, there seemed after all to be no way to arrive at such a de-idealization that matched the actual advance in the perihelion of Mercury. But the arrival of such an anomaly signals revolutionary change. On the other hand, if the imaginative leap succeeds in solving the originally posed problem, where de-idealization guided by an earlier analogy did not work, then again the success consists in arriving at a model of the situation that is not a counterexample to the background theory. Thus, as Suarez and Cartwright also emphasize, the London brothers’ model of superconductivity is still a model of Maxwell’s electrodynamics, the background theory within which they were working. So what accounts for the apparent tension in our two very different discussions of modeling and representing? We ﬁnd the answer in the precise complaint that Suarez and Cartwright voice. To display the differences, they give their own ‘‘structural account’’ of the superconductivity story, which is quite like that in various versions of the semantic approach and then write: this framework can account for the piecemeal borrowing characteristic of the practice of modeling precisely because it leaves out of the description everything that is of interest for our thesis in ‘‘The Toolbox’’, namely the actual reasons scientists advance for building a new model. The framework leaves out precisely the Londons’ reasoning in deriving the London model out of the acceleration equation theory—which is what we have been claiming is crucial to assess the theory-driven view. Hence we see that an appropriate structural characterisation of this practice can at best describe correctly the products of the modeling practice (i.e. the models themselves); while necessarily leaving out the intellectual processes that lead to those models. (loc. cit.)

Quite right! Structural relations among models form a subject far removed from the intellectual processes that lead to those models in the actual course of scientiﬁc practice. So what is important depends on what is of one’s interest. Accordingly, if one’s interest is not in those structural relations but in the intellectual processes that lead to those models, then the semantic view of theories is nowhere near enough to pursue one’s interests. Both interests are important, it seems to me; leaving out either one we would leave much of science un-understood. But the basic concepts of theory and model—as opposed to the historical and intellectual dealing with formulations of theories, or on the contrary, construction of models—do not seem to me very different in the two approaches to understanding science. In Part II I take up this divergence in interests with respect to measurement, which is also studied in two very different—not incompatible but complementary—ways: ‘from above’ and ‘from within’. That is a speciﬁc, and in some ways simpler, instance of the same issue, since measurement too is representation.

APPENDIX TO CHAPTER 6

Quantum peculiarities: fuzzy observables The conclusions about measurement in Part II are general, though quantum mechanical examples motivated the leeway to be left for general conditions of measurement. In the foundations of quantum mechanics we ﬁnd furthermore speciﬁc features that we cannot attribute to physical theory generally. They provide further grist for our mill if we want to press the point that a measurement outcome does not show what the target is like, as opposed to what it ‘‘looks like’’ in that particular measurement set-up. In classical contexts the signiﬁcance of this point is minimized, by the presumed possibilities of so varying and combining simple measurements into complex joint measurements that the limitations are effectively transcended. It is now customary in the literature on quantum theory to distinguish ‘‘sharp’’ observables from a generalization that encompasses also ‘‘fuzzy’’ or ‘‘unsharp’’ or ‘‘smeared out’’ observables.4 The former, which are the familiar sort, have (in their simplest form) real numbers as their possible values. We shall restrict ourselves to these for a bit, and then go to the more general category. We consider to begin an observable that is discrete: its set of possible values is not a continuum but a discrete set of values. If E is a set of possible values of sharp discrete observable A then there is a state s such that the conditional probability PsA (E) = 1. So if A can possibly have a certain value, or value in a given range, then there is a measurement situation in which that will appear with certainty. When E contains just a single possible value of A then such a state s is called an eigenstate of A. But it is quite easy to ﬁnd sharp observables A and B which are incompatible in the sense that they have no eigenstate in common. Then there is in that case no measurement apparatus or set up such that the Criterion for the Physical Correlate of both A and B measurements could be satisﬁed simultaneously—A and B are mutually incompatible.5 This incompatibility relation will have a familiar sound, because of the wellknown uncertainty relations for position and momentum which are often the ﬁrst curiosity of quantum mechanics anyone comes across. The familiar sound is somewhat deceptive. For position and momentum observables are indeed sharp observables with real number values; however, they are not

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discrete but continuous, they have no eigenstates. Still the somewhat weaker requirement: If E is an interval of possible values of sharp observable A then there is for each positive number δ < 1 a state s such that the conditional probability PAs (E) > δ holds for them as well. Their incompatibility means now of course that this will not be the case for both in the same states for the same probabilities—intuitively, δ will have to decrease for one if it increases for the other. Here we recall the notions of occlusion and ‘‘explicitly non-committal’’ picturing. The extent to which measurement of one observable can be revealing, however indirectly, will limit the extent to which it can be revealing about certain other observables. (The analogy is fragile, however: if the unmeasured observable has no value, then its value is not hidden from sight!6 ) That applies to both discrete and continuous sharp observables. But what is then actually, really measured in a measurement of position or momentum? It is entirely unrealistic to think that the outcome could be a real number clearly distinct from any nearby numbers, or that in a series of measurements there could be a clear distinction between a frequency of 1 and of δ for any and all δ < 1. So either the outcomes of real position or momentum measurements should be reconceived, or we should reconceive just what observables are being measured. It is the latter option that arrives with the generalized observables allowing for ‘unsharpness’. I will explain the basic idea here, but for the simple case of an observable whose possible values form a ﬁnite set RA = {v1 , . . . , vN }. The function PAs assigns a probability to each of these values (the probability that it would be found if A were measured on an object in state s). But now suppose that these values as not being sharply distinguishable in measurement, so that if a measurement outcome indicates value v3 , for example, we should conclude that we have ‘really’ found values v2 , v3 , or v4 , with probabilities p2 , p3 , or p4 . Or, to be more operational in the gloss we give it: if the outcome indicates value v3 , for example, we expect that immediately repeated measurements would ﬁnd v2 , v3 , or v4 , with probabilities p2 , p3 , or p4 . Let me put this a bit more generally. The function p, which is thus associated with value v3 , has brothers, so to speak, similarly associated with each of the other values in RA. Let’s therefore write, not simply p, but p[v3 ]. This association is a conﬁdence mapping. Then we can think of identifying another observable B such that PBs assigns to v3 the probability PAs (v2 ) + PAs (v3 ) + PAs (v4 ), and similarly for the other values and associated conﬁdence mappings. Then B is called a fuzzy version of A.7 For B to be a fuzzy version of A, the indicated relationship has to hold of course for each of the values in RA and their associated smearings out by a particular

314  conﬁdence mapping; and this has to hold for all states s. This may not make sense, intuitively. One way to think of it is to regard B as having values that are not numbers but probability distributions over numbers (possible values of A)—such values are sometimes called ‘‘fuzzy sets’’.8 The fuzzy versions of sharp observables are in general not sharp, and in quantum theory are represented by a different sort of operator. A theorem on this subject establishes that in any standard measurement of an observable, in the way that was described above, with the Criterion for a Physical Correlate of Measurement being satisﬁed, what is really measured is always a fuzzy version of an observable.9 To see this, the Criterion for the Physical Correlate of Measurement is applied in reverse, as it were. Imagine that the speciﬁcations of a measurement set-up, including the pointer observable B and the unitary operator that governs the measurement interaction, are given. Now ask: what is the observable A such that the equation B A Pﬁn (E) = Pinit (E)

is satisﬁed, for all initial states init? The found answer is that observable A, so identiﬁed, is then a fuzzy version of some sharp observable, and the two are the same only if A is discrete. In the case of continuous observables such as position and momentum, that fuzzy version can not be the same as the ‘original’, that is, as the observable that is putatively the target of the measurement.10

APPENDIX TO CHAPTER 7

Surface models and their embeddings As mentioned in the text, literature on the foundations of quantum mechanics typically focuses on small structures that represent data, and they do not in general have such a familiar form as e.g. Stevens was considering. They are not always algebras, but more generally partial algebras, or just partially ordered sets (posets)with some relations and/or operations. That generalization is needed because of the presence of incompatible observables, which is foreign to classical physics. The reason for the more general, more liberal, notion of the structures that can emerge from measurement is that here one is forced to attend to measurements which require incompatible set-ups and are not even in principle, not even in the ideal limit, combinable into single measurements.  (continued from the section The surface model of an EPR experiment in Chapter 7) One way to report the ﬁndings in the experiments above is to note that in no surface model so obtained do Lxa and Rxa both receive the score T when they can receive an informative score at all (i.e., when the preconditions Lx and Rx obtain). We then call Lxa and Rxa orthogonal. If we keep ﬁxed this generalization about all obtainable surface models, then Rxl must receive score T when Lx0 does (modulo probability zero), again when the informative scoring conditions obtain; and we call that implication. The latter is a partial ordering, naturally symbolized as ‘‘≤’’. So what we found above, the family of propositions that register the possible experimental outcomes, is a partially ordered set with an orthogonality relation. Reﬂection on this form of representation leads to assertions of the form: ‘‘all data models can take the form . . . ’’ and there are various proposals for what this mandatory general form must be. It is important to note that structures taking this form may not in general be ‘classical’. How can we view the measuring procedure, thus conceived, as locating the system investigated in a logical space? . A. R. Marlow (1978, 1980) used the concept of dual poset: a partially ordered set with zero element and a relation of orthogonality, plus a single operation, duality, symbolized as *. This operation has the properties that x** = x*,

316  x ≤ y only if y* ≤ x*, and x is orthogonal to y if and only if x ≤ y*. Duality is a generalization of the idea of a complement or negation. In the world of mathematical entities there are many dual posets. What must a theoretical framework be like if it is to provide models that can have even very strange dual posets of experimental propositions embedded? The embedding must be good, that is, we must be able to see in the theoretical model all the signiﬁcant features of those ‘empirical algebras’. Think back for example to the above type of experimental report, for an EPR experiment, labeled S. Such a report starts ‘‘With initial preparation X, . . . ’’ and then mentions probabilities. These probabilities characterize what is called the state prepared by procedure X. Such states look like fragments of ordinary probability functions, in that they assign probabilities only to propositions for which the informative scoring conditions obtain. (For example, a probability is assigned to L20 only conditional on L2.) If the theoretical model is to accommodate these probabilities then they must be ‘visible’ in a certain sense in the computational structure of that model.  . Here is Marlow’s theorem. It requires two preliminary deﬁnitions: A probability function on a dual poset is any function f with the properties that it assigns 0 to the zero element, that f (x*) = 1 − f (x), and that f (x) ≤ f (y) if x implies y. A base for the dual poset is a set of elements that contains either x or x* for each element x, but does not contain elements orthogonal to each other.11 The theorem says now that if we have a dual poset and a base, we can embed the poset in the algebra of projection operators on a Hilbert space, in such a way that: (i) duality becomes orthocomplementation (as deﬁned on these projections), (ii) the partial ordering is preserved for elements within the base, (iii) each probability function on the poset can be associated with a vector and becomes calculable by means of the familiar trace computation used in quantum mechanics. This is a beautiful result, and one could take it to provide good reason to think that we have here a plausible ﬁlling out of the contention ‘‘all data models can take the form . . . ’’. A good reason is not necessarily conclusive, though, and actually in some ways the theorem is less powerful than it looks (the implication order is preserved only within the base!) and in some ways less than informative (there are enormous Hilbert spaces with room to embed almost anything).12 What the theorem does illustrate very well is the form in which phenomena, depicted in data models, are located in logical spaces provided by theoretical modeling.

APPENDIX TO CHAPTER 13

Retreat (?) from The Scientiﬁc Image Readers of The Scientiﬁc Image may wonder how the view of science outlined in Part IV, on Appearance and Reality (but speciﬁcally as pertaining to quantum mechanics) relates to the view advanced there. Not surprisingly: it was with respect to probabilistic theories in science—and hence, with respect to recent physical theory überhaupt—that I had a change in view. At the earlier stage I subscribed to a moderate frequentist interpretation of probability. I was under the impression that empirical adequacy could be deﬁned for probabilistic theories more or less in the same way as for the rest. That impression was mistaken. Let me rehearse some of the details here. To begin there was the following general pattern, to which I still hold, if not taken too strictly: Science aims to give us theories which are empirically adequate; and acceptance of a theory involves as belief only that it is empirically adequate. This is the statement of the anti-realist position I advocate; I shall call it constructive empiricism. (1980: 10) What was meant by ‘‘empirically adequate’’ was explained ﬁrst of all for what had been the mainstay of modern physics until, arguably, the twentieth century: theories in which no probabilities occur. As main example I took Newton’s mechanics and theory of gravity, as depicted in the semantic approach. I will here quote the representative passage, but let’s remark at once that I did not at the time distinguish clearly the observable phenomena from the appearances: The ‘apparent motions’ form relational structures deﬁned by measuring relative distances, time intervals, and angles of separation. For brevity, let us call these relational structures appearances. In the mathematical model provided by Newton’s theory, bodies are located in Absolute Space, in which they have real or absolute motions. But within these models we can deﬁne structures that are meant to be exact reﬂections of those appearances, and are, as Newton says, identiﬁable as differences between true motions. These structures, deﬁned in terms of the relevant relations between absolute locations and absolute times, which are the appropriate parts of Newton’s models, I shall call motions, borrowing Simon’s term. (Later I shall use the more general term empirical substructures.) When Newton claims empirical adequacy for his theory, he is claiming that his theory has some model such that all actual appearances are identiﬁable with (isomorphic to) motions in that model. (This refers of course to all actual appearances throughout the history of the universe, and whether in fact observed or not.) (Ibid.: 45)

318  Indeed, Newton’s ‘apparent motions’ are relative motions, and they are precisely as they can be determined to be by measuring relative distances, time intervals, and angles of separation. So although we can draw the distinction between phenomenon and appearance at the conceptual level, that distinction does not in this case make a difference in practice. What then of the case where the theory involves probabilities? Thinking of the chi-squared tables and the like whereby statisticians assess the ‘ﬁt’ of probabilistic hypotheses to actual frequencies, I thought that we had to deal there with only a small and manageable extension of the same relationship. This ‘ﬁt’ extended ‘embed’, I thought. I proceeded to outline a ‘modal frequency interpretation’ of probability which related theoretical models to families of frequencies, so as to show how and where the ‘ﬁtting’ might be located as a relation between phenomena and theoretical model. The failure of this approach was brought home to me by the writings of Joseph Hanna (1983, 1984) and correspondence with Zeno Swijtink. During the next decade in Princeton I came to abandon any objective notion of probability, and to move toward something like Richard Jeffrey’s radical probabilism in epistemology. ‘‘Belief ’’ was no longer the operative term, but ‘‘opinion’’, the latter conceived of largely (though not entirely) in terms of subjective probability. But now, how to understand probability in physics? My understanding of the formal character of physical theory did not change: a physical theory may have in it irreducible probability, which is truly the new modality of science. What had to change rather was now to conceive of the epistemic attitude of acceptance of a theory as empirically adequate. The new understanding I proposed of how we relate epistemically to the probabilities offered to us by a physical theory appeared then in Laws and Symmetry. Belief must be replaced by the nuances of gradated opinion, modeled as personal probability. The question becomes then: if we accept a theory, how do the probabilities it offers guide our personal expectation? The answer I shall now begin to elaborate is: in the form of Miller’s Principle. That is what constitutes acceptance. For someone who totally believes the theory, that guidance will involve all theoretical probabilities. For the scientist qua scientist (as described by empiricism) only the theory’s probabilities for observable phenomena will play this guiding role. ‘Objective chance’ is in either case the honorary epithet we give to the probabilities in theories we accept. (1989: 194)

To formalize this, I took my cue from Chaim Gaifman’s introduction of the idea of expertise as guiding clue: If P is my personal probability function, then q is an expert function for me concerning family F of propositions exactly if P(A | q(A) = x) = x for all propositions A in family F. (Ibid: 198)

(This relationship is generalized in various ways for less simple cases.) The point is that an expert is someone or something that guides and constrains one’s opinion in a certain range. Thus, for example, to say that one accepts contemporary physics without qualiﬁcation, can now mean that one takes this physics as expert on

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the probabilities of observable events to which it assigns a probability. Outright implications about what observable entities are like or how they behave or interact can be treated as propositions which the theory ascribes probability 1, and thus covered as a limiting case.13 When, as in the case of quantum theory, the probabilities provided are for measurement outcomes, given that such measurements are made, there is nothing in the theory to guide one’s opinion about the observable phenomena as opposed to the appearances. That does not mean that someone who accepts quantum theory cannot have opinions about the phenomena! Quite the contrary, we have such opinions (and practices or policies to update them) already in place before we learn quantum theory. The guidance we get from the theory is only about how things will appear in situations classiﬁed relative to the theory as measurement set-ups of certain kinds. That guidance will be vindicated, in the context of that classiﬁcation, if the measurement outcomes comply with the induced expectations.

Notes to Appendices 1

The semantic approach, or the semantic view of theories, has a number of different versions and I try here for a weak formulation that does not distinguish between them. 2 The exact meaning to be attached to this phrase, which has also been subject to critique, is explored in Chapter 11. 3 This does not imply that there is, for each such problem, an exact solution that scientists can ﬁnd, even in principle, for there are mathematical structures not describable by any ﬁnite or recursively speciﬁable sets of equations. 4 The former, familiar, ones are those represented by self-adjoint operators, which correspond to projection valued measures, while the latter are represented by a larger class of operators corresponding to a normalized positive operator measure. For details see e.g. Busch and Lahti 1996; Bush, Lahti, and Mittelstaedt 1966; and more recently Dalla Chiara, Giuntini, and Greechie 2004. 5 We must be careful in how we understand this. Neither operational incompatibility nor operational compatibility offers us any logically sufﬁcient or necessary clue to theoretical signiﬁcance. In discussing incompatibility of observables, we are looking at the matter from above, that is, as classiﬁed within the theory. As Grünbaum 1957 begins by pointing out, both in Heisenberg’s and Margenau’s accounts we can ﬁnd consistent operational prescriptions for the simultaneous ascription of position and velocity. There is no logical contradiction, for example, in taking the data of a sequence of position and time measurements on particles emitted from a given source and ascribing a velocity during the relevant interval by means of the classical formula, dividing distance covered by time elapsed (‘‘time of ﬂight measurements’’). But such ascriptions do not have the value that assertions about physical magnitudes are meant to have, and those processes are not classiﬁed as measurement processes by the theory. The prediction of a future position on the basis of such an ascription of a current position and velocity will, according to the quantum theory itself , not be veriﬁed.

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Thanks to Angela-Adeline Mendelovici for insisting on this. The uncertainty principle bears a relation to occlusion, but only in the way that the quantum mechanical ‘correspondence principle’ ever does, rupturing the very concepts that it appeals to as analogues. 7 At this point, do not equate observables with what is represented by Hermitean operators: admitting unsharp observables requires admitting a larger class of operators for their representation (positive operator valued measures). 8 The term ‘‘fuzzy set’’ is due to Zadeh 1965. Cf. Heinonen 2005: 15–17. 9 Busch and Lahti 1996; see further Heinonen, Lahti, and Ylinen 2004, and the discussion in Heinonen, 2005: 18. 10 See especially Busch and Lahti 1996: section 3. 11 Note that 0 is orthogonal to itself, and so is not in a base. Every set with the second property can be extended to one that has the ﬁrst as well. In view of how we characterized orthogonality and implication at the outset, it follows that any set of elements that have all received score T on a particular occasion cannot have mutually orthogonal elements among its members. So what is obtained in any experimental set-up of the above sort is part of a base. 12 Marlow realized this very well and attempted to add postulates to narrow down this embarras de richesse so as to recover empirical content . My purpose here is not so to draw attention to his speciﬁc program, but to illustrate the general structure of measurement understood as the task of locating items (situations) in logical spaces supplied by theories. 13 If I were to expand on this view now, I would draw on Stephen Leeds’ insightful [1984], which develops a related way to think about the quantum theory.

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Notes

Introduction 1

As I shall characterize representation below, ‘‘mental representation’’ is an oxymoron. 2 Boltzmann’s way of thinking about this, and especially the various pragmatic aspects of representation with models, is continued and elaborated in Teller 2008. 3 This describes unqualiﬁed acceptance; in practice, acceptance will come with restrictions and qualiﬁcations, and belief will come in degrees. 4 While I will not rehearse or respond to arguments for or against empiricist views here, I want to thank the editors of two recent volumes to allow me my say: Andreas Berg-Hildebrand and Christian Suhm 2006, and Bradley Monton 2007.

Part I: Representation 1

Suarez 2004, 770. This appears to be a change of mind from his earlier ‘‘I take it that a substantive theory of scientiﬁc representation ought to provide us with necessary and sufﬁcient conditions for a source to represent a target’’ (Suarez 2003: 226). While Suarez 2004 contrasts his view to such approaches as R. I. G. Hughes’s inﬂuential (1997), his own ‘‘inferential conception’’ still came very close to providing just the sort of theory that he dismisses (in the passage I quoted) as not to the point. See further Ducheyne (2005) which emphasizes that a model represents for a person if that person accepts certain things. While not wishing either to propose rivals to their accounts or to adjudicate between them, I have learned much from these, as well as from such other recent writings as French 2003. 2 For the etymology, and for what became the stable concept in art history, see Panovsky 1991, 27. 3 We shall take a close look at how Alberti and Dürer, among others, developed this point, as well as how it was further developed in projective geometry.

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See for instance the photo of Mars made through the Hubble telescope, at http://www.jpl.nasa.gov/news/features.cfm?feature=533. Ian Hacking has recently studied the use of such terms in the history of science, and has emphasized that the term ‘‘phenomenon’’ typically denotes things classed as remarkable, unusual, or amazing. What amazes are often individual occurrences but also often processes with many instances. Some of this appears in Hacking 2006 (see p. 32, section C1 ‘‘The creation of phenomena’’) issued in connection with his Carl Friedrich von Weizs¨acker-Vorlesungen, Hamburg 2007. 1. Representation Of, Representation As 1

In Pliny the Elder’s Natural History, xxxv. Available on-line http://www. perseus.tufts.edu/cgi-bin/ptext?doc=Perseus%3Atext%3A1999.02.0137&query =toc:head%3D%232431 2 For the point that many recent objections to ‘resemblance’ or ‘isomorphism’ views of representation (or as necessary and/or sufﬁcient conditions for representation) were already offered in Plato’s Sophist and Cratylus, see my 2000b. 3 Once the Eleatic Stranger’s point is appreciated, examples abound. A painting or drawing, however realistic, achieves its aim by means that are strictly different from what it is a ‘likeness’ of—think only of the cross-hatching to render shadows, or of the strokes of thick pigment in a fresco, noticeable when inspected from close up. 4 We should add: and that he had as well correctly spelled out the distorting relationship that produces quite different appearances to the astronomer’s eye. 5 The issue of misrepresentation, and how it requires adjusting any view of representation that trades on resemblance, is dealt with for the case of sculpture (but with farther reaching conclusions) by Hopkins 1994. 6 For an exploration of how representation-as is crucial to how models in science represent what they model, see Hughes1997 and my 1994. 7 Reportedly in E. Spott, Bismarck: A Book of Mistakes. 1883; I have found no trace of this book. 8 For an analysis of the difﬁculty see Harms 1998: 482, 485. 9 For further development of Goodman’s view, with special application to scientiﬁc representation see Elgin 2006. 10 I leave aside Goodman’s attempt to accommodate them somehow in extensional discourse. 11 Goodman 1976: 26. This phrasing derives from his constant attempt to observe a strict nominalism: a depiction of Scott as wise he would call a ‘‘wise’’picture, and say that that was the kind of picture that it was.

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For a clear and balanced exposition of an alternative, including a defense of a moderate ‘resemblance’ view of pictorial content, see Files 1996. 13 The main moral I want to draw is that this account is not far from the view that sentences and pictures have after all the same kind of thing as content: propositions, to use the common term for the content of a sentence. Goodman’s nominalism would of course stand in the way of this sort of formulation, but it can be construed innocuously, without violence to anti-ontological scruples. 14 See further Elgin 2006, which explores this with special reference to scientiﬁc experimentation and models. 15 Bartels 2006. These terms are also context-sensitive: an isomorphism is a oneto-one onto function that preserves ‘‘pertinent’’ structure, while a homomorphism is a many-to-one onto function that does so. The term ‘‘pertinent’’ gets its content from the context, where one sort of structure or another is under study. 16 For the moment I am staying with representations of speciﬁc, real things; later we may have to look seriously at such examples as ‘‘X represents a man holding a candle/ Santa Claus delivering presents/phlogiston as escaping rapidly’’ and the like. 17 ‘‘Once we take on board the distinction between bare bones and ﬂeshed out content, we are in a position to notice an important feature of pictorial representations. In the late 70s, Sherry Levine produced a controversial series of photographs. She made photos of some Walker Evans photographs that were basically indistinguishable from Evans’ originals. This was controversial because she displayed her photos as her own work. In a sense, they are her own work: she made photos of Evans’ photographs while Evans made photos of life in the rural United States. In another sense, their similarity to Evans’ originals makes them seem like mere copies. Controversy aside, this sheds light on an interesting feature of pictures. Many different scenes, like a chair and a jumble of line segments, can result in the same kind of photograph or linear perspective picture from any given point of view. Levine’s photos show that one potential subject for any given picture is a plane that has shapes and colors indistinguishable from the picture itself. So, in a sense, the rural US, a well-designed Hollywood set, or simply a photo like Evans’s could result in a photo like Levine’s.’’ (John Kulvicki, ‘‘Any Way You Slice It: The Viewpoint Independence of Pictorial Content’’, with comments by Dilworth, on http://www.interdisciplines.org/artcognition/papers/9/version/ original). 18 This form is still restricted; as we will see, it needs extra contextual parameters, such as the purpose for which the representation is made or which it is made to serve. That is especially relevant for scientiﬁc representation; see e.g. Giere 2006: 60.

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See further sections 3 and 4 of Schwartz 1980. That what makes something a representation is the fact that it is pressed into representational service by representation users is also a theme emphasized in Paul Teller’s 2001a. 20 The standard reference here is Hagen 1986. 21 Compare Georgalis 2005: 128–9: ‘‘For any item r to represent a particular item t, there must be a conscious agent s to whom r represents t. I call this the fundamental fact of representation.’’ My statement goes beyond this, since I imply that there is also a representing by (and not just to) an agent. However, there is a limiting case, in which one and the same agent plays both roles, for example when someone spontaneously takes an encountered natural object or event to represent something. 22 cf. Georgalis 2005: 122: ‘‘Since there is no necessary connection between a physical item serving as a representation and that which it represents, if the representation is to do its job, . . . somehow the uniqueness of what is represented must be secured’’—and see his pages 123–9 for where this point leads. 23 I take this to be consonant with the literal use of ‘‘use’’. When we say that a car’s engine uses gasoline there is no implication of agency or community, and perhaps one part of a brain could be said to use another in that sense. But I take those uses of ‘‘use’’ to be at best derivative from the literal use. The narrow concept of representation, due to that narrowly construed use of ‘‘use’’ will not, in my view, hamper our discussion of scientiﬁc representation. 24 The emphasis on use, as here understood, implies community: there is no such thing as essentially private representation any more than private language, except in the sense in which private uses can exist as derived from or parasitic on communal practices. 25 The example is essentially Putnam’s. As to the main point, the Eleatic Stranger has a line on this: he has prefaced the list by saying that he regards the natural ones as divine workmanship—so perhaps he thought of shadows as really made in order to represent the objects casting them. What about things in dreams? It’s a nice conceit, that paintings are dreams made for people who are awake—a conceit echoed by the idea of ﬁlms as dreams that money can buy, and indeed by the ﬁlm ‘‘Dreams That Money Can Buy’’. But as an example to show that there are ‘representations in nature’ it begs the question of what dreams are. 26 This is part of an objection to the semantic approach. Suarez takes that approach to involve a view of what representation is—mistakenly, in my opinion. In this paper he also refers to my view of representation as I presented it in my 2000b, but sees that as an addition to what he gleans from the semantic approach. 27 Cf. Goodman 1968: 21–6. Since Goodman kept denotation at the heart of his account of representation, he devoted considerable attention to the case of

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representations that do not represent real things, hence have nothing to denote. His gloss on this will do for our purposes. 28 The term ‘‘relation’’ may in fact not be the most apt to explain intentionality. As Benoist 2006 emphasizes, Brentano who deﬁnes the intentionality of mental activity by the statement that it is characterized as such by ‘‘the relation to something as object’’ drew a distinction between this case and a genuine relation (which must be between real things), and says eventually that intentionality is not a relation but something relation-like: kein Relatives, aber ein Relativliches. 29 In more linguistic terms, the context created by ‘‘represents’’ is ‘referentially opaque’, for the premise that S and T are the same or apply to the same things does not license the inference from ‘‘X represents Y as being S’’ to ‘‘X represents Y as being T’’. 30 I use ‘‘intensional’’ for terms, expressions, forms of discourse, the criteria being essentially those discussed under this heading by Quine: opacity to reference, resistance to substitutivity of identicals, and so forth. See for instance The University of Alberta Dictionary of Cognitive Science, ‘‘ ‘Intentional’ is not to be confused with ‘intensional’ spelled with an ‘s’, the latter of which refers to the meaning of a term, (along with ‘extensional’)’’ (though that dictionary entry gives too narrow a meaning to ‘‘intentional’’). For ‘‘intention’’ see Dennett and Haugeland 1987. For a metaphysical approach, of the sort that I contest here, see Zalta 1988. 31 It is best and clearest to think of this as a point about language, displayed by focusing on what is and what is not to be inferred in particular examples. But the same examples, presented in the ‘‘material mode’’ (as Carnap would say) serve to make the point about what these assertions display as (putative) fact. I will not bother to obsessively distinguish formal and material mode. 32 Note however that even the regimented verbal distinction does not hinge on the word ‘‘of’’ alone. ‘‘His speech included a description of Mrs. Thatcher as draconian’’ is not covered by the convention, it is still ambiguous in the usual way, and the substitution of ‘‘the then Prime Minister’’ might or might not change the truth-value of this sentence. 33 In aesthetics, discernment of that level of meaning is the subject of iconography—see Panovsky, 1955: 26–54. 2. Imaging, Picturing, and Scaling 1

Even in Goodman’s theory, the important case of representation by exempliﬁcation hinges on a highlighted resemblance. The relevance of such sorts of representation for science as well as for art is strongly argued in French 2003. 2 See especially Sellars 1965: 180–2.

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For a detailed discussion of the use of visual imagery in the sciences, see Wimsatt 1990. 4 This does come up in debates in metaphysics: must there be a basis in nature for objective judgments of similarity, or does the similarity we privilege in our judgments relate to our concerns, values, and practice? That will be a topic for another time. 5 There are many paintings to which my description applies, but of course I mean Manet’s. Thus Zola 1867: ‘‘Les peintres, surtout Edouard Manet, qui est un peintre analyste, n’ont pas cette pr´eoccupation du sujet qui tourmente la foule avant tout; le sujet pour eux est un pr´etexte a` peindre tandis que pour la foule le sujet seul existe. Ainsi, assur´ement, la femme nue du D´ejeuner sur l’herbe n’est l`a que pour fournir a` l’artiste l’occasion de peindre un peu de chair. Ce qu’il faut voir dans le tableau, ce n’est pas un d´ejeuner sur l’herbe, c’est le paysage entier, avec ses vigueurs et ses ﬁnesses, avec ses premiers plans si larges, si solides, et ses fonds d’une d´elicatesse si l´eg`ere; c’est cette chair ferme model´ee a` grands pans de lumi`ere, ces e´ toffes souples et fortes, et surtout cette d´elicieuse silhouette de femme en chemise qui fait dans le fond, une adorable tache blanche au milieu des feuilles vertes, c’est enﬁn cet ensemble vaste, plein d’air, ce coin de la nature rendu avec une simplicit´e si juste, toute cette page admirable dans laquelle un artiste a mis tous les e´ l´ements particuliers et rares qui e´ taient en lui.’’ 6 Recall the earlier reference to Nelson Goodman’s discussion of this, his 1976. 7 Section 6.1 (pp. 112–17) of Lopes 1996. 8 Hyman 2000 and 1992; see further Derksen 2004. 9 Chapter 3 of Giere 2006; see speciﬁcally pp. 48–9. 10 Wittgenstein proposed the idea of cluster concepts to highlight classiﬁcation by ‘‘family resemblance’’: concepts characterized by a network of more or less loosely interconnected properties. Application of the concept draws on some parts of the cluster; there is no general limit to this conceptual ‘‘plasticity’’. Wittgenstein’s main example is the idea of a game: ‘‘How should we explain to someone what a game is? I imagine that we should describe games to him, and we might add: ‘This and similar things are called games’ ‘‘(Philosophical Investigations, para. 69) . . . . ‘‘But this is not ignorance. We do not know the boundaries because none have been drawn . . . We can draw a boundary for a special purpose’’ (Ibid). 11 Lopes: 118; Block 1983, and Introduction to Block 1981 where he discusses Fodor’s and Dennett’s views on this. 12 Although not willing to offer a precise characterization of pictorial as opposed to descriptional representation, Ned Block has an instructive example of how the distinction is made in practice: Consider what goes into a computer by way of storing a picture of, say, a vertical line. Consider a matrix n squares wide. Each of the squares can be light or dark. A vertical line

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would be represented if each lighted square is n squares distant from the ﬁrst lighted square (counting by row, as on a calendar). The descriptionalist sees the line representation as a set of sentences. If n = 7, the set of sentences could be: ‘1 is dark’, ‘2 is light’, ‘3 is dark’, . . . . ‘9 is light’, ‘10 is dark’, and so on . . . . The pictorialist, on the other hand, sees the representation of a line . . . as like the matrix itself, not the corresponding set of sentences. But what does the distinction come to between the set of sentences and the matrix itself? Aren’t they just different ways of inscribing the same representation with the same semantic properties? I say ‘‘No.’’ The key is the way the representations function. Consider how the descriptionalist would rotate his line. A small counterclockwise rotation could be accomplished if the ﬁrst lighted square stayed lit, the next number of a lighted square was increased by 1, the next by 2, the next by 3, and so on. In terms of the example, 2 would stay lit, 10 would be lit instead of 9, 18 instead of 16, 26 instead of 23, etc. So the descriptionalist’s new set of sentences would be ‘1 is dark’, ‘2 is light’, . . . ‘10 is light’, . . . ‘18 is light’, . . . ‘26 is light’, and so on. The important point is that the computer’s ‘‘rotation’’ calculation just involves the numbers, not the arrangement of the numbered squares. The matrix display is for us and plays no role in what the computer does. From the point of view of the computer’s calculations, the squares could as well be arranged in a line or a circle rather than in a matrix. Real live computer graphics works this way. The machine manipulates numbers. The programmers think of the numbers as numbers of cells in matrices, and they put in numbers that correspond to matrices of visual interest. They program the computer to operate on the numbers in ways that correspond to visually interesting or useful matrix changes. These correspondences are what makes the computer’s number crunching graphical, but the correspondences play no role at all in the number crunching itself. (1981: 53)

This is illuminating, though I don’t think it can stand without qualiﬁcation either. Note that the functioning in question is not what is meant when we say that a certain array of numbers or dots functions as a representation. The functioning on which Block focuses is relevant here, but must be distinguished: it concerns the transformation of one symbol in a system of symbols into another symbol. 13 Thanks to Isabelle Peschard for this criticism. 14 This insight may be best honored by attention to global differences between pictorial and other systems of representation. That is, in fact, how Goodman approaches the matter in his Languages of Art. According to Goodman, as we have seen, pictures denote and predicate. But what distinguishes pictorial systems from other denotational systems (such as systems of verbal description) are features such as denseness that make them more like analog systems, like diagrams and maps. Cf. Goodman 1976: 194–8; Goodman and Elgin, 1988 ch 7. Pictorial symbol systems are syntactically and semantically dense. That is, any two pictures, no matter how similar, could be depicting different things or depicting the same things in different ways. Cf. Goodman 1976: 226–7. That is not at all true of verbal description, where small differences typically make no difference at all. An analog measuring device is similar in that respect: two different pointer positions, no matter how

352  :   close together, indicate different values. In diagrams or maps too a small difference in lines or positions can be signiﬁcant. For example in maps made to scale the merest movement of a border or contour line depicts a different possible situation in the mapped region. Nevertheless, there is a distinction also between pictures and diagrams or maps. Goodman takes that difference to be syntactic, that is, it does not have to do with reference or attribution but solely with the composition. The difference is only a matter of degree, however: pictorial symbol systems are more replete. In a painting a much larger set of features—color, thickness, intensity, contrast, etc.—is relevant to what it denotes and what it predicates than in diagrams or even the most detailed topographical maps. This is a difference of degree that I will not build into my terminology here, for it will not be pertinent to distinctions I will need for different modes of scientiﬁc representation. So I will describe diagrams, maps, etchings, and paintings all as pictures, and their use as picturing. 15 Adam Elga has suggested an appropriate form of presentation: we can here write S = {Set of Points}, Assignment of co-ordinates (x,y,z), Content occupation function K . This is a model of an ‘occupied’ space, of a world [within just four parameters, of course, a bit skimpy]. 16 But they do assume, very unrealistically, that ordinary macroscopic objects are as sharply deﬁned as e.g. classical atoms. What is the shape of this beer glass? Descartes’s answer (translated into our current terms) would have been ‘‘an analytic function’’. Why do scientists and philosophers tend to be impatient with the point that such mathematical modeling is unrealistic? The answer is presumably that they have long since appreciated it, and prefer to focus on the practical usefulness of the models—a reasonable attitude, even if we can’t afford it when we study scientiﬁc representation as a subject in its own right. Below I will give an example from classical optics to show how this sort of idealization can lead to paradox, and thence to changes in theory. 17 See further my 1999. 18 I take this simple example from the history of geometric optics, on its way to becoming physical optics in the modern sense. For a fascinating study of this historical stage see Fokko Jan Dijksterhuis 2004. 19 Hero’s fundamental assumption about nature was not precisely true. It is true under rather limited conditions; correct in the very simple case of reﬂection by a plane mirror set in a homogeneous medium like ordinary room temperature air. Later it was modiﬁed: ﬁrst by suggesting that light takes the path that takes least time, and later still further. 20 This simple result has far reaching consequences too. Consider the following problem set by Aristotle: if a circular light set in the ceiling shines through a square

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hole in the ﬂoor, what is the shape of the light spot on the basement ﬂoor? (This question is, as it were, a precursor to the discussion of the camera obscura constructed by the Arab mathematician Alhazen in the tenth century.) 21 Willebrord Snel van Royen (Netherlands, early seventeenth century). It is also called the Law of Sines because, in modern terminology, the ratio of the sines of the angles of incidence and refraction equals the ratio of velocities in the two media. 22 For animated demo see http://micro.magnet.fsu.edu/primer/java/refraction/ criticalangle/index.html 23 Teller 2001a; see further his 2008. 24 I am staying here with classical physics, which sufﬁces to make the current point about vagueness in mathematical-physical representation; the same point could be made for later physics, mutatis mutandis. 25 i.e. locally determined by a convergent power series. 26 For Descartes, a geometrical problem is always a problem of construction with a ruler and compass; only when such a construction can be achieved with a continuous movement can we have a clear and distinct idea of the geometrical solution. The exclusion of the inﬁnite is a constant theme in Descartes; cf. Principles I, Proposition 26. In attributing a contrary opinion to Pascal we don’t mean of course that he had the conception developed in point set theory after 1900; though he went very far in his rejection of the apparent paradoxes of the completed inﬁnite. Cf. Chevalley 1995. 27 Birkhoff and von Neumann 1936 point out that areas that differ by measure zero are equivalent as far as anything instrumentally measurable is concerned. A quotient construction is a reduction modulo some equivalence relation, that is, replacement of elements by their equivalence classes. In the case of intervals, for example, (0,1) and [0,1], (0,1], [0,1) all differ just by measure zero, and would be treated as the same. 28 At that point we can either say or deny Birkhoff and von Neumann’s contention that the calculations can go on as usual, but the shape is correctly represented not by one region in geometric 3-space, but by an object in the quotient construction that identiﬁes regions modulo differences of measure zero. If we go on to meta-mathematics we can extend this critique much further: anything beyond ﬁnite, combinatorial mathematics is susceptible to multiple ‘‘unintended’’ and ‘‘non-standard’’ interpretations, introduced by pointing to very different sorts of mathematical structures. 29 We may be reminded here of Solovay’s 1965 ‘‘2ˆaleph0 can be anything it ought to be’’. The foundations of mathematics galloped away with its own subject, shanghaied it, abducted it (to hear some mathematicians’ complaints about

354  :   meta-mathematics). The categoricity proofs for geometry, in the early 1900s, suggest that at least your text will be invulnerable to deconstruction if you say: ‘‘the shape of this glass is a surface in a geometric 3-space deﬁned by these axioms’’. But if analytic geometry is conceived of as part of set-theory (in the general set-theoretic reconstruction of mathematics) then these categoricity proofs have a very limited bearing. They establish only that within each model of set-theory, those geometric 3-spaces are isomorphic. So the assertion which looked like a totally closed text, is once more to be regarded as open, vague, and subject to many alternative construals. 30 To make this concrete, imagine a very small situation, involving only 4 men and 7 women. Under good conditions the women produce 8 items per hour and the men 4. Under bad conditions the women produce 4 items per hour and the men 2. But two men and two women work in good conditions, with two men and ﬁve women assigned to bad working conditions. In the bad workplace, the production is 2(2) + 5(4) = 24 items per hour. In the good workplace, the production is 2(4) + 2(8) = 24 as well, precisely the same. So the abstract picture shows no correlation between production and working conditions. 31 For this topic I have beneﬁted greatly from Sterrett 2002, 2005, and 2006. 32 See Galileo 1964–1966, vol. iv, p. 756, cited in Torretti 1999: 3. 33 The following similar passage in Galileo’s Discourse on Floating Bodies shows well how easily and naturally he proposes experiments to be done on a small scale so as to test hypotheses that pertain as well to much larger bodies: ‘‘The third difﬁculty in the doctrine of Archimedes was, that he could not render a reason whence it arose that a piece of Wood, and a Vessel of Wood, which otherwise ﬂoats, goes to the bottom, if ﬁlled with Water. Signor Buonamico hath supposed that a Vessel of Wood, and of Wood that by nature swims, . . . goes to the bottom if it be ﬁlled with water . . . but I . . . dare in defense of Archimedes deny this experiment, being certain that piece of Wood which by its nature sinks not in Water, shall not sinke though it be turned and converted into the forme of any Vessel whatsoever, and then ﬁlled with Water: and he that would readily see the Experiment in some other tractable Matter, and that is easily reduced into several Figures, may take pure Wax, and making it ﬁrst into a Ball or other solid Figure, let him adde to it so much Lead as shall just carry it to the bottome, so that being a graine less it could not be able to sinke it, and making it afterwards into the form of a Dish, and ﬁlling it with Water, he shall ﬁnde that without the said Lead it shall not sinke, and that with the Lead, it shall descend with much slowness; & in short he shall satisﬁe himself, that the Water included makes no alteration.’’ (Galileo 2005: 21)

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Speciﬁcally the example of airplane modeling is held up by Boltzmann in his Encyclopedia article ‘‘Model’’, as illustrating how alteration in spatial dimensions can alter other physical characteristics: ‘‘a ﬂying machine, which when made on a small scale is able to support its own weight, loses its power when its dimensions are increased’’ (1904: 220). For extended discussion see Sterrett, Wittgenstein Flies a Kite. 35 For analysis and historical context, see Peterson 2002. 36 These calculations suggest that a nano-machine would not have to worry about gravity, almost like an ordinary machine in weightless conditions. 37 I refer here to Reichenbach’s striking and little appreciated demonstration that indeterminism was all along consistent with the success of classical physics. He gives an example from statistical mechanics to show that the classical deterministic laws can, with appropriate initial conditions, lead to behavior that is, by measurement, indistinguishable from genuine indeterminism. The example is exactly one in which what I call here the Approximation Principle fails. See Reichenbach 1991: 93–5. 38 Buckingham1914; Bridgman1916. 39 quote from sections 160–1 of Fourier 1952. 40 Dimensional analysis, which we can think of as a special application of symmetry arguments (see e. g. Hornung 2006) has the appearance sometimes of generating far-reaching empirical consequences from a priori reasoning—a beautiful topic but which we can only touch on here. For a literate and literary but rigorous treatment see Barenblatt, 2003; for rare philosophical discussions see e.g. Ellis1966: 127–51 and Sterrett 2006: 127–8, 135–8, 180–202; Lange [forthcoming]. 41 See further Hornung, op.cit.; for the dimensionless parameters as invariants of transformation groups see Barenblatt 2003: 94–6. The seminal paper relating dimensional invariance to more general notions concerning symmetry, with special reference to measurement scales, is Luce 1978. 42 The law is that force is proportional to mass times acceleration; the more usual speciﬁc form F=ma does depend on choosing the units such that one unit of force gives a unit mass unit acceleration. 43 For a precise exposition, see Barenblatt 2003: section 1.2.2. This requirement of scale invariance for what is signiﬁcant is generalized to invariance under any ‘‘rendition’’ or deﬁnition of some quantities in terms of others, such as ‘‘velocity’’ as ‘‘distance/time’’. Buckingham gave the example of two sets of quantities that can furnish the basic units for mechanics: Force, density, and linear speed can be expressed in terms of mass, length, and time; but the converse is also the case (Buckingham, op. cit: 348–9, with force to have dimensions MLT−2 , density ML−3 , and speed LT−1 ).

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3. Pictorial Perspective and the Indexical 1

‘‘Painting is a kind of natural philosophy, because it imitates the quantity, quality, form and character of natural objects’’, Pino, Dialogo di pittura, Trattati 1: 109; cited Gopnik1995: 99; see further his pp. 95–102 on how Leonardo da Vinci and his contemporaries saw the relation between the sciences and the visual arts. 2 For general studies on the development of geometry, optics, and perspective in art with reference to their role in the birth of the new sciences, see Edgerton 1975, 1991 and Freeland and Corones 1999. 3 For a synopsis of the relevant history from roughly Heron and Ptolemy to the end of the eighteenth century, see the introduction in Homann 1991, which is a translation, with an introduction and notes, of Hugh of St. Victor’s Practica Geometrica, dated circa 1120 . 4 The history of astronomical measurement as aid in navigation is part of the history of Perspectiva. Ptolemy’s planisphere is a geometric construction in which the eye is located at the South Pole, and instead of an intersecting plane just the Equator—an intersecting line—is used in the projection. The astrolabe improved on this by adding lines of latitude and longitude drawn from a second point of view, for an observer located at a certain latitude. This involved therefore a projection of the celestial sphere on the Equatorial plane. Hence astrolabes could be used to show how the sky looks at a speciﬁc place at a given time. Typical uses of the astrolabe include ﬁnding the time during the day or night, ﬁnding the time of a celestial event such as sunrise or sunset and as a handy reference of celestial positions. Cf. Veltman and Keele 1986: 42–4. 5 Since PT = h/A and QT = h/B, it follows that the measured distance PQ, which is just the difference between them, must be h times the difference between 1/A and 1/B. But both PQ and that difference are known from the measurement results, so h can be calculated directly from the known numbers. For example if PQ is 100 ft, A = 1 and B = 2 then h = 200 feet. As a historical note, Thales was credited with calculating the height of the pyramids from the lengths of their shadows, in the ﬁfth century . One suggestion, however, is that Thales probably did not prove any general theorem on similar triangles, but noticed (empirically) that at a given time of day, for every object, the ratio of height to shadow was the same. So he could measure a nearby stick and its shadow directly, and also the length of the pyramid shadow—and then the pyramid height would be the only unknown factor, so could be calculated. 6 In the case of Alberti too we see at least in practice a close connection between his studies of measurement and of perspective. His Ludi matematici applies mathematics to the measurement of distances, dimensions, and weights. The

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Elementa picturae describes some geometric ﬁgures and projections. Immediately after De Pictura Alberti produced his Descriptio urbis Romae in which he explained triangulation surveying and included a table of sightings he had made of monuments in Rome. There he details how he has used a surveying disk similar to an astrolabe. His De Statua deals with proportions in the human body and how to replicate them in sculpture. See further the Introduction by Martin Kemp to Alberti 1991. 7 Although Alberti begins with the modest disclaimer that he writes as a painter rather than as a mathematician, his monograph begins in Book I with a good bit of geometry and treats his subject in a mathematically sophisticated way. 8 Reputedly, Alberti went on to considerably more sophisticated examples. As was reported at the time, ‘‘the pictures, which were contained in a very small box, were seen through a tiny aperture. There you were able to see very high mountains and broad landscapes around a wide bay of sea, and, furthermore, regions removed very daintily from sight, so remote as not to be clearly seen by the viewer. He called these things ‘demonstrations’ ’’ (from the anonymous Vita in Opere vulgari, ed. Bounucci, pp. cii-ciii; cited in Alberti, p. 98, Explanatory Note 15). 9 In this passage and below—see also the passage quoted below from Leibniz for a comparison—I am using the term ‘‘appearance’’ judiciously, in a way that does not equate it with ‘‘phenomenon’’ but allows a distinction between an observable phenomenon and its appearance (the way it ‘looks’ in the content of a measurement outcome). Recall the remarks on this distinction in the Introduction to Part I. 10 See Panovsky 1956. While these mechanical aides and machines may or may not have been actually made or used, their design brings out clearly how the perspectival drawing technique is actually the recording, in a sophisticated visual display, of the outcome of a complex spatial measurement. 11 From Part I of Thomas 2006. 12 A crucial complication comes in when it is realized by the end of the seventeenth century that light rays do not provide an instantaneous connection. While this point goes to stage center with Einstein’s famous 1905 discussion, it is well enough developed in astronomy throughout the modern period. When focusing on the unmoving or instantaneous case, one does have the luxury of ignoring the ﬁnite speed of light and thinking of all the connections between object and screen effected simultaneously. Certainly inaccuracies due to this will not show up in a rustic landscape or family portrait, but it could even be ignored in the early debates of whether or not Copernicus saves the appearances of planetary motion. Descartes still offered the possibility that light is indeed an instantaneous connection. By the end of the seventeenth century, however, the ﬁnite speed of light was known, and was taken into account in the derivation. So by then

358  :   it is possible to speak, at least in principle, of true perspectival geometry and kinematics. 13 Bernard Williams 1978 introduced ‘‘the absolute conception’’; Tom Nagel’s phrase ‘‘the view from nowhere’’ is also the title of his 1989; Eddington coined ‘‘the point of view of no one in particular’’. 14 Notes for the letter to Des Bosses, February 5, 1712: 199–200 in Leibniz 1989, cited in Jauernig 2004, chapter 1. The passage begins with ‘‘If bodies are phenomena and judged in accordance with how they appear to us, they will not be real since they will appear differently to different people. And so the reality of bodies, of space, of motion, and of time seems to consist in the fact that they are phenomena of God, that is, the object of his knowledge by intuition.’’ 15 See Ryckman 2005: 128–35 on Weyl (especially the quoted passages on page 134), and on Eddington, 183–4. 16 Recall, from the Introduction to Part I, the distinction between ‘‘phenomenon’’ and ‘‘appearance’’. 17 I’ll keep within that context for now. The subject can certainly be pursued within Euclidean geometry, especially as rendered analytic by Descartes and supplemented soon thereafter by the inﬁnitesimal calculus of Newton and Leibniz. So it is perhaps not so surprising that after a good start by Desargues and Pascal the subject of projective geometry itself languished and was only completed more than a century later. 18 A few simple results will illustrate this. If two line segments, one k times as long as the other, lie on a line parallel to the painterly window then their projections on that window will also be in proportion k:. If two line segments parallel to the window are equal in size, with one edge on the central line of sight, but the one is at a distance from the eye m times that of the other, then its projection on the window will be /m as large as that of the other. 19 We focus here on the cross ratio of collinear points. There is also a dual, the cross ratio of a quadruple of lines intersecting in a point. 20 Also called anharmonic ratio, and anharmonic section. There are a number of different conventions and deﬁnitions in use in the literature; I use here the one of Coxeter and Greitzer 1967: Section 5.2, 107–8. None of the main points made here hinge on this choice. Also, the four points can of course be ordered in six different ways, so as a set they have six cross ratios. Given any one of these, the others can be calculated from it, so we can just concentrate on one. 21 Let the distance between the equidistant adjacent points be called 1 unit; then CA/CB = 2/1 and DA/DB = 3/2, so their ratio is 4/3. If the painter placed the points meant to correspond to these with e.g. distances AB = 9, BC = 3,

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CD = 1, the cross ratio on his picture plane would be 12/3 divided by 13/4, which is pretty close to 4/3, perhaps close enough for the purpose. 22 Because the cross ratio is equal to the same magnitude with the sines of the corresponding angles replacing the line segments: CA/CB : DA/DB = sin(CPA)/sin(CPB) : sin(DPA)/sin(DPB) But those angles, and hence the sines, are the same if we look instead at the projected points A , b, C, d for example. So that must also equal CA /Cb : dA /db. The proof is elementary, using the fact that the area of any triangle equals on the one hand half of the height times the base, and on the other hand the product of two sides times the sine of the angle between them. All the relevant triangles in the diagram with vertex P have the same height. 23 A demonstration experiment, to be distinguished from experiments that test hypotheses on the one hand, and from experimental exploration on the other. For Brunelleschi’s ‘experiment’ see for example Zajonc 1995. Whatever process Brunelleschi actually used, it produced a painting that provides a view of the Baptistery from a certain vantage point, and is thus properly conceived of as a measurement of the Baptistery. See also Feyerabend 2001: 89–115 in his chapter ‘‘Brunelleschi and the invention of perspective’’, and my review thereof 2000a. 24 In the case of Alberti too we see at least in practice a close connection between his studies of measurement and of perspective. His Ludi matematici applies mathematics to the measurement of distances, dimensions, and weights. The Elementa picturae describes some geometric ﬁgures and projections. In Descriptio urbis Romae he details how he has used a surveying disk similar to an astrolabe; De Statua deals with proportions in the human body and how to replicate them in sculpture (see Martin Kemp’s Introduction to Alberti 1991). 25 For this example see Geometry Forum Articles at http://www.geom.uiuc. edu/docs/forum/, ‘‘Photo Puzzles’’. 26 Having but three points A , B , C to relate to the ‘eye’ at P, we can take the degenerate cross ratio CR(A , B , C ) to be the limit of the numbers CR(A , B , C , D ) as D moves along the ground farther and farther away beyond C . That limit exists because the difference between segments D A and D B diminished then toward zero, so that the ratio D A /D B tends toward 1. CR(A , B , C ) turns out, on this construal, to be just C A /C B . 27 In this example the person is at A and the distance between B and C is one mile. The cross ratio CR(A, B, C, D) is found by measurements in the photo, and equals C A /C B , but C B = 1 (in miles). 28 To see this idea developed in a different key, see Ismael 2007. 29 For philosophical background see especially Giere 2006: ch. 4; my 1993, sect. 1; Sismondo and Chrisman 2001; Ismael 1999.

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http://www.hewett.norfolk.sch.uk/CURRIC/soc/theory.htm http://www.indiana.edu/∼intell/map.shtml 32 ‘‘Mediology: a metatheory for analyzing media and institutions’’; see http://www.georgetown.edu/faculty/irvinem/CCTP748/mediology-map.html 33 We can relate this to Lewis 1979, where he construes all assertion as selfascription of properties, and writes ‘‘What happens when he believes a proposition, say the proposition that cyanoacrylate glue dissolves in acetone? Answer: he locates himself in a region of logical space’’ (p. 518). This must be understood, in his context, in terms we need not share, e.g. that a region of logical space is a set of possible worlds; but it is not difﬁcult to assimilate his general conception of assertion and belief in our own terms. 34 In view of the strictures I will relate below, I try to be careful not to use ‘‘perspective’’ terminology in this section—while self-location is a crucial hallmark of perspective, the notion of perspective involves more than that, and we need to be careful to distinguish the precise characteristic under discussion. 35 There are obvious connections here with the literature begun with Perry 1979. Much of that literature focuses however on belief in general, which will not be the focus here; the closer relation is to Lewis 1979 which certainly concerns belief but only as one example in a large range that includes bare assertion. 36 See further Pooley 2003; van Fraassen and Peschard 2008: Part II. 37 Compare page 367 of the translation on pp. 361–72 of Kant 1992. The full text is this: ‘‘Wenn ich auch noch so gut die Ordnung der Abteilungen des Horizonts weiss, so kann ich doch die Gegenden darnach nur bestimmen, indem ich mir bewusst bin, nach welcher Hand diese Ordnung fortlaufe, und die allergenaueste Himmelskarte, wenn ausser der Lage der Sterne untereinander nicht noch durch die Stellung des Abrisses gegen meine H¨ande die Gegend determiniert würde, so genau wie ich sie auch in Gedanken h¨atte, würde mich doch nicht in den Stand setzen, aus einer bekannten Gegend, z. E. Norden, zu wissen, auf welcher Seite des Horizonts ich den Sonnenaufgang zu suchen h¨atte.’’ (Kant 1983: 995/6) 38 I hesitate to use such terms as ‘‘applied science’’ and was hesitant to insert the word ‘‘pure’’ into this discussion: these points about indexicality pertain equally well to experimental research in pursuit of theoretical goals. 39 We can think of Nelson Goodman as making this into a general point for the viewing of art works as well, with his introduction of the concept of languages of art. 40 To give but a few examples from the Oxford English Dictionary: ‘‘ F. MYERS Catholic Thoughts IV. xxxv. 359 Clearly no method can be satisfactory but that which preserves the perspective of history true.  H. DRUMMOND 31

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Lowell Lect. Ascent Man 11 Evolution . . . has thrown the universe into a fresh perspective.  J. GOULD & W. L. KOLB Dict. Social Sci. 262/1 There has been much discussion from many perspectives as to the origins and ‘‘causes’’ of fascism.  Review No. 53. 21/1 Aiming for a 100 per cent safety record is the right thing to do, not just from an ethical standpoint but also from a hard-nosed business perspective.  W. JAMES Let. 12 Dec. in R. B. Perry Thought & Char. W. James (1935) I. 727 Metaphors and epigrams which, witty and striking and perspective-suggesting as they often are, . . . may be in danger of having the changes rung on them too long.’’ 41 But note my qualiﬁers ‘‘in general’’ and ‘‘in their ofﬁcial formulation’’—I am not ruling out examples that bespeak the contrary of the general claim. 42 To give one example from epistemology: as De Finetti emphasized, subjective probability is more realistically modeled by ﬁnitely additive functions. But that is not a very tractable space, so one closes it by adding in limit points for certain kinds of sequences, thus embedding it in the much more tractable space of probability measures. 43 The tale I mean is largely told in Ryckman 2005, though from a transcendentalist rather than empiricist point of view. Part II: Windows, Engines, and Measurement 1

cf. Maddy 2007, p. 2, describing a character who is thoroughly at home in current science ‘‘She uses what we typically describe with our rough and ready term ‘scientiﬁc methods’, but again without any deﬁnitive way of characterizing exactly what that term entails. She simply begins from commonsense perception and proceeds from there to systematic observation, active experimentation, theory formation and testing, working all the while to assess, correct, and improve her methods as she goes.’’ 4. A Window on the Invisible World (?) 1

I have here beneﬁted greatly (although our use of the terms ‘‘representation[al]’’ and ‘‘image’’ is not quite the same) from Pitt 2005. 2 Quoted Alpers 1983: 7. The Latin text makes clear, in the preceding sentences, that Constantijn Huygens (not to be confused with his more famous son Christiaan Huygens) conceives of the experience with the microscope as a case of seeing. 3 Heidelberger 2003. For an illuminating discussion that relates Heidelberger’s conceptions to those of Baird and Harr´e, see Boon 2004. 4 This is precisely what Pitt 2005 discusses critically, addressing how the results are presented in nano-technology literature. Contrast his discussion with the view argued in Hacking 1985 and my reply to that paper in the same volume.

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See Boon, op. cit. Nancy Cartwright’s [1999] concept of a nomological engine was certainly also one of the factors inspiring me to see putative ‘observation by instruments’ as involving engines of creation, although her concept is at the same time different from, as well as related to, that of instruments as providing engines to produce phenomena. Speciﬁcally, her notion of a nomological engine cannot be understood apart from some grasp of her notions of capacity and necessity. But the examples are of the creation of processes that instantiate certain patterns with a faithfulness achievable in the laboratory and not found in an uncontrolled natural setting. 6 See further Anderson 1993. 7 In summary, Mieke Boon offers the following classiﬁcation that covers Heidelberger’s among others: ‘‘Simplifying somewhat, I assume that the categorizations proposed by these authors share enough features to be merged into three types, which I shall call Measure, Model, and Manufacture. Measure is a category of instruments that measure, represent or detect certain features or parameters of an object, process or natural state. Model is a type of laboratory systems designed to function as a model of either natural or technological objects, processes or systems. Manufacture is a type of apparatus that produces a phenomenon that is either conjectured from a new theory or not as yet theoretically understood.’’ (Boon 2004: 223) 8 The phrase ‘‘creating new phenomena’’ comes from Hacking 1983. But although his illustrations of this idea are grist to my mill, it is clear also that there are real differences in what we mean by this phrase. As I use the words, ‘‘observable phenomena’’ is just an emphatic way of saying ‘‘phenomena’’, for by that word I refer to all and only observable, i.e. perceptible, objects, events, and processes. See further the discussion of phenomena versus appearances in Part IV. 9 Gilbert is said to have demonstrated to Queen Elizabeth I his theory of the Earth’s magnetism: placing a small compass at various places around the terrella, Gilbert showed that it always pointed north–south—offering this as a model to explain why on Earth a compass points north–south. (Actually, even if the Earth’s core is iron, it is too hot to be a magnet.) 10 For discussion both of the experiment and complaints about its poor use in reﬂections on methodology, see Worrall 1989 and Cantor 1989. 11 Cantor 1989 remarks that the phenomenon had actually been observed almost a century earlier. Its use to illustrate how new phenomena are created by experiment does not depend on Poisson being the ﬁrst one to create it, of course. Even less does its being new have anything to do with the methodological value or lack of value of novelty in predictions, if only because being new does not imply that it should have been surprising.

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This way of presenting the story of an experiment, emphasizing creation rather than observation, may seem to underplay the role of measurement in empirical research. It does not: when such experiments are done—and today it is rare for them to consist of a single clearly recognizable observable phenomenon that can play such a role—the result takes the typical form of a summary presentation of measurement results. So we will not have a clear and synoptic view of experiment and its relation to theory until we have clarity on the topic of measurement itself. For further comparison and views on the roles of creation as well as inspection of phenomena see for example Galison and Assmus 1989. 13 The idea that scientiﬁc instrumentation undermines the observable/ unobservable distinction, touted especially with the advent of scientiﬁc realism in the 1960s, is thus not exactly new! 14 In addition to acknowledging my debt to Catherine Wilson’s The Invisible World, I would like to thank the University of Oklahoma, and especially Dr. Kerry V. Magruder, for access to the History of Science Collections to obtain material from Hooke 1665 and Power 1664. 15 Although the distinctions drawn to classify instruments in science seem usually to go against this assimilation, I ﬁnd this suggestion for the microscope also in Michael Heidelberger’s ‘‘Roentgen’s apparatus was, as we might say, unconditionally productive, but there are other productive instruments that produce known phenomena—although in circumstances where they have not appeared before. I am thinking of instruments, like microscopes or telescopes . . . .’’ (op. cit: 146/147). 16 The rainbow is unlike reﬂections in the water because it is not the image of some real arch. That is important to illuminate the point below. But the more important feature is the status they both share with mirages (and share, I will argue, with microscope images), which makes them ‘‘public hallucinations’’. 17 The qualities and structure of the rainbow do reveal something about the light source; yet the rainbow is not a copy or picture of that source. 18 Hence the anecdotal evidence, in ancient sources, of the appearances of gods and spirits, which could have been produced by projecting images made by concave mirrors into smoke in a darkened room. 19 In such ‘quantiﬁer’ locutions as ‘‘something’’ or ‘‘there is such a thing as’’ or ‘‘everything’’, the word ‘‘thing’’ does not occur with any substantive meaning, but is a sort of pronomial device. In elementary logic we paraphrase ‘‘Something is . . . ’’ as ‘‘There is x such that x . . .’’ and ‘‘Everything . . . is—’’ we render as ‘‘(All x)(if x is . . .then x is —)’’. The word ‘‘thing’’ has disappeared. Two of the three occurrences of ‘‘x’’ correspond there to the relative pronoun ‘‘it’’. But the ﬁrst occurrence of ‘‘x’’, corresponding to the ‘‘thing’’ part of ‘‘Everything’’,

364  :   does not play a different role from the others. (This is clearer in combinatory logic: a universally quantiﬁed sentence says that a certain predicate has universal application.) In venerable terminology, use of ‘‘thing’’ in ‘‘something’’ is not categorematic but syncategorematic. 20 Let me explain how I understand this. My experiences are the events that happen to me of which I am aware. Such events have two sides, so to say: what really happens to me and the spontaneous judgment I make in response, which classiﬁes that event in some way. In good cases the two coincide, but often they do not. For example, I trip over a marmot but take it to be a cat. What happened to me was that I tripped over a marmot, but I ‘experienced it as’ tripping over a cat. See further my 2002: 134–6. 21 The most sustained objection along this line is that of Hacking 1981, which actually also details tellingly the differences between ‘‘seeing through’’ an optical microscope and a magnifying glass, but still argues for the realist conclusion. See further my reply ‘‘Ad Ian Hacking’’ in the same volume. 22 Besides the above reply to Hacking 1985, see e.g. my 1982a, and the critique of Inference to the Best Explanation in my 1989, chapter 6, and in my review of Lipton 2005. 23 While I cannot go further with this here, I mean to echo at least some aspect of the transcendentalist story of the constitution of theoretical objects when such correlations appear. 24 The discussion of images, such as are projected on screens or by reﬂections in water, certainly introduces a sort of observation report that I did not have in mind while writing The Scientiﬁc Image. Then I was thinking quite simply in terms of a classiﬁcation of objects, events, and processes as observable and unobservable. 25 John Bell 1990 said famously that ‘‘ordinary quantum mechanics is just ﬁne FAPP’’ (introducing this abbreviation for ‘‘for all practical purposes’’), while criticizing interpretations of the theory as unsatisfactory if they remain at that level. 26 To continue the preceding footnote about experience: to have the experience of seeing a paramecium means here to have the spontaneous judgment that one is seeing a paramecium, in response to what is happening to one, namely to have one’s eye pressed to the eye-piece of a microscope. 27 His footnote at that point indicates, however, that he wrote this passage in response to a comment by a philosopher, Christopher Hitchcock. In any case, Weinberg’s opinion of philosophy should not make us discount his own philosophizing. 28 Once you do, however you do it, we will see that you have a view of science instantiating the same pattern as constructive empiricism. 29 I won’t elaborate on this here, but empiricist positions on philosophy are stances, and the lines drawn for any useful distinctions are sensitive to value,

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purpose, use, and other contextual factors. With respect to science, as to other such important, pervasive aspects of our culture and civilization, the values an empiricist pursues require rendering it intelligible without the need for metaphysical underpinnings. 30 For a contrasting, realist way of appreciating this harmony, see Norton 2000 and 2003. 5. The Problem of Coordination 1

The contrary opinion, that the new terms do admit of explicit deﬁnitions that use only old terms, was argued by David Lewis 1970; for critique see my 1997b. 2 For Mach see further below; the reference is to ch. II, especially sections 11 and 12, of Mach 1986. 3 Michael Friedman 2002 describes Moritz Schlick’s Space and Time in Contemporary Physics (1917) as follows: ‘‘Schlick portrays the variably curved space-time of general relativity as an entirely abstract, entirely non-intuitive ‘conceptual construction’, which can only be related to experience and to the physical world by an entirely abstract, entirely non-intuitive relation of ‘designation’ or ‘coordination’ by which the purely mathematical ‘conceptual construction’ . . . can then receive empirical content by being interpreted in terms of physical measurement.’’ (p. 136). See further the extensive discussion in Ryckman, 2005, sections 2.4.1 and 3.2. Thanks also to Anja Jauernig for helpful discussion of how Schlick used and understood ‘‘coordination’’ in his Algemeine Erkenntnislehre. 4 This well-known essay is illuminatingly analyzed in Ryckman (who cites this passage) 2005, section 3.3. 5 For a sustained critique of equivocations in Reichenbach’s discussion of coordination in The Philosophy of Space and Time, see Klein 2003. 6 The chapter from which I quote is called ‘‘Cognition as Co¨ordination’’. 7 The set-theoretic context he chose to make this point is not at all essential to the point. Suppose we use only algebraic terms and ask about two entities whether they are isomorphic, say, or whether the second can be homorphically embedded in the ﬁrst. As thus stated, the question has no sense. At best it is elliptic, for the terms are context-dependent. The question receives different answers depending on the parameters selected as contextually relevant. Take for instance a particular Hilbert space and ask whether the family of a certain kind of Hermitian operators on it is isomorphic to a given permutation group. The former may indeed form a group, under certain operations, and considered only as such, be isomorphic to the latter. But obviously by considering it only as such we are ignoring a great deal—the permutation group deﬁned in terms of shufﬂing operations on a domain has nothing to do with Hilbert space.

366  :   So the assertion or denial of isomorphism depends on a certain selection on our part. In the case of two mathematical objects we can make the selection in a straightforward way, since they are already ‘given’ in a format which lends itself to us. Given a particular Hilbert space and a family of operators on it singled out by some equations, the relevant questions can obviously be formulated: for example, does this family contain an element I such that for all its members X, IX = XI = X? But how do I formulate questions of this sort for a part of nature, without using a selective description of it that already rests on a ‘mathematization’? 8 I am deeply indebted to the writings of Michael Friedman, Flavia Padovani, and Thomas Ryckman for my understanding of this episode (though of course they are not to be reproached for any misunderstanding on my part). 9 In the case of classical mechanics, the axioms of coordination are Newton’s laws of motion, and his law of universal gravitation (which can only be formulated in the context where the former are given) is an axiom of connection. 10 It is instructive here to see how, almost just in passing, Reichenbach touches on ways to back this up: ‘‘Not only the totality of real things is coordinated to the total system of equations, but individual things are coordinated to individual equations. The real must always be regarded as given by some perception. By calling the earth a sphere, we are coordinating the mathematical ﬁgure of a sphere to certain visual and tactile perceptions that we call ‘perceptual images of the earth,’ according to a coordination on a more primitive level. If we speak of Boyle’s gas law, we coordinate the formula p.V=R.T to certain perceptions, some of which we call direct perceptions of gases (such as the feeling of air on the skin) and some of which we call indirect perceptions (such as the position of the pointer of a manometer). The fact that our sense organs mediate between concepts and reality is inherent in human nature and cannot be refuted by any metaphysical doctrine.’’ (op. cit. page 37). But under what conditions do we have a right to call those visual and tactile perceptions ‘‘perceptual images of the earth’’? It seems to me we cannot classify them as such unless we are already able to describe perceptual images by themselves on the one hand, and the earth on the other. If that condition is not met, coordinatization seems to amount to no more than equating one otherwise meaningless noise to another. 11 There is a good sidelight on the problem in Campbell 1943 where he discusses ‘‘on what experiments the Newtonian theory of dynamics is most suitably based, and in particular whether quantity of matter, mass, and force can be measured independently of that theory.’’ That the independent measurement procedures he discusses involve reliance on previous coordination of some parameters with measurement procedures is all too clear. Nor do his arguments meet the points made by Poincar´e and Sneed (see note below).

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Poincar´e pointed out that to measure the mass of an object, in the sense of Newtonian physics, we must presuppose the object to be a Newtonian mechanical system, to which Newton’s laws of motion apply. Joseph Sneed later made this insight the basis of his theoretical/non-theoretical distinction. 13 Mach provided what he called a deﬁnition of mass, for example, but to say that is to use his own words; by our present understanding of ‘‘deﬁnition’’ he did not do that—we should rather read it as another example of attempts to spell out a coordination. Patrick Suppes’ comment (1957: 298) is strictly speaking correct, but for a more balanced exposition of what Mach was up to, see Koslow 1968. 14 What about modality: the object has such and such a feature if a measurement would have outcome so and so, if the measurement were made? This pushes the problem back a step: how do such modal assertions receive empirical content? 15 Once that theory is accepted, it begins to infect the language in use; the language in which the measurement procedures themselves are described becomes new-theory-laden, and of course after a while what had been the new, novel theory sinks into the background of research (until and unless its empirical adequacy is drawn into doubt). 16 This sort of consideration has sometimes been seen to support scientiﬁc realism against empiricism. But that is a confusion: the eventually established theory classiﬁes the observable measurement processes and set-ups in theoretical terms, and that classiﬁcation is the assumed starting point when predictions are to be made via the theory from the measurement results obtained there. There is nothing un-empiricist in the remark that the ink in this book, for example, is classiﬁed in current physics as having a certain kind of molecular structure, and that predictions about its observable fading can be made on that basis. 17 Cf. Peschard 2007. 18 For the early history I take the main narrative details from Middleton 1966, which enlarges on and corrects the history that Mach himself provides in Mach 1986. There are now more up-to-date treatments (see for instance Chang 2004) but the main philosophical points I wish to take up are all in Mach’s treatise. As I will say also of Poincar´e, Mach is presenting us with a ‘‘just so’’ re-creation of the history, to make the philosophical points, even though as conscientious as he could be with respect to the historical details. 19 The entire passage from the records of the Accademia is quoted in Middleton, pp. 33–4. 20 That what is measured is still, in our terms, a combination of mutually isolable factors, remains true though. Middleton writes about Hooke’s liquid thermometers: ‘‘While it was an excellent attempt, it suffered from several disabilities: spirit of wine is not a well-deﬁned substance, its properties varying rapidly with the

368  :   amount of water in it; Hooke’s choice of the freezing point of water, rather than the melting point of ice, was unfortunate; and he did not make any allowance for the difference in expansion of brass—or silver—and glass. It may be noted that R´eaumur inherited the ﬁrst two of these sources of error.’’ (op. cit. page 46) 21 Liquid water has its maximum density at 3.98 ◦ C, and expands both if the temperature is decreased and if it is increased at that point. At exactly 3.98 ◦ C, the thermal expansion coefﬁcient of water is actually zero. Plotting density against temperature yields a graph that is a parabola. 22 Cf. Mach 1986: ch. II, section 8–10, 14. 23 quoted Mach 1986, ch. II, section 14, 54. 24 Amontons investigated the relationship between pressure and temperature in gases near the end of the seventeenth century, and his results led him to the speculation that a sufﬁcient reduction in temperature would lead to the disappearance of pressure. As Hasok Chang points out, this is a different notion of ‘‘absolute temperature’’ from that initially introduced by Kelvin, though later the two were assimilated. Kelvin was originally concerned to give an independent standard for equality of temperature intervals, not tied to the choice of one ‘‘standard’’ thermometric substance. For this purpose he drew on Carnot’s work, using the measure of work equated with the descent from a given higher temperature to a lower one: a speciﬁc amount of work could mark a difference of one degree, so that ‘‘all degrees have the same value’’. See Chang 2004: 173–86. 25 Poincar´ e, H. The Value of Science, ch. II. ‘‘The measure of time’’. The page numbers cited below are for the French edition, Flammarion 1970 (original 1905). 26 This begins section III; I am ignoring the passages in which Poincar´e concentrates on psychology. 27 Cf. my 1970: ch. III-2-c, pages 78–81 concerning Bosanquet, Russell, and Russell’s debate with Poincar´e on this subject. In 1897 Russell still states at least that ‘‘No day can be brought into temporal coincidence with any other day . . . .; we are therefore reduced to the arbitrary assumption that some motion or set of motions, given us in experience, is uniform’’ (Russell 1996: section 151, 155), but his views expressed in that debate shortly afterward were already quite different. See further Grünbaum 1968: 44f. 28 This is analogous to the question whether the spatial congruence relation is ‘‘objective’’ or ‘‘intrinsic’’, a ‘‘matter of fact’’, as e.g. Russell maintained at one point in exchanges with Poincar´e (as we will discuss in connection with the roots of Structuralism). 29 For an account of the entire expedition and its results, see Olmsted 1942–43. 30 ‘‘On admet, par un d´eﬁnition nouvelle substitu´ee . . . que deux rotations compl`etes de la terre autour de son axe ont même dur´ee.’’ (p. 43) This was in

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keeping with the ancient form of time reckoning that speciﬁed units of time in astronomical terms. Note that the 24-hour period thus deﬁned, in terms of passage of a ﬁxed star across the meridian is not equal to the older one deﬁned in terms of the sun’s passage across that meridian, with an average difference of almost 4 minutes. The latter, however, are not equal to each other if reckoned in sidereal time; ‘sun days’ are not the same during the different seasons of the year. 31 In section IV he considers the suggestion that the case of the pendulum illustrates that some general principle may be thought to justify a deﬁnition of this sort, namely the principle that identical phenomena have the same duration. But, as he points out, this is empty unless it has been spelled out what we count as identical phenomena, and if that has been speciﬁed then the ‘principle’ is an empirical assumption(Ibid., 44–5). 32 ‘‘ils d´ eﬁnissent la dur´ee de la fac¸on suivante: le temps doit être d´eﬁni de telle fac¸on que la loi de Newton et celle des forces vives soient v´eriﬁ´ees.’’ (Ibid. 46) This could perhaps be said more appropriately about Euler; see my 1985, section III-2-a. 73–4. 33 ‘‘Le temps doit être d´ eﬁni de telle fac¸on que les e´ quations de la m´ecanique soient aussi simple que possible’’ (Ibid.). 34 E. A. Milne ﬁrst made his suggestion in 1937; see his 1948: 22; Whitrow 1961: 46, 247; Roxburgh 1977. See also the discussion in Grünbaum 1952 followed by Grünbaum 1954 and Whitrow 1954: 151; Grünbaum 1968: 18–19. 35 As Fine 1986 showed clearly, Einstein’s views are not univocal; in this connection I would cite speciﬁcally his 2004a and the debate he imagined between Reichenbach and Poincar´e (Einstein 1988: 676–79). 36 See Ryckman 2005 and my 2007. 37 From the Latin ‘‘scrupulum’’ meaning a small sharp, or pointed, stone: an unfounded apprehension and therefore unwarranted fear that something is a sin when in fact it is not. 38 Would it be too polemical to call this ‘‘naïve scientiﬁc realism’’? 39 See further below for this problem in a more abstract setting, where it takes a form that threatens paradox for any Structuralist conception of science. 40 I have been using Putnam’s (perhaps somewhat derogatory) term  here, and I take the response that we are presently exploring to be precisely of the sort that Putnam called metaphysical realism. What I call Reichenbach’s problem is certainly not unrelated to the problems raised by Putnam in his celebrated ‘‘model-theoretic argument’’, which we shall have occasion to examine as well. 41 Not to mention the required previous coordination for the classiﬁcations of height and mercury, for example.

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In the technical sense of the logic of questions—the subject of what Nuel Belnap called the theorem of the ﬁfth gymnosophist: Ask a foolish question, get a foolish answer. 43 This is neo-Kantian terminology, of course, and could easily be misunderstood if taken out of context. The assertion that temperature, as we now understand it, is constituted in the historical development starting in the time of Galileo does not imply, for example, that it is not now correct to say that there were days before his time when the temperature in someone’s cellar dropped to 4 ◦ Celsius. Compare the carefully evenhanded treatment in Bitbol [forthcoming a] of Latour’s discussion of whether Ramses II died of tuberculosis. 44 This is not to deny that the theoretical representation goes beyond what can appear in any physically possible measurement. In a classical physics text there is no objection to stating a problem about a particle with mass 1/π grams. But there is no physically possible series of mass measurements that literally converges to that value, only ones that will not contradict it. 45 Just for now, let me remark on the Duhemian view that what theories provide are new classiﬁcations for observable things. To say, for example, that light is a wave disturbance in the ether is to classify observable optical effects in the taxonomy provided by classical electrodynamics. To say that these observable effects are thus classiﬁed (correctly, in that taxonomy) is not to imply that the ether is real. Peter Lipton seems to me to be confused on this point, when he suggests that a constructive empiricist must be involved in such simple self-contradictions as ‘‘my computer is on the table, which is a swarm of particles, but there are no particles’’ (2004: 146). 46 By writing in terms of a single theory cum measurement practice, stabilized in this historical process, I have also simpliﬁed the matter. There is typically a broad range of theories involved; when eventually the community achieves stability in the measurement of a theoretically identiﬁed parameter, that will generally involve concordance between a variety of different procedures, for which the harmony in their results is explained theoretically by their counting as measurements of the same quantity. The main points, about how theory and practice are entangled in this process, and how that involves elements of choice as well as appreciation of empirical regularities, remain the same. 6. Measurement as Representation: 1 1

Perhaps we’ll be less happy, however, to call it a painting of the Duke than a portrait of the Duke. On this distinction see Freeland 2007.

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Relevant to quantum mechanics is the question whether a theoretical description of this physical correlate must sufﬁce to tell us how the object ‘looked’ in the outcome. I’ll bracket that question for now, but return to it later. 3 That a claim is theory-laden does not mean that it presupposes the truth of a theory, only that theoretical terms are used in its formulation. For example, ‘‘this powder is classiﬁed in chemistry as sodium nitrate’’ is theory-laden but does not imply anything about whether the chemical theory is true or false. 4 Cf. Weinberg’s 1998 discussion of mass, and my 2002: 115–16. 5 Even if not equally salient, the situation is not in principle different in earlier physics. The thermometer, for example, is itself an object in the domain of thermodynamics, and the theory is required, for its coherence, to admit a description of the thermometer-object interaction that satisﬁes the relevant criteria for measurement. 6 Those questions can appear also even if the theory is well established and accepted; however, they will play a visible role mainly if the measurements are being made at the theoretical threshold of accuracy and precision. Examples include the gravitational disturbance by ‘test particles’ in general relativity, which can be lessened by reducing size, while in quantum mechanics reducing size increases uncertainty. 7 Perhaps I am being too charitable. If this extreme reaction ever had any plausibility, that may have been partly because it is generally only in a new theoretical context that the relevant interactions ﬁrst occur. But if the point needs to be hammered home, take a simple example: you and I inspect a paper with black marks on it, I recognize it as a Greek word and can read it while you have no idea what it is. Are we seeing the same thing? Undoubtedly. If there is any doubt, I will ask you to take a pin and trace the black marks. You may not do it in the order in which I would, but you certainly trace the same marks—so you are seeing those very same black marks. If it is objected that here I chose a context in which you and I do share pertinent concepts and apply them in the same way to the object before us, I answer that this is inevitable. No matter how far back we go, with examples of comparative ignorance or lack of similar education, there is a common background sufﬁcient to ensure communication. Or, if you like, if there is not, as with perhaps an alien intelligence, then the question of whether we are seeing the same thing doesn’t arise at all. There is no intelligible Robinson Crusoe state without prior knowledge or opinion, a tabula rasa waiting to be inscribed by bare experience. But that we see the same thing does not mean that you can see it as writing, or as Greek, let alone read it. So for me this object is classiﬁed, is assigned a location in a logical space, which may not be available to you.

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The question is rhetorical of course; in many discussions of quantum mechanics the main criterion applied is just that the outcome should predict what outcome a new measurement made immediately afterward will have. The desire to gather information about what preceded the measurement is sometimes ruled out of court altogether. My rhetorical question is not meant to indicate a taking of sides. 9 Cf. Busch, Lahti, and Mittelstaedt 1996 and my 1991, chapter 7. 10 This is admittedly a bit of an idealization, which we could ameliorate by limiting to certain ranges of states etc. 11 As Ronald Giere has perspicuously argued, the example of color vision should already tell you that things aren’t so simple. For do we really want something like this: if the object has the property that the color objectivist account mentions then the apparatus will end up with the property that the color subjectivist account mentions? One of the objections to the objectivist account is that there is no physical property, describable without reference to observers, which equates in this way to the subjectivist’s observer’s property. There is no physical property that the object has if and only if it looks blue to the observer. The most troubling phenomena for the color objectivist are the contextuality and constancy of color. Observers will report different colors when the object is placed against relevantly different backgrounds; on the other hand they will continue to report the same color while the object is subjected to different lighting conditions. There is a further difﬁculty in the common inability to identify colors independently of certain objects or kinds of objects that have them. Seeing a blue steel ball and a blue woolly sweater, we can always continue to doubt whether they really have the same color (shade of blue). Sean Kelly 1998: The second kind of dependency—the dependency of a perceived property on the object it’s perceived to be a property of—is shown by Peacocke’s example of the height of the window and the height of the arch . . . and also by Merleau-Ponty’s equivalent claim that ‘the blue of the carpet would not be the same blue were it not a woolly blue’. The basic idea is that when I perceive a property like height or color, what I see is not some independently determinable property that any other object could share; rather what I see is a dependent aspect of the object I’m seeing now. (paragraph 26). 12

I ﬁrst heard this sort of example from Simon Kochen in a seminar in Princeton. 13 Also referred to in the literature as ‘‘faithful measurement’’. 14 The ﬁrst sentence of this section was phrased more carefully than the second (which is a standard formulation), because of differences in interpretation with respect to the notion of state. (See for example my [forthcoming] on Rovelli’s Relational Quantum Mechanics.) It is more interpretation-neutral to discuss the issue in terms of physical quantities (observables) and their values only, than in terms of states. But since most discussions are in the standard formulation which

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takes the notion of physical state for granted, I will ignore this complication except when very pertinent. This remark applies also to subsequent chapters. 15 These factors can be identiﬁed in many alternative ways, when the mathematical foundations of a theory are presented. The state is typically something quite theoretical. In some presentations, however, the state s is simply identiﬁed by (or with, or through) the set {Psm : m is an observable}. On the other hand, sometimes the observables are simply identiﬁed by (or with, or through) the way in which states assign those probabilities. Also, a simpliﬁcation is typically achieved by thinking of all observables as ‘‘made up’’ of simple ones, whose possible values are real numbers, so that the variable E in Psm (E) can just range over the Borel sets of real numbers (that is, the sets formed from intervals by inﬁnitary intersections and unions). We can proceed on a level that abstracts from these different forms of representation of basic theory structure. 16 The theory must imply more than that the criterion is satisﬁed in a particular case; it must imply that this is always so for such cases. In quantum theory this means that the Hamiltonian which governs this sort of interaction is such as to guarantee that relation between initial and ﬁnal states. This point is crucial to eliminate limiting cases as counterexamples; see the discussion of an objection by Jon Dorling in my 1991: 221. 17 What I have here related in very general terms is provided in detail for quantum mechanics in the references above. The reader may naturally wonder whether our account of measurement so far helps to solve or dissolve the famous ‘‘measurement problem’’ of quantum mechanics. The answer is Yes and No. On the one hand, the account which takes for granted that surface models are produced from realizable experimental conditions, and views the empirical content of a theory as consisting in claims as to how and to what extent these surface models can be accommodated by the theoretical models, does not run into conceptual difﬁculties. On the other hand, in quantum mechanics there is a core problem—not touched here at all—that can be formulated for the physical correlate of measurement taken in and by itself (without regard to the ‘‘information gathering’’ connotation of the word measurement). This core problem is one to which we will return under the heading of ‘appearance and reality’ in theoretical description. 18 I call the replacements ‘‘surface models’’ in contrast to the data models; I will return to this subject later. 19 Veracity in Measurement does of course imply that Value Deﬁniteness is satisﬁed at least to the extent that any parameter has a deﬁnite value when it is measured, and that means always, if any occasion at all would in principle allow it to be measured. But the Criterion for the Physical Correlate of Measurement does not have any such implications by itself.

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Not without premises going beyond the one that the surface model is thus related to that state—what are needed are also premises about the sorts of systems involved, and their relations (general conditions formulated in terms of e.g. the Hamiltonians). On the assumption that the theory applies, however, these too are extrapolated from, or made with support of, previous measurement results in experimental situations. 21 By sequencing such Stern–Gerlach apparatus it is easily shown that the process does not leave the state unaffected; hence data have to be collected on different samples of the original beam. 7. Measurement as Representation: 2 1

Francis Bacon insisted, in his clarion call to the founders of the new sciences, that experience must become literate (Bacon 1994: Novum Organum I:101). By this he emphatically did not mean to include the sort of bias that practically every form of literacy has riding along with it. Yet bias is inevitable, both for good and for bad, and it is not incidental. If we can read at all, our responses are shaped by presuppositions and assumptions, prior opinion, conditioning, learning, not to mention strong intellectual commitments and norms that govern our selectivity. We read measurement outcomes through theoretically-schooled eyes; and this is appropriate as well as inevitable, while also inevitably hostage to the fortunes of later learning that shows those spectacles’ distortions. 2 For the purpose at hand we can ignore some complexities, but they will be remembered from the ﬁrst chapter. How the object is represented by the measurement outcome depends on the theoretical context in which it is read: meaning is not independent of reading. Recall the example of ‘‘burro’’, read as a word in Italian or as a Spanish word—a symbol has its meaning not absolutely but due to its role in a (contextually) given language, symbol system, or representational framework, which in scientiﬁc contexts is determined by a theory governing discourse and practice. 3 Both here and elsewhere I am using illustrations from the history of measurement; it is not to my purpose to present that history. The parts I draw on are chosen to serve as motivating illustrations for ingredients in the account of measurement that I am presenting, to some extent in contrast to views that have been proposed earlier. For the history of the concept of measurement, especially in the nineteenth century, see e.g. Michell 1993 and Darrigol 2003; but note the sometimes contentious philosophical differences in these and related historical literature. 4 Historically this may be said to begin with Ho ¨ lder in 1901; for deﬁnitive results see Suppes, Krantz, Luce, and Tversky 1989.

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At ﬁrst blush at least there are numerical scales which do not ﬁt in any of these four categories. For example, an earthquake ranked 6 in the Richter scale is 10 times stronger than one ranked 5, which in turn is 10 times stronger than one ranked 4. But this is a matter of presentation: the magnitude on the Richer scale is the logarithm (to base 10) of a magnitude on an underlying ratio scale, namely the logarithm of the combined horizontal amplitude of the largest displacement from zero on a seismometer output. Thanks to Richard Otte for raising this point. 6 The Mohs scale of hardness had its difﬁculties. Mohs, in developing his scale for the hardness of minerals, ranked minerals relative to each other by the relation ‘‘scratches’’. He selected ten minerals to represent particular points on the scale and assigned them numerals from 1 to 10. The operation of scratching does not give more than an ordinal signiﬁcance to these numbers. Mohs had assumed that his relation of ‘‘scratches’’ was transitive and asymmetrical and that the equivalence indicated by not being able to scratch one another was transitive and symmetric. This assumption ran into a problem when it was found that some minerals could not scratch each other, yet differed with respect to scratching a third mineral. But this does not refute that an ordinal scale could be deﬁned in terms of the classes of minerals scratchable and minerals not scratchable. That is, minerals could be classiﬁed as equally hard if they can scratch all the same minerals, and can be scratched by all the same minerals. Yet here too there is an empirical assumption, if it is asserted that this leaves no ambiguity in the ordering. Later attempts to arrive at a measurement scale for hardness by other operations such as microscopic measurement of the depth of a scratch made by a diamond under constant pressure or the amount of work done in grinding away a certain weight or volume of material, allowed for more quantitative comparisons. Note well: the real issue here was to ﬁnd ways to arrive at a stable ordering induced by outcomes of physical procedures applied to the minerals. 7 Suppose that x, y, z are three Fahrenheit temperatures, and x , y , z the corresponding three Celsius temperatures. If y = Nx, it certainly does not follow that y = Nx . But the ratio of |y-x| to |z-x| equals the ratio of |y -x | to |z -x |. So the usual measurement of temperature is on an interval scale, not a ratio scale. 8 For discussion see for example Darrigoll 2003, section 1.4 and 3.1.6. 9 On Stevens’s proposal, measurement scales are classiﬁed according to the mathematical group structure of the group of admissible transformations. For variants and later developments, see Ellis 1966: 58–67, Narens and Luce 1986. 10 See further Falmagne and Narens 1983. 11 This point is most salient in literature on the foundations of quantum mechanics, originally emphasized in Birkhoff, and von Neumann 1936. They also suggested the innovation of reducing modulo differences of measure zero. 12 Two qualiﬁcations: mechanics will provide different families of models for

376  :   systems with different numbers of degrees of freedom, and the relevant phase spaces are thus not common to all models. Models can be grouped by common state spaces (see Lloyd 1994: 19–20 and 35). Secondly, the conception of time, space, or space-time as a logical space for which I argued in my 1970 is of course controversial, being at odds with substantivalist theories in this area. 13 This becomes especially pertinent when the ascription of an ‘unsharp’ value in quantum theory does not arise simply because of ignorance of the ‘real’ sharp value (see Appendix pp. 312–314). 14 A well-known paper by James Bogen and James Woodward (1988)—but in a way that results in a terminology more confusing than the simpliﬁcations they criticize. It is not grammatical, it seems to me, to say that data are observed—a datum is the content of the outcome of a measurement. Nor does it seem to me to properly respect the history of the word ‘‘phenomenon’’ to say that phenomena are typically not observable. Admittedly, regimenting language always has its leeway; here I will concentrate on the word ‘‘data’’, and in a later chapter address usage of ‘‘phenomena’’ and ‘‘appearances’’. 15 The analysis may be made by the experimenter, or already automatically by the instrument which may issue e.g. a graph rather than a set of points. Thanks to Todd Harris and Paul Teller for pointing this out. 16 I am taking for granted here agreement that a frequency interpretation of probability—in contrast even to the modal frequency interpretation—is not feasible; see e.g. my 1980: ch. 6 section. 4 for critique. See further my 1979 for a discussion, in connection with Reichenbach’s view, of how geometric probability cannot be regarded as a simple extrapolation of relative frequency in the long run. 17 Latour 1999 provides an illuminating narrative for this process, from digging up the soil samples to disseminating the smooth graphs and annotated maps, in a study of the changing boundaries between savannah and forest. 18 The semantic view of theories to which this statement refers is easily misconstrued if conveyed by such a short, slogan-like statement. A model is a mathematical structure; but that is like saying that a sentence is a black mark on a page—true, but not a sufﬁcient guide to what is meant when we say that a theory provides a set of models for representation of the phenomena. I spelled this out to some extent in my 1991, sections I, 2–4; see further e.g. my 2006a. 19 Here, as almost everywhere in this book, I concentrate on the ideal case of exactness. For detailed considerations of how this can miscarry, and how we are to relate to the inexact representations we can have in practice, see Teller 2008, as well as his earlier 2001a. 20 This does not ignore the possibility that some measurement choices in PRC are mutually incompatible; in fact, that would make it easy for the family

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of conditional probability functions P(. . .|C), C in PRC, to be embedded in a classical probability function. 21 The design of the experimental or measurement set-up, in contrast to its physical construction, is however guided by theory: in that sense, theoretical models and data models constrain each other’s form. Thanks to Isabelle Peschard for insisting on this point. 22 For a more extensive discussion and references to the literature, see my 1974 and 1982b. 23 An algebra is a mathematical structure consisting of a set of elements and a collection of operators on those elements—though the term is variously deﬁned in different contexts in mathematics, so as to narrow the meaning (e.g. an algebra is a vector space with a bilinear multiplication operation). 24 As for representation in general, so for measurement: it would be useless, perhaps fatuous, to try for a deﬁnition. But in this case too, we can come to an understanding of the subject by eliciting its general features and placing them in context. 25 Weyl 1953: 8–9; quoted and discussed in Ryckman 2005: 134. 26 From section I of ‘‘The Relativity of Space’’ in Poincar´ e 1897. 27 Important to keep in mind that ‘‘invariant’’ is not an absolute term, that it is context-sensitive: invariant under what family of transformations? But the context may specify that, as it does in these examples. 28 At the risk of boredom, let’s remind ourselves that the discussion of measurement interactions in physics concern the physical correlates of measurement, and that the type of physical interaction there so characterized may well have, according to the theory, instances far beyond what human agents can construct, including e.g. interactions at the Planck level or in the stratosphere. The term ‘‘measurement’’ in that context refers solely to the general physical constraints required for an interaction to be of the sort that can be the physical correlate of a measurement properly so-called. 29 Cf. e.g. Dretske 1981 for an attempt to explicate information content in this way, and its severe critique by e.g. Timpson 2004: ch. 1; Loewer 1982; McGinn 1997. 30 When two terms with the same referent cannot be substituted salva veritate, the context is ‘referentially opaque’, a mark of intensionality, non-extensionality . 31 cf. Kroes 2000: 29; Kroes 2003: 74–8; Peschard, 2007b. 32 The point returns with a vengeance in many reactions (not only Bohr’s!) to the Einstein–Podolsky–Rosen critique of quantum mechanics. Lately this point has taken pride of place again in approaches to quantum theory such as Rovelli’s

378  :   relational quantum mechanics as well as information-theoretic approaches (Fuchs 2002, Bub 2005, Timpson 2004). 33 Thanks for this point to Angela-Adeline Mendelovici. 34 I am paraphrasing John Worrall here. 35 For exploration of this as an analogy to be drawn on in the philosophy of perception, see Peschard, 2007a. Part III: Structure and Perspective 1

This is the sense in which Charles Morris introduced the term ‘‘pragmatics’’ in the International Encyclopedia of Uniﬁed Science. 2 To call the old terms ‘‘already coordinated’’ is only a metaphor if they were not previously at some deﬁnite historical moment explicitly subject to that process. Further below I will take up Reichenbach’s problem of coordination in a more abstract form as well. 3 While perhaps every categorematic term we have today was new at some previous stage of our history, it does not follow that there was or could have been a moment when meaningful language was created from nothing. With scientiﬁc terms being our concern alone, we do not need to speculate on how language came into the world; we must only become clear on how new terms can ﬁnd their use when explicitly introduced for practical and theoretical reasons. For the way David Lewis responded quite differently to the basic problems in philosophy of science (though mainly as posed by Carnap and Putnam) see my detailed examination of his take on this issue in my 1997b. 4 Much of the probl´ ematique encountered here is found, mutatis mutandis, in the ‘Ramsey sentence’ literature. I will not take that up, but see my 1997b, where I examine David Lewis’ attempt to deﬁne theoretical terms by adapting Ramsey’s move. 5 My phrase ‘‘reach for the unconditioned’’ is of course meant to echo Kant’s diagnosis of what he calls the transcendental illusions, the illusions of reason: such illusions are made inevitable by reason’s tendency to seek the unconditioned, that is, to carry a series of ideas or questions or arguments to their ‘logical conclusion’ even when their completion would lie clearly beyond the bounds of sense. 8. From the Bildtheorie of Science to Paradox 1

For the name ‘‘Bildtheorie’’ and speciﬁcally the development of this view by Boltzmann see especially Blackmore1999; Henk de Regt 1999 and 2005; Visser 1999; section 3.4 of St¨oltzner 2003. See also D’Agostino 2004 and Leroux, 2001. The entire period with all the relevant dramatis personae is surveyed in chapter V

  BILDTHEORIE    

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of Cassirer 1950. Today the term seems to be used mainly for the study of visual media, both in aesthetics and in communication theory, but in philosophy also for Wittgenstein’s ‘‘picture theory’’ of meaning in the Tractatus. 2 Nyhof 1988 argues convincingly against the claim that Boltzmann later converted to, or at any point held, a purely positivistic or instrumentalist conception of science, as was sometimes claimed. (This article is otherwise permeated with a philosophy of science that, as I see it, tends to color his account of the period.) I would add here also that for constructive empiricism, which is a view of what science is and not a view about what exists, the question of whether atoms are real is a bit of a red herring, just as for Boltzmann’s philosophy of science. 3 Translations of the entire paper can be found in Planck 1994 and in Toulmin 1970. 4 This sort of conﬂation of the philosophical issue of scientiﬁc realism with the question of the reality of unobservable entities is a recurrent problem in the literature, and accounts for some of the puzzlement philosophers on both sides of the debate tend to express concerning the others’ views. 5 Early in the nineteenth century, the term still meant something like ‘‘look of the world’’, and not ‘‘worldview’’. But by the time Dilthey introduces its most famous use, he can only make a rather subtle distinction between Weltbild and Weltanschauung. (Thanks to Anja Jauernig for help with the history and etymology.) In Planck’s lecture it is not so clear whether he is attempting to depict the world as, according to him, it is depicted in physical theory or rather what he takes to be the Weltanschauung proper to the physical sciences. 6 Though Boltzmann was undoubtedly indebted to and inspired by Mach’s more radical view, he argues convincingly that his own ‘picturing’ view of science is more informative than Mach’s phenomenalism—see e.g., his 1902. 7 For Boltzmann’s attribution of this view to Maxwell, see especially Boltzmann 1974: 217–19; for his references to Hertz, 214, 225. 8 For an early appreciation of the role of higher-order similarities in structural representation, see Sellars 1965: 180–2. 9 ‘‘Wir machen uns Scheinbilder der a¨ ußeren Gegenst¨ande, und zwar machen wir sie von einer solchen Art, dass die denknotwendigen Folgen der Bilder stets wieder die Bilder seien von den naturnotwendigen Folgen der abgebildeten Gegenst¨ande’’ Hertz [1894/1956]: 1–2; see also p. 2–3 and 177, and discussion by Leroux 2001. 10 The diagram will, I hope, make clear what is meant, but see further my 1989: 258–61 concerning symmetry.

380  :   11

Schr¨odinger 1929: 16; cited D’Agostino 2004: 381; see also Schr¨odinger

1928. 12

Schr¨odinger 1951: 40; cited Bitbol 1996: 29; and see further Bitbol 1996, passim, and Schr¨odinger 1953. The ﬁrst part of this passage does not attribute continuous spatio-temporal trajectories for particles; it refers solely to the fact that the wave function of a system is deﬁned with spatial coordinates (though it is not a wave in 3-space). 13 The reference is to Aristotle, Meteorology I, 7. In the next section (proposition 205), Descartes submits that under these conditions we can have ‘‘moral certainty’’ (‘‘a certainty sufﬁcient for the conduct of life’’). This could certainly be cited as presaging (an admirably modest version of) the Rule of Inference to the Best Explanation. 14 In medieval paintings of the Annunciation we sometimes see an attempt to assimilate the action of the Holy Spirit to optical or mechanical interaction; see for example Steinberg and Edgerton 1987 on Filippo Lippi’s Annunciation in London. 15 For textual support see Blackmore 1999. Blackmore, a prominent Boltzmann scholar, is very unsympathetic to the anti-realist leanings and makes a point of showing that Boltzmann is careful to hedge his bets on this account. 16 Here I am much indebted to Hyder and Lübbig 2000 and Hyder 2003. 17 Heisenberg 1945, p. 36; cited in translation in Cassirer, op. cit., p. 117. There is a clear echo here of Bohr’s repeated insistence that the wave function is no more than a summary of what will be observed in measurement arrangements which themselves are described in our common language in use before the advent of atomic physics. 18 Hertz 1962: 20–1. 19 When Poincar´ e later says that Maxwell had shown that there must exist mechanical models of electromagnetism, he presumably thought that this subsumption under the generalized Lagrangian mechanics was successful. This already derives its sense at best from several successive weakenings of what can count as a mechanical model. 20 Note however the vagaries of linguistic change in science. Stein (1989: 57) makes a good case for holding that the retention of the word ‘‘atom’’ and discarding of ‘‘ether’’ was historically arbitrary: ‘‘our own physics teaches us that there is nothing that has all the properties posited by nineteenth-century physicists for the ether or for atoms; but that, on the other hand, in both instances rather important parts of the nineteenth century theories are correct.’’ 21 Hertz 1956: 1–3; see also p. 177. 22 Poincar´e 1952: 161; in view of Poincar´e’s literate and literary style we may well think of historical precedent for this sentiment in e.g. Condillac’s

  BILDTHEORIE    

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‘‘[C]omment nous assurer que les principes que nous imaginerions, sont ceuxmêmes de la nature? Et sur quel fondement voudrions–nous qu’elle ne sache faire les choses qu’elle nous cache, que de la mani`ere qu’elle fait celles qu’elle nous d´ecouvre ? Il n’y a point d’analogie qui puisse nous faire deviner ses secrets ; et, vraisemblablement, si elle nous les r´ev´eloit elle-même, nous verrions un monde tout diff´erent de ce que nous voyons. [. . .] C’est que l’imagination voit tout ce qu’il lui plaît, et rien de plus’’ (Condillac 1749/1798/1949: 197–8; cited and discussed in Vuillemin 2005). 23 My translation; ‘‘Nulle th´eorie ne semblait plus solide que celle de Fresnel . . . Cependant, on lui pr´ef`ere maintenant celle de Maxwell. Cela veut-il dire que l’œuvre de Fresnel a e´ t´e vaine? Non, car le but de Fresnel n’´etait pas de savoir s’il y a r´eellement un e´ ther, s’il est ou non form´e d’atomes, si ces atomes se meuvent r´eellement dans tel ou tel sens ; c’´etait de pr´evoir les ph´enom`enes optiques’’ Poincar´e 1968: 173. 24 This view, expressed by John Worrall, was named ‘‘epistemic structuralism’’ by James Ladyman. 25 There is an often mentioned bit of support for such views, already alluded to by Maxwell’s remarks about fruitful analogies across different areas of physics. Important equations tend to recur in many places. They tend to identify recurrent patterns in nature, found not once but many times. Often a new process is ﬁrst described in analogy to an old one, with the equations transposed or reinterpreted. Heat diffusion and gas diffusion are analogous, the harmonic oscillator crops up everywhere. . . . So the equations omit the distinguishing characteristics. As a reason for structuralism, this observation does not show much at all. For whenever we see the same equations describing two scientiﬁc subjects, we also see science describing the differentiating characteristics. If we didn’t, we wouldn’t have an example to give! The point that such equations describe at once many different processes needs serious reﬂection, but it is not much of an argument for any general view. 26 It is exactly the difference he outlines between the technical language of a practical profession or craft, and scientiﬁc language: only the former can we interpret by identifying the technical terms’ referents (Duhem 1962: 147–53). 27 As I see it, this applies to all the sciences, not just physics: biology and the social sciences too involve construction of models that are essentially abstract, hence mathematical, structures. I realize that this view rests on a view of what mathematics is, how it encompasses all abstract representation used in science. 28 This insight, and the problem it raised, is found quite explicitly in Carnap’s Aufbau (some ﬁve years before Weyl’s lecture) but with enormous resistance to its import (as we’ll see below). The famous objection by Newman to Russell’s

382  :   structuralism, at about the same time, trades on a speciﬁc instance of what Weyl states here with complete generality. 29 We will need to re-examine this point carefully below, since assertions of isomorphism are context-sensitive—a group isomorphism between two mathematical objects certainly does not imply that they are identical, for example, for although they are both groups, they may have other characteristics as well. Weyl’s point is telling only when there are no mathematically describable features left out of account in the isomorphic mapping. 30 In the novel it seems she has simply been told that her ﬁngernails are pink. She could have also done an experiment to see what a spectrometer would show about the light reﬂected from her ﬁngernails, so that she would know the wavelength of the reﬂected light. Such variations do not affect the problem. 31 The crucial clue here, as in our discussion of the use of maps and models in Part I, I see in the word ‘‘this’’ in Mary’s ‘‘this is pink’’. For a similar line, with respect to the philosophy of mind issues for which the Mary example was devised, see Ismael 1999. 9. The Longest Journey: Bertrand Russell 1

In this section I am drawing on the more extensive account by Torretti 1984: 52, 63–4, 74, 149–52. 2 The result was not perfectly satisfactory. Poincar´e’s models of the hyperbolic plane (see note below) are considerably more user-friendly. Beltrami himself asserted that no similar interpretation is possible for hyperbolic 3-space in Euclidean 3-space, presumably because he had provided a Euclidean surface extended in three dimensions to model the hyperbolic plane. The model also has singularities, and Hilbert proved later that the hyperbolic plane cannot be mapped isometrically onto a Euclidean surface with no such singularities. 3 Russell, 1897: chs II and III. At the end of ch. II he writes ‘‘Spaces without a space-constant, . . . spaces, that is, which are not homogeneous throughout, we found logically unsound and impossible to know, and therefore to be condemned a` priori. The constructive proof of this thesis will form the argument of the following chapter.’’ 4 Poincar´ e reviewed the book in 1899 and Russell replied (see reference in the next note). Russell’s reply (similar to Frege’s point of view, see below), is in effect that the axioms of geometry are statements capable of being true or false, which presupposes that their terms designate; what those designata are like determines the truth value. 5 ‘‘La question de M. Poincar´ e me place dans la situation d´esavantageuse d’un e´ colier a` qui l’on demanderait d’´epeler la lettre A, en lui d´efendant d’employer

  :  

383

cette lettre dans sa r´eponse. Si cet e´ colier e´ tait math´ematicien, il r´epondrait tout bonnement : A est la lettre qui pr´ec`ede B; et si on lui demandait d’´epeler B, il dirait que c’est la lettre qui suit A. Mais s’il sait vraiment ce que c’est qu’´epeler, il renoncera a` la tache, de d´esespoir.’’ (Russell 1899: 701) 6 Frege elaborated his similar response in a controversy with Hilbert during the years 1899–1906. (See Coffa 1986; Torretti 1984: ch. 3, section 2.10.) The theory of space must be non-vacuous and true. But the question of truth can’t arise unless the primitive terms have an independent meaning which ﬁxes their reference. They certainly cannot get that meaning from the axioms or theorems, as Hilbert asserted. For those axioms and theorems are incapable of having a truth value unless their terms have referents. Pressed to explain how we identify those real spatial relations, Frege also retreated to intuition or direct acquaintance: ‘‘I give the name of axioms to propositions which are true, but which are not demonstrated, because their knowledge proceeds from a source which is not logical, which we may call space intuition’’ (in correspondence with Hilbert; quoted Torretti 1984: 235). This was of course just the sort of thing which Poincar´e and Hilbert explicitly professed to not understand. 7 Russell 1901; see speciﬁcally pp. 313–14; this passage was spoofed in Philip Jourdain’s The Philosophy of Mr. B*rtr*nd R*ss*ll ( Jourdain 1918: 84–5). 8 As far as Russell’s view at this point goes for concrete entities, that includes only sense data (and possibly the self ), but that point does not play much of a role here. The objects postulated by physics are beyond the reach of experience; that is the only relevant point here. 9 The importance of this article, its devastating import, and its relation to more recent discussions of realism, was pointed out by Demopoulos and Friedman. 10 We can also easily put the point in terms of models and equations. Suppose some equations have a model in which there are N distinct entities (where N may of course be uncountably inﬁnite). Choose a set of the same cardinality in the world. Because same cardinality implies the existence of a correspondence, we have an implicit transfer of the relations in the model to that chosen set. Therefore the world satisﬁes those equations! Provided only the world’s size is large enough, experimentation is superﬂuous. As David Lewis realized (and as we shall take up below), Newman’s point becomes simply Putnam’s model-theoretic argument, once we phrase it in terms of models and equations. 11 Or so it seems. In the next chapter we will see how David Lewis denied this very assertion. 12 Since we are here speaking of a repair being carried out within the context of Russell’s own epistemology, note that the only things with which we are acquainted there are our sense data and those Universals (and possibly our selves).

384  :   But we can presumably adapt the same moves to a more liberal view according to which we are acquainted with all the observed phenomena as usually understood. 13 In the discussion of Putnam’s paradox, we will see this question returning in a more general way. The answer is yes, the same problem does return even if one retreats from Russell’s extremism with respect to non-logical terms. 10. Carnap’s Lost World and Putnam’s Paradox 1

Michael Friedman, addresses the same sections of the Logische Aufbau that I shall examine here, in Friedman 1987: sections II and III. Friedman’s purpose is not to examine the perils of structuralism but to explore Carnap’s epistemology and his relations to the neo-Kantian tradition (continued in ch. 6 of his 1999). Here he makes clear just why Carnap, in his pursuit of objectivity for knowledge, cannot take the option of allowing the reference or extension of theoretical terms in science to rest on ostension. That this option is excluded for Carnap I’ll take for granted here. 2 I want to thank Daniel Rothschild for directing me to this and for much helpful discussion. See his insightful paper ‘‘Structuralism and reference’’. 3 Here we see the pattern of reaction that Newman suggested to Russell: to view a scientiﬁc theory as quantifying only over a selection of relations, the ‘‘important’’ ones in some sense. At the same time we see Carnap, like Russell, indicating a selection that is somehow connected to the possibility of experience, though with so little elaboration that it does not do much more than pay lip service to what an empiricist should honor. 4 In his own commentary on this move, Michael Friedman calls it an ‘‘extraordinary suggestion’’ (1999: 103), and asks ‘‘But what can the ‘experienceable, ‘natural’ relations’ be except precisely those relations somehow available for ostension?’’ (ibid.). 5 I’ll quote here the argument as it was originally presented to the APA in 1976 (in which that core is not isolated): ‘‘So let T1 be an ideal theory, by our lights. Lifting restrictions to our actual all-too-ﬁnite powers, we can imagine T1 to have every property except objective truth—which is left open—that we like. E. g. T1 can be imagined complete, consistent, to predict correctly all observation sentences (as far as we can tell), to meet whatever ‘operational constraints’ there are . . . to be ‘beautiful’, ‘simple’, ‘plausible’, etc. . . . I imagine that   has (or can be broken into) inﬁnitely many pieces. I also assume T1 says there are inﬁnitely many things (so in this respect T1 is ‘objectively right’ about  ). Now T1 is consistent . . . and has (only) inﬁnite models. So by the completeness theorem . . ., T1 has a model of every inﬁnite cardinality. Pick a model M of the same cardinality as THE WORLD.

  

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Map the individuals of M one-to-one into the pieces of  , and use the mapping to deﬁne relations of M directly in  . The result is a satisfaction relation SAT—a ‘correspondence’ between the terms of [the language] L and sets of pieces of  —such that theory T1 comes out true—true of  —provided we just interpret ‘true’ as TRUE(SAT). So whatever becomes of the claim that even the ideal theory T1 might really be false?’’ (Putnam 1976: 485; Putnam 1978: 125–6) 6 This theorem was initially debated in the context of philosophy of mathematics, in connection with Skolem’s relativism. One spoke there also of the Loewenheim–Skolem paradox. 7 Thanks to Jenann Ismael for striking examples of this sort. 8 It might be objected that the map, like Paris itself, has much additional structure, while a model has only the structure it displays. That is not a good disanalogy, for the model also displays a selection of the structure it has. Suppose the model is D, F with D its domain and F a family of sets and relations on D. Then F is of course a selection from the set of sets of n-tuples, for various numbers n, of members of D. All those other sets of sets there are selectively excluded from F. 9 This is just the sort of example that must have led Leibniz to agree that abstract entities can violate his Identity of Indiscernibles principle. 11. An Empiricist Structuralism 1

I begin with the title of E. J Dijksterhuis’s 1969 magisterial work that stops with Newton; recall though that the ‘‘world-picture’’ phrase was current only in that later tradition. 2 See my 2006b for a discussion of the recent Structural Realisms, due to John Worrall and James Ladyman, and a defense of an empiricist structuralism with reference to the criteria that Worrall laid down for a successful realism. 3 That ability is according to Dieks and De Regt 2004 the hallmark of understanding a scientiﬁc theory. 4 For exposition and critique see my 2006b. See further Ladyman 1998a, 1998b; Da Costa and French 1990; Bueno 1999a, 1999b. 5 Newman’s argument against Russell’s structuralism, which we saw also at the heart of Carnap’s difﬁculties and of Putnam’s Paradox, is explicitly drawn on in the critique of the semantic view in general, and constructive empiricism in particular, in Demopoulos 2003. 6 The semantic view of theories runs into severe difﬁculties if these notions are construed either naively, in a metaphysical way, or too closely on the pattern of the earlier syntactic view. Constructive empiricism, which was formulated in the

386  :   context of the semantic view of theories, shares this vulnerability (cf. Demopoulos 2003). I began to face these difﬁculties in my 1997a, 1997b, 2001, and mean to complete their dissolution here. 7 Easy to see, I think, the echo of this passage in Reichenbach’s student Putnam’s ‘‘This simply states in mathematical language the intuitive fact that to single out a correspondence between two domains one needs some independent access to both domains’’ which I quoted above. 8 Mea culpa: in The Scientiﬁc Image constructive empiricism was presented in the framework of the semantic view of theories, but seemingly in the shape of the above ‘‘offhand’’ realist response. See for instance ch. 3 section 9, p. 64 where I deﬁne empirical adequacy using unquestioningly the idea that concrete observable entities (the appearances or phenomena) can be isomorphic to abstract ones (substructures of models). Demopoulos 2003 comments on this passage. That this ‘‘offhand’’ way of talking is most readily interpreted in metaphysical terms is clear also in comments by such acute and careful readers as Stathis Psillos (2006, remarking on correspondence and structures in re) and Michel Ghins (Ghins 1998: 328). Rather than try to excuse or explain my obliviousness to these issues at that time, I tried to do better in my 1997a, 1997b, 2001, and I will try to do better here. 9 Admittedly, this conception is quite a natural one to ﬁnd in the foundations of physics, for physical systems in the intended domain of a theory are conceived of as highly structured, and the entire discussion is targeted on mathematical models. 10 In his preface Whitehead says that he took the term from an earlier use: ‘‘The general name to be given to the subject has caused me much thought: that ﬁnally adopted, Universal Algebra, has been used somewhat in this signiﬁcation by Sylvester in a paper, Lectures on the Principles of Universal Algebra, published in the American Journal of Mathematics, vol. vi., 1884. This paper however, apart from the suggestiveness of its title, deals explicitly only with matrices.’’ (Whitehead 1898) 11 This point is made and explored by Psillos 2006. 12 The motto of the British Order of the Garter. When Edward III danced with the Countess of Salisbury her garter fell off. Edward said ‘‘Honi soit qui mal y pense,’’ (‘‘shame on whoever thinks ill of it’’) and tied the garter around his own leg. 13 It seems to me, though, that this line of thought is disappearing from the scene: Stephen Leeds (1978, 1994, 1995, forthcoming ) shows that the scientiﬁc realist does not need to rely on a correspondence theory of truth; lest we think that metaphysical realists must all need it, there is David Lewis 2001. 14 I won’t do much to argue that here; if a reference is needed beyond the above discussions of the correspondence theory of truth, my own favourite is Quine’s ‘‘On what there is’’.

   15

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See e.g. the papers by Leeds and Lewis cited above, or the enormous further literature on the subject. 16 This is essentially the point raised by Mauricio Suarez which I took up in Chapter I-1. 17 Giere 1988: chs. 3, 7, 8; Giere 1999: chs. 6, 7; Teller 2001, discussed in my 2001. 18 There is an echo here of the main point of virtue epistemology: for some information held to count as knowledge, not only its content but the history of its acquisition is crucial. The point is different but the re-direction of attention, to how the item is arrived at, is the same. 19 It may be objected that a given model may not have the requisite complexity to represent a particular phenomenon. But representation is selective: the most we can say here is that a given model may not have the requisite complexity to represent certain aspects of a particular phenomenon in a particular way. Nevertheless, that does qualify my assertion; take it as thus amended. 20 This assumes of course that we have ways of describing the phenomena—we already have a language that we live in, as I said. We must once again resist the ‘just born Robinson Crusoe self’ picture, in which we are confronted by a world which is neither thus nor so, as yet indescribable by us, etc. 21 Perhaps Reichenbach means something like this when he says that reality (i.e., the phenomenon that we are describing) is ﬁrst deﬁned by the coordination. But it could only be ‘something like this’, since this way of putting it seems to imply that what the models represent is something like the physical objects as described rather than the physical objects. The ‘‘as’’ in that phrase is the traditional ‘‘qua’’, and one is meant to both identify and distinguish the referents of ‘‘the X, which is F’’ and ‘‘the X qua F’’—not a resource the later empiricism can draw on, to put it mildly. I want to thank Anja Jauernig for help in my effort to understand Reichenbach’s 1920 attempt to forge a modiﬁed neo-Kantian view of the matter. 22 The simplifying assumptions are strong of course: consider just two bodies, the mass of one of them much less than the other’s, and with the distance between them remaining limited. Newton’s laws admit other solutions even for the twobody system and also quickly allowed corrections to Kepler’s laws, taking into account that the smaller mass is not negligible even for a planet in relation to the sun. 23 That is not to deny that there is a pertinent 3-place relation that can be described, assigned a set as its extension, and so forth: namely a relation between the scientist, the bacteria colony, and the data model this scientist constructed. The important point is that an indexical sentence is not meaning-equivalent to an

388  :   non-indexical one, except within or relative to a context of use, and it can have uses which cannot be served by any non-indexical sentence. 24 As Psillos 2006 also emphasized. 25 A response entirely at odds with empiricist scruples would be to escape into metaphysics, by postulating some nexus to link representations to what they are representations of, independent of our fragile practices, hoping to make our tenuous grasp on reality secure. 26 As analogy consider the two questions ‘‘Is snow white?’’ and ‘‘Is our sentence ‘Snow is white’ true?’’ They are not the same question. After all ‘‘Snow’’ could have been our word for grass, water, or wine. But given that this sentence is our sentence, as things actually are, the questions amount to the same thing for us. That is why for us, who speak this language, ‘‘The sentence ‘Snow is white’ is true if and only if snow is white’’ is a pragmatic tautology. 27 It may be easy to see the similarity to Descartes’s ‘‘I think’’, for which he demonstrates not the necessity or truth but the indubitability for the speaker of the ‘‘I’’, so that the sentence ﬁts well into the category of pragmatic tautologies. It seems to me that the same holds for Putnam’s ‘‘I am not a brain in a vat’’ in view of Putnam’s own form of refutation of this sort of scepticism. 28 Now we can see why the offhand realist responds sounds plausible at ﬁrst, because it does get something right—namely that in the end there is no problem, precisely because we can (a) correctly describe relevant parts of nature and mathematical objects, and (b) say how they are related to each other. But this plausibility hides the mistake of replacing ‘‘we can’’ with a relation independent of the user (the ‘‘we’’) and ignores the selectivity exercised by the user for the user’s speciﬁc purpose.

Part IV: Appearance and Reality 1

For a more detailed discussion of the parallels between Aristotle’s Poetics and Physics see my 2000b. 2 Physics I, 184a15–21, 194b19–20; the discussion of a poorly constructed theory that I mention next is found at 198b10–35. 3 Metaphysics N3, 1090b, tr.: Ross I want to thank Fran O’Rourke for pointing me to this passage. 4 I do not mean to underrate the way in which mathematical modeling too trades on resemblance. But the extent to which mimesis is involved at all diminishes continuously as we move our view successively from table-top models to textbook illustrations, from there to mathematical modeling, and eventually to the geometric spaces that appear in relativistic cosmology and quantum ﬁeld theory.

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12. Appearance vs. Reality in the Sciences 1

Cf. Margaret Wilson 1984. I gratefully acknowledge my debt to her analysis. For the seminal text for later discussions and the now standard terminology, see John Locke, Essay Concerning Human Understanding, bk. 2, ch. 8, sections 9, 10. 3 A distinction also made by Galileo’s contemporary Gassendi, with similar reference to ancient atomism. 4 The Assayer, section 48; p. 274 in Drake 1957. 5 Also not Galileo’s terminology. Galileo wrote ‘‘Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one ﬁrst learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles and others geometric ﬁgures without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth.’’ Il Saggiatore (1623), in Drake 1957: 237–8. Similarly, ‘‘The book of philosophy is that which stands perpetually open before our eyes, but because it is written in characters different from those of our alphabet it cannot be read by every body; and the characters of this book are triangles, squares, circles, spheres, cones, pyramids and other mathematical ﬁgures ﬁttest for this sort of reading.’’ Lettera a Fortunio Liceti, gennaio 1641 in Crombie 1994: vol. i, 585. 6 This quaint conviction beset much of early empiricism as well. Locke writes that the mind ‘‘hath no other immediate object but its own ideas’’ (An Essay Concerning Human Understanding, Bk IV, i, I); Berkeley that ‘‘the objects of human knowledge are either ideas actually imprinted on the senses, or else such as are perceived by attending to the passions and operations of the mind’’ (Principles of Human Knowledge, Part I, I); Hume that ‘‘[a]ll the perceptions of the human mind resolve themselves into impressions and ideas’’ (Treatise of Human Nature, I, i, I). 7 Meditation III, section. 19, tr. John Veitch (Descartes 1959). 8 See for example ‘‘Huygens and Leibniz on Universal Attraction’’, pp. 115–38 in Koyr´e 1965. 9 For the former cf. ch. 2 of Vargish and Mook 1999; for the latter, Cushing 1994. 10 That logically necessary connection may not be ﬁnitary, may in fact be inaccessible to any ﬁnite mind—as Leibniz made explicit (therefore not within even the potential reach of the physical sciences)—but be graspable only by the divine mind. There are many ambiguities in these developments. Sometimes Descartes and Leibniz do sound as if there can be only one world, of logical necessity. The popular version would go like this: from the concept of God it 2

390  :   follows that he would not create a world at all, if among all the conceivable ones there was not a best one (Leibniz’s Theodicy) or one uniquely transparent to the human mind (Descartes’s posthumous The World). But at other points the claim is that although the regularities derive with logical necessity from the laws of nature, those laws characterize a selection from the realm of conceivable possible worlds which has no further rationale, at least within the context of even these ‘‘theories of everything’’. Notice that we have here, in effect, the ﬁrst ‘‘supervenience without reduction’’ claim, for reduction would require ﬁnitary reasoning but the demonstrative link is claimed to be non-ﬁnitary. I will return to questions of supervenience below. 11 ‘‘Some seekers after the theory of Everything would seem to be hoping that the uniqueness and completeness of some Particular mathematical Theory will make it the only logically consistent description of the world . . .’’ (Barrow 1991: 202). In philosophy too there are arguments currently that the laws of nature are necessary after all (although necessity need not be thus related to logical consistency): cf. Bird 2007. 12 Boyle 1772: vol. v, 162. 13 See St¨ oltzner 1999. 14 Sociologists of science have (in)famously explored social and political factors in post-World War One Europe to explain the readiness of prominent physicists to embrace indeterminism. Looking back after four decades of discussion of Bell’s Inequalities and surrounding issues, we must certainly also see that readiness as (perhaps fortuitously) prescient. 15 Cf. my 1982a, and references therein. Under certain conditions this criterion actually demands determinism, as I show there. But from the example of Bohmian mechanics, we can see as well that satisfaction of this criterion is not logically implied by determinism; see further notes below. For more on Reichenbach’s conception see e.g. Graßhoff, Portmann, and Wüthrich, 2005. 16 Bohm’s mechanics is deterministic but also violates the sorts of locality constraints required by Reichenbach’s criterion; surprisingly it presents a picture of ‘determinism without causality’ so to speak. For early discussion of Bohm’s alternative, see Grünbaum 1957: 715–16n., and for an early sympathetic survey of this alternative and its links to other work then, see Freistadt 1957. For its more recent and quite spectacular revival in philosophy of physics see Cushing 1994. 17 This point, like so much else in the interpretation of quantum mechanics is tendentious and controversial. I do not want to enter into that debate just now. The controversies illustrate, in any case, the denouement of the historical pattern that I discern. The new and revolutionary success in science came with a ostensible rejection of a prevalent completeness criterion for science, but in the aftermath

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many strove (and strive!) valiantly to reinstate the criterion and to show that its rejection was not logically required. 18 There is some disparity between what Bohr and his colleagues, students, and followers originally professed and what almost the entire physics community came to subscribe to under the name of ‘‘Copenhagen interpretation’’; see Howard 2004. We could say the same for any revolutionary stance, whether social, religious, or intellectual. What I will insist on is that with all its ambiguity intact, the Copenhagen view was intellectually and scientiﬁcally revolutionary. 19 See further e.g. p. 80. After maintaining this view strongly throughout most of the book, Leplin includes a section in which he raises the possibility that recent and current theories in physics will not be in accord. 20 There is now a quite extensive literature on how this point about wavelength of reﬂected light is a far cry from the nuanced and delicate account of human visual experience. The critique I am in the course of offering here is independent of that; it targets not the degree of success in such explanations of speciﬁc appearances, but the conception of methodology that involves those putative criteria of completeness. 21 Attempts to provide such accounts which could perhaps provide such explanations even today include work by Stephen L. Adler and his colleagues on Generalized Quantum Dynamics, as well as earlier work by e.g. Nelson 1967. Such accounts, if successful, can satisfy the Appearance from Reality Criterion no less than deterministic theories. 22 This is too simple an idea of course, but is not in essence too far from the work by Adler and Nelson mentioned above. Besides these I am thinking here of the Ghirardi, Rimini, and Weber (GRW) version of quantum mechanics, which introduces a ‘Lucretian swerve’ into Schr¨odinger’s deterministic equation. 23 Wigner’s 1961 appeal to consciousness in quantum mechanics, to ‘explain’ collapse of the wave-packet falls under this heading, as do the idealist and (quasi-)instrumentalist accounts which Grünbaum depicted as prevalent forms of easy anti-realism among scientists (Grünbaum 1957: 717–19). 24 I have only slowly come to see the importance of marking such a distinction. In The Scientiﬁc Image I did not make this distinction either carefully or clearly. The chapter on saving the phenomena introduces ‘‘appearances’’ to denote what Newton called ‘‘apparent motions’’, identifying them as ‘‘relational structures deﬁned by measuring relative distances, time intervals, and angles of separation’’(p. 45). I would now refer to those relational structures as data models. Data models are the summarizing reﬁnement of the contents of a battery of measurements, typically, so this is not far from my present usage. But in the passages that follow there, the reference seems from time to time to be just to observable entities, i.e.

392  :   phenomena rather than appearances in my current stricter usage. Thus Paul Teller rightly writes ‘‘First is the idea of. . .phenomena (which means, the observable process and structures). . .’’ (SI 3). ‘‘I take van Fraassen to use ‘phenomena’ and ‘appearances’ interchangeably. (SI 45, 64) I will understand phenomena and appearances as that which we can observe, or that of which we can become perceptually aware, without the use of instruments’’ (Teller 2001: 125–6). 25 Kant, Critique of Pure Reason, ﬁrst para. of Transcendental Dialectic, Introduction section I. The quote is from Meiklejohn’s translation (Kant 1850: 209), which had many new editions well into The twentieth century. The translation by Francis Haywood (Kant 1848: 234) is practically the same. The German original is ‘‘Noch Weniger dürlen Erscheinung and Schein für einerlei gehalten Werden’’. 26 There is lots more to be said about the terms of course. The term phenomenon is often enough applied to microphysical processes that are not observable—against the explicit stipulations by Bohr, which Wheeler formulated as ‘‘No phenomenon is a phenomenon unless it is an observed phenomenon’’. We should also note that like many other nouns ‘‘phenomenon’’ has both a generic and a speciﬁc use—a speciﬁc effect produced in a laboratory at a particular time is a phenomenon, but so are oxidation, ebbing, planetary motion, and so forth. In the generic use, as I understand the term, it refers to classes of observable entities. See further Hacking 2006. 27 This makes the point I want to make simple. The entire discussion of modern science that follows here could be more informatively developed around e.g. Newton’s system of the world—but I think we would lose the forest for the trees. 28 Amazingly, this common sense realism is often lacking among the most soi-disant ‘‘realist’’ philosophers. In their wish to defend a requirement to believe in the reality of unobservable entities it has not been rare to see the tu quoque that we all believe in such things as mountains although, as Quine famously claimed, ‘‘physical objects are epistemologically, to the gods of Homer [. . .]Both sorts of entities enter our conception only as cultural posits. The myth of physical objects is epistemologically superior to most in that it has proved more efﬁcacious than other myths as a device for working a manageable structure into the ﬂux of experience.’’ (Quine 1951: 42). Surely this idea rests subliminally on the ‘homunculus’ view of the subject, modeling our epistemic and doxastic activity on the case of a little man in a booth reading only a ticker tape and inferring, postulating, speculating on market activity? 29 On the face of it, my introducing the noun ‘‘appearance’’ may suggest that I will postulate the existence of a special sort of entity, denoted by that noun. I am certainly not doing that; this sort of game in analytic metaphysics is very far

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from my mind. That a penny looks elliptical to me if I hold it up in a certain way does not imply that there exists besides the penny also an elliptical penny-look; it implies the existence of me and of the penny, and that is all. Nor does it imply any mistake on my part: I know very well that circular things can look elliptical if held at a certain angle. 30 For a clear and informative description of Copernicus’s geometric ‘‘transcription’’ see Barbour, 2001: 216. 31 Thus Regiomontanus published the Theoricae novae planetarum of his former master Georg Peurbach. Another example is Pedro Nunes (1502–78), Tratado da sphera com a theorica do sol e da lua (1537). An anonymous, Theorica Planetarum was recently offered for sale on the internet with the advertisement ‘‘One of evidently only three copies known of a richly illustrated astronomical handbook ﬁlled with colored diagrams and movable volvelles that show the persistence of the Ptolemaic world view as expressed in his Almagest, which remained a cornerstone of astronomical thought even after the discoveries of Nicolaus Copernicus (1475–1543). The present manuscript probably served as a demonstration text for a teacher-astronomer.’’ 32 We may reasonably suspect that the conviction, that this is so, helped to inspire the ‘‘construction’’ programmes of Russell’s Our Knowledge of the External World and Carnap’s Aufbau. 13. Rejecting the Appearance from Reality Criterion 1

Thus I agree with Nancy Cartwright (1983: 163–216) that the measurement problem is an artefact of the formalism (though we do not have quite the same diagnosis). 2 As with every viewpoint taken on this issue, it is possible in retrospect to see it presaged in earlier ones. Some of what I will argue is certainly along the line of how Michel Bitbol depicts Schr¨odinger’s responses to the measurement problem: ‘‘But what was really needed was a full acceptance of the parallelism between the time-development of the holistic wave-function (object + apparatus) and the sequence of macroscopic events, rather than an new blend of the old idea of a causal interaction which takes place between objects and apparatuses in order to produce the events.’’ (Bitbol 1996: 123). 3 I emphasize this because arguably, the Appearance from Reality Criterion is not violated in Bohmian mechanics, where position is the only genuine observable, evolution of states is deterministic, and all measurements, of any sort, are reconstructed as in the end just position measurements. Even if such a pattern should overtake all of physics in the coming century (however unlikely that may be), the methodological point would stand: the Criterion cannot be said to have been

394  :   in force throughout the history of science, hence is not one to be discerned as binding in scientiﬁc practice. 4 David Armstrong took a similar position in the 1960s; see Armstrong 1993. 5 There is an enormous literature on this subject, with different notions of supervenience distinguished, most of them intelligible only in the context of substantial metaphysical presuppositions. But there is a basic notion that relates languages (e.g. the language of physics and the language of psychology), which can be understood with more minimal background and without violating empiricist scruples. I would describe it roughly and informally as follows: L supervenes on L if for every set of sentences of L there is a set of sentences of L such that the truth values in the former being different requires some difference in the truth values in the latter. This implies an abstract form of translatability of L into L , but if we add non-reducibility, that means that there is a translation only for the angels, not by any humanly or mechanically feasible means. For a longer but still brief discussion see my 2004. 6 I am drawing here on some notions from logic and foundations of mathematics. Deﬁnable sets include here sets deﬁnable by recursion; weaker and stronger notions of deﬁnability are available to differentiate various strengths in claims of [non-]reducibility. 7 There are of course complaints from the side of philosophy of science about philosophy of mind shenanigans; see for instance Glymour’s gleefully critical 1999 review of Kim 1998. 8 As a metaphysical postulate this supervenience claim presumably gives some emotional comfort to the materialist/physicalist. While science is here admitted to be incapable of showing this, the world is still conceived as how the physicalist would like it to be. 9 Quoted by Mates 1986: 108–9. For this part, and for the following references, I am thoroughly indebted to Anja Jauernig’s dissertation (Princeton 2003). Leibniz’s On Freedom: But in the case of contingent truths, even though the predicate is in the subject, this can never be demonstrated of it, nor can the proposition ever be reduced to an equation or identity. Instead, the analysis proceeds to inﬁnity, God alone seeing—not, indeed, the end of the analysis, since it has no end—but the connexion of terms or the inclusion of the predicate in the subject, for he sees whatever is in the series (Parkinson 1970: 109).

The Monadology has: There are two kinds of truths, truths of reasoning and truths of fact. Truths of reasoning are necessary, and their opposite is impossible. Truths of fact are contingent, and their opposite is possible. When a truth is necessary, the reason for it can be found by analysis, resolving it into more simple ideas and truths until we reach the primitive. It is thus that speculative theorems and rules of practice in mathematics are reduced by analysis to deﬁnitions, axioms,

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and postulates. There are, ﬁnally, simple ideas which cannot be deﬁned, and there are also axioms and postulates, or, in brief, primitive principles, which cannot be proved and need no proof. And these are identical propositions whose opposites contain an explicit contradiction. (Loemker 1975: 1050). 10

For simplicity of exposition I assume that A is discrete and not degenerate: there is a single unit eigenvector for each eigenvalue, and the possible values constitute a countable set of eigenvalues. Let me repeat another caution: I am using a quite standard formulation here, which takes not only the notion of observable but also that of physical state for granted. This is the form in which the issues will be most familiar to most readers. It also seems to me that the same issues will arise with more or less the same impact in a more interpretation-neutral formulation, where it is allowed that local information (e.g. due to previous measurements) or even subjective probabilities play a part in the assignment of states. See for example Rovelli 1996 and my [forthcoming]. 11 Notice that I use ‘‘refer’’ to set up the dilemma; the function-dependence in meaning of these terms is left aside. 12 There is no way to rule out the possibility of inconsistency, of course: even the mathematics within which quantum theory is formulated does not allow of a consistency proof. 13 Evolution of an isolated system obeys Schr¨odinger’s equation: there is a group of unitary operators { Ud : d in R } such that the pure states ψ(t) evolve under the action of these operators: ψ(t+d) = Ud ψ(t). 14 We must distinguish here the actual historical development of the quantum theory from interpretative additions and extrapolations of recent years. The ‘‘bare’’ theory is empirically empty without the Born Rule, but there have been attempts to deduce the Born Rule from the basic theory supplemented with assumptions involved in certain interpretations of the theory, such as the ‘‘many worlds’’ interpretation, GRW, or Bohmian mechanics. These sorts of assumptions were either entirely absent in the development of quantum theory or roundly rejected by the main physicists involved, so this does not affect the claim that this historical episode in physics involved a rejection of the Appearance from Reality Criterion. 15 We may note here that there are certainly set-ups that are not neatly dissectible into object measured and measuring apparatus. Rom Harr´e 2003 aptly coined the terms ‘‘Bohrian artifacts’’ and ‘‘apparatus/world complexes’’ to designate the peculiarities of such set-ups: ‘‘Let us call the apparatus/world complexes that scientists, engineers, gardeners, and cooks bring into being Bohrian artifacts. Properly manipulated they bring into existence phenomena that do not exist as such in the wild [. . . .] In the famous Bohr–Einstein debate around the EPR paradox, it is possible to see the outlines of Bohr’s account of experimental physics.

396  :   While Einstein is insisting that for every distinct symbol in a theoretical discourse there must be a corresponding state in the world . . . Bohr . . . is concerned with the concrete apparatus and its relation to the world as part of the world. An apparatus is not something transcendent to the world . . . . The apparatus and the neighboring part of the world in which it is embedded constitute one thing.’’ (pp. 28, 29.) Despite this indissoluble entanglement, when the set-up is classiﬁed as a measurement, then its outcome is classiﬁed as representing something in that apparatus/world complex. 16 That is (another way to state) the Measurement Problem! This has seen many offered ‘solutions’ and ‘dissolutions’ and much debate between their advocates, in its now almost century long history. The literature on this subject is enormous. For an older detailed treatment see my 1991; for a perspicuous recent discussion that highlights the points that I will take up here, but in a general probabilistic setting, see Wilce, forthcoming. 17 See the caution in a previous note about the reliance here on a standard formulation of the subject matter. I meant to introduce all the basic notions needed to understand the Measurement Problem, but at this point (and some others) I included assertions that I am not justifying here. They are not egregious however; the justiﬁcation can be found at many places in the philosophical literature on quantum mechanics, at almost any level of (relative) (non-) technicality. 18 The quick argument sketch is as follows: if the apparatus ends up in one of the pure states |B,r> then it is not true that the Apparatus + Object ends up in a superposition involving more than one such eigenstate of B–which contradicts the conclusion about how the composite system has evolved. See for example Eugene Wigner’s seminal 1963, speciﬁcally pp. 11–12. 19 See for instance Greenstein and Zajonc 1997. 20 Wigner, op. cit.; see speciﬁcally pp. 10–12, where Wigner discusses the separation of an x-polarized beam of spin-1/2 atoms into two z-polarized beams by a Stern-Gerlach apparatus, and asks what would result if one subsequently merged the two beams, whether the reconstituted beam could be in a coherent, x-polarized spin state. 21 No such point as this is incontrovertible: the literature contains so many interpretations that almost all logically available niches are occupied and almost every point will fail on some offered interpretation of quantum mechanics. But there certainly are salient interpretations on which supervenience fails, and that without locating the measurement outcomes in consciousness or other non-physical realm. I am thinking here especially of the entire range of modal interpretations of quantum theory. See my 2005a. What is not satisfactory, it seems to me, is to appeal to decoherence, for that does not remove the problem in principle.

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As always, in practical contexts this is a matter of satisfactory approximations; as is customary, the condition here stated relates to the ideal case. 23 This was the insight that has driven all ‘hidden variable’ and ‘modal’ interpretations of quantum mechanics—though those also go beyond the minimum point that is needed for my argument (namely by offering a separate identiﬁcation of the events in the measurement situation in terms of certain theoretical parameters, whether taken from quantum theory itself (as in modal interpretations) or added as alien quantities absent from the theory altogether).

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Index abstract structures 120, 238, 240–1, 242–3 and phenomena 245–6, 248–50 see also data models; theoretical models abstraction 48–9, 251 accuracy 12, 15, 19, 139, 269, 309: see also approximation; distortion admissible rescaling 160–1 admissible transformations 55, 161–3, 174 air thermometers 125–6 Alberti, Leon Battista 62–4, 72 Alhazen 43 alidades 61 altimetry 61–2 Amontons, Guillaume 117, 128, 129 analogies 17, 25, 27, 101, 195, 380 n. 22, 381 n. 25 and disanalogies 219, 385 n. 8 maps 77–80 models and 310–11 and observables 184, 313 analytic geometry 41, 46, 66–7, 353 n. 29 anti-empiricism 307–8 anti-realism 198 apparatus 94–5, 112, 170–1, 362 n. 7, 393 n. 2, 395 n. 15 measurement 143, 144, 298–9, 304 measurement outcomes 180, 183 measurement theory 147–8, 150–3, 154 in quantum mechanics 300, 302–3, Stern–Gerlach 155, 179 thermometry 123, 125–6, 130 see also instrumentation Appearance from Reality Criterion 281–3, 291–2 and cognitive psychology 292–5 and deducibility 296–7 and philosophy of mind 292–5 and quantum mechanics 291–2, 297–300, 308 and reducibility 292–5 appearances 8–9, 29 Appearance from Reality Criterion 281–3, 291–300, 308 completeness criteria 276–83

determinism 278–80 necessity and 277–8 and phenomena 283–8, 317 primary/secondary qualities 271–6 and reality 270–6 approximation 52–3, 56, 57, 111, 128, 163 Arago, Dominique 98 Aristotle 7, 96, 200, 265–6, 283, 352 n. 20 primary/secondary qualities 271–3 rainbows 102 artifacts 25, 94, 238, 245, 265, 395 n. 15: see also models Aspect, Alain 170 Assmus, A. 94–5 astrolabes 61–2 astronomical clocks 132, 134 astronomy 8, 168, 214, 218, 257, 289 planetary motion 285–8 sidereal days 133–4 asymmetry 17–18, 189 atomic clocks 134 atomic theory 112–13, 191–2, 198, 200–1, 206 atomism 280 Aviation Model (AVN) (weather prediction) 77–8, 196–7 Avogadro’s number 112–13 Bacon, Francis 75, 273, 374 n. 1 Baird, David 165 barometers 126 Barrow, John D. 390 n. 11 Beauvoir, Simone de 27 Bell, John 364 n. 25 Bellarmini, Cardinal 281 Belnap, Nuel 370 n. 42 Beltrami, Eugenio 215 Benoist, Jocelyn 349 n. 28 Berkeley, George 389 n. 6 Bildtheorie Boltzmann 1–2, 195–6, 197 controversy over 191–204 Hertz 195–7 Mach 197–8

400  Bildtheorie (cont.) Maxwell 195 Planck 192–5, 196 Birkhoff, G. 47 Bitbol, Michel 370 n. 43, 380 n.12, 393 n. 2 Block, Ned 21–2, 38, 350 n. 12 Bogen, James 376 n. 14 Bohm, David 390 n.15 and 16, 393 n. 3, 395 n. 14 Bohr, Niels 76, 380 n. 17, 392 n. 26, 395 n. 15 Boltzmann, Ludwig 192, 200, 279 and Bildtheorie 1–2, 195–6, 197 scale models 355 n. 34 Boon, Mieke 95, 362 n. 7, 361 n. 3 Born rule 297–300, 308 Botticelli, Sandro 64 Boyd, Richard 199 Boyle, Robert 278 Boyle’s Law 128, 138, 242 Bradley, F. H. 270 Brahe, Tycho 289 Brentano, Franz 349 n. 28 Bridgman, Percy W. 53 Brunelleschi, Filippo 73, 183 Bub, Jeffrey 378 n. 32 Buckingham, E. 53, 55 Bueno, OtÆvio 385 n. 4 Campbell, Norman 158, 366 n. 11 Cantor, Geoffrey 362 n. 11 caricature 13–15 Carnap, Rudolf 225–9 cartography 75–82 Cartwright, Nancy 310–11, 362 n. 5, 393 n. 1 Cassirer, Ernst 123, 379 n. 17 Celsius scale 160, 161–2, 174 Chang, Hasok 368 n. 24 Charles’s Law 128 Charleton, Walter 99 chiaroscuro 36 classical mathematics 47 classiﬁcation 103, 113, 151, 155, 179, 203–4, 211 of experiences 106–8, language and 84, 206 and measurement outcomes 180–1, 182–3, 299, 305

measurement procedures and 123, 124, 128, 139, 164, 172 of psychological phenomena 292, 294–5 theory and 111, 139, 143–5, 164, 203, 246, 261, 319 Clausewitz doctrine of experimentation 112 Clausius, Rudolf 128 Clifton, W. K. 223 clocks 130–6 cloud chambers 94–5, 113, 242 cluster concepts 38 cognitive psychology 292–5 coherence 122–3, 136, 144, 145–6, 160, 184, 303 of experience 278–9 of measurement 153 coherence conditions 130, 131–3, 134, 136, 145, 152, 153 coherence constraints 129, 131, 132, 134, 136, 152–4 color vision 209, 372 n. 11 committal 38: see also non-committal common cause 108, 136, 279–80 ´ Condillac, Etienne Bonnot de 380 n. 22 congruence 13, 118, 130–1, 135, 216–17, 218 constructive empiricism 317, 385 n. 6, 386 n. 8 contingent truth 296, 394 n. 9 Convention of the Meter 135 conventions 19, 23, 43, 66, 82 choice and 128–30, 134–6, 158, 160, 162 and coordination 208 coordinating principles 117, 120 coordination 241, 244–5 historical context 116–21 and measurement 136–7 problem of 121–4 thermometers 125–30 time measurement 130–6 coordinative deﬁnitions 121 coordinatization 136, 175, 178, 234, 366 n. 10 Copernicus, Nicolaus 8, 13, 271, 286–8 cosmology 134, 277, 289 Cratylus 19–20, 22, 35 Criterion for Physical Correlate of Measurement 182–3, 302, 312, 314 cross ratios 66, 72–3, 74–5

 Dalton, John 117, 127, 130 data models 166, 167–8, 172, 251–2, 391 n. 24 and phenomena 252–9 David, Jacques-Louis 29 de Beauvoir, Simone, see Beauvoir, Simone de Dear, Peter 97 deducibility 296–7 Dennett, Daniel 38 denotation 16, 348 n. 27 depiction 16 Desargues, G´erard 72 Descartes, Ren´e 22, 30, 199–200, 269, 271, 278, 280 analytic geometry 41, 46, 66–7 frames of reference 66–70 logical necessity 389 n. 10 primary/secondary qualities 273–5 determinism 169, 292 and appearances/reality 278–80 in mechanics 29–30 dilation 161–2 dimensional analysis 53–5, 355 n. 40–3 distortion 12–15, 36–7, 38, 40, 183 Doisneau, Robert 20–21 dreams 24 Dretske, Fred 15 Duhem, Pierre 203, 206–7 D¨urer, Albrecht 8, 65–6, 142 Eco, Umberto 116 Eddington, Arthur 71, 136 Edgerton, S. Y. 63, 356 n. 2, 380 n. 14, 351 n. 14 Einstein, Albert 118, 134–5, 183, 395 n. 15 principle of relativity 69–71 Einstein–Podolski–Rosen (EPR) experiment 170–1, 299 n. 15, 315–16 Elga, Adam 352 n. 15 Elgin, Catherine Z. 17, 346 n. 9, 347 n. 14 embedding 29–30, 87, 168–72, 240, 247, 252, 316 empirical adequacy 3, 136, 199, 246, 249, 258, 317 empiricism 3, 304–6 empiricist structuralism 237–9 epistemology 222 Escher, M. C. 39 essential indexical 3, 83, 88 ether theory 100–1

401

Euclidean geometry 61, 66–7, 213–14, 215–16, 234, 285–6, 309 exempliﬁcation 17 Exner, Franz Seraﬁn 279 experimentation: roles of 111–13 explicitly non-committal representations 38–9, 40, 50, 313 Faraday, Michael 95, 101 Feyerabend, Paul 73, 150, 359 n. 23 Feynman, Richard 150 De Finetti, Bruno 361 n. 42 Fine, Arthur 369 n. 35 Fourier, J. B. J. 53 frames of reference 66–70 Frege, Gottlob 383 n. 6 French, Steven 345 n. 1, 349 n. 1, 368 n. 25, 385 n. 4 Fresnel, Augustin 98 Friedman, Michael 365 n. 3, 384 n. 1 and 4 Frigg, Roman 309 Fuchs, Christopher 378 n. 32 function 79, 181 of experimentation 111 indexicality and 182 instrumentation and 94–100, 157 mimetic 97 use and 21–2, 23, 25, 30 fuzzy observables 184, 312, 313–14, 321 n. 7 fuzzy values 154, 376 n. 13 Galileo Galilei 41, 67, 278, 280, 281 buoyancy experiment 50–1 primary/secondary qualities 34, 271–3 scaling 50–1 thermometry 117, 123, 125 Galison, P. 94–5 gas law 127–8, 129 Gassendi, Pierre 278, 280 geometric optics 42–5 geometry analytic 41, 46, 66–7, 353 n. 29 Euclidean 61, 66–7, 213–14, 215–16, 234, 285–6, 309 hyperbolic 213–14, 215 non-Euclidean 213, 215–16 projective 66, 72–3, 74–5, 215–16, 286 Georgalis, Nicholas 348 n. 21, 348 n. 23

402  Gerlach, Walther 155 Stern-Gerlach apparatus, 179, 305–6 Giere, Ronald 28, 168, 183, 309, 372 n. 11 Gilbert, William 97 Global Positioning Systems (GPS) 81–2 Glymour, Clark 394 n. 7 Golding, William 21 Gombrich, Ernst 12–13 Goodman, Nelson 11, 19, 351 n. 14 on art 16, 21 denotation 16, 348 n. 27 exempliﬁcation 17 GPS (Global Positioning Systems) 81–2 Gr¨unbaum, Adolf 166, 320 n. 5 Guericke, Otto von, see von Guericke, Otto Hacking, Ian 108, 346 n. 4 (Part I) hallucinations 101–5, 107–9 Hanson, N. R. 144 Harr´e, Rom 395 n. 15 Haupts¨atze 23–4, 28, 55 Heidelberger, Michael 94, 95, 363 n. 15 Heisenberg, Werner 201 Helmholtz, Hermann von 214, 229 hermeneutic circle 116 Hero of Alexander 43 Hertz, Heinrich 98, 192, 200–3, 284, 306 Bildtheorie controversy 195–7 Weltbild 193 hidden variables 30, 397 n. 23 higher order resemblance 33–4, 35, 182, 195 Hilbert, David 382 n. 2, 383 n. 6 Hipparcus 177 homomorphism 18 Hooke, Robert 99 horizon of alternatives 39, 59 Hughes, R. I. G. 345 n. 1, 346 n. 6 Hugh of St Victor 61 Hume, David 389 n. 6 Huygens, Christiaan 131–2, 135, 195 Huygens, Constantijn 93–4, 96 Hyman, John 37 hyperbolic geometry 213–14, 215 hyperrealism 11 ideal gas law 127–8 image categories 104–5 imagery

kinematic 34–5, 39, 182 mathematical 39–49 visual 11, 34, 35, 39, 40, 182 images 22, 35, 101–10, 198, 202, 204, 221 categories of 103–5 as copies 12–13, 18–21, 24, 97 distortion and 40 hallucinations 101–5, 107–9 instrumentation and 97, 105–6, 108–9, 110, 168 Loewenheim–Skolem–Tarski–Vaught theorem 230 measurement and 75 mental 24 mirror 214 in models 242–3 and perspective 39 perspectival 73 scientiﬁc 45, 47–8, 271, 275, 276 imaging: and scaling 56–7 implication 315–16 incoherence 133, 160, 256, 260–1 indeterminism 279–80, 355 n. 37 indexicality 59, 181–2, 239, 259–60, 261 and maps 77–8, 79–80, 257–8 and perspective 60, 85–6 information 21, 36, 75, 76, 80 data models and 166 experiments and 66 maps and 78–9, 82 measurement and 91, 143, 145, 146, 150–1, 155–6, 157, 179, 180–1, 182, 183–4 perspectival drawing and 8, 91–2 perspective and 68–9, 72–3 scale models and 50, 56 instrumentation 363 n. 15 astrolabes 61–2 barometers 126 microscopes 93, 99–107, 108–9, 110 observation by 93, 105–11 observation metaphors 93, 96–9 roles of 94–100, 157 thermometers 125–30, 144, 371 n. 5 intensionality 27–8, 181 intentionality 27–8, 181 interval measurement 159, 160, 161–2 invariance 52, 117, 136 cross ratios 72–3, 74–5 dimensions and 53–5 and perspective 72–3, 91–2, 175–9

 phenomena and 103, 108 and scaling 158, 160–3 invariants 70, 103, 161–2, 176, 177, 179 cross ratios 72–3, 74–5 irreducibility 304 Ismael, Jenann 359 n. 28 and n. 29, 382 n. 3, 385 n. 7 isomorphism 18, 214–15, 238, 247, 249, 365 n. 7 Weyl and 208–10, 211 Jackson’s Mary Problem 210–12 Jauernig, Anja 358 n. 14, 365 n.3, 379 n. 5, 387 n. 21, 394 n.9 Kant, Immanuel 8, 80, 257, 278–9, 283–4 Kelly, Sean 372 n. 11 Kelvin, Lord (William Thomson) 113, 158, 368 n. 24 kinematic imagery 34–5, 39, 182 kinematics 30, 34–5, 42, 274, 282, 285–6, 287–8 kinetic theory 127–8, 130 Klein, Felix 215, 229 Kuhn, Thomas 144 Kulvicki, John 347 n. 17 Ladyman, James 381 n. 24, 385 n.2, 385 n.4 Lambert, Joseph 117 Lange, Marc 355 n. 40 language 83–4, 86 scientiﬁc 206–8 theory-laden 84, 206 Latour, Bruno 370 n. 43, 376 n. 17 Law of Charles and Gay Lussac 128 Leeds, Stephen 321 n. 13, 386 n. 13, 387 n. 15 Leibniz, G. W. 30, 69, 296, 389 n. 10 Leplin, Jared 199, 281 Levine, Sherry 347 n. 17 Lewis, David 229, 231, 360 n. 33, 365 n. 1, 378 n. 3 and 4, 383 n. 10 light-clocks 132–3 linear perspective 63–4 linear one-point perspective 285, 286 Lipton, Peter 370 n. 45 liquid thermometers 126–7 Lobachevsky, Nikolai 214

403

Locke, John 389 n. 6 Lodge, David 210, 211 Loemker, L. E. 394 n. 9 Loewenheim–Skolem–Tarski-Vaught theorem 230 logical necessity 278 logical space 164–6, 172–9 perspectival effects in 173–5 Lopes, Dominic 36, 37, 38, 39 Luce, Duncan 162–3 Mach, Ernst 116–17, 123, 138, 191–2 Bildtheorie controversy 197–8 thermometers 125–30 Maddy, Penelope 361 n. 1 (Part II) maps 76–84 and indexicality 77–8, 79–80, 257–8 Margenau, Henry 166 marginal distortion 34, 38, 183 Marlow, A. R. 315–16 Masaccio 64 mathematical imagery 39–49 distortion and 40 geometric optics 42–5 mathematical statues 41–2 Simpson’s paradox 48–9 mathematical statues 41–2 Maxwell, J. C. 195, 202, 278 Mead, Herbert 25 measurement 2–3, 91–2 altimetry 61–2 approximative 163 coherence and 145–6, 152–4 coordination and 136–7 deﬁnition of 157 general theory of 147–56 interval 159, 160, 161–2 and logical space 164–6, 172–9, 179 nominal 159, 160–1 ordinal 159, 160 and perspective 8, 60–6, 73–5 and physical correlate 142–5, 156 in quantum mechanics 300–3 scale/scaling invariance 161–3 signiﬁcance 161–3 of time 130–6 types of 158–63 measurement, general theory of 147–56 coherence constraint 152–4 form of 150–2

404  measurement, general theory of (cont.) initial set-up 147–8 Value Deﬁniteness 149–50, 154 Veracity in Measurement 149–50, 154–6, 373 n. 19 measurement outcomes 91, 157, 179–84 measurement procedures 123–4 measurement scales 159 mechanics: determinism in 29–30 Mendelovici, Angela-Adeline 378 n.33, 321 n. 6 mental representation 24 mental acts as 27, 28 see also Bildtheorie Mermin, David 170, 205 Michelson-Morley experiments 133 microscopes 93, 99–107, 108–9, 110 Middleton, W. E. Knowles 367 n. 20 Millikan, Robert Andrews 112 Milne, Arthur 134 mimetic experimentation 94–5 Minkowski, Hermann 71 misrepresentation 13–15 modality 28, 119, 282–3, 318 models data 166, 167–8, 172, 251–9, 391 n. 24 geometric 285–6 theoretical 238, 240, 245–6, 248–50 and theories 309–11 surface 166–72, 240, 250–2, 257, 305, 315–16 Monton, Bradley 345 n. 4 Mohs hardness scale 160 Moore’s Paradox 260–1 Morrison, Margaret 310 Nagel, Thomas 358 n. 13 (p.69) nano-technology 94 navigation 177–8 necessary truth 296, 394 n. 9 necessity 277–8 neo-Kantian tradition 115, 120 Neurath, Otto 137, 190 Newman, M. H. A. 219, 222 Newton, Isaac 30, 52, 257, 278, 280–1, 317–18 nominal measurement 159, 160–1 non-committal 37, 38–9, 40, 50, 313 non-Euclidean geometry 213, 215–16

non-simultaneity 119 Nyhof, John 379 n. 2 observables 137–9, 297–8, 300–2, 305–6 fuzzy 184, 312, 313–14, 321 n. 7 and measurement 117, 134, 147, 150, 151–5, 182, 305 and models 169, 305 sharp 184, 312–13 observation 93 instrumentation and 96, 100 models and 87, 168–9 observation language 144 occlusion 34, 37, 39, 184, 313 one-point linear perspective 64 opacity 27–8 optics: geometric 42–5 ordinal measurement 159, 160 orthogonality 315–16 Otte, Richard 375 n. 5 Padovani, Flavia 366 n. 8 paintings 12, 141–2 Panovsky 345 n.2, 349, n. 33, 357 n.10 Parkinson, G. H. R. 394 n. 9 Pascal, Blaise 46, 72, 126, 353 n. 26 pendulum clocks 131–2 pendulums 29, 131–2, 135, 137, 310 Perrin, Jean Baptiste 113 Perry, John 182 Perspectiva 75, 285, 356 n. 4 perspectival drawing 62–5, 91–2 linear one-point 285, 286 as measurement 8 perspective 34 angle 38 and indexicality 60, 85–6 and invariance 72–3, 91–2, 175–9 linear 63–4 and logical space 173–5 marginal distortion 34, 38, 183 and measurement 60–6, 73–5, 183 occlusion 34, 37, 39, 184, 313 one-point 64 texture-fading 34, 38, 184 two-point 64 view from nowhere 69–72, 122 visual 84–6

 Peschard, Isabelle 351 n. 13, 360 n. 36, 367 n. 17, 377 n.21, 377 n.31, 378 n. 35 phenomena 8–9 and abstract structures 245–6, 249–50 and appearances 283–8, 317 and data models 252–9 outside experience 247–50 and theories 250–2, 259–61 philosophy of mind: and Appearance from Reality Criterion 292–5 photographs 18, 20–2, 178–9 physical correlates 118–19, 121, 136, 179, 298, 304, 305 Criterion for Physical Correlate of Measurement 182–3, 302, 312, 314 physical conditions for measurement 141–6 theory of measurement 147–56 picture plane 62–4, 65, 358 n. 21 picture theory of science, see Bildtheorie pictures 35–9 picturing 34, 182, 313 Pino, Paolo 356 n. 1 Place, U. T. 292 Planck, Max 192–5, 196, 279 planetary motion 8, 271, 286–8 planisphere 356 n. 4 Plato 12, 24, 101–2, 231 Pliny the Elder 11, 346 n. 1 Poincar´e, Henri 204, 279, 367 n. 12, 380 n. 19 and coordination 118, 208 and measurement 125, 130–6, 138, 176, 183 Poisson, Sim´eon 98 Power, Henry 99 pragmatic contradictions 259 pragmatic tautologies 259 pragmatics 3, 17, 21–2, 25, 82, 189, 190, 232–3, 259–60 predication 16 prediction 283 primary/secondary qualities 34, 271–6 Principle of Approximation 52–3 Principle of Similitude 51–2 probability 305, 317–19 projective geometry 215–16, 286 cross ratios 66, 72–3, 74–5 Psillos, Stathis 253–4 Ptolemy 43, 356 n. 4 Putnam, Hilary 229–35, 386 n. 7

405

Putnam’s Paradox (model-theoretic argument) 229–35 dissolution of 232–5 quadrants 61 quantum theory 144, 155, 164, 183, 197, 198, 199, 277 and Appearance from Reality Criterion 291, 297–300, 308 and data models 172, and determinism 279–80 and empiricism 306–7, 308 and fuzzy observables 184, 313–14 and measurement 147–8, 154, 166, 184, 291, 300–3, 319 and sharp observables 184, 312–13 and surface models 169, 170, 315, 316 and supervenience 304 and theoretical models 252 Quine, W. V. O. 392 n. 28 rainbows 102–3, 110 Raphael Sanzio 64 ratio measurement 159, 160 ratio scales 128, 160 re-scaling 160–1 realism 198–204, 229, 241–4 real property 220–1 scientiﬁc 198–9 structural 198 reality 270–6 Appearance from Reality Criterion 281–3, 291–300, 308 completeness criteria 276–83 determinism 278–80 necessity and 277–8 primary/secondary qualities 271–6 rectilinear propagation 42 reducibility 292–5 reﬂection 42, 43–5 in water 101–2, 103, 105 refraction 43, 44–5 Reichenbach, Hans 30, 52–3, 240–1, 279 common cause principle 280 and coordination 118–21, 123, 136–7, 387 n. 21 relationality 26, 225–9, 231–2, 242–3 relativity, principle of 69–71

406  Renoir, Pierre-Auguste 35 representation 7–9, 35 asymmetry of 17–18, 189 committal/non-committal 38, 39 and distortion 14 intentional 27 modes of 33–5 and referents 244–5 and resemblance 14, 17–19, 30–1, 33–4, 35 and structuralism 204–12 use and 22–6, 28 resemblance 11–13, 35 caricature and 14 and distortion 14 measurement and 192–3 misrepresentation and 14 paintings and 141–2 and representation 14, 17–19, 30–1, 33–4, 35 Socrates on 19–20 Richer, Jean 132 Richter scale 375 n. 5 Rosselli, Francesco 63 Rossetti, Dante Gabriel 142 Rothschild, Daniel 384 n. 2 Rovelli, Carlo 372 n. 14, 377 n. 32, 395 n. 10 Rubens, Peter Paul 35 Russell, Bertrand 213, 216–17, 229, 239, 368 n. 27, 368 n. 28 epistemology 222 and real property realism 220–1 and structuralism 217–23 Rutherford, Ernest 76 Ryckman, Thomas 135, 358 n. 15, 361 n. 43 Saliger, Ivo 35 Salmon, Wesley 108, 199 saving the phenomena 8 scale: as logical space 164–6 scale models 49–50 scale transformation 52, 53–5, 161, 162 scaling and imaging 56–7 and invariance 54–5, 160–3 as picturing 50–1 scaling invariance 54–5, 160–3 Scanning Tunneling Electron Microscope 94

Schlick, Moritz 117 Schr¨odinger, Erwin 198 scientiﬁc realism 8, 113, 184, 194, 198–9, 237, 266, 281 screw propeller 55–6 sculpture 12, 38, 41–2 selective likeness/unlikeness 7–8, 9, 14, 18, 23, 30–1, 33, 34, 141 drawing and 91 imagery and 39, 182 and measurement 60, 91, 179–80, 182–3 scale models and 49–50, 57 selectivity 7–8, 37, 76, 87 data models and 253–4 maps and 261 mathematical models and 242, 243, 247 self-ascription 79, 82, 83 self-location 78–84, 85 Sellars, Wilfrid 33–4, 84, 349 n. 2, 379 n. 8 semantics 17, 27, 84, 232, 258, 259–60, semantic view of theories 239, 247, 309–11 set theory 353 n. 29 sextants 178 sharp observables 184, 312–13 Shepard, Roger 13 Shomar, Towﬁc 310 sidereal clocks 132, 134 similitude, principle of 51–2 simplicity 133–4, 136, 138–9, 202, 203 Simpson’s paradox 48–9 simultaneity 37, 68, 70, 81, 165, 166, 312 sines, law of 353 n. 21 Sneed, Joseph 367 n. 12 Snel van Royen, Willebrord 353 n. 21 Snell’s (Snel’s) law 44 Socrates 12, 19–20, 24 Solovay, R. M. 353 n. 29 space of reasons 84 Spott, E. 14 standard meter 135 Stern, Otto 155 Stern–Gerlach apparatus 155, 179, 305–6 Sterrett, Susan 55 Stevens, S. S. 158, 159, 161 stochastic response function 151, 169, 300 structural realism 198 structuralism 191 Carnap and 225–9 empiricist 237–9

 Newman and 219 and representation 204–12 Russell and 217–23 Suarez, Mauricio 7, 25–6, 310–11 supervenience without deducibility 296–7 and irreducibility 304 without reducibility 292–5 Suppes, Patrick 167, 172, 309 surface model 166–172, 240, 250–252, 257, 305, 315–316 surface models 167, 168–72, 252, 315–16 symmetry 18, 40, 66, 163, 196, 211, 212, 226, 234 synchrony 131–3, 134, 136 syntax 26, 233, 294, 351 n. 14, 385 n. 6 Tarski, Alfred 260 Teller, Paul 45, 105, 107–8, 391 n. 24 temperature 53, 86, 116–17, 122–3, 124, 131, 138 Fahrenheit/Celsius scales 161–2, 174 of gases 138, 242 and length measurement 135 and mean kinetic energy 173 pendulums and 132 statues and 42 thermometers 125–30, 146, 160, 175, 180 weather forecasts 77, 166 texture-fading 34, 38, 184 Thales 356 n. 5 Theaetetus 12 theoretical models 238, 240 and phenomena 245–6, 248–50 Theorica 288–90, 308 theories de-idealization of 310–11 and models 309–11 and phenomena 250–2, 259–61 semantic view of 309–11 theory-laden language 84, 206 theory-ladenness 75, 94, 144, 181 thermodynamics 42, 279 thermometers 125–30, 144, 146, 160, 173, 175, 180, 371 n. 5 thermometry 116–17, 123 thermometers 125–30, 144, 146, 160, 173, 175, 180, 371 n. 5 Thomas, David A. 357 n. 11

407

Thompson, Paul 310 Thomson, William (Lord Kelvin) 113, 158, 368 n. 24 time 34, 68, 70–1, 87, 115, 156, 164, 242 in equations 247–8 self-location and 81, 178 and space 280, 288 space–time 71, 137, 221, 223, 289 unit of 119 see also kinematics; time measurement time measurement 118, 122, 125, 130–6, 176–7, 183, 317–18 clocks 131–4, 199 Timpson, Chris 377 n. 29, 378 n. 32 Tolman, Richard 51–2 Toulmin, Stephen 144 tragedy 7, 26, 265–6 transcendentalism 8, 266 transformations 55, 161–3, 174 translation 84, 161–2, 394 n. 5 truth 144, 200, 234, 246, 260, 283 contingent 296, 394 n. 9 correspondence theory of 244, 248–9, 252 images and 14, 20 necessary 296, 394 n. 9 secondary qualities and 274 two-point perspective 64 unsharp observables 312, 313–14, 321 n. 7 unsharp values 154, 376 n. 13 Value Deﬁniteness 149–50, 154, 373 n. 19 van Leeuwenhoek, Antonie 104 Vasari, Giorgio 64 Veracity in Measurement 149–50, 152, 154–6, 373 n. 19 verbal description 17, 86 view from nowhere 69–72, 122 visual imagery 11, 34, 35, 39, 40, 182 visual perspective 84–6, 182 von Guericke, Otto 94, 95 von Helmholz, see Helmholtz, von von Neumann, J. 47 Weinberg, Steven 110–11 Weltbild 141, 192–5, 237, 274, 275

408  Weyl, Hermann 71, 136, 175, 228–9 and isomorphism 208–10, 211 Whitehead, A. N. 243 Wigner, Eugene 303 Williams, Bernard 358 n. 13 Wilson, Catherine 94–5, 99 Wilson, Margaret 389 n. 1 Wittgenstein, Ludwig 38, 164 Woodward, James 376 n. 14 world-picture 141, 192–5, 237, 274, 275

Worrall, John 239, 362 n. 10, 378 n. 34, 381 n. 24, 385 n. 2 Young, Thomas 251 Zajonc, Arthur 359 n. 23, 396 n. 19 zero-point perspective 69 Zeuxis 11 ´ Zola, Emile 350 n. 5