On Massey's Explication of Grunbaum's Conception of Metric

On Massey's Explication of Grünbaum's Conception of Metric Bas C. van Fraassen Philosophy of Science, Vol. 36, No. 4. (Dec., 1969), pp. 346-353. Stable URL: http://links.jstor.org/sici?sici=0031-8248%28196912%2936%3A4%3C346%3AOMEOGC%3E2.0.CO%3B2-B Philosophy of Science is currently published by The University of Chicago Press.

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ON MASSEY'S EXPLICATION OF GR~NBAUM'S

CONCEPTION OF METRIC*

BAS C. VAN FRAASSEN1 University of Toronto

Professor Massey's exposition and analysis 151 of Professor Grunbaum's writings on metric aspects of space seem to me both very helpful in understanding those writings and to contain a considerable original contribution to the subject. Nevertheless I would like to argue that there is an alternative to Massey's explication which seems to me more faithful to Grunbaum's remarks; it seems at least to have the virtue of not forcing Grunbaum to reject the usual mathematical definitions of the notions used.

1. Preliminary remarks. Before turning to a main critical point made by Massey concerning denumerable spaces, let us introduce some terminological distinctions. As an example of a space let us take the set of couples of real numbers C = ((x,y) :x2 + y2 = 1). This is just the unit circle with the origin as center in the Euclidean plane, but it can of course be considered as a space by itself. If a, b and c are three points on C, there are two questions we may ask: what is the length of the segment of or interval on C which contains c and with a and b as end points? and what is the distance between a and b? The answer to the first question is straightforward: intuitively speaking, we measure the distance along the circle between Q and b taking care to pass through c. The answer to the second question is not straightforward: for now we can again measure along the circle, arriving at the same answer if we pass through c (as perhaps we should if that gives the smaller answer); on the other hand we can measure "across," the distance then being the usual Euclidean function of the coordinates a,,a2 of a and bl,b2 of b:

d(a,b)

=

d(b, -

+ (b,

-

So there can be a fairly direct relation between length and distance, but there need not be. This brings us to the generalizations of these two notions common in mathematics. The generalization of length is a measure (function) and the generalization of distance is a distance function. p is a measure on the set X if and only if p is defined for a 0-ring of subsets of X, and for all subsets A, for which p is defined, (i) p(Ai) is a non-negative real number (ii) p ( A ) = 0 (iii) if the sets (A,) i = 1,2, . . . are disjoint, then

* Received August, 1969. The author wishes to thank Professors Massey and Grunbaum for their helpful correspondence; and Professor P. Asquith, Michigan State University, and Professor C. Glymonr, Princeton University, for stimulating discussions.

346

ON M ~ S S E Y ' SEXPLECAEON 81:GRUNBAUM'S CONCEPTION OF IMElWC

347

d is a distancefunction on the set X if and only if d is defined for all couples of members of X and such that if x,yyzare in X then (a) d(x,y) is a non-negative real number (b) d(x,y) = 0 if and only if x = y Cc) d(x,y) = d(y,x)

( 4 d(x,y) + ~(YYz)2 d(x,z)

Pw common mathematical usage, "metric" is now synonymous with "distance function," but this has not been the case in the discussions of space and time initiated by Reichenbach. For Reiclzenbach divided questions concerning space and time into those relating to metric aspects and those relating to topological aspects, counting measure under the former and order under the latter. (Indeed, he has used "metric" in ways which could not be explicated with reference to distance functions at all, as when he uses "probability metrics" in [6] to refer to probability measures.) In this aspect of terminology, Griinbaum clearly follows Reichenbach, and so shall I. Professor Massey explicitly restricts hilaself to measures closely related to distance functions. Thus in the case of the Euclidean 1-space, the real line, the distance between x and y is and thatis also the length of the interval (x,y). Since some difficult questions about metric aspects of space concern the extension of our concepts beyond the nornial, familiar cases, it remains to be seen whether the familiar relations between measure and distance will carry over.2 In preparation for further discussion, we may present this subject with somewhat greater generality. Let us suppose that a certain family of measurable subsets of a set X with measure p are singled out as line segments. Assume furthermore that any unit set is a line segment. Then we maj7 say that a distance function d on X i s ilzdtdced by p if d(x,y) is the greatest lower bound of the measures p ( Y ) of the line segments Y which contain both x and y. It is very important to point out that, for a given measure, no such induced distance function may exist. For example, if there is a point a such that p({a)) > 0, then p does not induce a distance function d, because d(x,x) > 0 is ruled out by the definition of a distance function. To apply this to Professor h4assey7s singly extended manifolds (X, 5 ) , let us define the line segments to be the sets { x : a < x < b} for points a, b on X such that a 4 b. Such a set can be referred to equivalently as [a,b] or [b,a], the orientation being irrelevant for our purposes. We note then that [a,b] is the shortest line segment containing both a and 6 (in that it is contained in all other such segments), and that [a,c]is the union of [a,b] and [b,c]when a 2 b < c. Hence in this case, the equation d(a,b) = p([rr,b]) pairs distance functions with measure functions, but the pairing is neither one-to-one nor exhaustive. 2,. Measures on denumerable spaces. When measures and distances are related in the familiar way, the measure of a unit set is zero, as we pointed out. But then the measure of any at most denumerable set must be zero, by clause (iii) of the Hence we do not see here a logical relation between distance and length, as Massey characterizes it in section 2 of his paper, [5],but only a relation that happens to hold in familiar cases.

348

BAS C. VAN FRAASSEN

definition. So in a denumerable space, every subset, including the whole space, would have measure zero. In a continuous space, this consequence is avoided, because continuous intervals are not denumerable unions of denumerable subsets; accordingly Griinbaum emphasizes this in his discussioll of Zeno's paradoxes. But Massey sees this as an objection to the use of the concept of measure as commonly defined in the discussion of denumerable spaces. Accordingly he proposes a weakening of clause (iii) in the definition. Thus he says in section 2 of his paper:

. . . in the theories of discrete and denumerable dense manifolds, the postulates or axioms for a metric [i.e. distance function] do not allow one to endow length with the countable additivity just mentioned. Nonetheless, in the theories of such manifolds length may be considered countably additive in a weaker sense, namely: the length of an interval divided into a finite or denumerable sequence 2 of nondegenerate, nonoverlapping subintervals is equal to the sum of the lengths of the members of 2: ( [ 5 ] ,p. 333). So in order to preserve the usual relation between distance and length, and to allow nontrivial measures on denumerable spaces, Massey weakens one of the clauses in the usual definition of measure. There are certainly passages in which Griinbaum simply accepts the usual relation between distance and length (most notably [3], Chapter 111, section 2), as Massey indicates. But as far as I can tell these passages relate only to the familiar cases where this raises no problem (thus, in the section cited above, the reference is explicitly to Euclidean n-spaces). I fear that some confusion may result from Massey's designation of the relevant principle as "the logically overriding requirement that a measure function accord with but outstrip the length function." For there is no denying the fact that length is a specific instance of measure (the measure of an arc, or an interval on a curve); but this raises no difficulties of the kind noted by Massey, unless length must accord with distance in the usual way so that the measure of a unit set is zero. Let me hasten to add, however, that from Griinbaum's discussions it is equally clear that he does not regard distance and measure as totally unrelated subjects even in the general case; to this we shall return when we discuss alternative metrizability below. Far more important in this connection is Griinbaum's discussion of the discrete denumerable manifold; 1 do not think that it would be an overstatement to call this the paradigm non-standard case in Griinbaum's discussion of metric aspects of space. As Massey notes, here Griinbaum agrees explicitly with Riemann that the natural measure is the one which assigns to each subset its own ~ardinality.~ It is clear, first of all, that this measure is countably additive, and in fact a measure by the usual definition of that concept. But equally clearly this measure does not fit Massey's amended definition, since unit sets are not assigned zero. And finally, this also shows that Griinbaum does not accept what Massey calls the logical relation between distance and length, which implies that unit sets have measure zero. In other words, if Massey's concept of measure provides the only plausible See [4], p. 8; on p. 9 Griinbaum gave a tentative reformulatioil in terms of distance, but he explicitly corrects this later ([2], p. 147).

explication of Grunbaum's remarks on metric aspects of space, then the agreement with Riemann on the discrete denumerable case would have to be revoked. But 1 see this rather as a reason to expect that an alternative explication could be provided.

3. A modest proposal. When a writer does not explicitly list the standard definitions of relevant terms, the obvious first assumption is that he accepts those definitions. Accordingly, the most obvious conclusion to reach on the basis of Grunbaum's explicit remarks is that he does not wish to reject any clause in the usual definition of measure, although he does wish to add a further clause for the the general concept of measure includes for special case of a spatial m e a ~ u r e(Since .~ example probability measures, it seems reasonable that spatial measures should have characteristics not shared by measures in general.) The additional clause, the arguments for which we shall not repeat (see [2], p. 154, footnote 5), appears to be (iv) p({a)) = nz, where n7 = 0 or m = 1/12 for some positive integer 12, for all points a (given that p is defined for unit sets). So, when Grunbaum says that the meaning of such terms as "congruence" is pre-empted (so that the conventionality of metric aspects of a continuous space has nothing to do with trivial semantic conventionality) I read him as referring to the standard definitions of standard terms in measure theory and geometry, completed perhaps by special additional clauses such as (iv). 4. Implications of the proposal. Having already indicated, in section 2 above, why I think that Grunbaum's own remarks do not force Massey's explication, I would like now to examine the implications of the preceding proposal for an alternative explication. (a) The definition of measure with clause (iv) added implies that the measure p(X) = cardinality of X is essentially the only measure on a denumerable space with subsets X. By this I mean that any other measure on such a space which is defined for all unit sets will be a multiple of p. Since, moreover, Grunbaum himself calls such multiples trivial variants of p ([2], p. 154), I see in this a deduction of the claim that the measure p is intrinsic to a discrete denumerable space. For the case of a denumerable dense space I would say the same, but p is itself not a significant measure there; this problem I shall discuss again below. (b) Near the end of his paper, Massey decides to "stoically accept" the consequence of his analysis that practically all manifolds are metrically amorphous. In this particular case it is my feeling that he has focused on distance functions where instead we ought to focus on measures. (In section 1 I have stated my reason for counting both under the metric aspects of space, as that term is used in the tradition initiated by Reichenbach.) Without denying that metrical an~orphousnesshas to do also with distance functions (see (c) below), I offer the following explication:

A space X i s metrically amorphous if there are measures p, p' on X defined at least for all unit sets, for which there is no constant c such that p'(Y) = cp(Y) This applies to spaces with homogeneous elements, to which we restrict our discussion throughout.

3 50

BAS C. VAN PRAASSEN

for all subsets iY for which p and p' are both defined, or p(Y) all such subsets Y.

=

cpf(Y) for

1 consider it no small virtue of the above definition that it implies that discrete denumerable spaces are not metrically amorphous, while continuous spaces are. (For let X be denumerable, and let p, p' on X assign m and m', respectively, to unit sets. Then, by countable additivity, p assigns to a subset Y for which it is defined a measure of in times the cardinality of Y, and p' a measure of m' times this cardinality. Hence the measures are in proportion p(Y) =: (m/m')pf(Y) if tn' f 8, and pl(Y) = O p( Y) if m' = 0; in each case for all subsets Y for which both are defined.) The definition also implies that denumerable dense spaces are not metrically amorphous; that case we shall discuss in subsection (d) below. (c) As Mr. Glyrnour has pointed out to me, a space may not be metrically nrnorpbous in the above sense and yet have nontrivially different distance functions defined on it. This is so, certainly, for the discrete denumerable case. The first conclusion is of course that by having an intrinsic metric, Griinbaum does not mean that no alternative distance functions can be defined-for he asserts that the discrete denumerable space does have an intrinsic metric. Looking to what Griinbaurn himself says about distance in this case ([2], p. 147) we find that he defines the distance between two points to be the least number of "steps" linking the two points. Thus if s is the discrete linear ordering of a denumerable space X, d(a, b)

=

p((x:a I x 5 b})

-I

where p is the natural measure: p( Y) = the cardinality of Y, for subsets Y of X. Reinforcing some of our earlier conclusions we may note that d(a, a) # (:((a}) and that the function p' :p'((x: a 5 x I b)) = d(a, b) is not a measure by our definition. For if Y is the four-point interval [a, c, (1, b] then pf([a, b]) # pt([a, c3> 4 pl([d, b]) as required by clause (iii) in that definition. (Indeed, the first passage which P cited from Professor Massey's paper leads me to believe that p' is not even a measure by his weakened definition.) But more important is the fact that in the passage in question, Griinbaum appears to regard d as an intrinsic distance function because it can be defined in terms of the natural measure-or equivalently, in terms of the cardinality of the interval. Definability may here be the crucial notion; it does not seem to me to be too farfetched to take this to mean that d has to be a linear5 function of p, and that if d' m s a multiple of d , then d' is a trivial variant of d. If this is correct, it seems to me that there is a large class of cases in which being metrically amorphous is equivalent no not having an intrinsic metric. We shall now make this precise specifically for Professor Massey's singly extended manifolds, and then indicate briefly how the discussion might be extended to cover multiply extended manifolds. We shall refer to line segments in singly extended manifolds as at the end of our section 1.

I. We say that a distance function dl is a~sociatedwith a measure p if there is a linear f~mctisnf such that d(w, b) = f(r) where r is the lower bound of the 5 The requirement of linearity might perhaps be relaxed in various ways without affecting our conclusions.

ON MASSBY'S EXPLICATION OF GR~!BAUM'S COKCEPTION OF MTHRIC

351

measures assigned by p to the line segments containing a and b, for any two points a and b in the space. 2. We say that a space X i s alternatiz)eij, nzetrizable irz the wealc sense if there are distance functions d and d' on X such that d' is not a multiple of d. 3. We say that a space X is alternatittely metrizable iiz the strong senseGif there are distance functions d, d' associated with spatial measures p, p' 011 X such that d' is not a multiple of d. Suppose now that (X, 5 ) is a singly extended manifold; then we can demonstrate that if X is not metrically amorphous then X is not alterrzatiuely metrizable in the strong sense. For suppose that p, p' are measures on X (defined at least on the line segments [a, b] in X) and d, d' distance functions on X such that p = cp'

d(x, Y) = ~ ( F ( E~L1()[)~ ,

df(x, Y> = f'(pf([x, ~ 1 ) ) c is scalar f, f' are linear functions for all points x, y on A". Then d is a linear function of d': there are scalars nl, n swkl that d = md' + n. But then the equations

show that a = 0, so that d is a multiple of d'. U'e can now conclude that denumesable discrete singly extended manifolds are not alternatively metriza-ble in the strong sense (since they are not metrically amorphous), although continuous spaces are. In the above proof, nothing hinged on the fact that X was singly rather than multiply extended, since definitions I and 3 make only those measures relevant which are defined for the line segments in X. Turning now to the converse of the italicized proposition, we find that exceptions are more easily produced, but for rather trivial reasons. Let us continue to suppose that X is a singly extended manifold and let us further restrict this discussion to cases where all relevant measures are defined exactly on the o-ring generated by the line segments. Suppose X i s not alternatively metrizable in the strong sense. Then if EL, p' are spatial measures on X with associated distance functions d, d', then d is a multiple of d'. Now we note that if d is associated with p then there is a scalar nz such that for all x, y in X, d(x, y) = rnp([x, y]) n where n is exactly minus m times the measure that p assigns to unit sets (since d(x, x) = mp([x, x]) + az = 0). Setting p(ix1) = 4, p'((x1) = 4'3 we get d =~a' d(x, Y) = mp([x, YI) - mq d'(x, Y) = mlp'([x, YI) - m'4' and hence mp(Lx, yl) - mq = cmlp'(tx, YI) - cm'q'

+

Synonymously, X does not have an intrinsic metric.

352

BAS C . VAN FRAASSEN

so that I*, is a linear function of p' so far as the line segments are concerned. But then p is a multiple of ,u' for line segments in all significant cases. For then we have, p([x, y]) = epf([x, y]) + k for all points x, y and for certain scalars e, k, and by the definition of measure: if the space is dense or continuous, ,L is not defined for all line segments unless 1c = 0. And if the space is discrete, p(Xf) = /Xflqand p'(X') = jXr/q', but also ep'(X) + k = elX'lql+ /X1lk,so that / X'j = 1 or k = 0, for all line segments X' (by countable additivity). This result is of course limited by the fact that such pairs of measures might be extended to sets of points which are not line segments in significantly different ways. This is especially important when we go on to multiply extended manifolds: a measure on a singly extended manifold can be extended in many different ways to a measure on a Cartesian power of that manifold. But here too it is not difficult to single out as of main interest a large class of cases in which metrical amorphousness and alternative metrizability in the strong sense will be equivalent. For the generalization of the line segments {x: a 5 x 5 b} in a singly extended nlanifold (X, )I to its nth Cartesian power is the family of generalized rectangles.

And here the usual requirement is that the measure p of this figure be the product of the measures of its sides,

The u-ring generated by the generalized rectangles is called the family of elementary figures, and the extension of a measure p on (X, )I to these figures, subject to the above requirement, is unique (see [I], section 119). Extension of the measure to other sets of points might of course again show significant alternatives. To suin up, we have tried to show that there is a sense of alternative metrizability which is equivalent to our sense of metrical amorphousness in a large class of cases, that discrete denumerable spaces are neither metrically amorphous nor alternatively metrizable in these senses, while coiitinuous spaces are both. These consequences being in agreement with Grunbaum's claims, we offer them as evidence in favor of our explication. (d) We come then finally to the strange case of the denumerable dense space. By our definitions, this space is not metrically amorphous, nor is it alternatively rnetrizable in the strong sense. But it certainly is alternatively metrizable in the weak sense. And if physical space were thus, the distance function defined in terms of transported rods might well be a nontrivial distance function, assigning a finite positive distance to any pair of distinct points. This metric would not be associated with any spatial measure on that space, by our definitions, and could not be used to define one. But since metric geometries need not be developed with reference to measure theory-the notion of distance sufficing for the definition of all common geometric relations-I simply cannot see this as an unforgivable shortcoming.

ON MASSEY'S EXPLICATEOX OF G R ~ ~ N B A U M 'CONCEPTION S OF MtTRIC

353

REFERENCES

[ I ] Birkhoff, G., Lattice Tizeorj~,American Mathematical Society (Colloquium Publications XXV), New York, 1940. [Z] Griinbaum, A., Geometry and Chrononzetry iit Philorophical Perspectice, University of Minnesota Prcss, Minneapolis, 1968. '.~ Wesleyan University Press, Middle-131- Griinbaum. A.. Moderrz Scierzce a n d Z e f ~ oP~radoxes. town (Conn.), 1967. e43 Griinbaum, A., Philosoplzical Problems of Space and Time, Knopf, New York, 1963. [5] Massey, G., "Toward a Clarification of Griinbaum's Conception of an Intrinsic Metric," Philosophy of Science, vol. 36, no. 4 (this issue). [63 Reichenbach, H., The Ditection of Time. University of California Press, Berkeley, 1956.

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Toward a Clarification of Grünbaum's Conception of an Intrinsic Metric Gerald J. Massey Philosophy of Science, Vol. 36, No. 4. (Dec., 1969), pp. 331-345. Stable URL: http://links.jstor.org/sici?sici=0031-8248%28196912%2936%3A4%3C331%3ATACOGC%3E2.0.CO%3B2-F

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Toward a Clarification of Grünbaum's Conception of an Intrinsic Metric Gerald J. Massey Philosophy of Science, Vol. 36, No. 4. (Dec., 1969), pp. 331-345. Stable URL: http://links.jstor.org/sici?sici=0031-8248%28196912%2936%3A4%3C331%3ATACOGC%3E2.0.CO%3B2-F

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