Realism in Mathematics (Clarendon Paperbacks)

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REALISM IN MATHEMATICS

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REALISM I N MATHEMATICS PENELOPE MADD Y

CLARENDON PRES S OXFOR D

This book has been printed digitally and produced in a standard specification in order to ensure its continuing availability

OXTORD UNIVERSITY PRES S

Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department o f the Universit y of Oxford. It furthers the University' s objective of excellence i n research, scholarship, and education by publishing worldwide in Oxford Ne w York Auckland Bangko k Bueno s Aires Cap e Town Chenna i Dar es Salaam Delh i Hon g Kong Istanbu l Karach i Kolkata Kuala Lumpur Madri d Melbourn e Mexic o City Mumba i Nairob i Sao Paulo Shangha i Taipe i Toky o Toront o Oxford i s a registered trad e mark of Oxford University Press in the UK and in certain other countrie s Published in the United States by Oxford University Press Inc., New York © Penelop e Maddy 1990 The moral rights of the author have been asserte d Database right Oxfor d University Press (maker) Reprinted 2003 All rights reserved . No part of this publication ma y be reproduced, stored in a retrieval system, o r transmitted, in any form or by any means, without th e prio r permission i n writing of Oxford University Press, or as expressly permitted b y law, or under terms agreed with the appropriat e reprographics right s organization . Enquiries concerning reproductio n outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the addres s above You must no t circulate thi s book in any other binding or cover And you must impose this same condition on any acquirer ISBN 0-19-824035-X

For Dick and Steve

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PREFACE THE philosoph y o f mathematic s i s a borderlin e discipline , o f fundamental importanc e t o bot h mathematic s an d philosophy . Despite this , on e find s surprisingl y littl e co-operatio n betwee n philosophers an d mathematician s engage d i n it s pursuit ; mor e often, widesprea d disregard an d misunderstandin g are broken only by alarmin g pocket s o f outrigh t antagonism . (Th e gli b an d dismissive formalis m o f man y mathematician s i s offse t b y th e arrogance o f those philosophers who suppos e they can know wha t mathematical objects are without knowin g what mathematic s says they are. ) This migh t not matte r much i n anothe r age , bu t i t doe s today, whe n th e mos t pressing foundational problems ar e unlikely to be answered without a concerted co-operativ e effort . I have tried in this book t o do justice to the concerns of both parties, to present the background , th e issues , th e propose d solution s o n a neutra l ground where the two sides can meet for productive debate. For thi s reason, I'v e aime d for a presentation accessibl e to bot h non-philosophical mathematician s an d non-mathematica l philo sophers and , i f I'v e succeeded , student s an d intereste d amateur s should als o be served. As far as I can judge, very little philosophica l training o r backgroun d i s presuppose d here . Mathematica l pre requisites ar e mor e difficul t t o avoid , owin g t o th e relentlessl y cumulative nature of the discipline, but I'v e tried to keep them to a minimum. Som e familiarit y with th e calculu s an d it s foundation s would b e helpful, thoug h surely not necessary . And the relevant set theoretic concept s ar e reference d t o Enderton' s excellen t intro ductory textbook (se e his (1977)) , for th e benefi t o f those innocent of that subject. The centra l them e o f th e boo k i s the delineatio n an d defenc e of a version o f realis m i n mathematic s calle d 'se t theoreti c realism' . I n this, m y dee p an d obviou s deb t i s t o th e writing s o f th e grea t mathematical realists of our day : Kurt Godel, W. V. O. Quine , and

Vlll

PREFACE

Hilary Putnam (i n the early 1970s). Mor e personally, I have learned most fro m Joh n Burgess , Pau l Benacerraf , Hartry Field , an d Ton y Martin. Afte r these , i t woul d b e impossibl e t o mentio n everyone , but I can't overlook th e forcefu l criticism s of Charle s Chihara , th e insightful comment s o f Ani l Gupta , an d th e generou s correspond ence, assistance , an d advic e o f Phili p Kitcher an d Michae l Resnik . Most recently , Burgess , Field , Lil a Luce , Coli n McLarty , Martin , Alan Nelson , Resnik , Stewar t Shapiro , Mar k Wilson , an d Pete r Woodruff hav e al l don e m e th e servic e of readin g an d reactin g t o drafts o f various parts o f the manuscript. (Naturally , the remaining errors an d oversight s should b e charged t o m y shortcomings rathe r than t o thei r negligence. ) And finally , wha t I owe t o m y long-tim e companion Stev e Madd y i s to o comple x an d varie d t o b e summarized here . I a m gratefu l t o al l thes e peopl e an d offe r m y heartfelt thanks . Also to Angela Blackburn and France s Morphy o f Oxford Universit y Press. Much o f thi s boo k i s based o n a serie s of article s (Madd y 1980 , 1981, 1984rf , 1988a,b, forthcoming a , b) th e preparation o f whic h was supported , a t variou s times, b y th e America n Associatio n o f University Women , th e Universit y of Notr e Dame , th e Nationa l Endowment fo r the Humanities , th e National Scienc e Foundation , and th e Universit y of Illinoi s a t Chicago . Th e origina l publisher s kindly grante d advanc e permissio n t o reproduc e materia l fro m these pieces; in the end , onl y parts o f (forthcomin g a) (in chapter 5 , sections 1 an d 2 ) an d (forthcomin g b ) (i n chapte r 1 , sectio n 4 ) actually survived, so I am particularl y obliged t o Kluwe r Academic Publishers an d th e Associatio n fo r Symboli c Logic. Preparatio n o f the fina l draf t wa s supporte d b y Nationa l Scienc e Foundatio n Grant DIR-8807103 , a Universit y o f Californi a President' s Re search Fellowshi p i n th e Humanities , an d th e Universit y o f California a t Irvine . Th e hel p o f al l thes e institution s i s hereb y gratefully acknowledged . Finally, I feel compelle d t o ad d a personal not e on sexis t language . Some year s ago, whe n I first introduced th e idea s behind se t theor y realism, construction s lik e 'th e se t theoreti c realis t think s hi s entities . . .' struc k m e a s amusing , bu t sinc e the n I'v e discovere d that som e reader s an d editor s ar e legitimatel y disapprovin g o f this usage. O f the many alternatives available, I've chosen on e that does the leas t violenc e to th e standar d rhythm , that is , the us e of 'she '

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and 'her ' in place o f 'he' an d 'his ' i n neutral contexts . Som e migh t find thi s jus t a s politicall y incorrec t a s th e automati c us e o f th e masculine, bu t I sincerel y doub t tha t phrasin g lik e 'whe n th e mathematician prove s a theorem , sh e . . .' make s anyon e ten d t o forget tha t som e mathematician s ar e men . S o I'l l stic k wit h thi s policy. To those who find it distracting, I apologize; this is not, afte r all, a political treatise. At least you have my reasons. P.M.

Irvine, California June 1989

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CONTENTS 1. Realism 1 1. Pre-theoretic realism 1 2. Realism in philosophy 5 3. Realism and truth 15 4. Realism in mathematics 20 2. Perception and Intuition 36 1. What is the question? 36 2. Perception 50 3. Intuition 67 4. Godelian Platonism 75 3. Numbers 81 1. Wha t numbers could not be 8 1 2. Numbers as properties 86 3. Frege numbers 98 4. Axioms 107 1. Reals and sets of reals 107 2. Axiomatization 114 3. Open problems 125 4. Competing theories 132 5. Th e challenge 14 3 5. Monism and Beyond 150 1. Monism 150 2. Field's nominalism 159 3. Structuralis m 17 0 4. Summar y 17 7 References Index 19

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1

REALISM 1, Pre-theoreti c realis m Of th e man y od d an d variou s thing s w e believe , fe w are believed more confidentl y tha n th e truth s o f simpl e mathematics . Whe n asked fo r a n exampl e o f a thoroughl y dependabl e fact , man y wil l turn fro m commo n sense—'afte r all , the y use d t o thin k human s couldn't fly'—fro m science—'th e sun has risen every day so far, but it migh t fai l u s tomorrow'—to th e securit y of arithmetic—'bu t 2 plus 2 is surely 4'. Yet i f mathematica l fact s ar e facts , the y mus t b e fact s abou t something; i f mathematical truth s ar e true , somethin g mus t mak e them true . Thu s arise s th e firs t importan t question : wha t i s mathematics about ? I f 2 plu s 2 i s s o definitel y 4 , wha t i s i t tha t makes this so? The guileles s answer i s that 2 + 2 — 4 i s a fac t abou t numbers , that ther e ar e thing s calle d '2 ' an d '4' , an d a n operatio n calle d 'plus', and tha t th e result of applying that operatio n t o 2 and itsel f is 4 . ' 2 + 2 = 4 ' i s true becaus e th e thing s it' s abou t stan d i n the relation i t claim s the y do . Thi s sor t o f thinkin g extend s easil y t o other part s o f mathematics : geometr y i s the stud y o f triangles an d spheres; i t is the properties o f these things that make the statement s of geometr y tru e o r false ; an d s o on . A vie w o f thi s sor t i s ofte n called 'realism'. Mathematicians, thoug h priv y to a wider rang e of mathematical truths tha n mos t o f us , ofte n inclin e t o agre e wit h unsullie d common sens e on th e natur e o f thos e truths . The y se e themselves and thei r colleague s a s investigator s uncovering the propertie s o f various fascinatin g district s o f mathematica l reality : numbe r theorists stud y th e integers , geometer s stud y certai n well-behaved spaces, grou p theorist s stud y groups , se t theorists sets , an d s o on . The ver y experienc e o f doin g mathematic s i s fel t b y man y t o support thi s position:

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The mai n poin t i n favo r o f th e realisti c approac h t o mathematic s i s th e instinctive certaint y o f mos t everybod y wh o ha s eve r trie d t o solv e a problem tha t h e i s thinkin g abou t 'rea l objects' , whethe r the y ar e sets , numbers, or whatever . . . (Moschovakis (1980), 605)

Realism, the n (a t firs t approximation) , i s th e vie w tha t math ematics i s th e scienc e o f numbers , sets , functions , etc. , jus t a s physical scienc e i s th e stud y o f ordinar y physica l objects , astro nomical bodies , subatomi c particles , an d s o on . Tha t is , math ematics i s about thes e things , an d th e wa y thes e thing s ar e i s what makes mathematica l statement s tru e o r false . Thi s seem s a simple and straightforward view . Why should anyone think otherwise? Alas, whe n furthe r question s ar e posed , a s the y mus t be , embarrassments arise . Wha t sor t o f thing s ar e numbers , sets , functions, triangles , groups, spaces ? Where ar e they ? The standar d answer i s that the y are abstract objects , a s opposed t o th e concrete objects o f physica l science , an d a s such , tha t the y ar e withou t location i n spac e an d time . Bu t thi s standar d answe r provoke s further, mor e troublin g questions . Ou r curren t psychologica l theory give s the beginning s of a convincin g portrait o f ourselves as knowers, bu t i t contain s n o chapte r o n ho w w e migh t com e t o know abou t thing s s o irrevocabl y remot e fro m ou r cognitiv e machinery. Ou r knowledg e o f the physica l world, enshrine d in th e sciences t o whic h realis m compares mathematics , begin s in simple sense perception . Bu t mathematicians don't, indee d can't , observ e their abstrac t object s i n thi s sense . How , then , ca n w e kno w an y mathematics; ho w ca n w e eve n succee d i n discussin g thi s remot e mathematical realm? Many mathematicians , face d wit h thes e awkwar d question s about wha t mathematica l things ar e an d ho w w e ca n know abou t them, reac t b y retreatin g fro m realism , denyin g that mathematica l statements ar e about anything , even denying that the y ar e true: 'we believe i n th e realit y o f mathematics , bu t o f cours e whe n philosophers attac k u s with thei r paradoxes we rush t o hide behind formalism an d sa y "Mathematic s is just a combination of meaningless symbols " . . .'- 1 Thi s formalis t position—that mathematic s i s just a gam e with symbols—face s formidabl e obstacles of it s own , which I'l l touc h o n below , bu t eve n withou t these , man y math ematicians fin d i t involvin g the m i n a n uncomfortabl e for m o f 1

Dieudonne , as quoted in Davis and Hers h (1981) , 321 .

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double-think. Th e sam e write r continues : 'Finall y w e ar e lef t i n peace t o g o back to ou r mathematic s an d d o i t as we have always done, wit h th e feelin g eac h mathematicia n ha s tha t h e i s working on somethin g real ' (Davi s an d Hers h (1981) , 321) . Tw o mor e mathematicians summarize: the typica l workin g mathematicia n i s a [realist ] o n weekday s an d a formalist o n Sundays . Tha t is , whe n h e i s doin g mathematic s h e i s convinced tha t he is dealing with a n objective reality whose properties h e is attempting t o determine . Bu t then, when challenge d t o giv e a philosophica l account o f this reality, he finds it easiest to preten d tha t he does not believ e in it after all (Davis and Hersh (1981), 321 )

Yet thi s occasiona l inauthenticit y is perhaps les s troublin g t o th e practising mathematicia n tha n th e dauntin g requirement s o f a legitimate realist philosophy: Nevertheless, mos t attempt s t o tur n thes e stron g [realist ] feeling s int o a coherent foundatio n fo r mathematic s invariabl y lead t o vagu e discussion s of 'existenc e o f abstrac t notions ' whic h ar e quit e repugnan t t o a mathematician . . . Contrast wit h thi s th e relative eas e wit h whic h formal ism ca n b e explaine d i n a precise , elegan t an d self-consisten t manne r and yo u wil l hav e th e mai n reaso n wh y mos t mathematician s clai m t o b e formalists (whe n pressed } while the y spen d thei r workin g hour s behavin g as if they were completely unabashe d realists . (Moschovakis (1980) , 605-6)

Mathematicians, after all , have their mathematics to do, and they do i t splendidly . Dispositionall y suite d t o a subjec t i n whic h precisely state d theorem s ar e conclusivel y proved, the y migh t b e expected t o prefe r a simpl e and elegant , i f ultimately unsatisfying, philosophical position t o one that demands the sort of metaphysical and epistemologica l rough-and-tumble a full-blown realism would require. An d i t make s n o differenc e t o thei r practice , a s lon g a s double-think is acceptable. But to th e philosopher, double-think is not acceptable . If the very experience o f doin g mathematics , an d othe r factors , soo n t o b e discussed, favou r realism , th e philosophe r o f mathematic s mus t either produc e a suitabl e philosophical version of that position , o r explain away , convincingly, its attractions. My goal her e will be to do th e first, to develo p and defen d a version of the mathematician's pre-philosophical attitude. Rather than attempt to treat all of mathematics, to bring the project

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down t o mor e manageabl e size , I'l l concentrat e her e o n th e mathematical theor y o f sets. 2 I'v e mad e thi s choic e fo r severa l reasons, amon g the m the fact that, in some sense, set theory form s a foundation fo r the rest o f mathematics. Technically, this means tha t any objec t of mathematical study can be taken t o b e a set, an d tha t the standard , classica l theorems abou t i t ca n the n b e prove d fro m the axioms o f set theory.3 Striking a s thi s technica l fac t ma y be , th e averag e algebrais t o r geometer loses little time over set theory. Bu t this doesn't mea n tha t set theor y ha s n o practica l relevanc e t o thes e subjects . Whe n mathematicians fro m a fiel d outsid e se t theor y ar e unusuall y frustrated b y som e recalcitran t ope n problem , th e questio n arise s whether it s solutio n migh t requir e som e stron g assumptio n heretofore unfamilia r withi n tha t field. At thi s point, practitioner s fall bac k o n th e ide a tha t th e object s of thei r stud y ar e ultimately sets an d ask , withi n se t theory , whethe r mor e esoteri c axiom s o r principles migh t b e relevant . Give n tha t th e customar y axiom s o f set theor y don' t eve n settl e al l question s abou t sets, 4 i t migh t even turn ou t tha t this particular open proble m is unsolvable on the basis of these mos t basi c mathematical assumptions, that entirel y new set theoretic assumption s mus t b e invoked. 5 I n thi s sense , then , se t theory i s th e ultimat e cour t o f appea l o n question s o f wha t mathematical thing s there are , tha t i s to say , on wha t philosopher s call the 'ontology' of mathematics. 6 Philosophically, however , thi s ontologica l reductio n o f math ematics t o se t theor y ha s sometime s bee n take n t o hav e mor e dramatic consequences , fo r exampl e tha t th e entir e philosophica l foundation o f any branc h o f mathematics i s reducible to tha t o f set theory. I n thi s sense , comparabl e t o implausibl y strong version s of 2

A se t i s a collectio n o f objects . Amon g th e man y goo d introduction s to th e mathematical theory of these simple entities, I recommend Enderton (1977). 3 Se e e.g. th e reductio n of arithmeti c and rea l numbe r theory t o se t theor y in Enderton (1977) , chs . 4 an d 5 . Ther e ar e som e exception s t o th e rul e tha t al l mathematical object s ca n b e though t o f a s sets—e.g . prope r classe s an d larg e categories—but 1 will ignore these cases for th e time being. 4 Som e details and philosophica l consequences of this situation ar e the subject of ch. 4. s Eklo f an d Mekle r (forthcoming ) giv e a surve y o f algebrai c examples , an d Moschovakis (1980) doe s the same for parts of analysis. 6 I n philosophical parlance, 'ontology', th e stud y o f what there is, is opposed t o 'epistemology', th e stud y o f ho w w e com e t o kno w wha t w e do abou t the world . I will use the word 'metaphysics ' more or less as a synonym for 'ontology'.

REALISM I N PHILOSOPH Y 5

the thesi s tha t physic s is basic to th e natura l sciences, 7 I think th e claim tha t se t theory i s foundational canno t b e correct. Even if the objects of, say, algebra ar e ultimately sets, set theory itself does no t call attentio n t o thei r algebrai c properties , no r ar e it s method s suitable fo r approachin g algebrai c concerns. W e shouldn' t expec t the methodolog y o r epistemolog y o f algebra t o b e identical to tha t of se t theor y an y mor e tha n w e expec t th e biologist' s o r th e botanist's basi c notion s an d technique s t o b e identical t o thos e o f the physicist . Bu t again , thi s methodologica l independenc e o f th e branches of mathematics from set theory does no t mea n there must be mathematica l entitie s othe r tha n set s an y mor e tha n th e methodological independenc e o f psycholog y o r chemistr y fro m physics means there must be non-physical minds or chemistons. 8 But littl e hang s o n thi s assessmen t o f th e natur e o f se t theory' s foundational role . Eve n i f se t theor y i s n o mor e tha n on e amon g many branche s o f mathematics , i t i s deservin g o f philosophica l scrutiny. Indeed , eve n a s on e branc h amon g many , contemporar y set theory i s of special philosophical interest becaus e it throws int o clear relie f a difficul t an d importan t philosophica l proble m tha t challenges man y traditiona l attitude s towar d mathematic s i n general. I will raise this problem i n Chapter 4 . Finally, i t i s impossible to divorc e se t theor y fro m it s attendan t disciplines o f numbe r theor y an d analysis . Thes e tw o field s an d their relationshi p t o th e theor y o f set s wil l form a recurring theme in what follows, especially in Chapters 3 and 4.

2. Realis m in philosoph y So far, I've bee n usin g the ke y term 'realism ' loosely, withou t clea r definition. Thi s ma y d o i n pre-philosophica l discussion , bu t fro m 7

Thi s view is called 'physicalism'. I'll come back to it in ch. 5, sect. 1, below. Ther e was a tim e when th e peculiaritie s o f biologica l science led practitioners to vitalism , th e assumptio n tha t a livin g organis m contain s a non-physica l component o r aspec t fo r whos e behaviou r n o physica l accoun t ca n b e given . Nowadays, this idea i s discredited—simply becaus e it proved scientifically sterile — and, a s fa r a s I know , n o on e eve r urge d th e acceptanc e o f 'chemistons' . Today , psychology i s the special science that most often lay s claim to a non-physical subject matter, but a s suggested in the text, it seems to m e that a purely physical ontology is compatible with the most extreme methodological independence. For discussion, see Fodor (1975), 9-26 . 8

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now o n I will try t o b e more precise . Thi s doesn' t mean I'll succeed in definin g th e ter m exactly , bu t a t leas t I'l l narro w th e fiel d somewhat, I hop e helpfully . Le t m e begi n b y reviewin g som e traditional use s of the term i n philosophy. One o f th e mos t basi c ontologica l debate s i n philosophy concern s the existenc e o f wha t commo n sens e take s t o b e th e fundamenta l furniture o f the world: stone s an d trees , table s and chairs , mediumsized physica l objects . Realis m i n thi s context , ofte n calle d 'common-sense realism' , affirm s tha t thes e familia r macroscopi c things d o i n fac t exist . Bu t i t i s not enoug h fo r th e realis t t o insis t that ther e ar e stone s an d tree s an d suc h like , fo r i n thi s much th e idealist could agree , al l the whil e assumin g that a stone is a menta l construct o f some sort , sa y a bundle of experiences. However, suc h an idealist , lik e th e Bisho p Berkeley , wil l hav e seriou s troubl e agreeing wit h th e realis t tha t stone s ca n exis t withou t bein g perceived.9 Thus th e common-sense realis t can state her position in a wa y that rule s out idealis m b y claiming that stones etc . exist, an d that thei r existenc e i s non-mental , tha t the y ar e a s the y ar e independently o f ou r abilit y t o kno w abou t them , tha t thei r existence is, in a word, 'objective' . A more recen t opponen t o f the common-sense realist uses a mor e devious technique. 10 The phenomenalis t hopes t o sa y exactly wha t the realis t say s whil e systematically reinterpreting eac h an d ever y physical objec t claim int o a statemen t abou t wha t sh e call s 'sens e data', o r really , int o statement s abou t possibl e sens e data . Fo r example, m y overcoa t exist s i n th e close t thoug h unperceive d because par t o f th e translatio n o f 'th e overcoa t i s i n th e closet ' is something lik e 'i f I were in the closet an d th e light were on, then I' d have a n overcoat-lik e experience' , whic h is , presumably , true . Physical objects are no t take n t o consis t of ideas , a s with Berkeley, but physica l objec t statement s ar e take n t o mea n somethin g othe r than what we ordinarily take them to mean. 9

Berkeley' s notorious solutio n wa s t o suppos e tha t Go d i s perceiving the object even whe n w e aren't ; indee d h e use s thi s as a nove l argument fo r th e existenc e o f God. See , e.g. Berkeley (1713), 211-13 , 230-1. It's worth noting , however, tha t in earlier work , Berkele y (171 0 § § 3 , 58—9 ) include s a 'counterfactual ' analysi s tha t prefigures the Millian phenomenalism described in the next paragraph. 10 Thi s ide a too k shap e i n Mil l (1865) , ch . 1 1 an d it s appendix , an d wa s developed i n th e for m describe d her e b y th e logica l positivists . See Aye r (1946) , 63-8.

REALISM I N PHILOSOPH Y 7

This ambitious programme wa s a complete failur e for a number of naggin g reasons , onl y the firs t o f whic h i s our seemin g inability to specif y th e require d sense datum—the overcoat-lik e experienc e —without referenc e t o th e overcoa t itself. 11 Bu t whateve r th e failings of phenomenalism, the attempt itself show s that the realist, to state her position completely , must also rule out such unintended misinterpretations o f common-sens e statements ; sh e mus t insis t that thes e statement s b e taken 'a t fac e value' . Becaus e it is hard t o say exactl y wha t thi s come s to , apar t fro m repeatin g that i t rule s out phenomenalism , realism is in some ways more difficul t t o stat e than it s particular rivals. In any case, we can be sure that common sense realism is opposed to both idealism and phenomenalism. Our discussio n s o fa r ha s centre d o n th e proble m o f statin g common-sense realism ; we must now as k why we should believe it. The failur e o f heroi c philosophica l alternative s lik e idealis m an d phenomenalism i s some reassurance , bu t w e woul d lik e a positive argument. Admittedly, we find it difficult no t t o believe in ordinary physical objects ; o f hi s ow n philosophica l scepticism , th e grea t David Hume writes: since reason is incapable of dispelling these clouds, nature herself suffices t o that purpose . .. I dine, I play a game of backgammon, I converse, and am merry with m y friends ; and whe n afte r thre e o r fou r hours ' amusement , I would retur n to thes e speculations, the y appear s o cold, an d strained , and ridiculous, tha t I canno t fin d i n m y hear t t o ente r int o the m an y farther . (Hume (1739), 548-9)

But eve n if common-sense realis m is psychologically inevitable, we should stil l ask after it s justification. The repl y give n b y man y contemporar y philosopher s i s simply that th e existenc e o f ordinar y thing s provide s th e bes t accoun t of our experienc e o f th e world . I n hi s landmar k essa y o n ontology , 'On what there is', W. V. O. Quine puts the point this way: By bringin g togethe r scattere d sens e event s an d treatin g the m a s perceptions o f on e object , w e reduc e th e complexit y o f ou r strea m o f experience t o a manageable conceptual simplicity . . .. we associate an earlier and a later round sensum with the same so-called penny, or with two differen t so-called pennies, in obedience to the demands of maximum simplicity in our total world-picture. (Quine (1948), 17) 11

Se e Urmson (1956), ch. 10 , for a survey of this and other difficulties .

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Now w e ca n hardl y b e sai d t o mak e a n explici t inferenc e fro m purely experientia l statement s t o physica l objec t statement s tha t account fo r them, because (a s noted i n connection wit h phenomenalism) w e hav e n o independen t languag e o f experience . Wha t actually happen s i s a developin g neurologica l mediatio n betwee n purely sensor y input s an d ou r primitiv e belief s abou t physica l objects.12 Th e justificator y inferenc e comes later , whe n w e argu e that th e bes t explanatio n o f our stubbor n belie f in physical object s is tha t the y d o exis t an d tha t ou r belief s abou t the m ar e brough t about i n variou s dependable ways , fo r exampl e b y ligh t bouncin g off thei r surface s o n t o ou r retina s etc . Thu s th e assumptio n o f objectively existing , medium-size d physical objects plays a n indispensable role in our bes t account of experience. But, on e migh t object , didn' t th e god s o f Home r provid e th e Greeks with a n explanation of their experience? Here Quin e point s to an important difference : For m y part I do, qu a la y physicist , believ e in physica l objects an d no t i n Homer's gods ; an d I consider i t a scientifi c erro r t o believ e otherwise. , , , The myt h of physical objects i s epistemologicaliy superior to most i n that it has prove d mor e efficaciou s tha n othe r myth s as a devic e for workin g a manageable structure into the flux of experience. (Quine (1951), 44)

Physical objects, not Homer' s gods , for m par t o f our bes t scientifi c theory o f th e world , an d fo r tha t reason , ou r belie f i n th e former , but not th e latter, is justified. Notice, however , tha t thi s sor t o f answe r wil l no t satisf y th e philosophical scepti c wh o call s al l ou r belief-formin g techniques, including thos e o f natura l science , int o question . Ren e Descartes , for example , wa s wel l awar e tha t scienc e presupposed a n objective external world, bu t he wanted a justification fo r science itself. Ho w can w e know , Descarte s asked , tha t th e scientifi c world-vie w i s correct? Ho w d o w e know ou r sense s aren' t deceivin g us? How d o 12

Fo r mor e o n this , se e ch. 2 , sect . 2 , below . Student s o f Quin e ma y detec t a tension betwee n m y position i n this paragraph an d suc h Quinean remark s as 'From among th e variou s conceptual schemes best suite d to thes e various pursuits, one— the phenomenalistic—claim s epistemologica l priority ' ((1948) , p . 19) . Here an d i n what follows , I will ignor e thi s lingerin g trace o f positivis m i n th e master . I n fact , there i s no phenomenalisti c language o r theory , an d a goo d scientifi c explanatio n must d o mor e tha n accuratel y predic t sens e experiences . (Cf . Putna m (1971) , 355-6.)

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we kno w w e aren' t dreaming ? How d o w e kno w ther e i s no Evi l Demon systematicall y makin g i t appea r t o u s a s i f the worl d i s as we think it is?13 These Cartesia n challenge s depen d o n a conceptio n o f epistemo logy a s a n a priori 14 stud y o f knowledg e an d justification , a stud y above, beyond , outside , indee d prio r to , natura l science , a stud y whose aim is to establis h that scienc e on a firm footing. One might think that the justificatory practices of science itself ar e the bes t we have, bu t classica l epistemology appeal s t o highe r canon s o f pur e reason. Unfortunately , it s attempt s t o reconstruc t natura l scienc e on a n a priori, philosophicall y justifie d foundatio n have all failed, beginning with Descartes's own effort. 15 In ligh t o f thi s history , Quin e suggest s a radicall y differen t approach t o epistemology . Ou r bes t understandin g of th e world , after all , i s our curren t scientifi c theory , s o b y what bette r canon s can we hope t o judg e ou r epistemologica l claims than b y scientifi c ones? Th e stud y o f knowledge , then , become s par t o f ou r scient ific study o f the world, rathe r than a n ill-defined , pre-scientifi c enterprise: 'Epistemolog y . . . simpl y fall s int o plac e a s a chapte r o f psychology an d henc e o f natura l science . I t studie s a natura l phenomenon, viz. , a physical human subject ' (Quin e (19696), 82). Standing withi n ou r ow n bes t theor y o f th e world—wha t bette r perspective coul d w e have?—w e as k ho w huma n subject s lik e ourselves ar e abl e t o for m reliabl e beliefs abou t th e worl d a s ou r theory tell s u s i t is . Thi s descriptiv e an d explanator y projec t i s called 'epistemology naturalized'. Thus science is used to justify science , but thi s circle is not vicious once w e giv e u p th e classica l projec t o f foundin g scienc e o n something mor e dependabl e tha n itself . Fro m naturalize d perspect ive, ther e i s n o poin t o f vie w prio r o r superio r t o tha t o f natura l science, and Quine' s argumen t for common-sens e realism becomes perfectly reasonable : th e assumptio n o f physical objects is part o f our bes t theory , an d bein g par t o f ou r bes t theor y i s th e bes t justification a belief can have. 13

Descartes (1641), esp. Meditation One . 'A priori' means prio r t o experience, a s opposed to 'a posteriori'. Descarte s argue d tha t our perception s are reliable becaus e Go d is no deceiver . Serious objection s t o thi s approac h aros e immediately ; se e hi s 'Objection s an d replies', publishe d a s an appendi x to hi s Meditations. A more recent failed effor t t o found scienc e is that of the positivists. Se e Quine's discussio n (19696). 14

15

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M

A secon d for m o f philosophica l realis m concern s itsel f wit h th e more esoteri c object s o f science , wit h unobservabl e theoretica l entities like electrons , genes , an d quarks . Here th e scientifi c realis t asserts that our belie f i n such things is justified, a t least to the exten t that theorie s involvin g them provid e u s wit h th e bes t explanatio n we hav e fo r th e behaviou r o f observabl e objects . Onc e again , however, i t is not enoug h fo r the realist to sa y just this. While ther e can b e no idealis t here , analogou s t o th e Berkeleian , insisting tha t electrons ar e jus t bundles of sensory experiences, there i s a positio n analogous t o phenomenalism : instea d o f translatin g tal k abou t medium-sized physica l object s int o tal k abou t wha t sensor y experiences woul d occu r unde r wha t circumstances , th e opera tionalist woul d hav e u s translat e tal k o f unobservabl e theoretica l entities into talk about ho w observables would behav e under which circumstances.16 Thus , fo r example , par t o f th e translatio n o f 'there's a quark here ' might be 'i f w e set up a cloud chamber , we' d get this kind of track' and 'i f we prepared a photographic emulsion , we'd se e this kin d o f trace ' an d 'i f w e ha d a scintillatio n counter , we'd ge t thi s typ e o f signal' . Thi s projec t faile d a s resoundingl y as phenomenalism, 17 an d fo r som e of the sam e reasons, bu t again , its very existenc e show s tha t alon g wit h assertin g th e existenc e o f those unobservable s presuppose d b y our bes t theory , th e scientifi c realist must also insist that this assertion be taken 'at face value'. The scientifi c realist' s mos t conspicuou s opponen t i s the instru mentalist, wh o hold s tha t unobservable s ar e a mere 'usefu l fiction ' that help s u s predic t th e behaviou r o f th e observable . Thu s th e instrumentalist denie s jus t wha t th e scientifi c realis t asserts—that there ar e electrons etc.—but continues to us e the sam e theories the realist doe s t o predic t th e behaviou r o f observables. Fo r a practisin g scientist, instrumentalis m woul d see m a s dramati c a for m o f double-think a s the duplicitou s mathematical formalis m describe d earlier. Bu t ther e ar e wors e problem s tha n inauthenticity . Fo r example, th e distinctio n betwee n theoretica l an d observationa l turns ou t t o b e devilishl y hard t o draw. 18 An d eve n i f it coul d b e 16 Th e classi c statemen t o f operationalis m is Bridgman (1927) . It i s criticized by Hempel (1954) , bu t th e logica l positivist s foun d mor e subtl e form s (e.g . Carna p (1936/7)). 17 On e o f it s man y critic s i s Putna m (1962) . Other s ar e Maxwel l (1962 ) an d Achinstein(1965). 18 Se e the papers cited in the previous footnote.

REALISM I N P H I L O S O P H Y 1

1

drawn, i t i s unclea r wh y th e differenc e betwee n bein g humanly observable an d no t shoul d hav e suc h profoun d metaphysica l consequences.19 Finally , th e instrumentalisti c scientist , happil y using the false premisses of theoretical science to derive purportedly true conclusion s abou t observables , provide s u s no explanatio n of why a batch of false claims should be so dependable. So, onc e again , a s i n th e cas e o f common-sens e realism , th e failure o f th e oppositio n provide s som e negativ e suppor t fo r scientific realism . Th e positiv e argumen t i s als o analogous. 20 Afte r the above-quote d defenc e o f commo n sens e realism , Quin e con tinues: Positing doe s no t sto p wit h macroscopi c physica l objects . Objects a t th e atomic leve l ar e posite d t o mak e th e law s o f macroscopi c objects , an d ultimately th e law s o f experience , simple r an d mor e manageable. . . Science i s a continuatio n of commo n sense , an d i t continue s th e common sense expedien t o f swellin g ontolog y t o simplif y theory . . . . Epistemologically thes e ar e myth s o n th e sam e footin g with physica l objects an d gods , neither bette r no r wors e excep t fo r th e difference s i n th e degre e t o which the y expedit e ou r dealing s with sens e experiences . (Quin e (1951) , 44-5)

From th e poin t o f vie w o f epistemolog y naturalized , what bette r justification coul d w e hav e to believ e i n th e mos t well-confirmed posits o f ou r bes t scientifi c theor y tha n th e fac t tha t the y ar e th e most well-confirmed posits of our best scientific theory? The final realist/anti-realist controversy I want t o conside r is in fac t the oldes t debat e i n which th e ter m 'realism ' arises . We common sense realist s all agree tha t ther e ar e man y red things—re d roses , red houses, red sunsets—but the ancient question is whether or no t there i s also , ove r an d abov e thi s lo t o f particula r re d things , a further thin g the y share , namely , redness . Suc h a n additiona l thing—redness—would b e a universal . The mos t basi c differenc e between particulars and universals is that a universal can be present ('realized', 'instantiated', 'exemplified') i n more tha n on e place at a 19

Se e Devitt (1984), §§ 7.1 , 7.6 , 8.5 . Th e analog y i s no t perfec t becaus e wit h physica l objects , unlik e theoretica l entities, we believ e i n the m fro m th e start ; w e ar e neve r i n th e positio n o f decidin g whether o r no t t o begi n believin g i n them . M y poin t i s tha t th e for m o f th e justification fo r the belief , however it is arrived at, i s the sam e in both cases. 20

12

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time, whil e a particula r cannot. 21 Plat o originate d th e mos t dramatic versio n o f realis m abou t universal s i n hi s spectacula r theory o f Forms: Redness , Equality , Beauty, and s o on, ar e perfect, eternal, unchangin g Forms ; the y exist outside of time and space ; w e know them by means of the non-sensory intellect; ordinar y physica l properties, perceived by the usual senses, are but pale an d imperfec t copies.22 Aristotle , Plato' s student , too k direc t ai m a t th e mor e bizarre element s o f this view, and defende d a mor e modes t for m of realism, accordin g t o whic h universal s exist onl y i n thos e thing s that exemplif y them. 23 Their opponen t i s the nominalist, lik e John Locke, who holds that there is nothing over and above particulars.24 One classica l argumen t fo r realis m abou t universals , th e On e over Many , survive s in th e moder n debate . Davi d Armstrong , it s most vocal contemporar y advocate , put s i t like this: 'It s premis e is that man y differen t particular s ca n al l hav e wha t appear s t o b e the same nature' (Armstron g (1978), p. xiii); 'I would . . . draw th e conclusion that , a s a result , ther e i s a prima facie cas e fo r postulating universals ' (Armstron g (1980) , 440-1) . A simila r argument ca n b e given i n linguistic form, arguin g fo r exampl e tha t a universa l rednes s mus t exis t sinc e th e predicat e 'redness ' i s meaningful. Mos t contemporar y thinkers , includin g Armstrong , reject thi s secon d for m o f th e argument. 25 T o se e why, consider a scientific universa l lik e 'bein g a t a temperatur e o f 3 2 degree s Fahrenheit'. Scientist s tel l u s tha t thi s i s th e sam e universal , o r property, t o us e a mor e natural-soundin g word , a s 'havin g such and-such mea n kineti c energy'. 26 Bu t thes e tw o predicate s hav e very differen t meanings . S o properties ar e no t meanings . The y ar e 21 A particula r nee d no t b e spatiall y o r temporall y continuous—m y cop y o f Principia Mathematica i s in thre e volumes, on tw o differen t shelves—bu t even then, it i s only par t o f th e particula r that i s present i n eac h o f th e disparat e locations . A universal is understood to b e full y presen t in each of it s instances. 22 Th e standar d referenc e fo r Plato' s theor y i s his Republic, chs . 5-7 . Wedber g (1955), ch . 3 , give s a usefu l summary . H e cue s th e late r Timaeus (3 7 D-38 A ) as defining th e sens e i n whic h Form s ar e outsid e time . A metaphorica l passag e i n Phaedrus (247 ) declare s thei r locatio n a s 'th e heave n whic h i s above th e heavens' , indicating they are not spatial, and Aristotle' s commentary confirms this: 'the Form s are not outside, because they are nowhere' (Physics, 203 a 8). 23 Se e his Metaphysics, bk . 1 , § 9 fo r criticism s o f Plato, and Categorie s 2, fo r hi s own view. 24 Se e Locke (1690) , bk. 3, ch. 3,§ 1 . 25 Se e Armstrong (1978), pt. 4. Se e also Putna m (1970) , § 1. 26 Wilso n (1985 ) argue s that thi s frequentl y cite d example i s more comple x tha n philosophers ordinaril y appreciate , bu t I don' t thin k thi s importan t observatio n affects th e point at issu e here.

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3

individuated by scientific tests, such as playing the same causal role, rather than b y synonymy of predicates. But eve n Armstrong' s preferre d for m o f th e On e ove r Man y argument ha s bee n severel y criticize d from a numbe r o f differen t perspectives.27 On e stunningl y simple counter-argument , Quine's , goes lik e this. We want to sa y that Ted and Ed are white dogs. This is supposed t o commi t u s to th e universa l 'whiteness' . But for 'Ted and Ed are white dogs' to be true, all that is required is that there be a whit e do g name d 'Ted ' an d a whit e do g name d 'Ed' ; n o 'whiteness' or eve n 'dogness' i s necessary. If there is more tha n this to th e On e ove r Man y argument , th e realis t owe s a n accoun t o f what tha t is . I f not, th e realis t need s some othe r suppor t fo r th e existence o f these universals. Contemporary thinker s hav e propose d a mor e moder n argu ment, moder n i n th e sens e tha t i t partake s o f th e 'naturalizing ' tendency identifie d abov e i n recent epistemology . Notic e firs t tha t the On e ove r Man y i s presente d a s a n a prior i philosophica l argument fo r a n ontological conclusion : ther e ar e universals. No w epistemology naturalized , a s describe d above , ha s renounce d th e classical clai m t o a philosophica l perspectiv e superio r t o tha t o f natural science . A s a result , th e Cartesia n deman d fo r certaint y beyond th e scientifi c wa s als o rejected . Fro m thi s ne w naturalized perspective, th e considere d judgemen t o f scienc e i s th e bes t justification w e ca n have . Finally , we'v e see n tha t thi s shif t i n epistemological thinkin g produces a correspondin g shif t i n onto logical thinking ; fo r example , despit e philosophical qualm s abou t unobservable entities, we should admit they exist if our bes t science tells us they do. The moral for the defender of universals is clear: to show tha t ther e ar e universals , don' t tr y t o giv e a pre-scientifi c philosophical argument ; jus t sho w tha t ou r bes t scientifi c theor y cannot do without them . Much o f th e curren t debat e take s thi s form. 28 Th e questio n a t issue i s whether intensiona l entitie s lik e universal s are needed , o r whether scienc e can ge t by on extensiona l entities like sets.29 Mos t 27

Se e Quine (1948), Devitt (1980) , Lewis (1983). Wit h Putna m (1970 ) an d Wilso n (1985 ; an d forthcoming ) o n th e positive , Quine (1948 ; 19806) on the negative. 29 Universal s ar e intensiona l becaus e tw o o f the m ca n appl y t o th e sam e particulars withou t bein g identical , fo r example , 'huma n being ' an d 'featherles s biped'. Sets , b y contrast , ar e extensional ; tw o set s wit h th e sam e member s ar e identical. 28

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will grant cha t sets are more promiscuous tha n universals ; random elements ca n b e gathered int o a set even i f they have no propert y i n common. Furthermore, scienc e seems to need a distinction betwee n random collection s an d 'natural ' ones ; when w e notice tha t all the ravens we'v e examine d ar e black , w e conclud e tha t al l raven s belong to the set of black things, not tha t al l ravens belong to the set of thing s tha t ar e eithe r blac k o r no t examine d b y us . Th e ope n question i s whethe r th e nominalis t ca n dea l wit h thi s distinctio n between natura l an d unnatura l collection s withou t appealin g t o universals.30 To summarize , then , t o b e a realis t abou t medium-size d physical objects, th e theoretica l posit s o f science , o r universals , i s to hol d that thes e entitie s exist , tha t the y d o s o objectively—the y ar e no t mental entities , and the y have the properties the y do independently of ou r language , concepts , theories , an d o f our cognitiv e apparatus in general—an d t o resis t variou s efforts—phenomenalism, opera tionalism—to reinterpre t thes e claims . And , i n th e naturalize d spirit, the realis t assume s that the most strongl y hel d of our current theoretical belief s ar e probabl y a t leas t approximatel y correc t accounts o f wha t thes e thing s ar e like . Beyon d what' s sketche d above, I will pause n o furthe r ove r argument s fo r o r agains t thes e three forms o f philosophical realism, 31 bu t I will take somethin g of a stand . Most o f wha t follow s wil l presuppos e bot h common-sens e an d scientific realism. 32 Indeed , a s wil l com e ou t below , th e debat e about th e existenc e an d natur e o f mathematica l entitie s i s almos t always pose d b y comparin g the m wit h medium-size d physica l objects and/o r theoretica l entities ; the philosopher' s temptatio n i s to embrac e common-sens e an d scientifi c realis m whil e rejectin g mathematical realism. 3 ' I wan t t o remark , however , tha t I don' t 30

Lewi s (1983) provides a usefu l surve y of the debate. Devit t (1984 ) provide s a usefu l compendiu m o f argument s fo r common sense and scientifi c realism . 32 Som e argu e tha t common-sens e an d scientifi c realis m ar e incompatible , because physic s reveal s cha t medium-size d physica l object s ar e quit e differen t fro m our common-sens e conception . Bu t showin g w e ar e ofte n wron g abou t stone s an d tables i s not th e sam e a s showin g tha t thes e thing s don' t exist . I se e no proble m i n allowing tha t scienc e ca n correc t commo n sense . Devit t (1984 , § 5.10 ) sketche s a position of this sort. 33 Se e e.g. Putna m (19756) , 74. 31

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5

think th e rough-and-read y mathematica l realis m introduced i n the previous sectio n stand s o r fall s wit h thes e othe r realisms . I f it s central tene t i s tha t mathematic s i s a s objectiv e a scienc e a s astronomy, physics , biology, etc., then thi s might remai n true even if those natura l sciences turn out no t t o be as objective as the realist thinks the y are . I n th e lon g run , I' m muc h mor e intereste d i n blocking a seriou s metaphysica l o r epistemologica l disanalog y between mathematic s and natura l scienc e than I am in maintaining a strict realism about either. 34 Finally, because I think the issues involved are not nearly as clear, I will remain officially neutra l on universals. The questio n will arise (in Chapte r 2 , sectio n 4 ; Chapte r 3 , sectio n 2 ; an d Chapte r 5 , sections 2 and 3 ) but I think nothin g I say will hang on the ultimate resolution of the metaphysical debate outlined above. The proble m in questio n i s quit e general ; i t i s n o mor e a proble m fo r mathematics than it is for the rest of science.35 3. Realis m and truth In recent years, many philosophers have come to think that realism should b e understood, no t a s a claim about wha t ther e is , but a s a claim about semantics. 36 Whether one is a realist or not—about the objects o f commo n sense , theoretica l entities , universals , o r mathematical objects—i s sai d t o depen d o n wha t on e take s t o b e the condition s fo r th e trut h o r falsit y o f th e correspondin g statements. Now m y own pre-philosophica l statement s abou t mathematica l realism d o involv e what sound s like a semanti c element: I claimed that mathematic s i s about numbers , sets , functions , etc., and tha t the wa y thes e thing s ar e i s wha t make s mathematica l statement s true o r false . Thi s sor t o f tal k ca n b e rea d a s espousin g a 34 Fo r example , m y argumen t agains t Wittgenstei n (Madd y (1986) ) take s th e form: eve n i f hi s genera l anti-realis m i s correct , stil l hi s stron g maths/scienc e disanalogy need no t b e accepted. 35 Unless , o f course , al l mathematica l entitie s ar e universals . A versio n o f thi s view is considered i n ch. 5, sect. 3, below. 36 Th e influenc e here i s Dummett's; see Dummett (1978) , introd. an d chs . 1 , 10, and 14 , an d (1977) , ch . 7 . Devit t (1984 ) give s a mor e complet e discussio n o f th e relationships betwee n realis m an d semantics , an d hi s ch . 1 2 take s u p Dummett' s position i n particular .

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correspondence theor y o f truth , accordin g t o whic h th e trut h o f a sentence depend s partl y o n th e structur e o f the sentence , partl y o n the relation s betwee n th e part s o f the sentenc e an d extra-linguisti c reality,37 an d partl y o n th e natur e o f tha t extra-linguisti c reality. One definitiv e aspec t o f correspondenc e theorie s i s tha t wha t i t takes fo r a sentenc e t o b e tru e migh t wel l transcen d what w e ar e able to know . Semantic anti-realists , b y contrast , wan t t o identif y th e trut h conditions o f a sentenc e with somethin g close r t o ou r abilitie s to know, wit h tha t whic h justifie s th e assertio n o f th e sentence , wit h some versio n o f it s 'verificatio n conditions' . Notic e tha t phenom enalism coul d b e reinterpreted this way, as the clai m tha t th e trut h of 'm y overcoa t i s i n th e closet ' reduce s t o th e trut h o f variou s counterfactual conditional s lik e the one about what experiences I' d have i f I wer e i n th e close t wit h th e ligh t on . Move s t o verificationism ar e variousl y motivated—by th e hop e o f avoidin g scepticism, b y the desir e to eliminat e metaphysics, by a disbelief in the objectiv e realit y o f th e entitie s i n question , b y attentio n t o purported fact s o f languag e learning , b y scepticis m abou t th e notion o f correspondence truth itself. 38 Thus thes e semanti c thinker s identif y realis m abou t a certai n range o f entitie s wit h a correspondenc e theor y o f trut h fo r sentences concernin g thos e entities , and likewis e anti-realism wit h verificationism. Give n our previou s characterization of realism a s a position o n wha t ther e is , suc h a n identificatio n seem s wrong headed. Fo r example , a n idealis t lik e Berkele y coul d embrac e a correspondence theory ; fo r him , th e extra-linguisti c realit y tha t makes ordinar y physica l objec t statement s eithe r tru e o r fals e consists o f bundles of experiences. Being a correspondence theoris t doesn't mak e hi m a realis t i n ou r sens e becaus e hi s objects aren' t objective. On th e othe r hand , ou r realis t think s he r entitie s d o exis t objectively, which includes the belie f tha t the y exist and ar e as they are independentl y o f our abilitie s to kno w abou t them . Sh e holds, 37 Thi s formulatio n will hav e r o b e modifie d i n th e specia l cas e o f statement s explicitly about language, but I'll ignore this complication. 58 Example s o f each , i n order : th e phenomenalis t Mil l (1865) , chs. 1 an d 11 , inspired b y the idealis t Berkeley (1710; 1713) , (se e the preface t o Berkele y (1713) ); the positivists , for exampl e Aye r (1946 , ch . 1) ; th e idealis t Brouwer (1913 ; 1949) , see als o Heytin g (1931 ; 1966) ; th e thoroughgoin g verificationis t Dummet t (1975 ; 1977, ch . 7); and finally, Putnam (1977; 1980) .

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then, tha t ther e are , o r a t leas t ma y be , man y truth s abou t thos e entities tha t ar e beyon d th e reac h o f eve n ou r mos t idealize d procedures of verification. Thus she could hardly be a verificationist. So, even if holding a correspondence theor y isn' t th e sam e thing as bein g a realist , it migh t see m that realis m require s a correspon dence theory . Bu t thi s onl y follow s o n th e assumptio n tha t thes e two candidate s fo r a theor y o f truth—correspondenc e an d verifi cationism—exhaust the field. They do not. A thir d typ e o f trut h theor y i s based o n th e simpl e observation that 'so-and-so is true' says no mor e than 'so-and-so' , in particular that 'M y overcoat i s in the closet ' is true if and onl y if my overcoa t is in the closet. This sort of theory has various names—redundancy theory, disappearanc e theory, deflationar y theory—bu t I'll call it a disquotational theory. O n thi s view, truth i s a nothing more than a simplifying linguisti c device . In case s lik e that o f m y overcoat , i t does littl e mor e tha n stylisti c work . Whe n w e us e i t i n mor e complex contexts—fo r example , i f I clai m tha t everythin g in th e Bible is true—it works a s an abbreviating device that saves me a lot of time . With tha t on e sentence I'm assertin g that i n the beginning God create d th e heaven s and th e earth , an d no w th e eart h wa s a formless void . . . and God's spirit hovered over the water, and God said, 'Le t ther e b e light', an d ther e wa s light , and. . ., an d s o on through the many sentences in the Bible. In recen t years , ther e ha s bee n considerabl e debat e betwee n naturalized realist s ove r correspondenc e versu s disquotationa l truth.39 Th e issue , o f course , i s whethe r o r no t th e notio n o f correspondence trut h mus t figur e a t al l i n ou r bes t theor y o f th e world. Fo r man y purpose s tha t see m t o requir e a full-blow n correspondence notion , disquotationa l trut h ha s turne d ou t t o d o the jo b a s well . Thi s i s tru e eve n i n philosophica l contexts . Fo r example, I claimed earlier that mathematical statements are true or false independentl y o f ou r abilit y t o kno w this . O n th e disquo tational theory , thi s woul d com e t o a lon g (indee d infinite ) conjunction: whethe r o r no t 2 + 2 = 4 i s independent of whether or no t w e ca n kno w which , an d whethe r o r no t ever y non-empty bounded se t o f real s ha s a leas t uppe r boun d i s independen t o f whether no t w e ca n kno w which , an d whethe r o r no t ther e i s an inaccessible cardinal is independent of whether or not we can know 39 Se e e.g. Fiel d (1972) , Grove r e t al. (1975), Leed s (1978) , Devit t (1984) , Fiel d (1986), an d the references cited there .

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which, an d . . ., etc. S o the questio n i s whether ther e ar e an y jobs for which a notion o f correspondence trut h is actually indispensable. To se e what hang s o n this , consider : Alfre d Tarski' s celebrate d definition o f correspondenc e trut h reduce s th e proble m o f pro viding a n accoun t o f tha t notio n t o th e proble m o f providin g a n account o f th e word-worl d connections , tha t is , an accoun t o f th e relation o f referenc e tha t hold s betwee n a nam e an d it s bearer , between a predicat e and th e object s tha t satisf y it. 40 Th e disquo tational theory , on th e other hand , includes a theory o f reference as unexciting a s it s theory o f truth : 'Alber t Einstein ' refer s t o Alber t Einstein; 'gold' refers to gold. So what hangs on the debate betwee n correspondence an d disquotationa l truth i s the nee d fo r a substantive theor y o f wha t i t i s b y virtu e o f whic h m y us e o f th e nam e 'Albert Einstein ' manage s t o pic k ou t tha t certai n historica l individual. Thi s i s n o trivia l matter ; henc e th e livel y debat e ove r whether or no t a correspondence theor y is really necessary. I won't ge t into that debat e here, because it would tak e us too fa r afield an d I certainly have nothing helpfu l t o add , bu t I do wan t t o take note o f where the hunt has led these investigators. It's perhap s not surprisin g tha t wher e pus h seem s t o com e t o shov e o n th e question o f trut h an d referenc e i s in tha t portio n o f ou r theor y o f the world tha t treats the activities of human beings. Consider thi s case : D r Jobe heal s Isiah Thomas' s ankl e injury i n time for the big game. How d o we explain this phenomenon?41 As a first step, we notice that D r Jobe ha s a vas t number of tru e beliefs abou t sport s injuries , abou t th e rigour s of basketball, abou t the physical and menta l condition o f basketball players, and abou t Isiah i n particular . Wha t D r Job e think s abou t thes e thing s i s usually correct. How , then , are we to explain his reliability on these topics? I n an effor t t o answe r thi s question, we begin to detai l such things a s the doctor' s previou s experience, his medical training, his interactions wit h man y basketbal l players , an d hi s previou s interactions with Isiah himself . Now th e trut h o f th e doctor' s belief s migh t b e accounte d fo r disquotationally. Bu t th e correspondenc e theoris t wil l poin t ou t that amon g th e sorts o f connections between Jobe's belief s an d th e subject matter of those beliefs that we've been describing in order t o account fo r hi s reliabilit y ar e jus t th e sort s o f connection s tha t 40 41

Fo r Tarski's theory, seeTarski (1933) ; for the reduction, see Field (1972). Her e 1 follow Fiel d (1986) , though the particula r example is my own .

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might wel l se t up a robus t referentia l connectio n betwee n hi s uses of the predicate 'basketbal l player' and basketball players, betwee n his us e o f 'Isia h Thomas ' an d th e Pisto n guard. 42 S o th e debat e between correspondenc e an d disquotatio n come s dow n t o th e question of whether or not such a theory o f reference can or need be constructed fro m these materials. But notice, bot h partie s t o the debate agre e that Jobe's reliability needs explanation , an d the y agre e on th e sort s o f fact s tha t migh t provide one. When a person i s reliable on some subject—whether it be Jobe o n Isiah' s ankle , o r a geologis t o n Moun t S t Helens, o r a historian o n th e cause s of the Industria l Revolution, o r a ten-yearold ki d o n hi s favourit e roc k star—tha t reliabilit y need s a n explanation. W e look fo r an account of how th e person's cognitiv e machinery is connected bac k to what she' s reliabl e about, vi a what she's rea d an d th e source s o f tha t material , vi a he r conversation s with other s an d thei r sources, vi a her observation s o f indicators o r instruments, and vi a her actual experience with her subjec t matter . The disagreemen t i s onl y abou t whethe r o r no t thi s welte r o f material will produce a non-trivial theory of reference. As I'v e said, I hav e n o intentio n o f takin g side s o n thi s las t question; I' m perfectl y willin g t o le t th e participant s reac h thei r own conclusions . Th e poin t tha t need s making here is this: even if the disquotationalis t succeed s i n relievin g th e realis t o f he r dependence o n correspondenc e truth , an d thu s o n non-trivia l reference, th e matte r o f explainin g th e variou s 'reliabl e connec tions' wil l remain . Thus, i n case s wher e th e nee d fo r a referentia l connection seem s to involv e the mathematica l realist in difficulties , casting of f th e nee d fo r referenc e is not likel y t o help , becaus e the requirements o f 'reliabl e connection ' ar e almos t certai n t o lea d t o the same, or essentially similar, difficulties. 43 In wha t follows , then , I wil l sometime s phras e bot h th e challenges t o mathematica l realis m an d m y responses i n term s o f 42 Wha t th e correspondenc e theoris t ha s i n min d her e i s a versio n o f th e causa l theory of reference described in ch. 2, sect. 1, below. 43 A s Fiel d (1986 ) make s clear , fro m th e poin t o f vie w o f th e overal l realis t project, a theor y o f reliabl e connection woul d probabl y b e easie r tha n on e o f ful l reference; fo r example , referenc e i s compositional—th e referenc e o f a whol e i s thought t o depen d systematicall y o n th e referenc e of th e parts—whil e a theor y o f reliability might not requir e so much detailed structure. But the aspects of the theory of referenc e tha t ar e though t t o creat e difficultie s fo r th e mathematica l realis t ar e those i t seem s to shar e wit h th e theor y o f reliability , fo r example , concern s abou t causation. Se e ch. 2, sect. 1, below.

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correspondence trut h an d reference, but here, again for the record, I want to emphasiz e that thi s way of stating things is convenient bu t not essential . Thos e realist s who believ e tha t a robus t referenc e relation is not neede d in science, eithe r becaus e the disquotationalist is right, or fo r some other reason , are invited simply to recast th e discussions that follow in terms of 'reliable connection'. One final remark on realism and truth. Som e anti-realists, assuming the realis t i s wedde d t o correspondenc e truth , hav e argue d tha t realism i s unscientifi c becaus e i t require s a connectio n betwee n scientific theor y an d th e worl d tha t reache s beyond th e bound s of science itself. 44 Her e th e anti-realis t attempts t o saddl e th e realis t with th e now-familia r unnaturalized standpoint, th e poin t o f view that stand s above , outside , o r prio r to , ou r bes t theorie s o f th e world, an d fro m whic h i s posed th e question : what connect s ou r theories to the world? We've see n tha t i n epistemology , the contemporar y realis t ha s answered b y rejectin g th e extra-scientifi c challeng e itself , alon g with th e radica l scepticis m i t engenders . Th e sam e goe s fo r semantics. There i s no poin t o f view prior t o o r superio r to tha t of natural science . Wha t w e wan t i s a theor y o f ho w ou r languag e works, a theor y tha t wil l becom e a chapte r o f tha t ver y scientifi c world-view. I n orde r t o arriv e a t thi s ne w chapter , i t woul d b e madness t o cas t of f th e scientifi c knowledg e collecte d s o far . Rather, w e stan d withi n ou r curren t bes t theory—wha t bette r account d o w e hav e o f th e wa y th e worl d is?—an d as k fo r a n account o f ho w ou r belief s an d ou r languag e connec t u p wit h th e world a s tha t theor y say s i t is . Thi s ma y b e th e robus t theor y of reference required by correspondence truth. If the disquotationalist is right, i t may b e something less structured, an accoun t of reliable connection. But neither way i s it something extra-scientific. 4. Realis m in mathematics Let m e tur n a t las t t o realis m i n th e philosoph y o f mathematic s proper. Most prominent i n this context is a folkloric position called 'Platonism' b y analogy with Plato' s realis m abou t universals . As is 44 Se e e.g . Putna m (1977) , 125 . Or , fro m a differen t poin t o f view , Burgess (forthcoming a) .

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common wit h suc h venerabl e terms , i t i s applied t o view s o f ver y different sorts , most o f them no t particularly Platonic. 45 Here I will take i t i n a broa d sens e a s simpl y synonymou s wit h 'realism ' a s applied t o th e subjec t matte r o f mathematics : mathematic s i s th e scientific stud y o f objectivel y existin g mathematical entitie s jus t a s physics i s th e stud y o f physica l entities . Th e statement s o f mathematics ar e true o r fals e dependin g on th e properties o f thos e entities, independen t o f ou r ability , o r lac k thereof , t o determin e which. Traditionally, Platonis m i n th e philosoph y o f mathematic s ha s been take n to involve somewhat more than this. Following some of what Plat o ha d t o sa y abou t hi s Forms , man y thinker s hav e characterized mathematica l entitie s a s abstract—outsid e o f physical space, eternal and unchanging—and as existing necessarily— regardless o f th e detail s of th e contingen t make-u p o f th e physica l world. Knowledg e o f suc h entitie s i s ofte n though t t o b e a priori—sense experienc e ca n tel l u s how thing s are, not ho w the y must be—an d certain—a s distinguishe d fro m fallibl e scientifi c knowledge. I wil l cal l thi s constellatio n o f opinion s 'traditiona l Platonism'. Obviously, thi s uncompromising account of mathematical realit y makes the question o f how we humans com e to kno w th e requisite a prior i certaintie s painfull y acute . An d th e successfu l applicatio n of mathematic s t o th e physica l worl d produce s anothe r mystery : what d o th e inhabitant s o f th e non-spatio-tempora l mathematica l realm hav e to do with th e ordinary physica l thing s o f the world we live in ? I n hi s theor y o f Forms , Plat o say s tha t physica l thing s 'participate' in the Forms, an d h e uses the fac t o f our knowledg e of the latter, vi a a sort of non-sensory apprehension , t o argue that th e soul mus t pre-exis t birth. 46 Bu t our naturalize d realist wil l hardl y buy this package. Given thes e difficultie s wit h traditiona l Platonism , it' s no t surprising tha t variou s form s o f mathematica l anti-realis m hav e been proposed . I'l l pause t o conside r a samplin g o f thes e view s before describing the two main schools of contemporary Platonism . 45 Fo r example, thoug h th e term 'Platonism ' suggest s a realism about universals, many Platonist s regar d mathematic s a s th e scienc e o f peculiarl y mathematica l particulars: numbers , functions , sets , etc . A n exception i s the structuralis t approac h considered i n ch. 5, sect. 3, below. 46 Se e his Phaedo 72 D-7 7 A .

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In th e lat e 1600s , i n respons e t o a numbe r o f question s fro m physical science , Si r Isaa c Newto n an d Gottfrie d Wilhel m vo n Leibniz simultaneousl y an d independentl y invente d th e calculus . Though the scientist's problems wer e solved, the new mathematica l methods wer e scandalousl y error-ridden an d confused . Among th e most vociferous and perceptiv e critics wa s th e idealis t Berkeley, an Anglican bishop wh o hope d to silence the atheists b y showing thei r treasured scientifi c thinkin g to b e even less clear than theology . Th e central poin t o f contentio n wa s th e notio n o f infinitesimals , infinitely smal l amount s stil l no t equa l t o zero , whic h Berkele y ridiculed a s 'th e ghost s o f departe d quantities'. 47 Tw o centurie s later, Bolzano , Cauchy , an d Weierstras s ha d replace d thes e ghost s with the modern theor y of limits.48 This accoun t o f limit s required a foundatio n o f it s own , whic h Georg Canto r an d Richar d Dedekin d provide d i n thei r theor y o f real numbers , bu t thes e i n tur n reintroduce d th e ide a o f th e completed infinit e int o mathematics . N o on e ha d eve r muc h like d the seemingl y paradoxica l ide a tha t a prope r par t o f a n infinit e thing coul d b e i n som e sens e a s larg e a s th e whole—ther e ar e a s many eve n natural numbers as there ar e even and odd , ther e are as many point s o n a one-inc h lin e segmen t a s o n a two-inc h lin e segment—but th e infinit e set s introduce d b y Canto r an d other s gave ris e to outrigh t contradictions , o f whic h Bertran d Russell' s is the most famous: 49 conside r the set of all sets that ar e not member s of themselves. It is self-membered i f and onl y if it isn't. The openin g decades o f this century sa w th e developmen t o f three grea t school s of though t o n th e natur e o f mathematics , al l o f the m designe d t o deal in one way or another with th e problem of the infinite . The firs t o f thes e i s intuitionism, which deal t wit h th e infinit e b y rejecting i t outright . Th e origina l versio n o f thi s position , firs t proposed b y L . E. J. Brouwer, 50 wa s analogou s t o Berkeleia n 4 " Se e Berkele y (1734) , subtitle d 'A Discours e Addresse d t o a n Infide l Mathe matician. Wherein I t i s Examined Whethe r th e Object , Principles , an d Inference s o f the Moder n Analysi s ar e Mor e Distinctl y Conceived , o r Mor e Evidentl y Deduced , than Religious Mysteries and Point s of Faith'. The quotation is from p . 89 . 48 Fo r a more detailed descriptio n o f the developments sketched i n this paragraph and th e next, see Kline (1972), chs. 17 , 40, 41 , an d 5 1, or Boyer (1949) . 49 Th e parado x mos t directl y associate d wit h Cantor' s wor k i s Burali-Forti' s (1897). See Cantor's discussion (1899) . Russell's primary target was Frege , as will be noted below. 50 Brouwe r (1913 ; 1949) . Other , les s opaque , exposition s o f thi s positio n ar e Heyting(1931; 1966 ) and Troelstra (1969) .

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idealism: i t take s th e object s o f mathematic s t o b e menta l constructions rathe r tha n objectiv e entities . Th e moder n version , defended b y Michae l Dummett, 51 i s a bran d o f verificationism : a mathematical statemen t i s said t o b e tru e i f and onl y if it has bee n constructively proved. Eithe r way, a series of striking consequences follow: statement s tha t haven' t bee n prove d o r disprove d ar e neither tru e no r false ; complete d infinit e collection s (lik e th e se t of natural numbers ) ar e illegitimate ; muc h o f infinitar y mathematic s must eithe r b e rejected (higher set theory) or radicall y revised (real number theory an d the calculus). These form s o f intuitionis m fac e man y difficulties—e.g . does each mathematicia n hav e a differen t mathematic s dependin g o n what she' s mentall y constructed ? ho w ca n w e verif y eve n state ments abou t larg e finit e numbers ? etc.—bu t it s mos t seriou s drawback i s tha t i t woul d curtai l mathematic s itself . M y ow n working assumptio n i s tha t th e philosopher' s jo b i s t o giv e a n account o f mathematic s a s i t i s practised , no t t o recommen d sweeping refor m o f th e subjec t o n philosophica l grounds . Th e theory o f th e rea l numbers , fo r example , i s a fundamenta l component o f th e calculu s an d highe r analysis , and a s suc h i s fa r more firml y supporte d tha n an y philosophica l theor y o f math ematical existence or knowledge. T o sacrifice the former to preserve the latter is just bad methodology . A second anti-realis t position i s formalism, the popular schoo l of double-think mentione d above . Th e earlies t version s o f th e vie w that mathematic s i s a gam e wit h meaningles s symbol s playe d heavily o n a simpl e analog y betwee n mathematica l symbol s an d chess pieces, between mathematic s and chess, but even its advocates were uncomfortably aware of the stark disanalogies: 52 To b e sure, ther e i s an importan t differenc e betwee n arithmeti c an d chess . The rule s o f ches s ar e arbitrary , th e syste m o f rule s fo r arithmeti c i s such that by means of simple axiom s th e numbers ca n b e referred to perceptua l manifolds an d ca n thu s mak e [an ] importan t contributio n t o ou r knowledge o f nature. The Platonis t Gottlo b Freg e launche d a fierc e assaul t o n earl y formalism, fro m man y direction s simultaneously , bu t th e mos t 51

Dummet t (1975; 1977). Frege cite s thi s quotatio n fro m Thoma e i n hi s critiqu e o f formalism : Frege (1903), § 88. 52

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penetrating aros e fro m jus t thi s point . I t isn' t har d t o se e ho w various true statements o f mathematics can help me determine ho w many brick s i t wil l tak e t o cove r th e bac k patio , bu t ho w ca n a meaningless string of symbols be any mor e relevan t to th e solutio n of rea l worl d problem s tha n a n arbitrar y arrangemen t o f ches s pieces? This i s Frege's problem : wha t make s these meaningless strings of symbols usefu l i n applications? 53 Suppose , fo r example , tha t a physicist test s a hypothesi s b y usin g mathematic s t o deriv e a n observational prediction . I f th e mathematica l premis s involve d is just a meaningles s string o f symbols , wha t reaso n i s there t o tak e that observation t o b e a consequence o f the hypothesis ? An d i f it is not a consequence, i t can hardly provide a fai r test . I n other words , if mathematic s isn' t true , w e nee d a n explanatio n o f wh y i t i s al l right to treat i t as true when we use it in physical science. The mos t famou s versio n o f formalism , th e on e expounde d during th e perio d unde r consideratio n here , wa s Davi d Hilbert' s programme.54 Hilbert , lik e Brouwer , fel t tha t onl y finitar y math ematics was trul y meaningful, bu t h e considered Cantor' s theor y of sets 'on e of the suprem e achievement s of purely intellectual human activity' and promised , in a famous remark, that No on e shal l driv e us out o f the paradise which Canto r ha s created fo r us . (Hilbert (1926), 188,191 ) Hilbert propose d t o sav e infinitar y mathematic s b y treatin g i t instrumentally—meaningless statement s abou t th e infinit e ar e a useful too l in deriving meaningful statement s abou t th e finite—bu t he, unlik e th e scientifi c instrumentalists , wa s sensitiv e t o th e question o f how thi s practice could b e justified. Hilbert' s pla n wa s to giv e a metamathematica l proo f tha t th e us e o f th e meaningles s statements o f infinitar y mathematic s t o deriv e meaningfu l state ments o f finitar y mathematic s woul d neve r produc e incorrec t finitary results . Th e sam e lin e of though t migh t hav e applied t o it s use i n natura l scienc e a s well , thu s solvin g Frege' s problem . Hilbert's effort s t o carr y through o n thi s project produced th e rich 53 54

Se e Fregef 1903), § 91. Se e Hilbert (1926; 1928).

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new field of metamathematics, bu t Kur t Godel soo n prove d tha t its cherished goal could not be reached.55 For all the simplicity of game formalism and th e fame of Hilbert' s programme, man y mathematicians, whe n the y clai m to b e formalists, actually have another ide a in mind: mathematics isn' t a science with a peculia r subjec t matter ; i t i s th e logica l stud y o f wha t conclusions follo w fro m whic h premisses . Philosopher s cal l thi s position 'if-thenism' . Severa l prominen t philosopher s o f math ematics hav e hel d thi s positio n a t on e tim e o r another—Hilber t (before hi s programme) , Russel l (befor e hi s logicism) , an d Hilar y Putnam (befor e his Platonism) 56 —but all ultimately rejected it. Let me briefly indicate why. A numbe r o f annoyin g difficultie s plagu e th e if-thenist : whic h logical languag e i s appropriate fo r th e statemen t o f premisses an d conclusions? whic h premisse s ar e t o b e presuppose d i n case s lik e number theory , wher e assumption s are usuall y left implicit ? from among th e vast range o f arbitrary possibilities, why d o mathemat icians choos e th e particular axio m system s they do t o study ? wha t were historica l mathematician s doin g befor e thei r subject s wer e axiomatized? what are they doing when the y propose new axioms? and s o on. Bu t the question tha t seem s to have scotched if-thenism in th e mind s o f Russel l an d Putna m wa s a versio n o f Frege' s problem: how can the fact that one mathematical statement follows from anothe r b e correctly use d i n our investigatio n of the physical world? Th e genera l thrus t o f the if-thenist' s reply seem s to b e tha t the anteceden t o f a mathematica l if-the n statemen t i s treated a s a n idealization of some physical statement. Th e scientist then draws as a conclusion th e physical statement that i s the unidealization of the consequent.57 Notice tha t o n thi s picture , th e physica l statement s mus t b e entirely mathematics-free ; th e onl y mathematic s involve d i s tha t used i n movin g betwee n them . Unfortunately , man y o f th e 55 Se e Gode l (1931) . Enderto n (1972) , ch . 3 , give s a readabl e presentation . Detlefsen (1986 ) attempt s to defen d Hilbert' s programme against the challeng e of Godel's theorem. Simpson (1988) an d Feferma n (1988 ) pursue partial or relativize d versions within the limitations of Godel's theorem. 56 Se e Resnik (1980), ch. 3 , fo r discussion . There if-thenis m i s called 'deductivism'. See also Putnam (1979), p. xiii. Russell's logicism and Putnam' s Platonism will be considered below. 57 Se e Korner (1960), ch. 8. Cf. Putnam (1967b), 33.

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statements o f physical scienc e see m inextricabl y mathematical. T o quote Putnam , afte r hi s conversion: one wants to say that the La w of Universal Gravitation make s a n objectiv e statement abou t bodies—no t jus t abou t sens e dat a o r mete r readings . What i s the statement ? I t i s just tha t bodie s behav e in suc h a way tha t th e quotient o f tw o number s associated wit h th e bodie s i s equa l t o a thir d number associated wit h the bodies. But how ca n such a statement hav e any objective conten t a t al l i f number s an d 'associations ' (i.e . functions ) ar e alike mere fictions ? I t is like trying to maintai n tha t Go d doe s no t exis t an d angels d o no t exis t whil e maintainin g at th e ver y sam e tim e tha t i t i s a n objective fac t tha t Go d ha s pu t a n ange l i n charg e o f eac h sta r an d th e angels i n charge o f each o f a pair of binar y stars were alway s created a t th e same time ! I f talk o f number s and 'associations ' betwee n masses , etc . an d numbers i s 'theology ' (i n the pejorativ e sense) , the n th e La w o f Universa l Gravitation is likewise theology. (Putnam (19756), 74-5 )

In othe r words , th e if-thenis t accoun t o f applie d mathematic s requires tha t natura l scienc e b e wholl y non-mathematical , bu t i t seems unlikely that scienc e can be so purified. 58 The thir d an d fina l anti-realis t schoo l o f though t I wan t t o consider her e is logicism, or really , the version of logicism advance d by the logica l positivists. Frege's origina l logicist programme aime d to sho w tha t arithmeti c i s reducible to pur e logic , tha t is , that it s objects—numbers—are logical objects and that its theorems ca n be proved b y logi c aione.^ 9 Thi s versio n o f logicis m i s outrigh t Platonistic: arithmeti c i s th e scienc e o f somethin g objectiv e (be cause logi c i s objective), that something objective consists o f object s (numbers), and ou r logica l knowledg e i s a priori. I f this project ha d succeeded, th e epistemologica l problem s o f Platonis m woul d hav e been reduce d t o thos e o f logic , presumabl y a gain . Bu t Frege' s project failed ; hi s syste m wa s inconsistent. 60 Russel l an d White head too k u p th e banne r i n their Principia Mathematics bu t wer e forced t o adop t fundamenta l assumption s n o on e accepte d a s 58 Hartr y Field' s ambitiou s attemp t t o d o thi s wil l b e considered i n ch. 5, sect . 2 , below. See Field (1980 ; 1989) . 59 Se e Frege( 1884). 60 Th e troubl e wa s th e origina l versio n o f Russell' s paradox. (Se e Russell' s lette r to Frege , Russel l (1902), ) Frege' s number s wer e extension s o f concepts . (Se e ch . 3 below.) Som e concepts , lik e 'red' , don' t appl y t o thei r extensions , others , lik e 'infinite', do . Russel l considere d th e extensio n o f th e concep t 'doesn' t appl y t o it s own extension' . If it applies t o it s own extensio n then i t doesn't, and vic e versa. This contradiction wa s provable fro m Frege' s fundamental assumptions . There have been efforts t o revive Frege's system; see e.g. Wrigh t (1983 ) and Hode s (1984).

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purely logical. 61 Eventually , Ernst Zermel o (aide d b y Mirimanoff , Fraenkel, Skolem , an d vo n Neumann ) produce d a n axio m syste m that showe d how mathematics could b e reduced to set theory,62 but again, n o on e suppose d tha t se t theory enjoy s th e epistemologica l transparency of pure logic . Still, the ide a tha t mathematic s is just logic was no t dead ; i t was taken up by the positivists, especially Rudolf Carnap.63 For these thinkers, however, ther e ar e no logical object s of any kind, and th e laws o f logi c an d mathematic s ar e tru e onl y b y arbitrar y convention. Thu s mathematic s i s not, as the Platonis t insists, an objectiv e science. Th e advantag e o f thi s counterintuitiv e vie w i s tha t mathematical knowledg e i s easily explicable ; i t arises fro m huma n decisions. Question : Wh y ar e th e axiom s o f Zermelo—Fraenke l true? Answer: Because they are part of the language we've adopte d for usin g the word 'set' . This conventionalis t lin e o f though t wa s subjecte d t o a histori c series of objections by Carnap's student , W. V. O. Quine. 64 The key difficulty i s that bot h mathematica l an d physica l assumption s ar e enshrined in Carnap's officia l language . Ho w ar e we to separate the conventionally adopted mathematica l part o f the language from th e factually tru e physica l hypotheses ? Quin e argue s tha t i t isn' t enough t o sa y that th e scientific claims , not th e mathematical ones, are supported b y empirical data: The semblanc e of a difference i n this respect i s largely du e t o overemphasi s of departmenta l boundaries . Fo r a self-containe d theor y whic h w e ca n check wit h experienc e includes , i n poin t o f fact , no t onl y it s variou s theoretical hypothese s o f so-called natural scienc e bu t als o such portions of logic and mathematic s a s it makes us e of. (Quine (1954) , 367)

Mathematics i s part o f the theor y we test agains t experience, and a successful test supports th e mathematics as much as the science. Carnap make s severa l effort s t o separat e mathematic s fro m natural science, culminating in his distinction between analyti c and synthetic. Mathematical statements , he argues, are analytic, that is, 61

Se e Russell and Whitehead (1913). Zermelo' s firs t presentatio n i s Zermelo (19086) . Se e also Mirimanof f (1917d , b), Fraenke l (1922) , Skole m (1923) , an d vo n Neuman n (1925) . Th e standar d axioms are now called 'Zermelo-Fraenkel set theory' or ZFC (Z F when the axiom of choice is omitted). See Enderton (1977), 271-2. 63 Se e Carnap (1937; 1950) . 64 Se e Quine (1936; 1951; 1954) . 62

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true b y virtu e o f th e meaning s o f th e word s involve d (th e logica l and mathematica l vocabulary) ; scientific statements, o n th e othe r hand, ar e synthetic , tru e b y virtu e o f th e wa y th e worl d is . Quin e examines thi s distinctio n i n grea t detail , investigatin g variou s attempts at clear formulation, and concludes : It i s obviou s tha t trut h i n genera l depend s o n bot h languag e an d extralinguistic fact . The statemen t 'Brutu s kille d Caesar ' woul d b e fals e if the worl d ha d bee n differen t i n certain ways, bu t i t would als o b e fals e if the word 'killed ' happened rathe r t o hav e the sens e of 'begat'. Thus on e is tempted t o suppos e i n genera l tha t th e trut h o f a statemen t i s someho w analyzable into a linguistic component an d a factual component . Give n this supposition, i t nex t seem s reasonabl e tha t i n som e statement s th e factual component shoul d b e null ; an d thes e ar e th e analyti c statements . But , fo r all it s a prior i reasonableness , a boundar y betwee n analyti c and syntheti c statements simply has not bee n drawn. That there is such a distinction to be drawn a t al l is an unempirica l dogma o f empiricists, a metaphysical article of faith. (Quin e (1951), 36-7 )

Without a clea r distinctio n betwee n analyti c and synthetic , Carnap' s anti-Platonist version of logicism fails . I wil l leave th e thre e grea t schools a t thi s point. I don't clai m t o have refute d eithe r formalis m or conventionalism , though I hop e the profound difficultie s the y face hav e been drawn clearl y enough . Intuitionism I reject o n th e ground s give n above ; I assume tha t th e job o f th e philosophe r o f mathematic s i s t o describ e an d explai n mathematics, not t o reform it. Let m e retur n no w t o Platonism , th e vie w tha t mathematic s i s an objective science . Platonis m naturall y conflict s wit h eac h o f th e particular form s o f anti-realis m touched o n here—wit h intuition ism o n th e objectivit y of mathematica l entities , with formalis m on the statu s o f infinitar y mathematics , wit h logicis m o n th e nee d fo r mathematical existenc e assumption s goin g beyon d thos e o f logic— but th e Platonist' s traditiona l an d pures t opponen t i s the nominal ist, wh o simpl y hold s tha t ther e ar e no mathematica l entities . (Th e term 'nominalism ' ha s followe d 'Platonism ' i n it s migratio n fro m the debat e ove r universal s int o th e debat e ove r mathematica l entities.) Tw o form s o f Platonism dominat e contemporar y debate . The firs t o f thes e derive s fro m th e wor k o f Quin e an d Putna m sketched above—thei r respective criticisms of conventionalism and if-thenism—and th e secon d i s describe d b y Gode l a s th e philo -

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sophical underpinnin g fo r hi s famou s theorems. 65 A s Quin e an d Putnam's writings have just been discussed, let me begin with them . Quine's defenc e of mathematica l realis m follows directly on th e heels o f th e defence s o f common-sens e an d scientifi c realis m sketched above . O n th e naturalize d approach , w e judg e wha t entities ther e ar e b y seein g wha t entitie s w e nee d t o produc e th e most effectiv e theor y o f th e world . S o far , these includ e mediumsized physica l object s an d th e theoretica l entitie s o f physica l science, an d s o far , the nominalis t migh t wel l agree . Bu t i f w e pursue th e questio n o f mathematica l ontolog y i n th e sam e spirit , the nominalist seems cornered: A platonistic ontolog y . .. is, from th e point o f view o f a strictly physicaiistic conceptua l scheme , a s muc h a myt h a s tha t physicalisti c conceptua l scheme itsel f i s for phenomenalism . Thi s higher myt h i s a goo d an d usefu l one, i n turn , i n s o fa r a s i t simplifie s ou r accoun t o f physics . Sinc e mathematics is an integra l par t of this highe r myth , th e utilit y of this myt h for physica l scienc e is evident enough. (Quin e (1948) , 18 )

If w e countenanc e a n ontolog y o f physica l objects an d unobserv ables a s part o f our bes t theor y o f the world, ho w ar e we to avoi d countenancing mathematical entities on the same grounds? Carnap suggested wha t Quin e call s a 'doubl e standard' 66 i n ontology , according t o whic h question s o f mathematica l existenc e ar e linguistic and conventiona l and question s of physical existence ar e scientific and real, but we've already seen that this effort fails . We've als o see n tha t Putna m take s th e same thinking somewhat further, emphasizin g not onl y tha t mathematic s simplifie s physics , but tha t physic s can't eve n b e formulated without mathematics: 67 'mathematics and physics are integrated i n such a way that it is not possible t o b e a realis t wit h respec t t o physica l theor y an d a nominalist with respec t t o mathematica l theory' (Putna m (1975 b), 74). He concludes that talk about 68 mathematical entitie s i s indispensabl e fo r scienc e . . . therefore w e shoul d 65 Se e hi s letter s t o Wang , quote d i n Wan g (19746) , 8-11 , an d Feferman' s discussion (19846) . 66 Quin e (1951), 45. 67 Se e the long quotation fro m Putna m (19756 ) above . A more complete account appears in Putnam (1971), esp.§§ 5 and 7. 68 H e reall y say s 'quantificatio n over' , whic h derive s fro m Quine' s officia l criterion o f ontological commitmen t (1948), bu t I don't want t o get into the debat e over that precise formulation.

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accept suc h [talk] ; bu t thi s commit s u s t o acceptin g th e existenc e o f th e mathematical entitie s in question . Thi s typ e o f argumen t stems , o f course , from Quine , wh o ha s fo r year s stressed bot h th e indispensabilit y of [tal k about] mathematica l entitie s an d th e intellectua l dishonesty of denyin g the existence o f what one daily presupposes. (Putna m (1971), 347 )

We are committed t o the existence of mathematical objects becaus e they ar e indispensabl e t o ou r bes t theor y o f th e worl d an d w e accept that theory. The particula r bran d o f Platonis m tha t arise s from thes e Quine / Putnam indispensabilit y arguments has som e revolutionar y features . Recall tha t traditiona l Platonism takes mathematical knowledg e t o be a priori , certain , an d necessary . But , i f ou r knowledg e o f mathematical entitie s i s justifie d b y th e rol e i t play s i n ou r empirically supporte d scientifi c theory , tha t knowledg e ca n hardl y be classifie d a s a priori. 69 Furthermore , i f w e prefe r to . alter ou r scientific hypothese s rathe r tha n ou r mathematica l one s whe n ou r overall theor y meet s wit h disconfirmation , i t i s onl y becaus e th e former ca n usuall y be adjusted with les s perturbation t o th e theor y as a whole.70 Indeed, Putnam/} goes so far as to suggest that the best solutio n t o difficultie s i n quantu m mechanic s ma y wel l b e t o alter our logica l laws rather than any physical hypotheses. Thu s th e position o f mathematic s a s par t o f ou r bes t theor y o f th e worl d leaves it as liable to revision as any other part o f that theory , a t least in principle, so mathematica l knowledge i s not certain . Finally, the case o f necessit y i s les s clear , i f onl y becaus e Quin e reject s suc h modal notion s ou t o f hand , bu t th e fac t tha t ou r mathematic s i s empirically confirme d i n thi s worl d surel y provide s littl e suppor t for th e clai m tha t i t i s likel y t o b e tru e i n som e othe r possibl e circumstance. So Quine/Putnam Platonism stands a t some consider able remove fro m th e traditional variety. But whil e disagreemen t wit h a venerabl e philosophical theor y i s no clea r demerit , disagreemen t wit h th e realitie s of mathematica l practice is . First, notic e tha t unapplie d mathematics i s completel y without justificatio n o n th e Quine/Putna m model ; i t play s n o indispensable role i n our bes t theory, so it need not b e accepted:72 69 Se e Putna m (19756 ) fo r a n explici t discussio n o f a posterior i method s i n mathematics. Kitche r (1983 ) attacks th e ide a tha t mathematic s is a prior i fro m a different angle . 70 Se e Quine (1951), 43^. 71 Putna m (1968). 72 Se e also Putnam (1971), 346-7.

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So much of mathematics as is wanted for us e in empirical science is for m e on a par wit h the res t o f science. Transfinite ramification s ar e on th e sam e footing insofa r a s they come of a simplificatory roundin g out, but anythin g further i s on a par rather with uninterpreted systems. (Quine (1984), 788) Now mathematician s ar e not ap t t o thin k tha t th e justificatio n fo r their claim s wait s o n th e activitie s i n th e physic s labs . Rather , mathematicians hav e a whol e rang e o f justificator y practice s o f their own , rangin g from proof s and intuitiv e evidence, to plausibility argument s an d defence s i n term s o f consequences . Fro m th e perspective o f a pur e indispensabilit y defence , thi s i s al l jus t s o much talk; what matters is the application. If thi s weren' t enoug h t o disqualif y Quine/Putnamis m a s a n account o f mathematics as it is practised, consider one last point. In this picture of ou r scientifi c theorizing , mathematics enters only a t fairly theoretica l levels . The mos t basi c evidence takes th e for m of non-mathematical observation sentences—e.g . 'thi s chun k o f gol d is malleable'—an d th e initia l level s o f theor y consis t o f non mathematical generalizations—"gol d i s a malleabl e metal' . Math ematics only enters the picture at the more theoretical levels—'gold has atomi c numbe r 79'—s o i t i s o n a n epistemi c pa r wit h thi s higher-level theory.73 But isn't it odd to think o f '2 + 2 = 4 ' or 'the union of the se t of even numbers with the se t of odd number s is the set o f al l numbers ' a s highl y theoretica l principles ? I n Charle s Parsons's phrase , Quine/Putnamis m 'leave s unaccounte d fo r pre cisely the obviousness o f elementary mathematics'.74 By wa y o f contrast , th e Godelia n bran d o f Platonis m take s it s lead fro m th e actua l experienc e o f doin g mathematics , whic h h e takes t o suppor t Platonis m a s suggeste d i n sectio n 1 above . Fo r Godel, th e most elementar y axioms of set theory are obvious; i n his words, the y 'force themselve s upon us as being true'.75 He account s for thi s b y positing a facult y o f mathematical intuition that plays a role i n mathematic s analogou s t o tha t o f sens e perceptio n i n th e physical sciences, so presumably the axiom s force themselves upon us a s explanations o f the intuitiv e dat a muc h a s the assumptio n of medium-sized physica l objects forces itsel f upo n u s a s a n explana tion o f ou r sensor y experiences. To pus h thi s analogy , recal l tha t this styl e o f argumen t fo r common-sens e realis m migh t have bee n 73 74 75

Se e Quine (1948), 18-19. Parson s (1979/80), 151 . Se e also Parsons (19836). Gode l (1947/64), 484 .

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undercut i f phenomenalist s ha d succeede d i n givin g non-realisti c translations o f ou r physica l objec t statements . Similarly , Gode l notes that Russell's 'no-class' interpretation of Principia was an effort t o d o th e wor k o f se t theory , tha t is , t o systematiz e al l of mathematics, withou t sets . Echoin g th e common-sens e realist , Godel take s th e failur e o f Russell' s projec t a s suppor t fo r hi s mathematical realism: This whole schem e o f th e no-clas s theor y i s of grea t interes t a s on e o f th e few examples , carrie d ou t i n detail , o f th e tendenc y t o eliminat e assumptions about the existence of objects outsid e the 'data' and to replace them b y constructions o n th e basi s o f these data. 76 Th e resul t ha s bee n in this cas e essentiall y negativ e . . . Al l thi s i s onl y a verificatio n o f th e vie w defended abov e that logic and mathematic s (jus t a s physics) ar e built u p o n axioms wit h a rea l conten t whic h canno t b e 'explaine d away' . (Gode l (1944), 460-1)

He concludes tha t the assumptio n o f [sets ] is quite as legitimate as the assumptio n of physica l bodies and ther e i s quite a s much reaso n t o believ e i n their existence . They are i n th e sam e sens e necessar y t o obtai n a satisfactor y syste m o f mathematics a s physica l bodie s ar e necessar y fo r a satisfactor y theor y o f our sense perceptions . . . (Godel (1944), 456-7)

But thi s analog y o f intuitio n with perception , o f mathematica l realism wit h common-sens e realism , i s no t th e en d o f Godel' s elaboration o f th e mathematica l realist' s analog y betwee n math ematics an d natura l science . Just a s ther e ar e fact s abou t physical objects tha t aren' t perceivable , ther e ar e fact s abou t mathematica l objects tha t aren' t intuitable . I n bot h cases , ou r belie f i n suc h 'unobservable' fact s is justified b y their rol e i n our theory , b y their explanatory power , thei r predictiv e success , thei r fruitfu l inter connections wit h othe r well-confirme d theories , an d s o on . I n Godel's words : even disregarding the [intuitiveness ] of some new axiom , and eve n i n case it has n o [intuitiveness ] a t all , a probable decisio n abou t it s truth i s possible also i n anothe r way , namely, inductivel y b y studyin g it s 'success' . . . . There migh t exis t axiom s s o abundan t i n thei r verifiabl e consequences , 76 I n this passage , 'data ' mean s 'logic without th e assumptio n o f the existenc e of classes' (Gode l (1944) , 460 n . 22). Earlie r in this same paper, Godel refer s t o arithmeti c as 'th e domai n o f th e kin d o f elementar y indisputabl e evidenc e tha t ma y b e mos t fittingly compared wit h sense perception' (p. 449) .

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shedding s o muc h light upo n a whol e field , an d yieldin g suc h powerful methods for solving problems . . . that, no matter whether or not they are [intuitive], they would have to b e accepted at least in the same sense as any well-established physical theory. (Godel (1947/64), 477)

Quite a number of historical and contemporary justification s fo r set theoretic hypotheses tak e this form, a s will come out i n Chapter 4 . Here th e higher , les s intuitive , level s ar e justifie d b y thei r consequences a t lower , mor e intuitive , levels , jus t a s physica l unobservables ar e justifie d b y thei r abilit y t o systematiz e ou r experience o f observables . A t it s mor e theoretica l reaches , then , GodePs mathematical realism is analogous t o scientific realism. Thus GodeP s Platonisti c epistemolog y is two-tiered: th e simple r concepts an d axioms are justified intrinsicall y by their intuitiveness; more theoretica l hypothese s ar e justifie d extrinsically , b y thei r consequences. Thi s secon d tier leads t o departures fro m traditiona l Platonism simila r to Quine/Putnam's . Extrinsicall y justified hypo theses ar e no t certain, 77 and , give n tha t Gode l allow s fo r justification b y fruitfulnes s i n physic s a s well a s i n mathematics, 78 they ar e no t a prior i either . But , i n contras t wit h Quine/Putnam , Godel give s full credi t t o purely mathematical form s of justification —intuitive self-evidence, proofs, an d extrinsi c justifications withi n mathematics—and th e facult y o f intuitio n doe s justic e t o th e obviousness of elementary mathematics. Among GodeP s staunches t critic s i s Charle s Chihara. 79 Eve n if Godel ha s succeede d i n showin g tha t th e cas e fo r th e existenc e of mathematical entitie s runs parallel t o th e cas e fo r th e existenc e of physical ones, Chihar a argue s that h e has by no means show n tha t the tw o case s ar e o f th e sam e strength , an d thus , tha t h e ha s no t established tha t ther e i s as much reaso n t o believ e in the on e a s t o believe i n th e other. 80 Furthermore , Chihar a argues , th e existenc e of mathematical entities is not require d to explain the experience of mathematical intuition and agreement : I believ e i t i s a t leas t a s promisin g t o loo k fo r a naturalisti c explanation based o n th e operation s an d structur e of th e interna l system s of human beings. (Chihara (1982), 218) 77

Gode l (1944) , 449 . Gode l (1947/64) , 485 . Se e Chihara (1973), ch. 2; (1982). 80 Chihar a (1982) , 213-14.

78

79

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. . . mathematicians , regarde d a s biologica l organisms , ar e basicall y quit e similar. (Chihara (1973) , 80)

And finally , h e question s whethe r Godel' s intuitio n offer s an y explanation a t all:81 the 'explanation ' offere d i s s o vagu e an d imprecis e a s t o b e practicall y worthless: al l w e ar e tol d abou t ho w th e 'externa l objects ' explai n th e phenomena i s that mathematicians are 'i n som e kin d of contact' with thes e objects. Wha t empirica l scientis t woul d b e impresse d b y a n explanatio n this flabby? (Chihara (1982), 217)

Now th e Godelia n Platonis t i s not entirel y defenceless in the fac e of thi s attack . Fo r example , Mar k Steiner 82 point s ou t tha t Chihara's 'explanation ' i s likewis e lackin g i n muscl e tone : th e similarity o f huma n being s a s organism s ca n hardl y explai n thei r agreement abou t mathematic s whe n i t i s consistent wit h s o muc h disagreement o n othe r subjects . Still, most observer s ten d t o agre e that no appea l t o purporte d huma n experience s o f xs that underli e our theor y o f xs ca n justif y a belie f in the existenc e o f xs unless we have som e independen t reaso n t o thin k ou r theor y o f xs i s true. 83 Thus the purported huma n dealing s wit h witche s that underli e ou r theory o f witche s don' t justif y a belie f i n witche s unles s w e hav e some independen t reaso n t o thin k tha t ou r theor y o f witche s i s actually correct . But notice : w e hav e recentl y rehearse d jus t such a n independen t reason i n th e cas e o f mathematics , namely , th e indispensabilit y arguments o f Quin e an d Putnam . Unles s endorsing thes e commit s one t o th e vie w tha t ther e i s n o peculiarl y mathematica l for m o f evidence—and I don' t se e why i t should 84 —there i s room fo r a n attractive compromis e betwee n Quine/Putna m an d Godelia n Platon ism. I t goe s lik e this : successfu l application s of mathematic s give us reason t o believe that mathematics i s a science, that much o f it at least approximate s truth . Thu s successfu l application s justify , i n a general way , th e practic e o f mathematics . But , a s we'v e seen , thi s isn't enough t o giv e an adequat e accoun t o f mathematical practice , K1 Thes e remark s of Chihara's are actuall y addresse d t o a quotation fro m Kreisel , but i t is clear from th e contex t tha t h e think s the sam e objection applies to Godel' s intuition. 82 Steine r (19756), 190 . 8 -' Se e Steiner (19756), 190 . Fo r a similar sentiment, see Putnam (19756), 73-4. 84 No r does Parson s (19836) , 192-3.

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of ho w an d wh y i t works . W e stil l ow e a n accoun t o f th e obviousness o f elementary mathematics, which Godel's intuition is designed t o provide , an d a n accoun t o f other purel y mathematical forms o f evidence , lik e proof an d variou s extrinsi c methods . Thi s means we nee d t o explai n what intuitio n is and ho w i t works; w e need t o catalogu e extrinsi c method s an d explai n wh y the y ar e rational methods in the pursuit of truth. From Quine/Putnam , this compromise takes the centrality of the indispensability arguments ; fro m Godel , it takes the recognition of purely mathematica l form s o f evidenc e an d th e responsibilit y for explaining them . Thu s i t avert s a majo r difficult y wit h Quine / Putnamism—its unfaithfulnes s t o mathematica l practice—an d a major difficult y wit h Godelism—it s lac k o f a straightforwar d argument fo r th e trut h o f mathematics . Bu t whateve r it s merits , compromise Platonis m doe s nothin g t o remed y th e flabbines s o f Godel's accoun t o f intuition . And i t i s in thi s neighbourhood tha t many contemporary objections to Platonism are concentrated.85 I opene d thi s chapte r wit h th e hop e o f reinstatin g th e mathemat ician's pre-philosophica l realism , of devisin g a defensibl e refinemen t of that attitude that remains true to the phenomenology o f practice. Along th e way , I'v e side d wit h common-sens e realism , scientifi c realism, an d philosophica l naturalism , an d seconde d man y o f th e advances of Quine/Putnam and Godelia n Platonism. It will come as no surprise, then, tha t the position t o b e defended her e is a version of compromise Platonism . I'll call it 'set theoretic realism'. Chapter 2 outline s a naturalisti c epistemology for item s locate d on the lower tier of Godel's two-tiered epistemology , a replacement for Godel' s intuition . Th e ontologica l questio n o f th e relationshi p between set s an d othe r mathematica l entities, particularl y natural and rea l numbers, i s the subjec t o f Chapte r 3 . Chapte r 4 contain s some preliminar y spadewor k o n th e proble m o f theoretica l justification, th e secon d o f Godel' s tw o tiers . I argu e tha t thi s illunderstood proble m i s th e mos t importan t ope n questio n o f ou r day, no t onl y fo r se t theoreti c realism , bu t fo r man y othe r mathematical philosophies a s well. Chapte r 5 take s a final look a t set theoretic realism from physicalist and structuralist perspectives. s5

Se e ch. 2, sect. 1 , below.

2

PERCEPTION AN D INTUITION 1. Wha t is the question? The genera l outline s o f th e epistemologica l challeng e t o Platonis m have alread y bee n hinte d at , bu t I' d lik e now t o plac e th e proble m in the context of contemporary philosophy . The sens e that there i s a problem goe s back , a s we've seen , t o Plato himself, but th e moder n form, th e on e exhaustivel y discusse d i n th e contemporar y litera ture, derive s fro m Pau l Benacerra f s 'Mathematica l truth' , whic h appeared i n th e earl y seventies. 1 Sinc e then , i t ha s becom e commonplace fo r scholarl y writing s o n th e philosoph y o f mathe matics to begin by dismissing Platonism on the basis of Benacerraf s argument. Benacerra f himself draws n o suc h dogmati c conclusion , but hi s successors, even those with generally realistic leanings, have scorned Platonism. 2 The Benacerrafia n syllogis m rests o n tw o premisses . The secon d is a traditiona l Platonisti c accoun t o f th e natur e o f mathematica l entities a s abstract, i n particular, a s non-spatio-temporal. Th e first premiss concern s th e nature of human knowledge : wha t i s it for me to kno w something ? I t wa s originall y suggested, agai n b y Plato, 3 that it is enough that I believe it, that my belief be justified, and that the belie f be true. Thoug h Plat o raise d som e objection s o f his ow n to thi s 'justified , tru e belie f accoun t o f knowledge , i t wasn' t unti l 1963 tha t Edmund Getde r pointe d ou t wha t i s now considere d it s fatal weakness. 4 Suppose I see Dick driving a Hillman; suppose he offers m e a ride 1

Benacerra f (1973). Th e anti-Platomsm s o f Fiel d (1980) , Boneva c (1982) , Gottlie b (1980) , an d Hellman (1989) , ar e al l a t leas t partl y motivate d by Benacerrafia n considerations . This styl e of argument is also noted wit h approval by Kitcher (1983), 59, and Resm k (1981), 529, (1982) , 95, and (forthcomin g a, b}. 3 Se e his Theaetetus, 202 c. 4 Gettier(1963) . 2

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to wor k i n thi s car . O n th e basi s o f thi s experience , I com e t o believe tha t Dic k own s a Hillman . M y belie f tha t Dic k own s a Hillman i s surel y justified—h e gav e m e a lif t i n one—an d le t u s further suppos e tha t i t i s true—tha t Dic k doe s indee d ow n a Hillman. But—an d here's th e catch—he doesn't own this Hillman . The Hillma n Dic k actuall y owns i s in th e shop , a s it ofte n is , an d this one, the one I saw, the one I rode in, was borrowed fro m Frank. In this case, though I have a justified, true belief that he does, I can't be sai d t o know Dic k own s a Hillman . Fo r knowledge , ther e i s some further requirement. Some year s afte r Gettier' s pape r cam e a respons e fro m Alvi n Goldman,5 diagnosin g the proble m i n case s lik e mine and Dick's , and proposing a fourth clause in the definition o f knowledge to cure it. Th e difficulty , accordin g t o Goldma n an d man y other s wh o largely agree d wit h him, 6 i s tha t Dick' s Hillma n wa s no t th e ca r that caused me to believe as I did. For a justified, tru e belief to count as knowledge , wha t make s th e belie f tru e mus t b e appropriately 7 causally responsibl e for tha t belief . Thi s idea , in it s many versions, is called the 'causal theory of knowledge'. The tw o premisses , then, o f ou r Benacerrafia n argumen t are th e causal theor y o f knowledg e an d th e abstractnes s o f mathematical objects. Wha t make s '2 + 2 = 4 ' tru e i s the natur e of the abstract entities 2 an d 4 an d the operation plus ; fo r me to kno w tha t ' 2 + 2 = 4' , thos e entitie s mus t pla y a n appropriat e causal rol e i n th e generation of my belief. But how ca n entities that don't even inhabit the physica l univers e take par t i n an y causa l interactio n whatso ever? Surely to be abstract i s also to b e causally inert. Thus, if Platonism is true, we can have no mathematica l knowledge. Assuming that we do have such knowledge, Platonism must be false. This dramati c conclusio n ca n b e pushe d furthe r b y recen t progress i n th e theor y o f reference. 8 Ho w doe s a name pic k ou t a thing? I n thi s field , th e classica l theor y i s Frege's: 9 a nam e i s associated wit h a descriptio n tha t uniquel y identifies th e thin g th e name names ; fo r example , 'Isia h Thomas ' i s associate d wit h th e description 'best friend o f Magic Johnson'. Numerous variations on 5

Goldma n (1967) . Se e e.g. Skyrm s (1967) o r Barman (1973). 7 Suppose , by some neural fluke, m y justified tru e belief i s caused by my being hit on the head b y Dick's Hillman. This would be inappropriate, 8 Se e Lear (1977). 9 Se e Frege( 18926). 6

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this ide a hav e bee n proposed—tha t ther e ar e i n fac t man y descriptions associate d wit h th e name , tha t som e o f thes e migh t even b e false, that wha t count s is the truth o f a sufficien t numbe r of the more importan t ones , an d s o on 10 -—but the centra l descriptive character o f the referring relation remains . In 1972, Saul Kripke 11 called this account int o question . Suppos e I hear the name 'Einstein ' use d repeatedly i n discussions t o whic h I am no t ver y attentive , and I come t o believ e only on e thin g abou t him, namely , tha t h e invente d the ato m bomb . O f course , Einstein didn't inven t the ato m bomb , bu t thi s i s nevertheless th e on e an d only descriptio n I associat e wit h th e name . O n th e descriptio n theory, m y us e of th e nam e shoul d refe r t o someon e else , or t o n o one, i f no singl e person invente d the bomb. That this isn't th e case is made clea r b y th e reactio n o f m y physicis t friend wh o insist s that I'm dea d wron g i n m y belie f abou t Einstein . I f th e descriptio n theory wer e correct , I' d have made a true statement abou t someon e else—the perso n wh o di d inven t th e bomb—o r a truth-valueless statement abou t nobody—i f n o on e perso n invente d th e bomb — but i n fact I made a false statement about Einstein himself. Kripke an d others 12 reac t t o thi s proble m b y proposin g a ver y different pictur e o f ho w w e refer . M y us e o f th e nam e 'Einstein ' picks out Einstein , not b y virtue of my knowledge o f some uniquely identifying descriptio n o f th e man , but becaus e m y usag e i s borrowed fro m thos e I heard usin g it, theirs in turn borrowe d fro m their teacher s o r fro m books , th e usag e ther e borrowe d fro m someone else's , i n a chain leading back, ultimately, to someone wh o was i n a positio n t o du b th e actua l individual . Thu s m y us e o f 'Einstein' refers to Einstein , despite my ignorance, becaus e it is part of a networ k o f borrowe d usag e tha t extend s fro m m e bac k t o a somewhat imaginar y even t calle d a n 'initia l baptism ' i n whic h Einstein himself participated. A simila r stor y work s fo r scientifi c genera l term s lik e 'gold' : a chain o f communication lead s bac k to a n event in which th e baptis t isolated som e sample s of the meta l and declare d tha t 'thi s and stuf f like i t i s gold'. Thu s th e scientifi c communit y wa s abl e t o refe r t o 10

Se e e.g. Searle (1958), Strawson (1959) , ch. 6. Kripk e (1972). For furthe r discussio n of the followin g and othe r examples , see Devitt (1981), 13-20, and Salmon (1981), 23-32. 12 Mos t notably , Putna m (1975*) , chs . 11-13 . Se e als o Devit c (1981 ) an d references cited there. 11

WHAT I S TH E QUESTION ? 3

9

gold vi a a direc t connectio n wit h sample s eve n befor e i t kne w enough abou t atomi c weight s t o giv e a uniquel y identifyin g description. An d successive , very different scientifi c theorie s ca n b e about th e sam e things , becaus e th e referent s ar e picke d ou t b y chains leadin g bac k t o th e dubbin g o f samples , no t b y th e ver y different, ofte n erroneous , description s th e competin g theorie s espouse. Of cours e no t al l genera l term s fi t thi s picture . 'Bachelor' , fo r example, refer s t o whateve r satisfie s th e descriptio n 'unmarrie d male', not t o thing s more o r les s like Marcel Prous t i n some yet-to be-discovered respect . Th e theor y works , no t whe n w e hav e a n explicit descriptio n o r definitio n i n min d for whic h ou r ter m i s an abbreviation, bu t whe n w e notic e a similarit y betwee n variou s things, du b thes e an d thing s lik e them b y some term , an d the n se t out t o discove r th e underlyin g traits tha t mak e thes e thing s wha t they are . Suc h groupings , commo n i n science , ar e calle d 'natural' , as opposed t o 'nominal', kinds.13 The kind consisting of all mathematical object s seem s unlikely to be nominal , becaus e availabl e description s ten d t o b e blatantl y circular one s lik e 'what mathematician s study' . Rather , i n picking out th e kind , w e ge t ou r poin t acros s b y examples: mathematica l objects ar e numbers, sets, functions , Hilbert spaces , an d thing s like that. But , som e migh t argue , i f al l mathematic s i s reducible to se t theory, o r i f we simply restrict our attentio n t o se t theory, there is a simple definition afte r all , namely, that a set is a thing that occurs i n the iterative hierarchy.14 Two thing s scotch this suggestion. First, the definition i n terms of the iterative conception i s still circular; we have to know wha t a set is, indeed wha t a n arbitrary subset is, before w e can understan d it. Considering thi s problem, I suppose there's littl e need fo r a secon d objection, bu t I want t o poin t out tha t th e iterative conception i s a 13 Thes e natura l kind s are th e natura l collection s mentione d i n connection wit h universals in ch. 1, sect 2, above. See Quine (I969d), and Ayers (1981) for very different discussion s of these ideas. 14 Th e iterativ e hierarch y i s arrange d i n stages . Th e firs t stag e consist s o f whatever individual s we begi n with . (I n pure se t theory , thi s i s the empt y set. ) Th e second stag e consists of the subsets of the first; the third of the subsets of the first and second; and s o on. Th e first infinite stage , stage o> , consists of all the sets generated a t the finit e levels . Stag e c o + 1 include s al l subset s o f stag e
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theory o f wha t set s ar e i n muc h th e sam e sens e a s 'havin g atomi c number 79 ' is a theory of what gol d is . Sets of natural numbers an d point sets wer e considere d lon g befor e Zermelo propose d hi s 190 8 axiomatization, an d th e iterativ e pictur e wa s firs t describe d onl y years afte r that , i n 1930. 15 I n bot h cases , w e star t wit h samples , dub th e kind , the n g o o n t o investigat e the natur e o f that kind ; i n both th e cases , th e natura l kin d a s identifie d b y samples pre-date d the scientifi c theory an d woul d surviv e its demise. I f our theor y o f the atomic structure of gold turne d ou t t o b e incorrect, we could go on t o for m anothe r theor y o f th e sam e stuff ; similarly , theoretical considerations migh t lead us to drop the claim that all sets occur in the iterative hierarchy and to study non-well-founded sets as well.16 If th e mos t basi c mathematica l kind s ar e natura l rathe r tha n nominal, th e referentia l difficult y fo r Platonis m arise s whe n w e consider th e natur e o f th e require d initia l baptism . I n th e simples t case, w e imagin e th e baptis t standin g i n fron t o f a numbe r o f samples o f th e natura l kin d i n questio n an d declarin g tha t 'thes e and thing s lik e them ar e . . .'. I n th e cas e o f gold, ther e i s a direc t causal interaction with th e samples , that is , they are touched, seen, perceived. Becaus e th e link s i n th e chai n o f communicatio n ar e causal, an d th e direc t connectio n betwee n th e baptis t an d th e samples i s causal,17 this theory i s often calle d 'th e causal theory of reference'. Runnin g paralle l t o Benacerraf' s epistemologica l di lemma, w e have two premisses—the causal theory o f reference and the abstractness , an d henc e causa l inertness , o f mathematica l objects—that lea d t o anothe r unpalatabl e conclusio n fo r th e Platonist: w e can' t refe r t o mathematica l objects. I f mathematica l reality i s a s th e Platonis t say s i t is , w e ar e doome d no t onl y t o 15 Se e Zermel o (1908fr ) fo r th e firs t explici t statemen t o f se t theoreti c axioms. The iterativ e conceptio n i s describe d in Zermel o (1930) , though som e (se e Wang (1974a)) give prior credit to Mirimanof f (1917<2 , b). 16 Indeed , th e axio m o f foundation , whic h restrict s the rang e o f se t theoretic study to th e members of the iterative hierarchy, is often take n not a s a truth, but a s a simplifying assumption . See Maddy (1988a) , § 1.2. O n th e possibilit y o f non-wellfounded sets , see Aczel (1988). 17 Kripk e allow s fo r baptis m b y descriptio n a s wel l a s b y ostension—fo r example, 'Gol d i s th e stuf f foun d i n For t Knox'—bu t a n attempte d descriptiv e baptism o f sets would necessaril y ru n i n a circle—'Sets ar e thing s like the se t o f my two hands' . Beside s which , Devit t ha s argue d (1981 , pp . 36—42 ) tha t whe n a description i s use d referentiall y rathe r tha n attributively—a s i t i s i n a descriptive baptism—a causa l groundin g i s stil l required . (Kripke' s ow n attitud e i s no t altogether clear. See Kim (1977), 615-17.)

WHAT I S TH E Q U E S T I O N ? 4

1

ignorance, bu t t o silenc e as well. And again, assuming that we can talk about mathematical objects, Platonism must be false. These, then , ar e th e seriou s epistemologica l challenge s man y philosophers tak e t o hav e sun k th e Platonist' s ship . Som e hav e replied tha t th e causa l theorie s ar e irrelevan t t o mathematics , because they are theories of a posteriori, contingent knowledge and mathematical knowledge i s a priori and necessary , but thi s sort of response i s o f n o us e t o th e compromis e Platonis t wh o follow s Quine i n questionin g thes e distinctions. 18 Le t m e tak e a momen t now t o examin e th e cogenc y o f th e causa l argument s fro m thi s perspective. We are face d wit h tw o anti-Platonis t arguments . These arguments seem t o depen d o n causa l theorie s o f knowledg e an d reference , along wit h th e traditiona l Platonisti c accoun t o f th e natur e o f mathematical entities . Thu s on e pro-Platonis t approac h woul d b e to cal l the causal theories into question . In fact, thi s approach wa s adopted earl y on, b y Mark Steiner , in a paper tha t appeare d in the same year as Benacerraf's. 19 Steiner's argumen t i s i n tw o parts . First , h e find s faul t wit h several particular formulations of th e causa l theory an d conclude s that eve n its best formulation is implausible. He the n argue s that a suitably generalize d 'causa l theory ' allow s tha t a fac t abou t numbers, fo r example , migh t play a role i n the causa l explanation of my belie f i n the corresponding axiom of number theory after all , simply becaus e the axiom s of numbe r theor y an d analysi s will all figure i n an y suc h explanation. 20 H e summarizes : 'th e mos t plausible versio n o f th e causa l theor y o f knowledg e admit s Platonism, an d th e versio n mos t antagonisti c t o Platonis m i s implausible' (Steiner (197 5 a), 116) . Now ther e i s room fo r rebutta l to bot h part s o f Steiner' s claim. For example , th e cas e Steiner uses to sin k the 'best ' version of th e 18

Se e Wright (1983) , § xi, an d Hal e (1987) , 86-90 , fo r th e a prioris t lin e of thought, an d Lewi s (1986) , 108—15 , fo r th e necessitarian . I t migh t als o b e argue d that the causal conditio n isn' t require d fo r mathematical knowledg e becaus e ther e is no suc h thin g as a mathematical Gettie r case , that is , a mathematical cas e i n which the subjec t has a justifie d tru e belie f without knowledge . Bu t suc h cases ca n arise , if not fo r al l compromis e Platonists , a t leas t fo r th e se t theoreti c realist . Se e sect. 2 below. 19 Se e Steiner (1973) , which later became ch . 4 of his (1975a). 20 Thi s thinkin g rest s o n th e Quinea n ide a tha t mathematic s wil l b e part o f th e overall theory that causally explains my belief i n the axioms of number theory.

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causal theor y involve s inferentia l knowledge , bu t Benacerraf s version o f th e causa l theor y doesn' t requir e tha t inferentia l knowledge mee t th e causa l conditio n place d o n mor e basic , non inferential knowledge. 21 O n th e othe r hand , agai n fo r example , there i s no guarante e tha t th e particula r axiom o f number theor y I believe wil l i n fac t figur e i n th e causa l explanation o f tha t belief, 22 and eve n i f i t did , woul d i t figur e 'appropriately' ? Bu t mos t interesting fo r ou r assessmen t of th e proble m i s the stron g reactio n of one reviewer: it i s a crim e agains t th e intellec t t o tr y t o mas k th e proble m o f naturalizing the epistemolog y o f mathematic s wit h philosophica l razzle-dazzle . Super ficial worries abou t the intellectual hygiene of causal theorie s o f knowledge are irrelevan t to an d misleadin g fro m thi s problem , fo r th e proble m i s no t so much abou t causality as about th e very possibility of natural knowledg e of abstract objects . (Har t (1977) , 125-6 )

If th e causa l theor y i s no t th e problem , the n attackin g th e causa l theory doesn' t help . Bu t i f th e causa l theorie s o f knowledg e an d reference ar e remove d fro m th e premisse s o f th e anti-Platonis t arguments, wha t will take their place? The ide a tha t th e causa l theorie s ar e no t th e proble m gain s support fro m th e historica l facts: th e causa l theor y o f knowledg e has graduall y los t favou r i n th e year s sinc e th e appearanc e o f Benacerrafs article , whil e th e sentimen t tha t ther e i s a persuasive Benacerraf-style argumen t agains t Platonis m remain s strong . On e might tr y t o pi n dow n thi s ne w argumen t b y combin g th e contemporary epistemologica l literatur e fo r a descendan t o f th e causal theory tha t coul d tak e it s place i n the first premiss. The bes t candidate would b e reliabilism: for my justified, tru e belief to coun t as knowledge, i t must be generated by a reliable process. Dependin g on ho w thi s accoun t i s spelle d out , i t ma y o r ma y no t involv e a causal constraint strong enoug h t o do th e causal theory's jo b i n the argument agains t Platonism. 23 I won' t g o int o th e detail s o f curren t reliabilis t epistemolog y because i n fac t I don't thin k th e forc e o f Benacerraf-styl e thinking depends o n thi s particula r philosophica l epistemolog y an y mor e than i t depended o n th e causa l theory . T o th e extent tha t thes e are 21 22

SeeBenacerraf(1973),413 .

Har t makes this point (1977, p. 124) . I n Madd y (1984a) , I indicat e tha t reliabilis m migh t no t d o thi s job . Casull o (forthcoming) gjve s a more refined accoun t that suggests otherwise. 23

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3

intended a s a prior i philosophica l theorie s o f wha t knowledg e o r justification consist s in , an y broa d sceptica l conclusio n base d o n them—e.g. tha t mathematic s i s no t a science—err s agains t th e tenets o f epistemolog y naturalized . To th e exten t tha t reliabilis m and th e res t ar e proposal s fo r naturalize d account s o f wha t knowledge is , give n th e overwhelmin g evidenc e i n favou r o f mathematical knowledge, they will not last long as parts of our best theory i f they purport t o rule it out.24 Indeed, for all we know, fro m the naturalize d perspective , th e ver y notion s o f knowledg e and/o r justification migh t be ultimately dispensable. But for all that, I think a Benacerraf-style worry would remain . To se e this , recal l th e discussio n o f trut h an d referenc e i n th e third sectio n o f Chapte r 1 . The curren t debat e betwee n supporter s of correspondence an d disquotationa l truth suggest s that even if the correspondence theorist' s substantiv e notion s o f trut h an d refer ence tur n ou t t o b e scientificall y dispensable , ther e wil l remai n a problem of explaining the reliability of an expert's beliefs abou t th e field of her expertise. Now I want to add a similar assessment of the situation i n th e theor y o f knowledge . Eve n i f reliabilism turn s ou t not t o b e th e correc t analysi s o f knowledg e an d justification , indeed, eve n i f knowledge an d justificatio n themselve s turn ou t t o be dispensable notions, ther e will remain the problem o f explaining the undeniabl e fac t o f ou r expert' s reliability . In particular , eve n from a completel y naturalize d perspective, th e Platonis t stil l owe s us an explanatio n o f ho w an d wh y Solovay' s belief s abou t sets ar e reliable indicators of ihe truth abou t sets. 25 The nominalis t Hartr y Field , realizin g that th e causa l theory of knowledge i s something 'almost no on e believe s any more', 26 ends up rephrasin g th e Benacerra f worry i n ver y simila r terms : ' . . . Benacerraf s challenge . .. is to provide a n accoun t o f the mechanisms that explain how ou r belief s about thes e remote entities can so well reflec t th e fact s abou t them. ' Fiel d combine s thi s wit h th e traditional Platonist's conceptio n o f mathematical entities: The relevan t fact s abou t ho w th e platonis t conceive s o f mathematica l objects includ e thei r mind-independenc e an d language-independence ; th e fact tha t they bea r no spatio-tempora l relation s to us ; the fac t tha t they d o 24 25 26

Se e Burgess (1983), 101 . R . M, Solovay is one of our leading contemporary set theorists. Al l the quotations in this paragraph come from Fiel d (1989) , 25-7 .

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not underg o an y physica l interaction s (exchange s o f energy-momentu m and the like) with us or anything we can observe; etc.

From thes e tw o premisses , h e draw s a guarde d pro-nominalis t conclusion: The ide a i s that i f i t appears i n principle impossible t o explain this, the n that tends to undermine the belief in mathematical entities, despite whatever reason w e migh t have for believin g i n them. . . . Like Benacerraf, I refrain fro m makin g any sweepin g assertion about th e impossibility of th e required explanation . However , I a m no t a t al l optimisti c abou t th e prospects of providing it.

Here we have a statemen t of th e problem tha t make s n o appea l t o theories o f truth , reference , justification , o r knowledge . I t simply, naturalistically, asks for an explanation o f a purported fact . Of course , there i s more t o i t than this . If this new versio n of th e Benacerrafian syllogis m is to shar e the anti-Platonisti c moral of th e original, ther e mus t be strong reason s t o suppos e tha t th e require d explanation wil l no t b e forthcoming . Field' s formulatio n surel y implies this , an d i n m y discussio n of truth , I als o suggeste d tha t whatever difficultie s th e theor y o f reference was suppose d t o cause the Platonist , th e obstacle s t o explainin g reliabilit y woul d b e similar, perhap s identical . But o n th e surface , thi s ne w argumen t hardly seem s t o shar e th e knock-dow n conclusivenes s o f it s predecessor. Give n tha t th e firs t premiss , th e replacemen t fo r th e causal requirement , include s n o claus e tha t stand s i n explici t contradiction t o th e traditiona l Platonisti c assumption s o f th e second, w e mus t as k wh y th e tas k o f providin g th e require d explanation shoul d still seem so daunting. Obviously, what w e ar e u p agains t her e i s another, les s specific , version o f th e sam e vagu e conviction that make s th e causa l theor y of knowledg e s o persuasive: in order t o b e dependable, the proces s by whic h I com e t o believ e claim s abou t x s mus t ultimatel y b e responsive in some appropriate way to actua l jcs. And, invoking the second premiss , nothing ca n b e responsive to non-spatio-temporal, unchanging, acausal , unobservabl e Platoni c entities . How , then , can Solovay's reliability be anything more tha n a fluke? Ho w ca n it possibly be explained? I won't try t o mak e this vague conviction an y mor e definite , nor will I try t o refut e it . Bu t I do wan t t o poin t ou t i t i s not, b y itself , enough t o caus e a proble m fo r traditiona l Platonism, Eve n i f ou r

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5

most basi c reliabl e beliefs , fo r exampl e perceptua l beliefs , ar e directly conditioned by the objects of those beliefs , many other, less basic belief s ar e inferre d fro m these . A physicist , fo r example , needn't see , o r causall y interact with , a molecul e o f wate r o n th e other sid e o f th e moo n i n orde r t o reliabl y believ e tha t i t ha s a certain structure ; sh e nee d onl y hav e goo d reaso n t o believ e tha t there i s water o n th e othe r sid e of the moo n an d a well-established theory o f th e structur e o f wate r molecules . I n othe r words , an y reasonable theory of reliability will have to allow fo r various forms of inference as reliable belief-forming mechanisms. Why the n couldn' t Solovay' s belief s abou t set s als o b e reliably inferred? O f course , man y o f the m are , assumin g deductio n i s reliable, but what abou t the axioms from whic h these are deduced? In fact , thi s possibilit y i s rarel y considered ; underlyin g th e persuasiveness o f al l thes e Benacerraf-styl e arguments—fro m th e original one based on the causal theory of knowledge to the current one base d o n th e nee d fo r a n explanatio n o f reliability—i s th e unspoken assumptio n tha t som e mathematica l belief s ar e no t inferred. Thi s omissio n i s especiall y glarin g give n tha t on e prominent form of Platonism, namely Quine/Putnamism, does treat all mathematic s a s inferred : mathematics is a collectio n o f highly theoretical hypotheses , justifie d b y thei r indispensabl e rol e i n science. As this sort of hypothetical inferenc e might well qualify a s a reliabl e process, Quine/Putnamis m shoul d a t leas t b e considere d as a possible reply to Benacerraf-style worries.27 But a s I say , i t ofte n isn't, 28 a fac t fo r whic h I offe r thi s explanation. Mathematician s aren' t th e onl y one s swaye d b y th e pre-theoretic realis m describe d a t th e beginnin g o f Chapte r 1 . Many of us tend to think of mathematics, not a s a highly theoretical adjunct to physical science, but a s a science in its own right , with its own subjec t matte r an d it s ow n methods . O n thi s view , math ematics i s parallel to , no t subservien t to, natura l science , s o i t i s natural to suppose that Platonistic epistemology should run parallel to scientifi c epistemology , an d fro m thi s i t follow s tha t som e mathematical belief s shoul d b e basi c an d non-inferential , just a s 27 O f course, as indicated i n ch. 1 , sect. 4, above, I don't thin k Quine/Putnamism offers a n acceptabl e accoun t o f mathematica l knowledge (o r th e reliabilit y o f mathematical beliefs) , bu t m y point here is that Benacerra f an d thos e who cit e him rarely even consider it. 28 Fiel d (1989) , 28—30, i s an exception . He reject s a Quine/Putna m solutio n for some of the reasons rehearsed in ch. 1, sect. 4, above.

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some scientifi c belief s are . Furthermore , a s th e mos t fundamental belief-forming mechanis m i n physica l scienc e i s perception , th e corresponding facult y i n mathematica l scienc e i s expecte d t o b e 'perception-like', an d hence , mos t likel y causal. 29 An d her e th e seeming impossibility arises. But i f this pre-theoretic science/mathematic s analogy lie s behind the stubborn Benacerraf-style worries of some philosophers , I don't want t o sugges t tha t i t account s fo r all . Eve n a n advocat e o f a Quine/Putnam Platonism , wh o hold s tha t al l mathematica l belief s are gaine d b y th e reliabl e proces s o f inferenc e t o th e bes t explanation, migh t worr y al l th e sam e becaus e sh e hold s tha t al l explanations ar e ultimatel y causal. Fo r another , th e rol e o f th e science/mathematics analog y migh t b e playe d instea d b y a stron g form o f physicalism tha t requires every legitimate entity to b e part, as Armstron g put s it , o f ' a single , all-embracin g spatio-tempora l system'.30 And ther e ar e doubtles s other possibilities . Thus, I think it's bes t t o se e these seriou s philosophical worries abou t Platonism as a syndrome with more than one aetiology. One las t question: does th e nagging Benacerraf-style problem fo r Platonism constitut e a n argumen t i n favou r o f nominalism ? Of course, man y anti-Platonist s hav e claime d tha t i t does , bu t Joh n Burgess strenuously disputes this.31 Fro m a naturalized perspective, epistemology i s a descriptiv e and explanator y enterprise; its goal i s to describ e an d explai n th e belief-formin g mechanism s of huma n knowers. So , Burges s considers th e actua l practice s o f scientists. Here w e fin d well-confirme d affirmation s o f mathematica l knowledge and no attributions of causal powers t o mathematical entities. Thus, a causa l requirement on knowledg e o f th e sor t enshrine d in Benacerraf's firs t premis s simply doesn't tur n u p i n th e descriptiv e phase o f epistemolog y naturalized . H e conclude s tha t ' a causa l 29 I' m assumin g her e tha t perceptua l belief s aren' t inferential , i n particular, tha t they aren' t inferre d fro m sens e data . Thi s doesn' t rul e ou t 'inferenc e t o th e bes t explanation' accounts of perception (e.g . Harma n (1973) , ch. 11 , or Gregor y (1970 ; 1972)) becaus e ther e th e inference s ar e fro m state s o f th e nervou s syste m t o beliefs , and I' m reservin g the word 'inference ' fo r inference s fro m belief s t o beliefs . Also, ther e ar e thos e wh o hol d tha t perceptio n needn' t requir e causatio n (Ki m (1977)), bu t I se e n o nee d t o quibbl e abou t this . Th e issu e i s whethe r o r no t mathematical object s ca n participat e i n interaction s suitabl y simila r t o th e participation o f m y han d i n th e formatio n o f m y perceptua l belie f tha t ther e i s a hand befor e m e when I look at i t in good light . Se e Grice (1961), and sect . 2 below. 30 Armstron g (1977), 149 . I'l l return to thi s ide a i n ch. 5 , sect. 1 , below. 31 Se e Burgess (forthcomin g ft) .

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criterion fo r knowledg e [is ] problematic, whethe r regarde d a s par t of a propose d analysi s of th e meanin g of "know " o r a s par t o f a proposed analysi s o f scientifi c standard s o f justification ' (Burges s (forthcoming b) ). Presumabl y the same could b e said for th e requirements o n reliabilit y implicit i n Field' s version of th e argument , an d even fo r th e variou s other assumption s o n whic h I have suggested that Benacerraf-style worries might rest. Field naturall y see s th e situatio n somewha t differently . H e doesn't clai m tha t Benacerraf-styl e worries provid e a conclusive argument agains t Platonism; he recognizes that the y do nothing t o disarm the otherwise powerful arguments for Platonism: Of course , th e reason s fo r believin g in mathematical entitie s (in particular , the indispensabilit y arguments ) stil l nee d t o b e addressed, bu t th e rol e of the Benacerrafia n challeng e (a s I see it) i s to rais e the cos t o f thinking tha t the postulation o f mathematical entitie s is a proper solution, an d to thereb y increase th e motivatio n fo r showin g tha t mathematic s i s no t reall y indispensable after all. (Field (1989) , 26)

That i s Field's plan : t o undermin e the indispensabilit y arguments by showin g ho w mathematic s coul d b e usefu l i n application s without bein g true. This project requires that science be rewritten in nominalistically acceptable terms, contra Putnam' s claims, and tha t the us e of mathematics in nominalized science be justified nominalistically, withou t appea l eve n t o Hilbert-styl e finitisti c metamath ematics. If thi s were possible, 32 Fiel d would b e presenting an alternative overall theory , accordin g t o whic h ther e i s n o mathematica l knowledge, bu t th e us e o f mathematic s i n scienc e i s nevertheless justified. Th e advantag e of this picture over the Platonist' s i s that it doesn't leave a puzzling open questio n i n the psychological part of our theor y abou t ho w peopl e com e t o hav e reliabl e beliefs abou t Platonic entities . Thu s I tak e i t Fiel d woul d b e arguing , i n goo d naturalized form, that hi s overall theory of the world i s better than the Platonist's . A t thi s point , then , th e Benacerraf-styl e proble m would becom e an argument in favour o f nominalism. Even then , though , I suspec t Burges s would doub t tha t Field' s theory i s actually preferable. After all , i t require s a revisio n i n th e 32 Numerou s difficultie s wit h Field' s projec t hav e emerged . Fo r a sampling , see Malament (1982) , Resni k (19850 , b) , Shapir o (19836) , Burges s (1984) , an d Detlefsen (1986) , ch. 1 . For my own discussion, see ch. 5, sect. 2, below.

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standard canon s o f scientifi c practic e t o balanc e a worry generate d by a vague and surel y less well-established psychological convictio n about th e nature of reliable processes. Perhap s he would argu e tha t no gain i n the psychological portio n o f our theor y coul d justif y th e rejection o f otherwise effectiv e scientifi c methods . A t this point, th e form o f Field' s repl y woul d depen d o n th e actua l detail s o f hi s proposed nominalis t theory. This i s an important debate , one I won't attempt t o resolve, but I do want to point out that, despite appearances, i t is irrelevant to my project. I f Burges s i s right , hi s argument s woul d undermin e th e effectiveness o f Benacerraf-styl e worrie s a s justification s fo r nom inalism. Howeve r welcom e thi s conclusio n migh t be , i t doe s nothing t o tak e th e compromis e Platonis t of f th e epistemologica l hook. B y rejectin g pur e Quine/Putnamism , b y embracin g som e version o f Godel' s science/mathematic s parallelism , th e com promise Platonis t incur s th e ver y rea l deb t detaile d i n th e las t chapter: within the bounds of epistemology naturalized, she owes a descriptive an d explanator y accoun t o f mathematica l knowledg e (or mathematical reliability) tha t does justic e to th e actua l practic e of mathematics , a n accoun t o f bot h intuitio n an d othe r peculiarly mathematical justifications . An d t o provid e suc h a n accoun t i s t o meet th e Benacerraf-styl e worrie s hea d on , whethe r o r no t the y constitute an effectiv e argumen t for nominalism. So, despite al l thi s talk abou t th e exac t (o r inexact! ) nature o f th e causal premis s t o th e Benacerrafia n argument , i t i s not m y pla n t o attack th e proble m a t tha t point . Instead , I inten d t o rejec t th e traditional Platonist' s characterizatio n o f mathematica l objects; I will brin g them int o the world w e know an d int o contac t wit h ou r familiar cognitiv e apparatus . A n accoun t o f ou r 'perception-like ' connection wil l be the goa l of this chapter. T o giv e a taste o f how i t will go , le t m e return t o th e (convenient , but inessential ) language of the causal theories . Consider agai n the initia l baptis m o f gold : th e baptis t stand s i n front o f an arra y of samples, look s a t them , an d declare s tha t thes e and thing s lik e them ar e gold. A n analogous baptis m of sets woul d go like this: our baptist , at her desk, declares, These three things— the pape r weight , th e globe , an d th e inkwell—take n together , regardless o f order , for m a set' , o r 'Th e individua l books o n thi s shelf, take n together, i n no particular order, form a set.' In this way,

WHAT I S TH E QUESTION ? 4

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she isolates samples of the kind 'set', and the word the n refers to the kind of which these samples are members. The obviou s objectio n i s that , whil e th e gold-dubbe r causall y interacts wit h he r samples , th e set-dubbe r causall y interact s onl y with th e member s o f he r samples . Her e th e Platonis t migh t respond:33 th e exten t o f th e causa l interaction s o f bot h th e gold dubber an d th e set-dubbe r i s somethin g lik e ligh t bouncin g of f certain objects and bringin g about som e retinal changes. In the case of the gold-dubber, strictl y speaking, the interaction is actually with the fron t surfac e of a time slic e of th e sample . I n other words , th e thing whose kin d w e count th e baptis t a s having dubbed is not th e thing with which the dubber has causally interacted; her interaction is onl y wit h a fleetin g aspec t o f th e temporall y extende d sample . Similarly, the set-dubbe r has onl y aspects o f her sampl e sets within her causa l grasp. But , if the interactio n o f the gold-dubbe r wit h a n aspect o f her sample s is enough t o allo w he r t o pic k ou t a sampl e and dub a kind, why shouldn't th e set-dubber's interaction with a n aspect o f her sample s accomplis h thos e sam e things? The Platonis t could argu e tha t th e relatio n o f elemen t t o se t i s n o mor e objectionable tha n th e relatio n o f fleetin g aspec t t o temporall y extended object. 34 If a n argumen t o f thi s sor t coul d b e fille d in , the n i t seem s th e Platonist migh t adop t a causa l theor y o f referenc e afte r all . An d since th e bar e causa l interaction s describe d for m th e basi s fo r a perceptual connectio n betwee n th e baptis t an d he r samples , a causal theor y o f knowledg e i s withi n reac h a s well . Bot h causa l theories requir e mor e tha n th e mer e causa l interaction—th e bouncing o f ligh t ray s of f object s an d ont o retinas—the y requir e that th e baptis t an d th e knowe r perceive a physica l object . Tha t these tw o ar e no t th e sam e i s clear fro m experiment s o n patients , blind fro m birth , whos e sens e organ s ar e restore d t o perfec t operating condition , bu t wh o canno t properl y perceiv e the objects around them. 35 Th e Platonist' s hop e i s tha t a n accoun t o f wha t

33 Tha t a n argumen t o f thi s sor t migh t b e availabl e t o th e Platonis t wa s firs t suggested t o me by John Burgess. 34 O f course, the set-dubber's interactio n i s also with mer e fleetin g aspect s of the members o f he r sets , s o th e relatio n betwee n wha t sh e interact s with , an d what' s kind sh e dubs, is the compositio n o f the aspect/objec t an d th e element/se t relations . This adde d complexit y coul d b e eliminate d b y imaginin g tha t th e set-dubbe r use s sets o f aspects, rather tha n set s of objects, a s samples, bu t a more reasonable cours e would b e t o assum e tha t i f both th e relation s i n questio n ar e legitimate , the n s o i s their composition. )5 Se e Hebb (1949), ch . 2.

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makes on e patter n o f sensor y stimulatio n int o a perceptio n o f a physical objec t migh t als o provid e a n accoun t o f wha t make s another patter n o f stimulation into a perception o f a set of physical objects.

2. Perceptio n The questio n i s wha t bridge s th e ga p betwee n wha t i s causall y interacted wit h an d wha t i s perceived , an d th e hop e i s tha t something lik e what doe s th e bridgin g in the case of physical object perception ca n b e see n t o d o th e sam e jo b i n th e cas e o f se t perception.36 Notic e tha t thi s wa y o f puttin g th e proble m alread y assumes that w e do i n fac t perceiv e physical objects, as opposed t o sense data , o r percepts , o r representation s o f som e kind . I n th e theory o f perception, thi s is called 'direc t realism' . Psychologists o f perception ar e generall y direc t realist s i n thi s sense, 37 an d thoug h some philosopher s ar e stil l incline d t o debat e th e issue , I'l l rely implicitly o n th e persuasiv e counter-arguments t o b e foun d i n th e literature. 38 What w e want her e i s a stron g sens e of 'perceives ' that rule s ou t illusions; what i s perceived in this sense is really there. Bu t that isn't all. W e als o insis t that, fo r example, a hiker doesn't perceive a leaf dwelling insect on a bus h sh e passes i f the bu g blend s to o perfectl y with it s surroundings for he r t o distinguis h it fro m them . I n such a case, eve n thoug h ligh t fro m th e bu g register s a patter n o n th e hiker's retina , sh e gain s n o belief s abou t it . I t isn' t enough , however, t o requir e tha t sh e gai n belief s i n orde r t o perceive , o r even tha t sh e gai n belief s visually , becaus e i t i s possibl e t o gai n beliefs, eve n visually , tha t clearl y don't coun t a s perceptual belief s about th e bug ; she could , fo r example , com e t o believ e there i s a Ceylonese leaf insect on th e bus h b y reading a sign. This belie f isn' t a perceptua l belie f that there i s a Ceylones e lea f insec t o n th e bus h because i t doesn't involv e it looking t o th e hike r (i n a phenomena l or non-metaphorica l sense ) a s i f ther e i s a bu g o n th e bush . Fo r 36 I will concentrat e her e on visua l perception . Naturall y the blin d ca n know and refer a s well a s the sighted , bu t I make th e customar y philosophica l assumptio n tha t what's true of vision can be adapted t o the other senses. 37 Fo r history , se e Hebb (1980) , 19 . Fo r agreemen t fro m variou s psychologica l camps, see). Gibson (1950) , 26-7, Neisse r (1976) , 16 , and Gregory (1972) , 220. 38 Se e Pitcher (1971) , ch. 1 , or th e reference s cite d i n Machamer (1970) , § II.

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perception, then , w e requir e tha t th e perceive r gai n appropriat e perceptual beliefs . But thi s stil l isn' t enough . Conside r a n exampl e analogou s t o Gettier's: a n illusionist arranges a system of mirrors s o that it looks to Stev e as if there i s a tree i n front o f him. Suppos e that th e actual source o f the image Steve sees is a tree behind him. Suppose, finally, that ther e is in fact a tree in front o f him, where he sees the illusion, though tha t tre e i s hidden . Her e Stev e gain s a tru e perceptua l belief—there i s a tre e i n fron t o f me—bu t i t doesn' t coun t a s a perception tha t ther e i s a tre e i n fron t o f hi m becaus e it s causa l genesis i s faulty . Clearly , w e mus t insis t tha t whateve r make s th e belief true—i n thi s case, th e tre e i n front o f Steve—b e responsible for hi s belie f i n som e appropriat e way . Pau l Grice' s solutio n i s t o insist that th e thing perceived mus t play the same sort o f role i n the causation of the perceiver's perceptual state as my hand plays in the generation o f m y belief that ther e i s a han d befor e me when I look at it in good light. 39 In sum, then, for Steve to perceive a tree before him i s for ther e to b e a tree befor e him, for him t o gai n perceptua l beliefs, i n particular that ther e i s a tree befor e him, an d fo r the tre e before hi m t o pla y a n appropriat e causa l rol e i n th e generatio n of these perceptual beliefs. 40 Notice tha t th e conten t o f a perceptua l belie f stat e i s extremely rich an d varied. For Steve to acquir e the perceptual belief that there is a tre e befor e him, h e mus t als o acquir e a grea t variet y o f othe r perceptual beliefs , dependin g on the occasion, suc h as, that th e tre e is roughl y s o big , so fa r away , tha t i t i s i n leaf , swayin g i n th e breeze, an d s o on. 41 Suc h belief s ar e non-inferential, 42 an d no t necessarily conscious or linguistic. 43 When th e various components of a perceptual belief state arise as a body, o n a given occasion, the y often influenc e eac h othe r non-inferentially , as , fo r example , a belief about th e identity of an object can influence perceptual beliefs about its shape and size, and obviously, vice versa. 39

Se e Grice{ 1961). Thi s accoun t i s a crude version of Pitcher' s (1971) , ch . 2. I t i s similar to Armstrong's (1961) , and t o variou s psychological theorie s of perceptio n a s information acquisition, lik e Gregory' s (1972) . Pitche r an d Armstron g insis t tha t belie f conten t exhausts th e perceptua l state , whil e Goldma n an d other s aren' t s o sure . (Se e Goldman (1977) , § 6.) For our purposes , thi s issue is beside the point. 41 Se e Pitcher (1971) , 87-9 . 42 Excep t fo r th e wea k sense , note d earlier , in whic h the y ca n b e considered a s 'inferred' fro m state s of the nervous system. 43 Se e Armstrong (1973), chs. 2 and 3. 40

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Beliefs i n genera l ar e psychologica l states . I assum e tha t a person's behaviou r gives good evidenc e fo r ou r hypothese s abou t her psychologica l state , thoug h I won' t g o s o fa r a s t o assert , a s some philosopher s would , tha t bein g i n a certai n psychologica l state simpl y i s behavin g (or bein g dispose d t o behave ) i n certai n ways. I f thi s behaviouristi c position i s incorrect , the n behavioural evidence i s no t conclusive , bu t i t i s stil l ofte n th e bes t available . Sometimes, there might also be introspective evidence for or against the clai m that a person i s in a given psychological state , an d i f there is a correspondenc e (no t necessaril y the identity ) betwee n psycho logical state s an d brai n states , a s I suppos e ther e is , the n neurophysiological evidence would als o be relevant.44 The questio n befor e u s no w i s this : ho w d o w e manag e t o perceive physical objects? Assuming 45 tha t t o hav e a concep t i s to have th e capacit y fo r belief s o f a certai n sort , w e ca n rephras e th e question: ho w d o we come to have the concept of a physical object? Psychologists an d neuropsychologist s hav e produce d variou s re sults an d theorie s t o answe r th e questio n o f ho w conceptua l elements ente r huma n perceptua l states . I'l l revie w som e o f thei r work here before returnin g to the philosophical issues involved. There i s considerabl e experimenta l evidenc e tha t th e abilit y t o perceive a primitive distinction between a figure and it s backgroun d is inborn i n humans and man y laboratory animals. 46 The structur e of th e retin a i s probabl y responsibl e fo r th e presenc e o f thi s conceptual informatio n in the human perceptual state , much as it is in th e frog . Warre n McCulloc h an d hi s co-worker s hav e isolate d various structure s i n th e frog' s retin a whic h sen d impulse s to th e frog's brai n onl y unde r certai n comple x set s o f conditions , independent o f the leve l of general illumination, for example, i n th e presence of sharp boundaries between relativel y light and relatively dark patches , o r dar k area s wit h curve d edges , o r movemen t o f such edges. In fact, one fibre responds best when a dark object, smaller than a receptive field, enters that field, stops, an d move s about intermittently thereafter. The respons e i s not affected i f the lightin g changes or i f the backgroun d (sa y a picture of gras s 44 O f cours e i t will tak e some substantial progres s in neuroscience befor e thi s i s a real possibility, but m y point is that it isn't ruled out a priori. 45 Wit h Armstrong (1973), ch. 5, § 1. 46 Se e Hebb (1949), 19-21.

PERCEPTION 53 and flowers) is moving, an d i s not ther e i f only the background , moving o r still, i s i n th e field . Coul d on e bette r describ e a syste m fo r detectin g a n accessible bug? (Lettvin et al (1959) , 254) As might b e expected, the researcher s cam e t o thin k of these fibre s as 'bug-detectors' , an d th e frog' s behaviou r certainly suggests tha t this mechanis m enable s i t t o acquir e perceptua l belief s abou t nearby bugs. Simila r mechanisms in humans are probably respons ible fo r perceptua l belief s concerning figur e an d background , an d perhaps some concerning distanc e and size.47 But beyon d this fairl y simpl e level, the evidence indicate s tha t the capacity t o acquir e perceptua l belief s o f th e familia r sor t i s no t present a t birth. 48 Psychologist s tal k o f a phenomeno n calle d 'identity' i n perception . A figur e i s sai d t o b e see n wit h identit y when i t appear s simila r t o som e othe r figure s bu t no t t o others , when i t is seen as falling int o some categories and no t i n others, when it i s easily recalled , recognized , o r named . When I see a triangular figure, fo r example , I automaticall y se e i t a s mor e lik e othe r triangles tha n lik e squares , I ca n recal l it , recogniz e it , an d cal l i t and othe r simila r figure s 'triangles' . I n th e terminolog y we'v e adopted here , I acquire the perceptual belie f tha t ther e i s a triangle before me . Experiments o n newl y sighted huma n patients wh o ha d been blin d fro m birth , and o n chimpanzee s raised in total darkness , demonstrate that the capacity to acquire such a belief—what we'v e also calle d havin g th e concep t o f a triangle—i s presen t onl y afte r considerable perceptual experience. For example, Investigators (o f visio n followin g operatio n for congenita l cataract ) ar e unanimous i n reporting tha t th e perception o f a square, circle , o r triangle , or o f spher e o r cube , i s very poor. To se e one o f thes e a s a whol e object , with distinctiv e characteristic s immediatel y evident , i s no t possibl e fo r a long period . Th e mos t intelligen t an d best-motivate d patien t ha s t o see k corners painstakingl y even to distinguish a triangle from a circle . .. A patient was traine d to discriminat e square from triangl e over a period o f 1 3 days, and ha d learne d s o littl e i n thi s tim e 'tha t h e coul d no t repor t thei r form withou t countin g corner s on e afte r anothe r . . . And yet it seems that the recognitio n proces s wa s beginnin g alread y t o b e automatic , s o tha t some da y th e judgemen t "square " woul d b e give n wit h simpl e vision , 47

Se e e.g. Bowe r (1966) . Thi s has little t o do with the philosophical controversy over innateness because even thei r defender s admi t tha t somethin g sensor y i s neede d t o 'dra w out ' o r 'awaken 1 innate ideas . 48

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which woul d the n easil y lea d t o th e belie f tha t for m wa s alway s simultaneously given', {Heb b (1949), 28, 32)

Similar results were obtained with the chimpanzees. Given tha t a capacit y a s simpl e as the abilit y t o se e a triangle a s more lik e anothe r triangl e tha n lik e a squar e i s th e produc t o f considerable sensor y experience , i t i s t o b e expecte d tha t s o complex a talen t a s tha t o f seein g a serie s o f differen t pattern s a s aspects of one thing—tha t is, as a sequence of views of one physical object—is not presen t at birth . This expectatio n i s substantiated by the experiment s o f Jea n Piage t an d hi s colleagues. 49 Th e child' s ability t o acquir e perceptua l belief s abou t physica l objects , a s judged fro m behaviour , develop s betwee n th e age s o f on e an d eighteen months . At the beginnin g o f this period, the child's world is a welter of isolated incidents . Then the behavio r of the child begins to b e centered o n objects; bu t t o hi m there is no objective reality—no general space or time, no permanence of objects. There ar e onl y events —i.e. component s o f th e child' s ow n functioning . When a n objec t i n his field of vision disappears, i t ceases t o exist. {Phillip s (1975), 28)

Some months later , objects begi n to enjoy a sort o f permanence : there i s a shif t i n th e child' s conceptualizatio n fro m objec t realit y dependent o n hi s own action s t o objec t reality dependent on th e surround. The resul t is a kind of 'context-bound object permanence'. (Phillips (1975), 38)

At this stage, th e objec t is associated wit h a particular location ; th e child expect s t o find it there eve n when i t is clearly hidden i n a ne w location. Similarly , we find that the infan t doe s no t realiz e that a movin g object i s the sam e objec t whe n i t becomes stationar y . . . [or] that a stationary objec t tha t begin s t o mov e is still the same object after i t starts moving. (Bower (1982), 206) 50

The abilit y t o distinguis h the objec t by features such a s size, shape , and colour , i n additio n t o it s locatio n o r trajectory , i s a majo r 49

Se e Piaget (1937 ) and Phillip s {1975). Bowe r an d hi s colleagues woul d disput e som e detail s i n the Piagetia n accoun t of th e developmen t o f th e objec t concept , bu t th e genera l ide a o f a non-trivia l developmental period i s all that matters here. Even psychologists who pla y dow n th e role o f learnin g i n perceptio n agre e tha t a developmen t o f thi s genera l sor t take s place. SeeE. Gibson (1969),ch . 16 . 50

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development, followe d b y th e abilit y t o distinguis h object s tha t share a commo n boundar y an d a stepwis e improvemen t i n searching behaviours . By the en d o f th e developmenta l period, th e child possesse s ou r familia r concep t o f a n independentl y existin g physical objec t an d i s full y capabl e o f acquirin g perceptual belief s about them. Supposing then , a s th e evidenc e suggest s w e should , tha t th e concept o f a triangle or o f a physical object—that is , the abilit y to acquire beliefs , includin g perceptua l ones , abou t suc h things—i s acquired over some period of time, what ca n be said about how thi s is accomplished ? Presumably some chang e mus t take plac e withi n the brain , some alteratio n i n the neural structuring, that enable s us to se e things wit h identity , that fills the gap betwee n th e pattern of light on our retina s and the perceptual beliefs we acquire. To fix our ideas, I'l l sketc h a particula r neurophysiologica l theor y o f wha t goes o n i n thi s development , a theor y du e t o Donal d Hebb. 51 I doubt tha t th e philosophica l moral s I'l l eventuall y dra w actuall y depend o n th e correctness o f exactly this scientific theory, bu t i n an area o f abstrac t epistemology , I fin d i t reassuring , eve n useful , t o have at leas t one fairl y specifi c exampl e o f how a naturalistic story of our knowledge might go. That par t o f th e brai n involve d in perception, th e visua l cortex, can b e divide d int o fou r layers , an d th e patter n o f retina l stimulation i s topologically equivalen t to th e patter n o f activit y in the first of these only . Afte r that , topological characteristic s ar e no t preserved, and th e neural connections betwee n th e inner layers, and between thes e an d th e oute r layer , ar e s o comple x a s t o see m random. Excitatio n o f a small part o f the initial layer can stimulate widely separate d area s i n al l thre e inne r layers ; widel y separate d parts o f th e initia l laye r ca n stimulat e neighbourin g cell s i n th e inner layers ; inne r area s ca n stimulat e othe r inne r area s an d eve n outer ones . For any one cell to fire and pass alon g its excitement t o the cells connected wit h it, it must be stimulated by many other cells simultaneously. Bu t an y cel l in the visua l cortex i s connected wit h many, man y others, s o th e firin g o f a numbe r o f cell s i n th e firs t layer wil l usuall y result i n a convergenc e o f sufficien t stimulatio n for firin g o n a numbe r o f cell s i n variou s othe r layers , i f fo r statistical reason s alone . Thu s an y visua l stimulu s create s a 51

Se e Hebb (1949), esp. chs. 4 and 5, and (1980), ch. 6.

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veritable hu m o f activit y throughou t th e visua l cortex , an d th e hums correspondin g t o differen t retina l stimul i ar e globall y th e same. The question, then , i s how al l this activity is organized. To answe r it , Heb b make s a simpl e theoretica l assumption , namely, tha t i f on e cel l repeatedl y play s a rol e i n th e firin g o f another, the n a change takes place increasing the first cell's efficac y in firin g th e second . H e suggest s tha t thi s phenomeno n coul d b e produced b y th e growt h o f synapti c knobs , bu t fo r theoretica l purposes, an y plausibl e mechanis m wil l do . No w suppos e tha t someone unfamilia r wit h triangle s fixes her gaz e o n th e ape x o f a given triangula r figure . Thi s generate s a certai n patter n o f stimulations whic h recur s a s lon g a s she stares a t tha t corner , an d occur s again ever y tim e sh e look s a t it . A s a resul t o f thi s repeate d exposure, group s o f cell s a t variou s cortica l level s ar e repeatedl y efficacious i n firing one anothe r (a t convergences of various kinds). As a result , i t become s easie r fo r thes e interconnecte d cell s to fir e one another, the y become mor e interdependent , eventually forming what Hebb call s a 'cell-assembly'. It will respond t o the apex of any similar triangle. Analogous processe s naturall y result i n assemblie s fo r th e bas e angles o f th e triangle , but thi s leaves open th e questio n o f ho w th e triangle a s a whole, a s a unit, is perceived with identity . Behavioural experiments sugges t tha t acquiring the ability to recognize triangle s depends essentiall y on successiv e eye movements an d fixation s o n various part s o f th e figure . Th e ey e i s especiall y inclined to trac e lines because all the movement-inducing peripheral stimulations are urging it in one direction . Thus angle s are frequen t fixation points, lying a s they d o a t th e intersectio n of straigh t lines, an d whe n th e eye i s fixate d o n on e angle , i t i s stimulate d t o mov e toward s another. Because o f th e numerou s interconnection s betwee n neuron s i n various part s o f th e visua l cortex , cells i n one assembl y happen b y chance t o b e connecte d wit h variou s cells i n the othe r two . Whe n the motor stimulation s describe d abov e induc e repeated movemen t from on e angle to another, th e neurons in these chance connection s become increasingl y efficacious , an d th e cell-assemblie s fo r th e three corner s ar e integrated int o a second-order cell-assembly . Th e individual angl e assemblie s can stil l work independently , s o actua l perception o f a triangle involves what's calle d a 'phase sequence' of excitations o f th e corne r assemblie s and th e integrate d assembly . If

PERCEPTION 5

7

the corner assemblie s are tf, b, and c, and the integrated assembly t , then a phase sequence is something like a-b-t-a-c-t-c-t-b.52 Once a n integrate d cell-assembl y o f thi s sor t ha s bee n formed , looking a t a triangle will cause it to reverberate for half a second o r more. Thi s represent s a considerable gain bot h i n organization an d in duration over the random hum o f activity brough t abou t by the same visual stimulation before th e formatio n of the assembly . This longer, repeatabl e trac e shoul d persis t lon g enoug h t o allo w th e structural change s require d for long-term memory . In other words, the cell-assembl y i s what permit s th e subjec t t o se e a triangle wit h identity, t o acquir e perceptua l belief s abou t it : i t i s a triangle detector i n much th e sam e sense as the fibre located b y McCulloc h is a bug-detector; i t provides the subject with her concept of triangle.53 The ability to perceive physical objects is not unlik e the ability to perceive triangula r figures , thoug h i t i s more complex . Th e tric k is to se e a series of patterns a s constituting views of a single thing. Just as th e abilit y t o se e triangle s develop s ove r time , throug h a painstaking proces s o f seekin g ou t corner s an d comparin g on e triangle wit h another , th e abilit y to se e continuing physical objects develops ove r a perio d o f experienc e wit h watchin g and manipu lating them . The clos e resemblance i n the structur e o f the learnin g processes suggests that what i s involved i n the physica l object case is just a mor e elaborat e versio n o f th e cell-assembly , i n particular , the development of higher-order cell-assemblie s which respond t o a 52 Thi s account i s based o n Heb b (1949) , chs . 4 and 5. Heb b (1980 ) cites evidenc e suggesting that the basic assemblies may respond to the sides of the triangle rather tha n its angle s (se e pp . 89 , 98—102) . Thi s modificatio n doesn' t affec t th e upsho t o f m y discussion. Th e 198 0 work als o summarize s ne w support s fo r Hebb' s theor y discovered i n th e year s sinc e 194 9 (se e pp. 98-100), an d develop s variou s applications o f th e view , includin g a fascinatin g accoun t o f scientifi c problem solving (pp. 119-21). 53 Philosopher s o f mind sometimes worr y abou t a blanket objectio n Dennet t ha s raised t o attempt s t o locat e concept s i n the neurons: 'Suppos e your "grandmother " neuron died ; no t onl y could yo u no t sa y "grandmother" , yo u couldn' t see her if she was standin g righ t i n fron t o f you . . . you woul d hav e a complet e cognitiv e blin d spot . . . Nothing remotel y lik e that patholog y i s observed, o f course , an d neuron s malfunction o r die with depressing regularity, s o ... theorie s that requir e grandmothe r neurons are in trouble' (1978, p. xiii). I don't think Hebb's theory require s anythin g so specific a s a grandmothe r neuron , bu t i t doe s posi t shap e an d othe r genera l detectors. Heb b (1949 ) alread y contain s a respons e t o objection s lik e Dennett's : 'The assembl y i s thought o f a s a syste m inherentl y involving some equipotentiality , in th e presenc e o f alternat e pathway s eac h havin g the sam e function , s o that brai n damage migh t remov e som e pathway s withou t preventin g th e syste m fro m functioning . . .' (p. 74).

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series o f aspect s o f a singl e object , an d fire , a s before , i n a complicated phas e sequence. 54 Whe n a n objec t stimulate s a phas e sequence o f such assemblies , i t participates i n the generatio n o f th e subject's perceptua l belie f stat e i n th e appropriat e causa l way , i n the wa y m y han d participate s i n th e generatio n o f m y belie f that there i s a hand befor e me when I look a t it in good light . Crudely put , huma n being s develo p neura l object-detector s which allo w the m t o perceiv e independent , continuin g physica l objects. I t i s thes e comple x cell-assemblie s tha t bridg e th e ga p between wha t i s interacted with an d wha t i s perceived. The objec t I perceive o n a given occasion , o r mor e precisely , the fron t sid e o f a time slic e of that object, i s causally responsibl e only for th e patter n of retina l stimulations , whil e th e unifyin g concep t o f a familia r physical objec t i s contributed by m y physica l object-detector. Th e presence o f th e object-detector , i n turn , i s partly th e resul t o f th e structure o f m y brai n a t birt h (conditione d b y th e evolutionar y pressures of the environment on m y ancestors) an d partly th e resul t of m y earl y childhoo d experience s wit h physica l objects . Al l this , while undeniably complex, i s still naturalistic, and causal . Let m e no w retur n t o th e subject that inspire d thi s detour int o th e theory o f perception i n the first place, namely my claim that w e ca n and do perceive sets, and tha t our abilit y to do so develops i n much the sam e wa y a s ou r abilit y t o se e physical objects. Conside r th e following case : Stev e needs tw o egg s fo r a certai n recipe . Th e eg g carton h e take s fro m th e refrigerato r feel s ominousl y light . H e opens th e carton an d sees , to his relief, three eggs there. M y clai m is that Stev e ha s perceive d a se t o f thre e eggs . B y th e accoun t o f perception jus t canvassed , thi s require s tha t ther e b e a se t of thre e eggs i n th e carton , tha t Stev e acquir e perceptua l belief s abou t it , and tha t th e se t o f egg s participat e i n th e generatio n o f thes e perceptual beliefs in the same wa y that my hand participate s i n the generation o f m y belie f tha t ther e i s a han d befor e m e when I look at it in good light. This clai m will doubtless elici t a clamour o f objections, an d eve n if i t doesn't , i t need s elucidation , so le t m e begi n a t th e beginning , with th e assertion that there is a set of eggs i n the carto n i n front of Steve. Th e simples t wa y t o resis t thi s ide a i s to den y tha t ther e ar e sets. To thi s I reply with a version of the Quine/Putna m argument s 54

Se e Hebb (1980), 107 ,

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sketched toward s the end of Chapter 1—mathematica l entities are indispensable fo r ou r bes t theor y o f th e world—supplemente d b y the observatio n tha t ou r bes t theor y o f mathematica l ontolog y i s that (at least some) 55 mathematical entities are sets. A second mod e o f resistance is to joi n the traditiona l Platonist in denying that set s have location i n space or time. But notice: there is no rea l obstacle t o th e position tha t th e se t of eggs come s int o an d goes ou t o f existenc e whe n the y do , an d that , spatiall y a s well a s temporally, i t i s locate d exactl y wher e the y are . A se t o f highe r order, lik e the se t consisting of the se t of eggs and th e se t of Steve's two hands , would agai n b e located wher e it s members are, tha t is, where the set of eggs and the set of hands are, which is to say, where the eggs and hand s are. In this way, even an extremely complicated set would hav e spatio-temporal location , a s long a s it has physical things i n it s transitiv e closure. 56 An d an y numbe r of differen t set s would b e located i n the same place; for example, the set of the set of three eggs an d th e se t of tw o hand s i s located i n the sam e place a s the set of the set of two eggs and the set of the other eg g and the tw o hands.57 Non e o f thi s i s an y mor e surprisin g tha n tha t fifty-tw o cards ca n b e locate d i n th e sam e plac e a s a deck . I n an y case , I hereby adop t thi s vie w a s par t o f se t theoreti c realism . O n som e terminological conventions , thi s means that sets no longer coun t as 'abstract'. So be it; I attach no importance to the term.58 More controversia l is the second part of the claim that Steve sees a set: the contention tha t he gains perceptual beliefs about the set of eggs, in particular, that this set is three-membered. Let me break my defence of this idea into two parts . First , and leas t controversially, I contend tha t th e numerica l belief—ther e ar e thre e egg s i n th e carton—is perceptual, 59 tha t is , tha t i t look s t o Steve , i n a non metaphorical sense , a s i f ther e ar e thre e egg s there . Ther e i s 55 A s indicate d i n ch . 1 , sect . 1 , I' m no t assumin g tha t al l mathematica l entitie s are sets, but ou r standar d mathematica l theories indispensably refer t o sets of points, sets of numbers , etc . Se e ch. 3 belo w fo r th e relationshi p of th e familia r natura l an d real numbers to sets. 56 Th e transitiv e closur e o f a se t consist s o f it s members , th e member s o f it s members, th e member s o f th e member s o f it s members , an d s o on . Fo r a forma l definition, se e Enderton (1977) , 178 . 57 Se t theorist s wil l notic e tha t thi s mean s ther e ar e prope r class-man y set s located wher e any physica l thin g is . (They stac k neatly! ) Pur e set s have no location . For more on pure sets, see ch. 5, sect. 1, below. 58 Se e Kat z (1981) , 20 7 n . 29 . Parson s use s th e ter m 'quasi-concrete ' fo r mathematical objects of this sort. See his (1983(2) , 26 . 59 Se e Kim (1981) for a similar claim .

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empirical evidence, based o n reaction times , that suc h beliefs abou t small number s ar e non-inferential. 60 Furthermore , thi s belief abou t the numbe r of egg s can non-inferentiali y influenc e an d b e influence d by othe r clearl y perceptual belief s acquire d o n thi s occasion ; fo r example, th e welcome fac t tha t there are enough egg s for the recipe can mak e th e egg s themselve s loo k larger. 61 Thi s particula r perceptual belie f abou t th e numbe r o f egg s i s thu s par t o f a ric h collection o f perceptua l belief s acquire d o n thi s occasion , belief s about th e siz e and colou r of th e eggs, th e fac t tha t tw o egg s can b e selected fro m amon g th e thre e in various ways, th e location s o f th e eggs in the nearly empty carton, an d s o on. So fa r s o good . No w le t m e tak e u p th e questio n o f whethe r o r not thi s perceptual belief i s a belie f abou t a set. What i s a numerical belief about , afte r all ? Th e easies t answe r woul d b e tha t Steve' s belief i s about th e eggs , th e physica l stuff ther e i n th e carton , bu t Frege lon g ag o demonstrate d th e inadequac y o f tha t response. 62 The troubl e i s tha t th e physica l stuf f i n th e carto n ha s n o determinate numbe r property : i t i s thre e eggs , bu t man y mor e molecules, eve n more atoms , an d onl y a quarter of a carton o f eggs. For a give n mass o f physica l stuff, ther e i s no predetermine d wa y that it must be divided up, and withou t this, there i s no determinate number property. S o the physical stuff b y itself cannot b e three. If no t th e physica l stuf f makin g u p th e eggs , the n wha t i s th e subject of a number property? Some would say , 'the eggs', meaning by thi s th e physica l stuff a s divided up b y th e propert y of bein g a n egg, wha t I'l l cal l a n 'aggregate 1.63 Frege' s answe r i s tha t a numerical statemen t i s a statemen t abou t a concept , fo r example , the concep t 'eg g i n th e carto n i n fron t o f Steve' . Other s migh t choose th e extensio n o f Frege' s concept , tha t is , wha t i s usuall y 60

Se e Kaufman etal. (1949). Fo r another exampl e o f such non-inferentia l influence , suppos e a majority vote from a three-membere d pane l wil l defea t you r motion . Whe n tw o panellist s rais e their hand s t o vot e no , you r numerica l belief tha t ther e ar e tw o o f them (enoug h t o dash you r hopes ) ma y weli influenc e you r perceptio n o f their facia l expressions (ho w malevolent the y look!). If only one ha d vote d negatively , she might only have looke d dense. 62 Freg e (1884) , §§22-3. 63 Som e would sa y that 'ther e are thre e eggs i n the carton' is properly analyse d as saying 'ther e is an x, there i s a y, and ther e i s a z, al l distinct, such tha t x i s an egg in the carton , y i s an eg g in the carton , z i s an eg g in the carton , an d anythin g that's a n egg in the carton is either x o r y or z 1.1 take this to b e a variant of the aggregat e view : some physical stuff i s divided u p i n a certain way. 61

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called the 'class' of eggs in the carton. 64 And the set theoretic realist opts for the set of eggs in the carton. But o n wha t grounds ? If Stev e is suppose d t o se e a se t o f eggs , shouldn't th e set theoretic realis t hold tha t h e can see, for example , that it is a set and no t a n aggregate ? Attractive as this move might seem, I think it is not correct . Notic e tha t the various candidates for the beare r o f th e numbe r property—th e set , the aggregate , th e concept, th e class—have their most basic properties, any properties that migh t coun t a s perceptual, i n common ; fo r example , the y all have subcollection s (e.g . the egg-stuf f unde r a mor e exclusiv e property), they are al l capable o f combination (e.g . the disjunctio n of tw o concepts) , an d s o on . Thes e similaritie s ar e wha t make s them al l potentia l candidate s fo r number-bearing . The propertie s that separat e the m ar e theoretica l properties , lik e extensionahty . Asking Stev e to loo k an d se e whether he' s perceiving a se t o r a n aggregate i s like asking him to look an d se e whether th e egg is solid or mainly an empty space littered with atoms . What I' m getting at is this: the amount we know abou t things by perception i s very limited. Abou t physical objects, for example, w e know littl e mor e tha n tha t the y are , i n Hebb' s words , 'space occupying and sense-stimulating something[s]\65 Beyond that, the bulk o f ou r knowledg e abou t the m i s theoretical : tha t the y ar e made up of atoms, o f this and tha t sort , arranged in such-and-such a way , and s o on . Th e sam e goe s fo r sets . Wha t w e perceiv e i s simply somethin g wit h a numbe r property, somethin g tha t ca n b e combined wit h other s o f it s ilk , and s o on . Nailin g dow n thi s number-bearer's more esoteric properties i s a theoretical matter . So, to decid e the cas e between sets, aggregates, concepts , classes , and whateve r else , w e nee d t o look , no t t o ou r perceptua l experiences, bu t t o ou r overal l theor y o f th e world , an d w e mus t ask whic h o f thes e i s bes t suite d t o playin g th e rol e o f th e mos t fundamental mathematica l entity . (Compare : decidin g whethe r Berkeleian bundle s of God' s experiences o r th e physicist's bundle s of atoms are best suited to playing the role of the most fundamenta l 64 A class differ s fro m a se t i n that i t is essentially dependen t o n a property , lik e 'being a n eg g i n th e carto n i n fron t o f Steve' . Sets , b y contrast , ar e generate d iteratively, b y takin g a t eac h stag e ever y subse t o f what' s bee n generate d before , regardless o f whether o r no t th e member s of that subse t ca n b e singled out b y som e property. See ch. 3, sect 3, below. 65 Heb b (1980) , 109. I'l l consider this passage in more detai l in the next section .

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physical entity. ) O n thi s score , set s wi n goin g away ; the y ar e extremely simpl e an d manageabl e entities tha t for m th e basi s for a surprisingly effective an d efficien t mathematica l theory . In contrast , properties, o n whic h bot h aggregate s an d classe s depend , ar e har d to handle—no comparably flexibl e and complete theory is known66 —and pron e t o paradox—fo r example , conside r th e propert y a property ha s whe n i t doesn' t hav e itself. 67 An d classe s ar e littl e better.68 Th e elementarines s o f th e notio n o f set , it s eas e o f manipulation, an d th e immens e succes s o f se t theory , bot h a s a foundation fo r othe r branche s o f mathematic s an d a s a math ematical theor y i n its own right , all help t o mak e the se t of eggs the most attractive candidate fo r the role of number-bearer. I tak e al l this t o suppor t th e se t theoretic realist' s clai m tha t th e bearers o f numbe r propertie s ar e sets , an d thus , tha t Steve' s perceptual belie f i s a belie f abou t a set . Bu t befor e leavin g thi s point, I want t o cal l attention t o the contingenc y of this conclusion. In it s support , I depend o n th e ide a tha t mathematica l entitie s ar e indispensable to physical science; i f they weren't, ther e would b e no reason t o includ e sets in our overal l theory. Thus m y preference fo r sets is contingent on the way the world is , to the extent that our bes t theory seem s to demand them . To appreciat e th e forc e of this fact , consider , for example , on e of the fundamenta l differences betwee n set s an d aggregates , namely , that ther e ar e set s of highe r rank—-sets o f sets , set s o f set s o f sets , and s o on—whil e ther e ar e n o aggregate s o f aggregates . I f th e physical world were simpler , allowing for a simpler physical theory with n o continuous phenomena, the n our overall theory might have no nee d fo r rea l numbers , an d consequently , fo r set s o f highe r rank. 69 Sinc e we'r e engage d i n scienc e fiction , w e migh t imagine that ou r perceptua l experience s o f discret e object s i n thi s simpler 66 Beale r (1982) , ch . 5 , suggest s a propert y theor y tha t essentiall y mimic s se t theory, bu t a s Anderso n (1987 , p . 151 ) point s out , th e plausibilit y o f som e se t theoretic assumptions does not carr y ove r to their property theoretic translations. 67 Th e relate d 'paradoxe s o f se t theory ' presen t n o proble m fo r set s o n th e iterative conception . Fo r example , th e Russel l set , th e se t o f al l non-self-membere d sets, cannot be formed a t any stage , because new non-self-membere d set s (al l set s ar e non-self-membered o n th e iterativ e picture ) wil l b e availabl e a t th e nex t stage . See Godel (1947/64) , 474-5. 68 Se e ch. 3 , sect. 3, below. 69 Reals , a s standardl y constructed , occu r a fe w stage s afte r w . Se e Enderton' s construction (1977 , ch . 5) . Th e relationshi p o f continuou s phenomen a wit h set s of higher ran k wil l be considered i n more detail i n ch. 4 an d ch . 5, sect. 2, below.

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world ar e exactl y lik e ou r experience s o f discret e object s i n thi s world. Still, in the simpler world, we might have no justificatio n fo r including highe r rank s i n ou r overal l theory , an d thus , muc h les s justification fo r taking our numerica l perceptions t o b e perception s of set s rathe r tha n aggregates . S o m y clai m tha t set s ar e th e bes t candidates for the bearers of number properties depends on the fac t that the y ar e th e bes t mathematica l entitie s fo r th e mathematica l theory thi s particula r world—wit h it s continuou s phenomena — requires. Let's grant, then, that there is a set of eggs in the carton, an d tha t Steve gains th e perceptua l belie f tha t thi s se t is three-membered. 70 For thi s t o coun t a s Steve' s perceivin g a set , only on e furthe r condition mus t b e satisfied : th e se t o f eggs mus t participate, i n a n appropriate causa l way , i n th e generatio n o f Steve' s belief . Appropriate participatio n i s exemplified by the rol e of m y hand i n the generatio n of m y perceptua l belie f tha t ther e i s a han d befor e me, an d tha t i n tur n come s dow n t o m y hand' s stimulatio n o f a phase sequence of cell-assemblies. So our questio n is: could a set of eggs do the same? The behavioura l evidence of Piaget and hi s colleagues suggests that the abilit y to gai n perceptual belief s abou t set s develops in a series of stage s paralle l t o thos e fo r perceptua l belief s abou t physica l objects, thoug h a t a somewhat late r age. 71 At the beginnin g of this period, a chil d ma y b e abl e t o classif y object s int o group s i n a consistent way—say , triangle s wit h triangle s an d square s wit h squares—but she does not correctl y grasp th e inclusion relation— in a grou p o f tw o blac k square s an d fiv e blac k circles , th e chil d thinks ther e ar e mor e roun d thing s tha n black . Fo r th e younge r child, a set ceases to exist when its subsets are attended to . 70 Stev e needn't expres s hi s belief i n this way to himself ; implicit in the word 'set ' is a mor e refine d theor y tha n mos t peopl e ar e awar e of . Whe n I sa y h e gain s a perceptual belie f abou t a set , I mean h e gains a perceptual belief abou t a somethin g with a number property, whic h we theorists kno w to b e a set. Analogously, when he perceived th e tre e in fron t o f him, h e gained a perceptua l belie f wit h les s theoretica l content tha n a botanis t coul d provide . Nevertheless , wha t h e perceive d wa s th e botanist's tree . 71 Piage t and Szemirisk a (1941 ) woul d pu t tha t ag e between seven an d fourteen years. Se e als o Phillip s (1975) , ch . 4. Mor e recen t wor k suggest s tha t thes e developmental period s occu r somewha t earlier , betwee n tw o an d a hal f an d fiv e years. See Gelman (1977) . Once again, the details are less important than the parallel between development of the physical object and set concepts.

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A simila r confusio n i s observe d i n connectio n wit h th e set' s number properties. The younger chil d imagines that th e number of elements i n a se t change s when i t i s rearranged, particularly when its element s ar e move d close r togethe r o r furthe r apart . Fo r olde r children, o n th e othe r hand , onc e a one-to-on e correspondenc e between tw o set s ha s bee n established , the belie f i n thei r equi numerosity canno t b e shaken; indeed the ver y question seem s sill y to them . Onc e a perceptual belief abou t a set is gained, the though t that th e set could change its number property when it s elements are moved about (barrin g mishap) appears preposterous . So, just as the concept o f an independent and continuin g physical object i s acquire d i n stages , th e concep t o f a se t wit h inclusion s and a constan t numbe r propert y i s itsel f gaine d ove r time , an d depends o n experienc e with group s o f objects . I t shoul d b e note d that th e child' s developmen t o f th e se t concep t i s no t a linguisti c achievement. O f course , childre n ar e rarel y taugh t th e wor d 'set' , but the y ar e taugh t numbe r words , an d i t migh t b e though t tha t their earl y error s ar e primaril y verbal , an d tha t i t i s verba l instruction tha t correct s them . Th e evidenc e is heavily agains t this assumption:72 The ma/or point is that the development of the concept of number begins in infancy, lon g befor e speech o r forma l instructio n play any part. The infan t is force d to generat e numbe r concept s b y the requirement s of it s everyday activities—activities so commonplace tha t th e fondest parent barel y thinks them worth y o f comment . The y ar e wort h mentionin g becaus e these ar e the simpl e beginning s fro m whic h th e whol e structur e o f mathematica l thinking takes root. (Bower (1982), 250)

One mus t expec t tha t th e se t concept , lik e th e physica l objec t concept, could be developed i n the complete absence of language. And ho w doe s th e se t concep t develop ? W e hav e see n th e evidence tha t i t develop s ove r a perio d o f time , lik e th e objec t concept, an d tha t th e determinin g facto r i n bot h thes e develop ments i s repeate d exposur e t o th e sor t o f thing s i n question . Th e development o f th e objec t concept i s brought abou t b y th e child' s experiences wit h variou s physical objects in he r environment , an d the se t concep t b y experience s wit h set s o f physica l objects , fo r example b y forming one-to-one correspondence s betwee n them , by regrouping them to form salient subsets, and so on.73 72 73

Se e also Phillips (1975), 145 . Thes e manipulation s are crucial . Kitche r (1983 , p . 103 ) ha s suggested tha t my

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Hebb's theor y o f th e formatio n o f th e neura l triangle-detecto r made essential use of the behavioural evidence that development of the abilit y to se e triangles wit h identit y requires repeated fixations on corners of triangular figures, ey e movements from on e corner t o another, an d even , i n som e cases , activ e seekin g ou t o f corners . Because of the behavioral similaritie s betwee n thi s process and that leading to the development of the object concept, it is theorized that an object-detecto r develop s in a simila r way, a s a result of various experiences wit h physica l object s i n th e environment . Give n th e evidence tha t th e se t concep t require s a simila r developmenta l period involvin g repeated experienc e with set s i n the environmen t parallel t o th e require d experience s wit h triangle s an d physica l objects, i t seems reasonable to assum e that these interaction s with sets o f physical objects brin g about structural changes in the brain by some complex process resembling that suggested by Hebb,74 and that th e resultin g neura l 'set-detector ' i s wha t enable s adult s t o acquire perceptual beliefs about sets . This assumption provides a solution to anothe r difficulty. Recal l that man y things— a mas s o f physica l stuff , an d man y differen t sets—occupy the same spatio-temporal location . These things also produce the same retinal stimulation, bu t on one occasion the stuf f is seen , o n anothe r on e set , o n anothe r a differen t se t altogether , and so on. We've al l been amused by the psychologist's example s in which we see a single picture first as an undifferentiated mass, the n as representin g a definit e numbe r of distinc t objects, or th e child' s puzzle in which th e homogeneou s jungl e foliage resolves itself int o a pac k o f ferocious beasts. Wher e a bookbinde r see s a larg e se t of individual book s (s o man y perhap s tha t sh e ha s n o perceptua l access to th e exact number), the encyclopaedia salesman sees three account i s inconsistent wit h th e vie w o f a grou p of perceptua l theorists called th e 'ecological realists'. He says that their 'genera l claim that the information whic h we gain in perception concerns transformations of the sensory array caused by events in which perceive d object s participat e seem s t o b e at odd s wit h the ide a tha t w e ca n acquire perceptua l information abou t unchanging abstrac t objects'. I merel y point out tha t th e 'abstract' object s involve d here ar e not unchanging ; in particular, they can be moved about for purposes of more ready classification an d t o display one-toone correspondences. The account in the text also seems to m e in harmony with the ecological realist' s emphasi s o n invariant s (th e chil d learn s tha t th e numerabl e collection i s invariant under transpositions of it s elements) an d affordance s (Steve' s three-membered set of eggs affords cake-making) . 74 Heb b (1949 ) doesn' t mentio n se t perception , bu t h e doe s conside r th e perception o f numbe r properties of collection s t o b e par t o f hi s theor y i n Heb b (1980), 122-4.

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of his rival's product; where I see a set of fou r shoes, yo u migh t see a se t o f tw o pairs . A microscopi c imag e look s t o m e lik e a n unorganized mes s of Jackson Polloc k drips , while the biologis t see s three paramecia and an amoeba . Hebb's theor y provide s a key to thes e phenomena . The y involve a chang e i n perceptio n o n th e par t o f a singl e subjec t (th e child' s puzzle), a differenc e i n perceptio n betwee n tw o roughl y compar able" subjects (the bookbinder and th e encyclopaedia salesman), an d a differenc e i n perceptio n betwee n tw o subject s wit h differen t training (m e an d th e biologist) . The las t cas e i s easiest: perceptual development continue s pas t childhood ; th e biologis t acquire d further perceptua l abilitie s (furthe r cell-assemblies ) durin g he r education i n la b techniques . I n th e othe r tw o cases , precedin g experiences an d neura l activit y (thoughts ) influenc e th e eas e wit h which a give n patter n o f stimulatio n wil l trigge r a give n cell assembly: th e bookbinder an d the salesman have different interests ; they notic e differen t things . I n th e puzzl e cases , som e shif t i n ou r attention cause s a ne w cell-assembl y t o com e int o pla y quit e suddenly. I f there wer e n o suc h se t assemblies , the phenomenon o f seeing differen t grouping s wit h differen t numbe r properties woul d have t o b e explaine d b y som e additiona l organizin g event s occurring afte r th e initia l stimulatio n o f th e ordinar y physica l object assemblies. This approac h i s less true to the phenomenon.75 On thi s account, then , whe n Stev e looks i n the eg g carton, ther e is a se t o f egg s there , h e acquire s perceptual belief abou t tha t set , namely, that it has three members, 76 and th e set of eggs participates in th e generatio n o f hi s perceptua l belie f i n th e appropriat e way , that is , i n th e wa y m y han d participate s i n th e generatio n o f m y belief tha t ther e is a hand befor e me when I look a t i t in good light , which is , b y som e aspec t o f th e objec t i n questio n causall y interacting wit h th e retin a i n suc h a wa y a s t o brin g abou t th e stimulation o f the appropriate detector . In the case of sets, jus t as in the cas e o f physica l objects, i t i s the presenc e o f a comple x neura l development tha t bridge s th e ga p betwee n wha t i s causall y interacted wit h and what is perceived. 75

Se e Hebb (1980), 83-7. Als o Bruner (1957) , 241-4. H e undoubtedl y acquires other perceptual beliefs abou t th e se t o f eggs a t th e same time , e.g . tha t i t ha s variou s two-elemen t subsets . Notic e als o that , a s promised, Gettier-styl e case s ar e easil y constructed ; th e sam e illusionis t who se t u p the tree exampl e described earlie r could arrange his mirrors to create the illusion of a set where there i s actually another, hidden, set. 76

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Thus, th e Hebbia n neurophysiologica l accoun t o f what bridge s the ga p betwee n wha t i s causall y interacte d wit h an d wha t i s perceived i n th e cas e o f physica l objects can als o provid e fo r th e perception o f sets. I think this lend s considerable credibility to th e set theoreti c realist' s clai m tha t set s ar e perceivable , but , a s mentioned earlier , I don't inten d t o ti e set theoretic realism to thi s particular neurophysiologica l theory . Benacerraf-styl e worrie s ar e based o n a dee p convictio n tha t a certai n kin d o f explanatio n cannot be given; see Putnam's rhetorical outburst: What neura l process , afte r all , coul d b e describe d a s th e perceptio n o f a mathematical object? (Putna m (1980), 430)

My goal in this section has been to indicate that there is at least one plausible answer to this question. 3. Intuitio n So far , I hav e argue d tha t sets , suitabl y understood , ca n b e perceived. While this may be enough to answer the Benacerraf-style worries abou t compromis e Platonism , i t provide s onl y th e bares t beginning o f a n accoun t o f se t theoreti c knowledge . Wha t i s th e relation, fo r example , betwee n ou r knowledg e o f particula r fact s about particula r sets of physical objects, and ou r knowledg e o f the simplest se t theoreti c axioms ? How, fo r example , d o w e com e t o know tha t an y two object s can b e collected into a set with exactly those tw o members , o r tha t th e member s of an y tw o set s ca n b e combined int o a se t tha t i s thei r union ? Thes e genera l belief s underlie two of the most elementar y set theoretic axioms—Pairing and Union—and our epistemology must account for them. Given ou r analog y betwee n mathematic s an d natura l science , let's firs t as k fo r th e sourc e o f comparabl e belief s i n th e physica l sciences. Rudimentar y accounts of elementar y physical knowledge most often begi n with simple, enumerative induction: every swan in my sampl e i s white; therefore , I conclude , al l swan s ar e white . I t might be argued that our most basi c general set theoretic beliefs are justified i n the sam e way, but thi s line is unconvincing. 77 Do I test 77 Th e ide a tha t mathematic s i s a simpl e inductiv e scienc e goe s bac k t o Mil l (1843), bk . 2 , chs . 5 an d 6 . Objection s hav e bee n mounte d b y man y writers , including Frege (1884), §§9-10, Hempel (1945), and Kim (1981).

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to se e whether o r no t th e tw o set s o f finger s o n m y righ t an d lef t hands ca n b e combine d t o for m a large r se t of fingers? No, onc e I am able to understand th e question, th e answer i s obvious. Would I be mor e sur e tha t tw o set s ca n b e combine d i f I successfull y combined a wid e variet y o f set s containin g differen t kind s o f objects? No, 78 but th e observation of white swans i n a wide variety of differen t environment s does add suppor t fo r th e clai m tha t the y are al l white . Evidently , particula r observation s provid e a ver y different typ e o f suppor t fo r genera l hypothese s lik e 'any tw o set s can b e combined ' tha n the y d o fo r genera l hypothese s lik e 'al l swans are white'. Does thi s mea n tha t se t theor y i s dramaticall y differen t fro m physical science, after all ? Where ther e i s an analogy , ther e mus t be differences a s well a s similarities, but I think we'v e no t ye t reache d the poin t a t whic h ou r compariso n betwee n mathematic s an d natural science is of no furthe r use . Rather, I think tha t we've filled in th e detail s incorrectly , tha t ou r primitiv e general se t theoreti c beliefs actuall y correspond, no t t o simpl e enumerative inductions, but t o primitiv e general belief s abou t physica l object s tha t ar e n o more subjec t t o simpl e inductiv e support tha n thei r se t theoreti c counterparts. Consider, fo r example, th e child's hard-won belief s tha t physical objects exis t independentl y o f th e huma n viewer , tha t the y ar e independent o f thei r stat e o f motion . Ca n thes e b e teste d b y observation o f particula r examples ? Doe s th e observatio n tha t a teacup persist s whe n it' s move d acros s th e tabl e ad d suppor t ove r and abov e tha t provide d b y simila r observation s o f coffee-cups ? Evidently not . Thes e ar e primitiv e general belief s abou t physica l objects that are not supporte d b y simple enumerative induction. We cannot chec k t o see whether o r no t physical objects persist when n o one i s observing them, bu t w e believ e it nevertheless, and belief s of this sort appea r a t the most elementar y levels of our physica l theory of the world. Hebb's analysi s o f neura l operation s provide s on e possibl e account o f their source. Recall that repeate d viewing s of a triangle's apex lea d t o th e developmen t o f a first-orde r cell-assembl y tha t 78 I n childhood , suc h manipulation s wit h a variet y o f set s helpe d engende r m y ability t o se e sets i n th e firs t place , bu t onc e I have thi s ability , m y convictio n that two set s ca n b e combine d doesn't depen d o n m y testin g a variet y o f th e set s I ca n now see .

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responds t o angle s of like magnitude and orientation. Assemblie s at this leve l respond t o particula r contour s i n th e visua l field, simple tastes, localize d tactil e pressures , an d th e like . Th e sam e mechan isms tha t produc e assemblie s a t thi s simpl e leve l ar e capabl e o f producing higher-orde r assemblie s as well , suc h a s th e integrate d triangle assembly . Similarly , repeated viewin g o f a singl e physica l object fro m on e perspectiv e produce s a second-orde r assembl y which integrate s th e first-orde r assemblie s fo r th e contour s o f th e object's parts . And finally, manipulation o f th e object , or seein g it in motion , permit s th e developmen t o f a third-orde r assembl y integrating th e assemblie s for th e object' s variou s perspectives. A t this point , perceptio n o f th e objec t involve s a comple x phas e sequence of stimulations of all these assemblies.79 With these mechanisms in place, th e subject is able to perceive an independent, continuin g physica l object. But ther e i s no reaso n t o suppose tha t neura l development s com e t o a n en d here . Heb b suggests: Fundamental . .. i s th e generalize d ide a o f a thing , a n object , a space occupying an d sense-stimulatin g something, a s th e activit y o f a higher order cell-assembl y mad e up of neurons that ar e usually or always active in the relatively small number of different situation s of infancy when a visible, tangible objec t attract s attention . Thos e neuron s mus t b e a smal l proportion o f th e tota l numbe r excite d o n an y on e o f suc h occasion s bu t may stil l be a large number i n absolute terms. The theoretica l possibilit y of such a n assembl y is clear an d th e psychologica l suppor t fo r it s existence is also clear. (Hebb (1980), 109)

This fourth-orde r assembl y woul d correspon d t o th e genera l concept o f a physica l object . I t woul d b e stimulate d durin g th e phase sequenc e associate d wit h perceptio n o f a particular physical object when attention i s drawn to its more general features. A subjec t wit h suc h a n assembl y woul d automaticall y hav e various general beliefs about th e nature of the objects that stimulate it; w e migh t sa y that thes e belief s are 'buil t into' th e cell-assembly much a s three-sidednes s i s buil t int o th e triangle-detecto r i n th e form of mechanisms stimulating eye movements fro m on e corner t o another, o r three-anglednes s i n th e for m o f th e thre e first-orde r corner components . Crudel y put , th e ver y structur e o f one' s 79 Se e Heb b (1980) , 107-8 . Heb b cite s wor k o n hierarchie s of neuron s tha t provides physiological support for the idea of higher-order assemblies.

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triangle-detector guarantee s tha t on e will believe any triangl e to be three-sided. Similarly , anyon e wit h a genera l physica l objec t assembly woul d believ e that physica l objects are 'space-occupyin g and sense-stimulating' , t o us e Hebb' s examples , o r observation and trajectory-independent , t o us e example s mentione d earlier . These ar e primitive , ver y genera l belief s abou t th e natur e o f whatever stimulate s th e appropriat e higher-orde r assembly . I cal l them 'intuitiv e beliefs 1. What goe s fo r physica l object s shoul d als o g o fo r sets : th e development o f higher-orde r cell-assemblie s responsive t o particu lar sets gives rise to an even higher-order assembl y corresponding t o the general concept of set. The structure of this general set assembly is the n responsibl e fo r variou s intuitiv e belief s abou t sets , fo r example tha t the y hav e numbe r properties , tha t thos e numbe r properties don' t chang e whe n th e element s ar e move d (barrin g mishap), that they have various subsets, that the y can be combined, and s o on . An d thes e intuition s underlie the mos t basi c axioms of our scientifi c theory of sets.80 I've suggeste d tha t th e ver y structur e o f one' s genera l physica l object assembl y give s on e som e intuitiv e belief s abou t physica l objects, fo r exampl e tha t object s can loo k differen t fro m differen t points o f view or tha t the y don' t ceas e t o b e when w e ceas e t o see them, an d tha t one's genera l set assembly gives one intuitiv e beliefs about sets , fo r example that any two object s can b e collected int o a set. O f course , it' s deceptive t o describ e thes e belief s i n thi s way, because they are not, in fact, linguistic. A child of two, for example , is perfectl y capabl e o f perceivin g particular physical objects , an d thus has or will soon hav e intuitive beliefs abou t physical objects in general, bu t sh e lacks the vocabular y to expres s the m a s I have. In short, suc h belief s ar e accessibl e t o thos e wh o lac k th e linguistic terms, but not t o those who lac k the concept. 81 When a term fo r the concept i s introduced, linguistic expressions o f intuitiv e beliefs will naturally see m obvious , to o obviou s t o benefi t fro m simpl e enumeration. Granting, then , tha t ther e ar e suc h primitive , genera l belief s about physica l objects and abou t sets , w e should inquir e int o thei r epistemological status . It' s alread y bee n note d tha t the y ar e no t 80 Heb b (1980 , esp . pp . 122-4 ) discusse s mathematica l case s a s par t o f hi s theory. 81 Se e Hebb (1980), 108-9 .

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inductively supported , bu t thi s shoul d no t b e take n t o impl y tha t they are, in any sense , infallible. Th e first and mos t obviou s source of potential erro r i s in the uncertai n transitio n fro m intuitiv e belief to linguistic formulation. Because all but th e most severel y disabled eventually attai n th e requisit e cell-assemblies, widespread agree ment abou t a n attempte d linguistic formulation constitute s one of its best claims to correctness . O n thi s account, then, i t is legitimate to suspec t th e clai m of a singl e scientist as to th e intuitivenes s of a certain principle if few others share this opinion.82 A secon d sourc e o f potentia l error i s the distinc t possibility that the intuitiv e belief itsel f i s false. 83 W e migh t b e radicall y mistaken in th e concept s w e form ; perhaps stimulatio n by aspect s o f things causes u s to for m assemblies whic h embod y feature s very differen t from thos e o f th e thing s themselves ; I suppos e i t i s possibl e tha t physical object s do i n fac t disappea r when n o on e i s watching, o r that set s actuall y don't hav e subsets, hard a s it is for u s to imagine such things . Som e intuitive belief s hav e in fac t bee n falsified b y th e progress o f science , fo r exampl e th e belie f that , a t an y give n moment, a physical object is in a certai n location an d movin g a t a certain speed, or that every property determine s a set of things with that property. 84 Thus, in scientifi c contexts, intuitive beliefs mus t be teste d lik e any othe r hypothesis , an d lik e any othe r hypothesis , they can be overthrown. 82 Fo r example , th e belief s ofte n describe d a s Godel' s 'intuitions ' abou t variou s consequences o f th e continuu m hypothesi s {1947/64 , pp . 479-80) ar e no t widel y shared. I shoul d als o not e tha t thes e judgement s of Godel' s ar e mor e esoteri c tha n the primitiv e proposition s I'v e bee n characterizin g a s linguisti c formulation s o f intuitive beliefs . I n man y suc h cases—whe n advance d idea s ar e calle d 'intuitive' — what i s really at wor k i s a hunch or conjectur e based o n mathematica l experience, a theoretical judgemen t of the sort a natural scientist makes when he r familiarit y with the field suggests tha t this , no t that , i s the sor t o f theory likel y t o work . Fo r th e set theoretic realist, thi s counts as a theoretical, rather tha n a n intuitive, justification. I'l l come back to Godel's cas e briefl y i n ch. 4, sect.3, below. 83 O r 'incorrect ' i n some sense . There i s some difficult y wit h classifyin g intuitiv e beliefs a s tru e o r fals e becaus e the y ar e non-linguistic , an d probabl y non propositional. Still , the y can b e classified a s correct o r incorrect , a s tending toward s success o r failur e i n thei r behaviour-guidin g function. (Se e Goldman (1977) , 276. ) I'll equivocat e a bit and continue to us e the words 'true ' and 'false' . 84 Th e se t theoreti c principl e that ever y propert y determine s a se t of thing s that have tha t propert y i s called 'unlimite d comprehension' . I t i s false becaus e it would allow th e formatio n o f Russell' s set , whic h the n lead s t o paradox . I t ha s bee n replaced i n axiomatic se t theory b y Zermelo's Separation Axiom , which assert s onl y that ever y propert y determine s th e se t o f thing s in a previousl y given se t wit h tha t property; withi n a fixe d set , we'r e allowe d t o 'separate ' th e element s wit h tha t property from the rest. See Enderton (1977), 4-6, 20-1.

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This highlight s th e centra l epistemologica l question : doe s th e intuitiveness o f a belie f coun t i n its favour ? Th e exten t t o whic h a claim strikes us as obvious, and th e degree of community agreemen t on thi s degre e o f obviousness , bot h constitut e evidenc e tha t th e claim i s a goo d linguisti c versio n o f a primitiv e intuitiv e belief. We've alread y see n tha t intuitiv e beliefs themselve s ca n b e false , that intuitivenes s i s no t conclusiv e evidence , tha t i t ca n b e outweighed b y opposin g theoretica l evidence , bu t w e no w as k whether i t provide s an y suppor t a t al l fo r th e trut h o f th e claim . There is no doub t tha t suc h evidenc e has bee n counted i n favour of various axiom s i n th e histor y o f se t theory , bu t i s ther e an y rationale for this practice? In lin e wit h th e earlie r discussio n o f Gettier' s proble m an d th e causal theory , w e migh t rephras e ou r questio n as : doe s a tru e intuitive belie f coun t a s knowledge ? Thi s renditio n i s deceptiv e because we'v e alread y granted tha t intuitiv e support i s not enoug h in itsel f t o full y justif y a claim, but I think this formulation can stil l be use d t o focu s th e mai n problem . Recal l tha t Goldman , i n response t o Gettier , adde d a causa l requiremen t to th e traditional , justified-true-belief accoun t o f knowledge , an d tha t thi s require ment i s see n b y man y a s fatall y damagin g t o Platonism . But , ou r intuitive belief s ar e product s o f ou r cell-assemblies , an d th e processes responsibl e fo r generatin g those— a combinatio n o f evolutionary pressure s on ou r ancestor s tha t determin e ou r initia l brain formatio n an d th e su m o f ou r childhoo d interaction s wit h physical object s an d sets—ar e causal . The y ar e suitabl y causa l because it is the corresponding genera l facts abou t th e environment that both exer t the evolutionary pressure and provide the childhoo d interactions. Th e proble m no w come s no t fro m Goldman' s addition, bu t fro m on e of the traditional requirements, namely, that the belie f b e justified : Steve' s belie f i n th e Pairin g Axiom ma y b e intuitive—the axio m ma y see m obviou s t o hi m an d t o us—bu t what if he can offe r n o justificatio n beyon d those feelings ? Turning thi s questio n int o a n outrigh t objectio n t o intuitiv e evidence woul d requir e th e assumptio n tha t a justificatio n mus t always tak e the for m o f a convincin g series of reason s available to the knower . I n contemporar y epistemology , thi s is called 'internalism',85 Th e 'externalist' , b y contrast , insist s tha t a belie f ca n b e 85 Thi s approac h goe s bac k t o Descarte s (1641) . For th e contemporar y debat e between internalist s and externalists , see Bonjour (1980 ) an d Goldma n (1980).

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justified eve n thoug h th e knowe r i s ignoran t o f tha t justification . Consider, fo r example , Steve' s perceptual belie f tha t ther e i s a tree in front o f him. It is generated by a suitable, causal process, bu t let's suppose tha t Stev e has n o knowledg e o f optics , retinas , o r brai n function, tha t h e ca n produc e n o reason s fo r hi s belief . Doe s thi s mean that Steve doesn't know ther e is a tree in front o f him? Or, t o take a more exoti c exampl e fro m Goldman, 86 conside r th e cas e of the professional chicken-sexer. The man looks at a chick and come s to believ e that i t is male, but h e has n o awareness of the process by means o f whic h h e come s t o thi s judgement . Give n tha t h e invariably prove s righ t i n his classifications, we ar e incline d to sa y he know s th e chic k t o b e a male . Bu t he ca n offe r n o reasons , n o arguments, no explicit justifications . I side here with the externalist, rejecting the demand that Stev e be prepared t o justify his belief in the Pairing Axiom. On thi s view, it is enough that the causal process tha t generates the belief be 'reliable', that is, the sort of process that generally leads to true beliefs. This is true i n th e perceptua l case , i n th e chicken-sexin g case, and , i f ou r assumptions ar e correct , i n th e intuitiv e cas e a s well . Thu s th e strength o f Steve's convictio n tha t Pairin g is obviously true, alon g with th e prevalenc e of simila r convictions i n others , support s th e claim tha t Pairin g i s a goo d linguisti c renderin g o f a n intuitiv e belief, an d th e fac t tha t a belie f i s intuitiv e lend s prima-faci e support t o th e clai m that i t i s true. I f Pairing is i n fac t true , an d i f further theoretica l suppor t i s forthcoming—for example , evidence that th e axio m i s consistent, tha t i t produces theorem s o f th e sor t expected, an d s o on—the n i t seem s Steve' s belie f ca n amoun t t o knowledge. One peculiarit y o f intuitiv e evidenc e shoul d b e noted . Th e acquisition o f intuitiv e belief s doesn' t depen d o n an y particula r experience, tha t is , an y sufficientl y ric h cours e o f experienc e wil l produce the required cell-assemblies: In th e cas e o f visua l perception, wha t assemblie s develo p an d becom e th e basis o f perceptio n i s full y dependen t o n a n innat e propert y o f th e organism, th e refle x responsivenes s o f th e ey e muscles , a s wel l a s o n th e innately determine d structur e o f the striate an d peristriate cortex , which— according t o th e theory—i s wha t make s th e formatio n o f cell-assemblie s 86 Goldma n (1975) , 114 . H e als o defend s externalis m i n Goldma n (197 6 and 1979).

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inevitable, give n exposure t o a norma l visual environment . (Heb b (1980) , 105)

So, thoug h experienc e i s neede d t o for m th e concepts , onc e th e concepts ar e i n place , n o furthe r experienc e i s needed t o produc e intuitive beliefs . Thi s mean s tha t i n s o fa r a s intuitiv e belief s ar e supported b y thei r bein g intuitive , that suppor t i s what' s calle d 'impurely a priori 1. Notice , however , tha t i t doesn' t follo w tha t even thes e primitiv e mathematical belief s ar e a priori. Withou t th e corroboration o f suitabl e theoretical supports , n o intuitiv e belief can count as more than mer e conjecture . I shoul d als o emphasiz e tha t th e particula r mathematica l intuitions discusse d her e ar e b y n o mean s th e en d o f th e story . Because o f m y interes t i n se t perception , I'v e concentrate d o n th e intuitions involve d i n th e perceptio n o f smal l set s o f medium-size d physical objects , bu t I don' t wan t t o sugges t tha t thes e discret e intuitions, th e ones that underli e number theory an d the beginnings of set theory, ar e the only intuition s relevant to a complete accoun t of se t theoreti c knowledge . T o begi n with , par t o f perceivin g a physical objec t i s perceiving it as existin g in external space , tha t is , perceiving a boundary betwee n i t and th e space surroundin g it . The concept o f thi s spac e an d th e abilit y t o perceiv e a boundar y o r shape develo p alon g wit h th e concep t o f physical object itself, a s a result o f th e child' s interaction s wit h th e environment. 87 A s i n th e case o f th e triangle-detector , neurologica l correlates o f boundarie s are partl y constitute d b y moto r stimulation s fo r eye-movement s along edges of a figure ; perceived shape s ar e closel y related t o act s of tracing. Thereafter, jus t a s childre n graduall y develo p th e idea s o f inclusion, collection , an d numeratio n leadin g t o th e discret e se t concept, the y als o develo p paralle l ideas base d o n part/whol e an d enclosure relation s rathe r tha n inclusion , proximit y an d distanc e rather tha n collection , measuremen t rathe r tha n number. 88 Thes e developments, beginning , as thei r se t theoreti c counterpart s do , i n perception an d action , lea d eventuall y t o th e perception o f lines— edges, intersection s o f planes , trajectories—a s continuou s struc tures. B y th e ag e o f te n o r twelve , th e chil d expresse s intuitiv e ^ Se e Piaget (1937), or Phillips (1975), chs. 2-$. Se e Piaget and Inhelde r (1948).

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beliefs abou t geometri c figure s tha t revea l a primitiv e notio n o f continuity. These intuition s pla y a rol e i n ou r systemati c thinkin g i n geometry an d analysi s that i s analogous t o th e role of intuition s of discrete collection s i n arithmeti c an d tha t o f intuition s o f physical objects i n natura l science . I n part , fo r example , the y lea d t o th e obviousness o f density , an d eve n th e Dedekind-styl e continuit y axiom.89 Th e abilit y o f se t theoreti c method s t o provid e a consistent renderin g o f ou r confuse d intuitiv e belief s abou t th e relation o f th e lin e t o it s smalles t part s i s on e o f it s greates t achievements, an d one of its strongest supports. 90 Finally, let m e repeat: I' m no t suggestin g tha t all , or eve n most, epistemic support fo r ou r theor y o f set s i s intuitive. In many cases , set theoreti c methodolog y ha s mor e i n commo n wit h th e natura l scientist's hypothesis formation and testing than wit h the caricature of th e mathematicia n writin g dow n a fe w obviou s truth s an d proceeding t o dra w logica l consequences . As the science/mathematics analogy woul d indicate , ou r se t theoreti c hypothese s deman d theoretical o r extrinsi c support , tha t is , support , a s i n natura l science, i n term s o f verifiabl e consequences , lac k o f disconfirma tion, breadt h an d explanator y power , intertheoreti c connections , simplicity, elegance , an d s o on . A preliminar y description o f th e important rol e o f suc h non-intuitive , non-demonstrativ e justifica tions in modern set theory will be sketched in Chapter 4.

4, Godelia n Platonis m In thi s chapter , I'v e sketched th e epistemologica l beginning s of se t theoretic realism , a versio n o f compromis e Platonism . The differ ences betwee n thi s vie w an d Quine/Putna m Platonis m shoul d b e clear enoug h fro m th e discussio n i n the fina l sectio n o f Chapte r 1 , but it s relationship t o Godelia n Platonis m ha s bee n les s explicitl y 89 Densit y i s the clai m that betwee n an y tw o point s there is another. Dedekind' s axiom is a bit mor e complex. If the points on a line are divided into two (non-empty ) groups, an d th e points o f one grou p ar e al l to th e lef t o f all points i n the other , the n there i s either a right-mos t point i n on e group , o r a left-most point i n th e other . I n other words , i f you cu t a lin e i n two , there' s alway s a point a t whic h yo u cu t it . See Dedekind(1872). 90 Se e ch. 3, sect. 1 , below .

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drawn. I'l l conclud e thi s chapte r b y furthe r detailin g m y consider able debt to Godel an d by locating our major disagreement.91 Godel's Platonism rest s on a n analogy betwee n mathematic s an d natural science , a n analogy h e traces bac k t o Russell. 92 Mathemat ical thing s ar e take n t o b e as objective as the object s o f the natura l sciences: '[Sets ] ma y . . . als o b e conceive d a s rea l object s . . . existing independently o f our definition s an d constructions ' {Gode l (1944), 456) . Th e next stag e of the analogy i s epistemological: 'The analogy betwee n mathematic s an d a natura l scienc e . . . compare s the axiom s o f logi c an d mathematic s wit h th e law s o f natur e an d logical evidenc e wit h sens e perceptio n . . .' {Gode l (1944) , 449) ; 'we d o hav e somethin g lik e a perceptio n als o o f th e object s o f se t theory . . .' (Gode l (1947/64) , 483-4) . The problem , fo r Gode l a s for th e se t theoreti c realist , i s t o explicat e thi s perception-lik e connection. To fles h ou t hi s proposal , Gode l consider s th e detail s o f ou r ordinary perceptua l experience and concludes tha t in the case of physical experience, w e form ou r ideas . . . of . .. objects on the basis of something els e which is immediately given, . . . That somethin g besides th e sensation s actuall y i s immediately given follow s . . , fro m th e fact tha t . . . ou r idea s referrin g t o physica l object s contai n constituent s qualitatively different fro m sensation s or mer e combination s o f sensations , e.g., the ide a of object itsel f . . . (Godel (1947/64), 484)

Here the set theoretic realist agrees, an d proposes that the source of this extr a constituent , wha t bridge s th e ga p betwee n retina l stimulation an d perception , i s the neura l cell-assembly . Sh e agrees also tha t thes e conceptua l element s o f perceptua l experienc e 'represent a n aspec t o f objectiv e reality , but , a s oppose d t o th e sensations, thei r presenc e i n u s ma y b e du e t o anothe r kin d o f relationship betwee n ourselve s and reality' (Gode l (1947/64), 484) . For th e se t theoreti c realist , this 'othe r kin d o f relationship ' i s th e complex causa l proces s tha t produce s th e cell-assembly , namely, 91 Ther e ar e others . First , a s wa s emphasize d i n ch . 1 , sect. 4, an y compromis e Platonist will place more weight tha n Gode l doe s on th e indispensabilit y arguments . Second, as remarke d i n sect. 3 above , Godel' s 'intuitions ' cove r mor e esoteri c cases than th e se t theoreti c realist's , case s involvin g wha t 1 would tak e t o b e theoretica l rather tha n intuitiv e evidence . I n fact , Godel' s tex t doesn' t explicitl y coun t thes e a s intuitions, s o o n thi s poin t I ma y b e disagreein g mor e wit h hi s reader s tha n wit h Godel himself. 92 Se e Godel (1944) , 449 . Fo r Russell' s views , se e Russel l (1906) , (1907) , an d (1919), esp. p . 169 .

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the evolutionar y pressure s o n ou r ancestor s an d ou r childhoo d experiences wit h objects , Now wha t o f mathematics ? Gode l goe s o n t o sugges t tha t 'Evidently th e "given " underlyin g mathematics is closely related t o the abstrac t element s containe d i n ou r empirica l ideas ' (Gode l (1947/64), 484) . Agai n the se t theoretic realis t agrees, understand ing th e 'given 1 her e t o mea n th e intuitiv e beliefs tha t underli e th e simplest se t theoretic axioms . Thes e intuitions ar e 'closel y related ' to th e cell-assemblie s responsible for th e 'abstrac t elements 1 of ou r perceptual beliefs : th e objec t an d se t concepts . I n suc h cases , indeed, 'th e axioms forc e themselves upon u s as being true'.93 This sort of obviousness is evidence for the intuitive basis of the axiom in question. Finally, Godel an d I agree tha t no t al l axioms ca n b e justified o n intuitive grounds , tha t th e science/mathematic s analog y shoul d b e extended on e step further, to the level of scientific hypotheses: the axiom s nee d no t necessaril y be eviden t in themselves, bu t rathe r thei r justification lie s (exactly as in physics) i n the fact that they make it possible for thes e 'sense perceptions' t o be deduced . . . (Godel (1944), 449) besides mathematical intuition, there exists another (though only probable) criterion o f the trut h o f mathematica l axioms, namel y their fruitfulnes s i n mathematics and, one may add, possibly also in physics. (Godel (1947/64) , 485).

He eve n supplie s a lis t o f variou s particular form s these justifica tions might take: Success here means fruitfulness i n consequences, i n particular i n 'verifiable ' consequences, i.e. , consequence s demonstrabl e withou t th e ne w axiom , whose proof s wit h th e hel p o f th e ne w axiom , however , ar e considerably simpler an d easie r t o discover , an d mak e i t possible t o contrac t int o on e proof man y differen t proofs . Th e axiom s fo r th e syste m of rea l numbers, rejected b y the intuitionists , have in this sense been verified t o some extent , owing t o th e fac t tha t analytica l number theor y frequentl y allows on e t o prove number-theoretica l theorem s which , i n a mor e cumbersom e way , can subsequentl y b e verifie d b y elementar y methods . (Gode l (1947/64) , 477)

And in a passage quoted earlier: There migh t exis t axiom s s o abundan t i n thei r verifiabl e consequences , 93

Gode l (1947/64), 484.

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shedding s o muc h ligh t upo n a whol e field , an d yieldin g suc h powerfu l methods fo r solvin g problems . . . that, n o matte r whether o r no t the y ar e [intuitive], they would hav e to b e accepted a t leas t i n the same sens e a s any well-established physical theory. (Godel (1947/64) , 477 )

Examples ar e describe d i n Chapte r 4 answerin g t o eac h o f thes e forms of justification an d other s like them. This quic k surve y o f Codel' s writing s surel y suggest s tha t se t theoretic realis m shoul d b e understoo d a s a furthe r developmen t along th e sam e lines ; indee d i t wa s th e passag e abou t th e relationship betwee n 'th e "given" underlying mathematics' and 'the abstract element s containe d i n ou r empirica l ideas' tha t se t m e o n this road in the first place. Still , Chihara has rightly pointed ou t that my quotation s an d reference s ar e highl y selective , sometime s deceptively so. 94 I n particular , I neglected t o sit e Godel's assertio n that 'the objects of transfinite set theory . . . clearly do not belon g to the physical world an d eve n their indirect connection wit h physical experience i s ver y loose , . . " (Gocle l (1947/64) , 483) . And , i n featuring Godel' s clai m tha t 'w e d o hav e somethin g lik e a perception als o o f th e object s of se t theory', I omitted th e qualifie r 'despite thei r remotenes s fro m sens e experience'. 95 Godel' s insist ence on th e traditional Platonisti c characterization of mathematica l objects a s non-spatio-tempora l clearl y disqualifie s se t theoreti c realism as a straightforward developmen t of his thinking. Of course , m y motivatio n fo r bringin g set s int o th e physica l world an d fo r tyin g mathematical intuition so closel y t o ordinar y perception i s naturalism; set theoretic realism seem s to m e the most promising approac h fo r bringin g mathematica l ontolog y an d epistemology int o lin e wit h ou r overal l scientifi c world-view . Godel, b y contrast , no t onl y characterize s set s a s traditiona l Platonistic entities , h e als o goe s o n t o postulat e a real m o f non physical, non-spatial , mentalisti c monads , a s well . I t i s wort h asking why. Consider th e followin g problem : th e huma n min d i s finit e an d 94

Se e Chihar a (1982) , writte n partl y i n repl y t o Madd y (1980) . {Joh n Burges s once warne d m e tha t I wasn't distinguishing clearl y enough betwee n m y ow n idea s and Godel's , bu t thi s wis e counsel fell o n dea f ears.) I n the cours e of arguing that my view i s no t Godel's , Chihar a raise s tw o difficultie s fo r se t theoreti c realism . Th e first, concernin g th e statu s o f th e continuu m hypothesis , wil l b e touche d o n i n ch. 4, sect . 3, below . Th e second , concernin g singletons , wa s th e inspiratio n fo r ch, 5 , seer . 1 . 45 Bot h quotations in this sentence comes fro m Gode l (1947/641 , 483-4.

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the se t theoreti c hierarch y i s infinite . Presumabl y an y contac t between m y min d an d th e iterativ e hierarchy ca n involv e at mos t finitely much of the latter structure . Bu t in that case , I might just as well be related t o any one of a host o f another structure s tha t agre e with th e standar d hierarch y onl y o n th e minuscul e finite portion I've managed to grasp. Accordin g to Benacerraf , Gode l fel t tha t his monads wer e immun e t o thi s finiteness problem, tha t the y coul d somehow gai n unambiguou s access to the ful l hierarchy. 96 Thus his monadology woul d succee d where naturalism purportedly fails . The question o f how ou r finite minds make contact with th e infinit e is give n a ver y general formulatio n b y Kripke in hi s interpretatio n of Ludwi g Wittgenstein. 97 In it s strongest form , this argument, th e so-called 'rule-followin g argument' , applie s not onl y to th e cas e of the mathematica l infinite , bu t als o t o an y rul e wit h a n indefinit e number o f potentia l applications . Consider , fo r example , ou r ordinary us e of the word 'triangle' . Kripke' s Wittgenstein 98 argue s that al l our training, our past usages , our mental images, our state d and unstate d intentions , ou r associations , an d s o on , canno t predetermine whethe r i t is correct or incorrec t t o call a given figure a triangle . This i s because, for example, thes e constraints can al l be perversely interprete d t o confor m t o th e se t o f thing s tha t ar e triangular u p t o no w o r squar e afte r no w a s completely as they do to th e set of triangular things. So nothing we've associated with th e word 'triangle ' ca n predetermine whethe r 'triangle ' should no w be applied t o thi s figure A rathe r tha n tha t D . The argumen t doesn' t depend o n the fac t tha t ther e ar e indefinitel y man y triangles; it can also b e applie d t o sho w tha t nothin g predetermine s whethe r thi s person o r tha t i s correctl y referre d t o a s 'Kripke' , becaus e everything I'v e associate d wit h th e nam e 'Kripke ' ca n no w b e interpreted a s applyin g a s wel l t o a bein g consistin g o f Kripke' s space-time worm u p till now and Putnam's afterwards. These conclusions ma y seem outrageous, an d indee d they should, but I will leave the intereste d reader to mor e complet e exposition s 96 Benacerra f mad e thi s observatio n i n a n ora l presentatio n tha t becam e Benacerraf (1985) . A s far as I know, th e details of Go'del's monadology an d ho w i t is to overcome this problem ar e as scant as his account of intuition . 97 Se e Kripke (1982) . Fo r Wittgenstein' s views , se e Wittgenstein (1953 ) an d th e posthumous (1978) . 98 I pu t i t thi s wa y becaus e ther e i s some debat e ove r whethe r o r no t Kripke' s argument i s really Wittgenstein's .

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of thi s fascinating paradox . Al l I want t o d o her e is to indicate—i n light o f Godel' s motivations—wha t I tak e t o b e th e mai n ingredients of a naturalistic solution." It begins, of course, wit h th e idea that m y linguistic training sets up a neural connection betwee n the word 'triangle ' an d my triangle-detector. This i s not a complet e reply, however, becaus e of the inevitable physical limitations on my neural processors . Th e rang e o f th e ter m 'triangle ' canno t b e identified wit h th e se t of thing s that stimulat e my detector becaus e my detecto r i s sometimes wrong, an d eve n if it weren't, i t couldn' t respond t o triangles too small, too large, or too far away. At thi s point , th e realis t mus t appea l t o th e objectiv e fac t tha t triangles are mor e lik e one anothe r tha n lik e squares, that is , to th e fact tha t ther e i s an objectiv e difference, note d i n the second sectio n of Chapte r 1 , between rando m an d natura l collections.100 Our bes t theory o f wha t m y detecto r respond s t o involve s not th e sceptic' s random collections , bu t thos e collection s scienc e take s t o b e th e natural ones . Triangles , then , ar e thos e thing s belongin g t o th e natural collection tha t include s most of the things that stimulate my detector. Thi s distinctio n betwee n natura l an d rando m ma y b e difficult t o pin down ; i t may b e a matter o f degree rather than al l or nothing, i t ma y ultimatel y require a n ontologica l commitmen t t o properties o r universals , but i t i s a crucia l part o f th e naturalist' s world view , and i t serves to rul e out th e gerrymandered interpretations o n which the sceptic's arguments depend.101 I have tried i n this section t o clarif y th e natur e and th e origi n of m y central disagreemen t wit h Godel . Thi s shoul d no t b e allowe d t o obscure m y obvious debt to his thinking, both i n the formulation of mathematical realis m i n Chapte r 1 abov e an d i n th e accoun t o f mathematical perceptio n an d intuitio n offere d i n thi s chapter . Hi s influence wil l agai n be prominent i n Chapte r 4 , whe n 1 take up th e subject o f theoretical evidenc e and justification , bu t befor e turnin g to tha t mor e esoteri c epistemologica l study , I wan t t o touc h o n another Benacerrafia n worr y abou t Platonism , a n ontologica l on e this time. 99 Fo r a fulle r accoun t o f bot h th e parado x an d it s naturalisti c solution , se e Maddy (1984b) . 100 An d between natural and unnatura l individuals, to deal with the 'Kripke' case. 101 Putnam' s (1977) an d (1980 ) present a puzzle in som e ways similar to Kripke' s version o f th e rule-followin g argument . I n repl y t o Putnam , Merril l (1980 ) an d Lewis (1983 ; 1984 ) us e natura l kind s muc h a s Madd y (1984b ) doe s i n repl y t o Wittgenstein.

3

NUMBERS 1. Wha t numbers could not be The widel y cite d epistemologica l challeng e t o Platonis m ha s bee n my focu s s o far , an d I wil l retur n t o epistemolog y i n th e nex t chapter t o tak e u p a n equall y important , thoug h les s talked-o f aspect o f mathematical knowledge. Bu t befor e doin g that , I would be negligen t no t t o discus s a celebrate d ontologica l challeng e t o Platonism which some observers consider nearly as daunting as the epistemological.1 Th e se t theoreti c realist' s answe r i s implici t i n what ha s alread y bee n said, bu t i t will take som e effor t t o se e this and to draw out the details. Let m e fram e th e proble m wit h anothe r touc h o f simplifie d history. The emergenc e o f se t theory a s a foundationa l theory ca n be trace d t o tw o separat e line s o f developmen t tha t eventuall y converged.2 Th e firs t o f thes e bega n wit h concer n ove r th e foundations o f th e calculus. 3 The much - and deservedly-criticized infinitesimals were replaced by Weierstrass's theory of limits, which in tur n depende d o n a theor y o f th e rea l number s propose d i n different form s b y Canto r an d Dedekind. 4 Thes e account s ar e thoroughly se t theoretic; Dedekind's , fo r example, makes essential use of infinit e set s of rational numbers. By these means, he clarifie s the notio n o f continuity, defines th e rea l numbers, and prove s that they are , i n fact , continuous . Thus , b y identifyin g rea l number s with certai n set s (calle d 'Dedekind cuts') , Dedekind obtain s a rich 1 Amon g thos e bothere d ar e Benacerra f (1965), Jubien (1977) , Kitche r (1978 ; 1983, p . 104), Fiel d (1980, p. 126, n . 66; 1989, pp . 20-5), and Resnik (1981, p. 529). 2 I'l l concentrat e her e o n tw o foundationa l motivations. Cantor' s mor e purel y mathematical interests will be discussed in the next chapter. 3 Thi s wa s touche d o n i n ch . 1, sect . 4, above . Th e resultin g worry abou t th e infinite produce d th e thre e grea t school s o f though t i n th e philosoph y o f mathematics that flourished in the early decades of our century. 4 Cantor' s theor y appear s i n Canto r (1872) . Se e Daube n (1979) , ch . 2, fo r discussion. Dedekind' s account , develope d somewha t earlier , firs t appeare d i n Dedekind (1872) . Enderto n (1977) , ch . 5, present s Dedekind' s mor e elegan t formulation.

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and explanator y theor y of their nature and behaviou r that puts the calculus an d highe r analysi s o n a consisten t foundation . T o thi s day, set theory provide s our bes t account o f mathematical analysis, which i n tur n play s a centra l rol e i n ou r mos t successfu l physica l theories. Thi s achievemen t i s on e o f th e stronges t theoretica l supports fo r th e mathematica l theory o f sets ; i t plays a n indispens able role in our bes t theory of the world. Meanwhile, fro m a mor e philosophica l orientation , Freg e wa s concerned t o provid e a foundatio n fo r ordinar y arithmetic. 5 H e was scandalize d b y th e lac k o f understanding , eve n amon g mathematicians, o f th e fundamenta l concepts o f thei r subject , i n particular, th e concep t o f a natura l number . Frege' s ai m wa s t o show tha t arithmeti c is in fac t a branc h o f logic, bu t i n doing so he made us e o f extension s o f concepts , tha t is , ofte n infinit e collections. Thi s projec t failed , a s ha s bee n noted, 6 bu t it s cor e has bee n incorporate d int o moder n se t theory . A s wit h th e reals , the natura l number s ar e identifie d wit h certai n sets , an d al l their basic properties, onc e assumed as axioms, becom e provable , explicable.7 Thu s moder n se t theor y als o provide s ou r bes t accoun t of th e natura l numbers , anothe r ke y ingredien t i n ou r overal l theorizing. Here w e hav e a scientifi c succes s story . I t depends , however , o n identifying number s with sets . I n the openin g pages o f Chapte r 1 , I suggested tha t th e philosophy an d foundation s ot se t theory ar e of interest regardles s o f whethe r o r no t al l mathematica l object s ar e properly take n t o b e sets , bu t i n thi s case , wher e th e theoretica l justification o f th e subject , indee d th e ver y motivatio n fo r th e subject, depend s s o centrall y o n thes e particula r identifications , philosophical question s abou t th e natur e an d propriet y o f se t theoretic reductio n canno t b e put aside . I t i s this question—of th e relationship between set s an d numbers—tha t I want t o conside r i n this chapter . 5

Se e Frege (1884), introd, Se e ch. 1 , sect. 4, above . I don't mean to suggest that all proofs are also explanations, but some clearly are. Fo r example , th e se t theoreti c accoun t o f natura l number s tell s u s wh y multiplication i s commutative : becaus e th e cros s produc t A X B i s equinumerou s with B x A . (Se e Drake (1974) , 52. ) Thi s sam e fac t ca n b e prove d fro m th e Pean o axioms, but th e proof require s a series of clever lemma s and shed s little light on wh y the theorem i s true. (Se e Enderton (1977) , 81-2. ) Sreiner (1978 ) make s a n effor t t o characterize mathematical explanation. 6

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To se e the sourc e o f th e problem , let's retur n fo r th e momen t t o Frege an d reconside r hi s centra l ontologica l query : wha t i s a number? Take a numerica l statement lik e 'ther e ar e tw o boy s playin g in the garden' . I'v e already reviewed 8 Frege' s observatio n tha t th e subject o f thi s numerica l ascriptio n canno t b e th e mer e physical stuff tha t make s u p th e boys , becaus e that physica l stuff coul d b e divided into units in various different ways—two boys, twenty boyparts (heads , torsos, arms , hands , legs , and feet) , million s of cells, even mor e molecules , stil l mor e atoms , an d s o on . Wha t ha s changed, Freg e argues , fro m th e occasio n whe n w e judg e tha t th e stuff i s two boy s to tha t when we judge it to b e millions of atoms is the substitutio n of the concep t 'ato m in this mass of physical stuff for th e concept 'boy in this mass of physical stuff. This leads him to the conclusio n tha t 'th e conten t o f a statemen t o f numbe r i s a n assertion about a concept'.9 But to identif y th e conten t o f a statement of number is not ye t to identify th e numbe r itself. Frege' s analysi s suggests that numbe r is something variou s concept s ca n share , fo r exampl e 'bo y i n th e garden' and 'han d on my keyboard'. In Fregean terms, this is to say that two i s a concept under which other concepts can fall, a secondorder concept , a s it were. But, howeve r straightforward this might sound, Frege insists, for reasons I'll consider later, tha t a number is a thing , a 'self-subsisten t object', 10 rathe r tha n a concept . So , instead of saying that two is the concept that applies to any concept equinumerous 11 with 'bo y i n the garden', he identifie s th e number with the extension o f that concept, tha t is , with the collection of all concepts equinumerous with 'boy in the garden'. The troubl e with thi s account i s that tw o turn s ou t t o b e a very large collection indeed . In a set theory like Frege's, with th e (false ) principle o f unlimite d comprehension, we coul d for m this set, but alas, w e coul d als o for m th e paradoxica l Russel l set. When thi s inconsistent naiv e se t theor y i s replace d b y th e contemporar y 8

I n ch. 2, sect. 2, above. Freg e (1884), § 46. 10 Freg e (1884), § 55. 1 ' Thi s sound s circular , definin g th e number two i n terms of equality in number, but i n fact , Frege define s 'equinumerous ' withou t referenc e t o numbers . Tw o concepts ar e equinumerou s i f ther e i s a one-to-on e correspondenc e betwee n th e things fallin g unde r th e firs t an d th e thing s fallin g unde r th e second . Se e Frege (1884), § 63, or Enderton (1977) , 128-9. 9

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axiomatic version based on the iterative conception, two , a s defined here, n o longe r exists . T o se e this, notic e tha t ne w two-membere d sets are formed a t each stage , so there is no stage at which the set of all two-membered set s is formed. Thus, i n contemporar y se t theory , tw o i s customarily identifie d not wit h the (non-existent ) collectio n o f all two-membered sets , bu t with som e particularly convenient example of a two-membered set . Standardly, the natural numbers are taken to b e the set of finite von Neumann ordinals , tha t is , 0 fo r 0, {0} for 1, (0, {0} } for two , {0, {0} , (0 , {0}} } fo r 3 , an d s o on. 12 Other , similar , identifications woul d d o as well, fo r example Zermelo' s 0 fo r 0, {0} for 1 , { {0} } for 2, { { {0} } } for 3, and so on.13 And it is this fact that generates an ontological question about numbers. Once again , the version of this problem most exhaustively cited an d discussed in the contemporary philosophical literature derives from a pape r b y Pau l Benacerraf , this tim e 'Wha t number s coul d no t be'.14 Benacerra f ask s u s t o conside r th e educatio n o f tw o hypothetical youn g children, Ernie (for Ernst Zermelo) an d Johnn y (for John vo n Neumann). Ernie an d Johnn y ar e bot h brough t u p o n se t theory. Whe n th e time come s t o lear n arithmetic , Erni e is told, to hi s delight, tha t he already know s abou t th e numbers; they ar e 0 (calle d 'zero'), {0 } (called 'one') , (0 , {0} } (calle d 'two') , an d s o on . Hi s teacher s define th e operation s o f additio n an d multiplicatio n o n thes e sets , and whe n al l th e relabellin g i s done , Erni e count s an d doe s arithmetic jus t lik e hi s schoolmates . Johnny' s stor y i s exactl y th e same, excep t tha t h e i s tol d tha t th e Zermel o ordinal s ar e th e numbers. He als o counts and does arithmeti c in agreement with his schoolmates, an d wit h Ernie . The boy s enjoy doin g sums together , learning about primes , searching for perfect numbers, and so on. But Erni e and Johnny ar e curious little boys ; the y want t o kno w everything they can abou t thes e wonderful things, the numbers . In 12 Se e e.g . Enderto n (1977) , 68 . Vo n Neumann' s proposal i s containe d i n hi s (1923), 347 . 0 Se e Zermelo (19086) , 205 . I n fact , ther e ar e reasons wh y Zermelo's versio n isn't as good as von Neumann's. Fo r example, von Neumann's accoun t work s just as well fo r infinit e number s as for finite, and it s 'less than' relation i s extremely simple: membership. 14 Benacerra f (1965). See also Parson s (1965). The argumen t is further develope d bvKitcher(1978).

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the process, Erni e discovers the surprising fact that on e is a member of three . In fact , h e generalizes , i f n i s bigger tha n m, then m i s a member o f n . Fille d wit h enthusiasm , h e bring s thi s fac t t o th e attention o f hi s favourit e playmate . Bu t here , sadly , th e buddin g mathematical collaboratio n break s down . Johnny no t onl y fail s t o share Ernie' s enthusiasm , h e declare s th e prize d theore m t o b e outright false! He won't even admit that three has three members! According to Benacerraf , the mora l o f this sa d story—or on e of them—goes like this: i f numbers are sets , the n the y must b e som e particular sets . Any choice o f particular set s will exhibit propertie s that g o beyon d wha t ordinar y arithmeti c tell s u s abou t th e numbers. (Ordinar y arithmeti c i s mut e o n th e subject s Erni e an d Johnny debate . Their classmate s ar e puzzled by the very questions these boys take s o much t o heart.) If one of these particular choice s is th e correc t one , that is , i f on e sequenc e o f set s reall y i s th e numbers, the n ther e ough t t o b e argument s tha t tel l u s whic h sequence that is. 15 (This doesn't see m to be the sort of question that requires som e further , dee p numbe r theoreti c theorem. ) But there are no such arguments. Therefore, numbers are not sets . Friends o f number s migh t b e prepare d t o fal l bac k o n th e position tha t whil e number s aren' t sets , stil l the y ar e object s o f some othe r kind . Suppose , then , tha t we'v e identifie d som e sequence o f object s suitable fo r countin g an d arithmetic , an d w e claim that th e fourt h o f these (w e started fro m zero) is the number three. Benacerra f argues tha t thi s objec t play s the rol e o f thre e i n our sequenc e b y virtue of its relations t o th e othe r member s of th e sequence, bu t i f i t i s t o b e single d out , independentl y o f th e sequence, a s thi s objec t o r that , i t mus t hav e som e additiona l properties. And , he continues , thes e additiona l propertie s wil l b e superfluous t o the object's numerical functioning, in the same sense that th e propertie s Erni e an d Johnn y debate d wer e superfluous , which lead s t o th e question : wh y shoul d thre e hav e thes e superfluous propertie s an d no t som e others ? If this object reall y is three, ther e shoul d b e argument s t o sho w tha t thes e superfluous properties ar e th e correc t ones , bu t ther e ar e n o suc h arguments . Therefore, numbers are not objects at all. 15 Th e metaphysicia n wants more here than the previousl y remarke d arguments from convenience . The fac t tha t vo n Neuman n ordinals are mor e convenient than the Zermelo ordinals , and henc e standard in contemporary set theory, doesn't giv e us any further reaso n to think that they really are the numbers.

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This secon d conclusio n i s less firmly supported tha n th e first; the second argumen t leave s room fo r the position tha t number s are the sort o f object s whos e non-superfluou s propertie s ar e al l th e properties the y have. 16 I f thes e non-set s ar e connecte d closel y enough wit h sets , perhap s eve n th e explanator y virtue s of th e se t theoretic reductio n ca n b e preserve d withou t th e actua l identifi cation o f number s wit h sets . Thus , fo r example , Canto r suggest s that natura l number s ar e separat e entitie s 'abstracted ' fro m equinumerous sets , an d Dedekin d tha t th e real s ar e 'associated ' with th e correspondin g cuts . Thi s sor t o f mov e obviousl y flaunt s ontological economy—it' s inefficien t t o overburde n ou r theor y with mor e thing s tha n w e nee d t o mak e i t wor k effectively—bu t worse than that , i t requires an account of the sort of 'abstraction' or 'association' involved. Neither Cantor no r Dedekind provid e this.17 Of course , i f w e tak e Benacerraf' s argumen t tha t natura l numbers aren' t set s t o b e persuasive , a s I thin k w e should , a n analogous lin e of though t show s tha t th e real s can't b e set s either . We coul d tel l a stor y o f Georgi e (fo r Georg Cantor ) an d Ric h (for Richard Dedekind}, one of whom learn s that the reals are Dedekind cuts an d th e othe r o f who m tha t the y ar e Cantor' s fundamenta l sequences.18 Th e res t o f th e stor y follow s a s before , an d th e conclusion: rea l numbers aren't sets. 19 If number s aren' t set s afte r all , th e stor y o f scientifi c succes s recited a t th e beginnin g of thi s section i s called int o question . I f its illumination o f the theorie s o f natural and rea l numbers is to coun t as evidenc e fo r th e theor y o f sets , w e nee d t o understan d th e tru e nature o f the ontologica l relationship between number s and sets . If not identity, then what?

2. Number s a s properties Assuming that number s aren't sets, the set theoretic realist faces the 16 Steine r (1975tf) , 88-92 , suggests a move of this sort in his reply t o Benacerraf , but onl y i n a n epistemologica l sense : 'w e accep t mathematica l objects , contr a Benacerraf, bu t w e agre e tha t th e onl y thing s t o kno w about thes e object s o f an y value are their relationships with other things' (p. 134). 17 Fo r a scorchin g attac k o n Cantor' s notion o f abstraction , se e Freg e (1979) , 68-71. 18 Thes e depend on an idea that goes back to Cauchy. See Enderton (1977), 112. 19 Th e sam e goe s fo r othe r se t theoreti c reductions . Se e e.g. Kitche r (1978 ) o n ordered pairs . Th e proble m fo r ordere d pair s carrie s ove r t o functions , understoo d as sets of ordered pairs, and s o on.

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prospect o f addin g a ne w typ e o f entit y t o he r ontolog y an d a n extra epicycl e to he r epistemology . And , to preserv e th e explanat ory forc e o f th e standar d se t theoreti c reduction , sh e mus t als o describe a relationshi p betwee n set s an d number s tha t make s th e behaviour of , fo r example , th e vo n Neuman n ordinal s someho w relevant t o ou r understandin g o f numbers . Finally , th e effort s t o naturalize th e epistemolog y o f se t theor y wil l b e waste d i f th e account of numbers isn't also naturalistic. Let's begi n the n wit h th e suggestio n tha t th e epistemolog y fo r numbers shoul d b e a s simila r a s possible t o tha t give n fo r sets . I n that case , numbers must also be located i n space-time. Where, then , is the number ten? The eas y answe r is : te n i s located wher e th e se t of m y fingers is located, i n motion ove r the key s of my word processor . Bu t if this is right, the n te n i s als o locate d wher e th e startin g line-u p o f an y American Leagu e basebal l tea m i s located, an d o n th e Times best seller list , and man y other places . No w a set of physical objects can have a discontinuou s location—th e se t o f Ange l startin g baseball players i s located i n lef t field , righ t field , secon d base , an d eve n i n the dugout , wit h th e designate d hitter—bu t onl y part o f th e se t is located i n each of these places, while the number ten is fully presen t in each and ever y ten-element set. By traditional criteria, this makes the se t o f basebal l player s a particula r an d th e numbe r te n a universal.20 In term s o f th e science/mathematic s analogy , then , th e ide a i s this: se t theor y i s th e stud y of set s an d thei r properties , o f whic h number i s one, just a s physics i s the stud y o f physical objects an d their properties , o f which lengt h (fo r example) i s one. To se e ho w this works i n a bi t mor e detail , conside r th e forma l feature s o f th e quantity 'length'. 21 First, object s wit h a give n physica l quantity , lik e length , ar e comparable; the y for m a linea r orderin g wit h respec t t o tha t quantity. Fo r example, ther e is a simple linear ordering o f mediumsized, easily movable objects tha t goe s lik e this: A is shorter than B if on e en d o f B extends beyon d the en d o f A when they are lai d side by sid e wit h th e othe r end s coincident . A method tha t als o work s for stationar y object s migh t g o lik e this : A i s shorte r tha n B i f a string that lie s straight wit h on e en d a t eac h en d o f A won't reac h both end s of B. Obviously, mor e sophisticate d test s will extend th e 20 21

Se e ch. 1 , sect. 2 , above. Her e I follow Elli s (1966) .

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linear ordering further . At this point, a scale of measurement can be assigned, a s lon g a s i t agree s wit h th e establishe d linea r ordering , that is , as long a s the orderin g o f the numerica l assignments agrees with th e linear ordering . Thi s is not difficult : compariso n with an y fair rule r measure s length i n yards . And finally, the sam e quantit y can b e measure d o n differen t scales ; lengt h i s detecte d b y metr e sticks as well. Now compar e th e cas e o f numbe r properties . Set s o f easil y movable objects are directly comparable with respect t o number; A is les s numerou s tha n B i f eac h membe r o f A ca n b e se t besid e a unique member o f B in such a way tha t ther e ar e element s o f B lef t over. Th e linea r orderin g o f simpl e set s thi s produce s ca n b e extended t o al l set s usin g th e mathematica l ide a o f a one-to-on e correspondence: A i s less numerous tha t B if there i s a one-to-on e correspondence betwee n th e member s o f A and a proper subse t of the member s o f B . Scal e ca n b e assigned , fo r example , b y comparison wit h the English number words, that is , by counting i n English.22 Or, more elaborately, in terms of one-to-one correspond ence with th e set of von Neumann ordinals. The onl y disanalog y i s that ther e i s no roo m fo r measurin g sets on differen t scales , because there is no room for an arbitrary choice of unit : sets come with thei r elements alread y individuated. This is how set s avoi d Frege' s objectio n t o physica l masse s a s bearer s o f number properties . Whil e i t i s arbitrar y t o assig n tw o t o th e physical mas s tha t make s u p th e boy s playin g in th e yard , ther e is no arbitrarines s i n assignin g two t o th e se t o f boy s playin g in th e yard. 23 This suggestion—tha t numbers are properties of sets, analogou s to physica l properties , an d i n particular , t o physica l quantities— meets ou r epistemologica l desiderata wit h admirabl e economy; th e naturalistic epistemology previousl y described fo r se t theor y need s no elaboration . Jus t a s th e perceptio n o f physica l objects includes the perception o f their properties, s o the perception o f sets doe s th e same. Indeed , ou r accoun t o f se t perceptio n develope d fro m th e observation tha t w e perceiv e thei r numbe r properties ; al l tha t i s new her e i s the furthe r clai m that thes e numbe r propertie s ar e th e numbers. 22 Compar e Benacerra f (1965) , 292 : 'Th e centra l ide a i s tha t [th e sequenc e o f .number words] is a sort of yardstick which w e use to measur e sets.' 23 Yourgra u (1985 ) disagrees, but fo r a cogent reply, see Menzel (1988) .

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Knowledge o f number s i s knowledg e o f sets , becaus e numbers are propertie s o f sets . Conversely , knowledg e o f set s presuppose s knowledge o f number ; fo r example , Piaget' s studie s indicat e tha t subset relation s canno t b e properl y perceive d befor e numbe r properties.24 From thi s perspective, arithmeti c is part, perhap s th e most importan t part , o f th e theor y o f hereditaril y finit e sets. 25 Neither arithmeti c nor thi s finite set theory enjoy s epistemologica l priority; th e two theories aris e together. Arguments that arithmeti c should not be reduced to set theory because set theory is less certain than arithmeti c mis s the fac t tha t i t make s little sens e t o separat e the epistemologica l basi s o f arithmeti c fro m tha t o f finit e se t theory.26 Highe r se t theor y i s admittedl y les s certai n tha n th e theory of hereditarily finite sets, but this is irrelevant. Furthermore, th e significanc e o f th e se t theoreti c reduction s i s now clear . Th e vo n Neuman n ordinal s ar e nothin g mor e tha n a measuring rod agains t whic h sets ar e compared fo r numerical size. We lear n abou t number s b y learnin g abou t th e vo n Neuman n ordinals becaus e the y for m a canonica l sequenc e tha t exemplifie s the propertie s tha t number s are . The choic e betwee n th e vo n Neumann an d th e Zermel o ordinal s i s n o mor e tha n th e choic e between tw o differen t ruler s tha t bot h measur e i n metres . Th e debate between Ernie and Johnny i s like an argument over whether an inch is wooden o r metal . Some versio n o f th e identificatio n of th e natura l numbers wit h properties o f set s i s considered b y bot h Freg e and Benacerraf , and both writer s ultimatel y rejec t position s o f thi s sort . Benacerra f s conclusion is drawn with little strong conviction—he says only that '"seventeen" need no t b e considere d a predicat e of [sets]' 27—but 24

See Piaget and Szemiriska (1941), ch. 7. A se t i s hereditaril y finit e i f i t i s finite , an d it s member s ar e finite , an d th e members o f it s members ar e finite , an d s o on , (Se e Enderto n (1977) , 256. ) I t might seem tha t arithmeti c onl y treat s finit e set s o f physica l objects, but th e mos t natura l way o f understanding such example s a s 'here are thre e pairs of shoes' involve s a set of thre e two-membered sets . The restriction t o hereditaril y finite sets, rather tha n t o finite sets simpliciter, rule s ou t suc h things a s the singleto n containin g th e se t of all von Neumann ordinals , hardly the sort of thing covered b y ordinary arithmetic. 26 Notice , fo r example , tha t recursio n theor y ca n b e develope d wit h equa l naturalness fro m number s o r fro m hereditaril y finite sets. A n argumen t o f th e sor t considered i n th e tex t appear s i n Steine r (1975a) , ch . 2. Sentiment s related t o m y own are voiced by Parsons (1965) , 173 . 27 Benacerra f (1965), 284 . 25

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Frege's i s harde r t o evaluate . I wil l briefl y conside r a fe w o f thei r reasons. The most conspicuously cite d ground s fo r their opposition to th e property vie w are grammatical : compariso n o f th e rol e o f number words wit h ordinary adjective s and predicates, 28 the use of number words wit h th e definit e article , thei r immunit y t o pluralization, 29 and s o on. O f course , gramma r i s no infallibl e guid e to th e actua l structure of the world, s o such evidence must be taken wit h a grain of salt . Freg e admit s a s muc h whe n h e dismisse s th e contrar y grammatical evidence: our concer n her e i s to arriv e at a concept o f number usable for the purpose s of science ; w e shoul d not , therefore , b e deterre d b y th e fac t tha t i n th e language of everyday life numbe r appear s als o i n attributive constructions . (Frege (1884), §57)

Benacerraf als o ignore s stron g grammatica l evidence, th e evidenc e that numbe r word s ar e names , whe n h e late r denie s tha t number s are objects. 30 In fact , Frege's commitmen t t o the objecthoo d o f numbers seem s to wave r a t th e ver y momen t whe n h e presents hi s own definition . He writes: 31 'th e numbe r whic h belong s t o th e concep t F i s th e extension o f the concept "[equinumerou s with] the concept F" '. To the wor d 'extension 1 h e append s a footnote : ' I believ e tha t fo r "extension o f the concept " w e could write simpl y "concept" / Thi s surely sound s lik e a suggestio n tha t number s coul d b e concept s rather tha n objects . Freg e immediatel y reject s hi s proposal : 'Bu t this woul d b e ope n t o th e tw o objection s . . .'. Th e firs t i s th e inconclusive grammatical considerations. Th e second, considerabl y more interesting , i s 'tha t concept s ca n hav e identica l extension s without themselve s coinciding'. This suggest s the fact tha t concept s are intensiona l rathe r tha n extensional . But whateve r th e forc e o f 28

Benacerra f (1965), 282-4; Frege (1884), § 57. Freg e (1884) , §38 . I n defence o f his own view , Frege admits tha t i t leads t o some unusua l ways of speaking—e.g. that a number is 'wider or less wide than th e extension o f some othe r concept'—but insists that there i s nothing 'to prevent us speaking i n this way' (1884, §69). Benacerra f als o speak s somewha t oddl y whe n h e say s tha t 'any objec t [including Laurence Olivier ] can play th e role of 3' (1965 , p. 291) . Unles s it is argued that thes e odditie s ar e semanti c rathe r tha n grammatical , a distinctio n notoriously hard t o draw , thes e ar e furthe r example s o f bot h writers ' willingnes s t o ignor e grammatical evidenc e when nee d be. 31 Al l the quotations in this paragraph come fro m Freg e (1884), §68. 29 30

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this objection , Frege doe s no t regar d i t as decisive. He concludes: ' I am, as it happens, convince d tha t both thes e objections can be met; but t o d o thi s woul d tak e u s to o fa r afiel d fo r presen t purposes. ' Thus the proposal i s rejected on grounds of convenience rather tha n principle. It may seem obvious that Frege is leaving open th e possibility that numbers ar e concepts , an d dependin g o n th e relationship betwee n concepts an d properties , perhap s th e possibility of a property vie w as well , bu t readin g Freg e i s never a simpl e matter . H e holds , fo r example, tha t 'th e concep t horse ' mus t refe r t o a n object , rathe r than a concept, 32 s o it can b e argued tha t thi s tantalizing footnot e suggests n o mor e tha n tha t 'th e concep t "equinumerou s wit h F" ' actually refer s t o th e extensio n o f the concep t 'equinumerou s with F'. On thi s reading, 'concept' could b e substituted for 'extensio n of the concept' in his original definition becaus e the two actuall y refer to the same thing, which isn't , b y the way, a concept a t all. And the reason th e concept s 'huma n being ' an d 'featherles s biped ' aren' t identical isn' t tha t thes e expression s hav e differen t meaning s o r stand fo r differen t properties , bu t tha t identity , and difference , ar e relations between objects, not between concepts. 33 I hav e nothin g t o contribut e t o th e debat e ove r wha t Freg e actually had i n mind here, but leavin g Frege himself behind , I'd lik e to examine the idea that numbers might be concepts an d the exten t to whic h th e intensionalit y of concept s stand s i n the wa y o f such a view. T o se e what's a t stak e here , conside r a simpl e arithmetical identity: 2 = S(S(0)). 3 4 I f Frege' s definitio n ha d rea d 'concept ' i n place of 'extension of concept', 2 would b e the concept 'equinumer ous with th e concept "identica l with 0 or 1"' , and the successor of the successor of 0 would be : the concep t 'equinumerou s wit h th e concep t "membe r o f th e serie s o f natural numbers ending with S(0} " '

which is the concep t 'equinumerou s wit h th e concep t "membe r o f th e serie s o f natural numbers ending with th e concept 'equinumerou s with th e serie s of natural numbers ending with 0'"'. 32

Se e Frege (1892^), 45. Thi s lin e o f interpretatio n appear s i n Resni k (1965) . Fo r othe r relevan t discussions of Frege, see Hodes (1984) and Luc e (1988) . 34 'S 1 here means 'the successor of . 33

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Now there' s n o doub t tha t th e extensio n o f 2 , s o defined , i s th e same a s th e extensio n o f S(S(0)) , s o defined—eac h involve s being equinumerous wit h a concep t unde r whic h tw o thing s fall—but if coextensive concept s ca n nevertheles s differ , ou r simpl e arithmeti cal identit y is in jeopardy. Thi s i s a clea r difficult y fo r th e concep t view. Now let' s se e how thi s works out o n the property vie w proposed here. If 2 is the numbe r property of , for example, the von Neuman n ordinal (0 , {0}} , the n t o hav e th e propert y 2 i s to b e equinumerous wit h thi s set . Whe n successo r i s define d fo r vo n Neuman n ordinals, S(S(0) ) turn s out to be the same set as {0, {0} } itself, so being equinumerous wit h eithe r one i s the sam e as bein g equinumerous wit h th e other. 35 Th e sam e goe s fo r Zermelo' s version . Bu t unless we're able to affirm , fo r example, th e identity the property 'equinumerou s with {0, {0}, (0, {0} } }' =5

the property 'equinumerou s with {0, {0} { {0} } }' a ne w versio n o f th e ol d Benacerra f proble m wil l arise : whic h o f these i s 3? Which propertie s reall y ar e th e numbers ? Those defined in term s of equinumerosity wit h particula r vo n Neuman n ordinal s or thos e define d in terms of equinumerosity with initial segments of the Zermel o ordinals ? Thi s i s th e analogou s proble m fo r th e property view . Now let' s compar e th e tw o problems . A Fregea n concep t i s closely connecte d t o a predicate ; indeed , i t i s th e referen t o f a predicate. Leavin g Frege's view s o n identit y aside , let u s ask whe n two predicate s ar e th e same . W e migh t insis t this i s only s o whe n they are typographically identical , but a more flexible notion allow s for trivia l grammatical transformations , fo r example, tha t 'i s Sam' s only friend ' i s the sam e predicat e a s 'is the onl y frien d o f Sam' . O n the other hand, anyon e would admi t that 'is a featherless biped' is a different predicat e fro m 'i s human' . This leave s intermediat e cases like 'is a bachelor' and 'i s an unmarried male'. If we think these tw o phrases mea n th e sam e thing , becaus e 'bachelor ' an d 'unmarrie d male' mea n th e sam e thing , the n a natura l accoun t o f identit y between predicate s equates it with synonymy. '5 Th e successor of x i s x u {x}.

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Now wha t o f properties ? W e migh t simpl y specif y tha t tw o properties ar e the same jus t when th e predicates tha t pic k the m ou t are synonymous , bu t we'v e alread y seen 36 tha t thi s approac h doesn't square wit h ou r ordinar y scientifi c thinking . Recal l that the same lengt h propert y ca n b e measure d o n variou s yardsticks ; th e property 'measure s thre e inche s o n thi s yardstick ' i s th e sam e a s 'measures thre e inche s on tha t yardstick' . Furthermore, lengt h ca n be measured in metres as well as yards, s o the same property can be expressed b y a predicate mentionin g yards and another mentionin g metres. Ye t I think n o on e woul d sugges t tha t al l thes e predicate s are synonymous . Fo r mor e dramati c examples , w e ca n tur n t o ordinary scientifi c identitie s lik e tha t o f 'temperature 1 wit h 'mea n molecular motion' . Thes e coul d hardl y res t o n samenes s o f meaning. For scientifi c properties , then , synonym y i s to o stron g a condition. Somewher e betwee n predicates—individuate d b y sameness of meaning—and sets—individuated by sameness of membership—there i s a n intermediat e categor y o f properties. 37 On e suggestion curren t amon g philosopher s o f scienc e i s tha t th e appropriate mod e o f individuatio n migh t b e specifiabl e i n terms , not o f coextensiveness , bu t o f law-lik e coextensiveness. 38 Thu s 'temperature* = 'mea n molecula r motion ' follows from th e laws of physics. B y contrast, 'featherles s biped ' = 'human ' i s tru e b y th e accident o f what specie s happen t o exis t i n our world ; n o physica l law would b e violated if there were bipedal fish. If somethin g alon g thes e line s can b e made t o wor k fo r physical properties, ou r analog y suggest s a similar course for mathematical properties, an d i n particular , fo r number . Conside r agai n th e predicates 'equinumerou s wit h (0 , {0} , {0 , {0}}} ' an d 'equi numerous wit h (0 , {0} , {{0}}}' . The y aren' t synonymous , but they are coextensive, s o they are different predicate s that determine the same set. Bu t our rea l concern i s whether the y express th e same 36

I n ch. 1, sect. 2, above. I n th e cours e o f hi s attac k o n Catnap' s distinctio n betwee n analyti c an d synthetic—see ch . 1 , sect . 4, above—Quin e cast s seriou s doub t o n th e notio n of synonymy a s well. I don't mea n to diffe r wit h Quine here. M y poin t is simply that scientific propertie s aren' t individuate d b y synonym y eve n i f concept s are . If synonymy i s a bankrup t concept, s o muc h th e wors e fo r concepts ; my concer n is with properties. 38 Se e Putnam (1970), 321 . 37

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scientific property ; for that they mus t b e coextensive b y law rathe r than b y accident . Woul d a law o f mathematic s b e violated i f thes e two faile d t o b e coextensive ? O f course ! Thei r coextensivenes s i s provable fro m the axioms o f set theory. Thu s the neo-Benacerrafia n dilemma fo r propert y theory—whic h propertie s ar e reall y th e numbers?—dissolves; understoo d a s scientifi c propertie s rathe r than set s o r predicates , th e vo n Neumann-styl e number s an d th e Zermelo-style numbers are in fact identical. I think th e part o f this story aime d mos t directl y at th e problem o f multiple reduction s fo r arithmeti c carrie s ove r t o th e analogou s problem fo r the real numbers. That is: what makes on e set theoretic version o f the reals preferable to the others? Answer : nothing; eac h version serve s to detec t an d measur e the sam e underlying properties. But whe n i t come s t o th e questio n tha t inspire d Benacerraf' s discussion—what are the natural numbers? or in this case, what ar e the rea l numbers?—th e answe r is , perhaps no t surprisingly , a bi t more complex . In fact , I thin k 'wha t ar e th e rea l numbers? ' i s no t a s directl y analogous t o 'what are the natural numbers?' as it at first seems. To see this, compare Dedekind' s projec t with Frege's . Freg e was face d with a firmly entrenched linguistic practice which strongly favoured the view that numbe r words were names and numbers were objects, and hi s job was to clarify th e nature of those objects. I n Dedekind's case, on th e other hand, there was no pre-existing systematic use of real numbers ; tha t wa s th e problem ! What pre-existe d i n this case was th e intuitive , geometri c line , an d Dedekind' s jo b wa s t o produce a syste m o f number s tha t woul d mimi c it s properties , particularly continuity . S o th e questio n Dedekin d face d wa s no t 'what ar e th e rea l numbers?' , analogou s t o Frege' s 'wha t ar e th e natural numbers?', but rather, 'what is continuity?' The answe r Dedekin d gav e was : continuit y i s wha t Dedekin d cuts have . And , as the Benacerraf-styl e argumen t points out , s o d o Cantor's fundamenta l sequences , and other se t theoretic versions of the reals . So there is after al l a single underlying property that all set theoretic version s o f th e real s serv e t o detect , a singl e propert y shared b y all the particula r disparate phenomena the y are use d t o measure, namel y continuity. Thus, i f ther e i s a prope r answe r t o 'what ar e the reals?' , a n answe r that run s parallel t o ou r answe r t o 'what ar e th e naturals?' , tha t is , parallel t o 'tha t whic h the various

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set theoreti c version s o f th e natural s serv e t o detect' , the n tha t proper answe r is: the real numbers are the property of continuity. But thi s sound s odd , an d I think th e reaso n i t does i s clear fro m what's alread y bee n said . Th e reals , unlik e th e naturals , ar e no t what need s t o b e accounte d for . Continuit y i s wha t need s explication, an d th e variou s set theoreti c version s of th e real s d o that. But our intuitiv e ontology is not th e reals an d th e various set theoretic versions, as it is the naturals and their various set theoretic versions; rathe r w e hav e th e phenomeno n o f continuit y an d w e have th e se t theoreti c reals , whic h al l explicat e tha t propert y b y exemplifying it . Bu t ther e aren' t an y pre-theoreti c real s t o b e identified with anything. Of al l th e man y mathematica l thing s whos e relationshi p t o set s might b e questioned , I'v e considere d onl y numbers , natura l an d real, becaus e they are mos t intimatel y connected wit h th e foundations an d justificatio n o f se t theor y itself . Fo r se t theoreti c purposes, th e onl y othe r essentia l i s th e notio n o f function , fo r which I propos e a simila r treatment . A functio n i s a relatio n between sets , an d a relatio n i s jus t a two-place d versio n o f a property.39 Jus t a s vo n Neuman n ordinal s giv e u s a wa y o f detecting the number property o f a set—we ask whether or not th e set is equinumerous with th e appropriate ordinal—th e set theoreti c version of a function should give us a way o f detecting whether o r not tw o give n sets stand i n the appropriate relatio n o f argument to value. In fact , w e d o thi s b y identifyin g th e functio n wit h a se t o f ordered pairs. 40 Given two sets , x an d y , we can tell if they stand in the functiona l relatio n b y askin g whether , i n ou r chose n se t o f pairs, x i s the firs t membe r of som e ordere d pai r of which y i s the second member . Naturally , w e coul d d o th e sam e job wit h many other particula r sets—e.g. a set of pairs of pairs (e.g . with ((0, jc), ({0}, y)) in place of (x, y)) o r a closely related set of ordered triples (e.g. with (0 , y , x) i n place of (x , y))—and similarly , we could use various substitute s fo r th e standar d se t theoreti c versio n o f th e 39

Thi s i s a n oversimplificatio n becaus e th e mathematica l notio n o f a functio n shifted ove r th e centurie s fro m a rule-like relation t o a n arbitrar y mapping . (Se e ch. 4.) Still , even th e mos t genera l mappin g ca n b e detected b y a set , a s described i n the text. 40 Se e Enderton (1977) , ch. 3.

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ordered pair. 41 Al l that matters , her e a s with numbers , i s that th e set theoreti c versio n giv e u s a convenien t wa y o f detectin g th e relation, the function, that we're intereste d in. I've argue d tha t number s ar e propertie s o f sets , tha t elementar y arithmetic i s th e stud y o f th e numbe r propertie s o f hereditaril y finite sets, that our knowledg e o f arithmetical facts is of a piece with our knowledg e o f these finite sets,42 and suggested simila r accounts for real numbers and fo r functions. Thi s leaves open th e metaphysical question o f whether or no t propertie s (an d relations) shoul d b e included a s a separat e categor y i n th e se t theoreti c realist' s ontology. An d this , obviously , is jus t a specia l cas e o f th e age-ol d debate ove r universals. 43 Recall tha t i n ou r naturalize d context , thi s questio n reduce s t o that o f whether o r no t ou r bes t overall theory of the world requires us t o spea k o f propertie s i n additio n t o ordinar y physica l object s (common-sense realism), various unobservables (scientifi c realism), and set s (se t theoretic realism) . Th e questio n i s just a s pressing for physical propertie s a s i t i s for mathematica l ones : shoul d physical properties (lik e being gold) or physical quantities (like being an inch long) b e included , along wit h physica l objects, i n th e ontolog y o f the natura l sciences ? Putnam, fo r one , say s yes . Fo r example , h e argues, a scientis t may conjectur e that 'ther e i s a singl e property , not ye t discovered , whic h i s responsibl e fo r such-and-such', 44 a statement fo r whic h Putna m see s n o property-fre e translation . Nominalistic philosophers of scienc e ma y eithe r doubt th e centrality o f thes e locutions , o r disagre e abou t th e prospect s fo r translation. Lewis , for example, suggest s tha t th e nominalis t might treat indispensabl e statement s abou t propertie s a s assertin g th e existence o f what I'v e called natura l collections o r kinds . The ide a of naturalnes s might b e taken as primitive, or i t might b e parsed in terms o f variou s objectiv e similaritie s betwee n things , withou t appeal to universals. 45 41 Th e standar d versio n nowaday s i s the Kuratowsk i ordere d pai r (x , y ) whic h i s just th e se t {{x}, {x, y}}, bu t ther e ar e man y othe r possibilities . Se e Enderto n (1977), 35-8 . 42 I don't mean t o suggest that w e can't count hereditaril y infinite sets , but again , I den y tha t thi s i s par t o f th e elementar y arithmeti c Freg e hope d t o reduc e t o pur e logic. 43 Se e ch. 1 , sect. 2 , above . 44 Putna m (1970), 316. 45 Lewi s doesn' t advocat e eithe r o f thes e views , bu t h e does argu e tha t the y ar e live possibilities. See Lewis (1983), 347-8.

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Without pretendin g t o resolv e thi s issue , le t m e conside r a n analogous questio n fo r ou r chose n branc h o f mathematics: shoul d natural numbers , as well as sets, be included i n the ontology o f the theory o f sets ? Thi s questio n ca n b e take n i n tw o senses . First , if we're asking whether there is anything of mathematical significance that can' t b e said withou t explici t referenc e t o numbe r properties , then I think th e answe r i s no. Thi s i s exactly th e mora l o f th e se t theoretic reductions : everythin g we wanted ou t o f numbers can be got out o f von Neumann ordinals (o r Zermelo ordinals , or . . .}. To say that 2 < 3 is to say that if x is equinumerous with the (0, {0} } and y i s equinumerou s wit h {0 , {0} , {0 , {0}}} , the n x i s equinumerous with a proper subse t of y. To say that 2 + 2 — 4 is to say tha t i f tw o disjoin t set s x an d y ar e equinumerou s with (0 , {0} }, then thei r union is equinumerous with (0, {0}, (0, {0} }, {0, {0} , (0, {0}}}} - 'Ever y natura l numbe r ha s a successor' is 'if x i s a vo n Neuman n ordinal , th e unio n o f x an d {x} i s a vo n Neumann ordinal' . '2 is prime' says 'if x i s equinumerous with (0, {0} }, the n ther e ar e no t tw o set s o f cardinalit y les s tha n 2 bu t greater tha n 1 whose cros s produc t i s equinumerous with x\ And so on. I n practice, thes e locutions ar e ofte n simplifie d eve n further , but the fact that number theorists have no problem operating inside set theory demonstrate s tha t nothin g mathematicall y important i s sacrificed b y such translations. Th e sam e goes for real numbers and functions. But it isn't enoug h simply to do arithmetic and mathematics ; ou r overall theor y o f th e worl d mus t als o contai n a chapte r tha t tell s what w e ar e doin g an d wh y i t works th e wa y i t does. This i s the descriptive an d explanator y theor y o f ou r practic e require d b y epistemology naturalized, just the sort of theory, i n fact, tha t we'r e now tryin g t o construct . Withi n tha t theory , explanation s o f knowledge an d referenc e (o r reliability) , of puzzle s like Wittgen stein's,46 o f th e notio n o f law-lik e coextensiveness , an d s o on , might wel l appea l t o objectiv e similaritie s betwee n individua l objects: tw o triangle s are more alik e than a triangle an d a square ; two sample s of gold ar e more similar than either one is to a sample of aluminium; ordered pairs of the form (x , x + 2 ) are more similar to eac h othe r tha n t o a pai r o f th e for m (y , y + 1) . I t remain s a n open question—fo r natura l scienc e jus t a s fo r mathematics — whether thi s distinction betwee n natural an d unnatura l collection s requires a full-blown realis m about universals. 46 Se e ch, 2, sect. 4, above.

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To summarize : fo r th e se t theoreti c realist , set s hav e numbe r properties i n the sam e sens e tha t physica l objects have length . Th e further questio n o f th e ontologica l statu s o f thes e propertie s i s again thoroughl y analogous ; th e mathematica l and physica l sciences are facin g th e sam e metaphysica l question . Th e differenc e i s that the mathematica l science s ca n offe r a n answe r t o par t o f tha t question; th e se t theoreti c reduction s o f arithmeti c sho w tha t al l that i s mathematically important about number s ca n b e sai d usin g only sets , whil e Putna m an d other s stil l debat e th e analogou s question i n physica l science . A s fo r th e secon d question , th e metascientific question of what needs to be said i n our theor y o f our respective sciences , th e issu e i s common t o both : ho w ar e natural kinds, or perhaps objectiv e similarity to be treated? Thus, I take the problem t o b e a general one, no t a t al l special t o th e philosophy of mathematics, and tha t i s why I feel justifie d i n leaving it unresolved here.

3. Freg e numbers The previou s sectio n o n number s a s propertie s mor e o r les s completes wha t I have to sa y here on my own vie w of numbers, bu t I'm no t quit e ready t o leav e the topi c entirely . So far, in evaluating the effor t t o identif y number s with sets , I'v e concentrate d o n vo n Neumann an d Zermelo ordinals , giving little consideration to Frege numbers: fo r example , th e Freg e numbe r thre e i s the extensio n o f the concep t 'equinumerou s wit h th e serie s o f natura l number s ending wit h 2' . I' d lik e t o paus e a momen t ove r Frege' s idea , no t because I intend to modif y th e theory o f numbers already proposed, but becaus e I thin k attentio n t o Freg e number s wil l cas t som e helpful ligh t on what our theor y of sets is and i s not. To focu s the issue , le t me reconsider Benacerra f s argument on e last time . W e hea r o f Erni e an d Johnny , wh o lear n th e vo n Neumann an d Zermel o ordinals , respectively. Each claim s that his sets ar e th e natura l numbers . Eithe r versio n wil l do : ther e ar e n o conclusive arguments to decide between them. Assuming there is no third candidat e fo r whic h ther e ar e decisiv e arguments , th e conclusion follow s that number s are not sets . But is there another candidate ? Benacerraf mentions a Frege-style proposal, that the numbe r three is the set of all three-element sets. If numbers ar e actuall y properties o f sets , a s I've suggested , Benacerra f

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fears thi s would coun t a s a compellin g argumen t i n favou r o f thi s set theoretic account ove r von Neumann's an d Zermelo's; thus he is motivated to rejec t the property view , even on slender grammatical grounds. Onl y th e convictio n tha t number s ar e no t propertie s o f sets allows hi m to put th e Frege-style option aside , and onl y then is he satisfie d tha t h e nee d loo k n o further: 47 'Ther e i s little nee d t o examine all the possibilities in detail, once th e traditionally favored one o f Freg e . . . ha s bee n see n no t t o b e uniquel y suitable ' (Benacerraf (1965) , 284). There is a missing premiss in this reasoning, something alon g the lines tha t i f a propert y i s t o b e treate d se t theoretically , the n i t should b e identified wit h the set of things that exemplify it . There is something t o this ; th e se t theoris t (usuall y presupposin g vo n Neumann's numbers ) speak s no t o f 'evenness' , bu t o f th e se t o f even numbers, not o f 'primehood', bu t o f the set of primes. In terms of this methodology, Frege-styl e sets are the natural choice. But, though I'v e embraced the property theory , I'v e advocate d a different lin e on how thes e properties ought t o be treated withi n set theory, namely, in terms of comparisons wit h a standard measuring rod lik e the von Neumann ordinals . While this approac h allow s us to sa y everything we want t o say , i t leaves us without anythin g in set theor y t o properl y cal l 'th e numbers'. 48 I thin k thi s i s a s i t should be . I'v e argue d tha t number s ar e reall y properties , i n th e sense tha t th e theory o f numbers, arithmetic, i s a theory o f number properties. I f 'numbers' are 'that which form s th e subject matter of arithmetic', the n number s are properties . Bu t i f 'numbers' ar e 'th e referents o f numbe r words' , the n ther e ar e n o numbers . 'One' , 'two', 'three' , an d s o on , ma y enjo y th e superficia l gramma r o f names, bu t the y ar e reall y jus t anothe r measurin g ro d lik e th e Zermelo ordinals. 49 We shoul d resis t the urg e t o find referent s fo r 47 Notic e tha t Benacerra f doesn' t rul e ou t a candidat e fo r identificatio n a s th e numbers simpl y becaus e i t ha s superfluou s properties ; th e Frege-styl e set s hav e those. Rather h e rule s out candidate s (wit h superfluou s properties ) fo r whic h ther e are no good arguments . 48 Grante d the usual conventions, the set theorist will call {0, {0} } 'two', but on on my view, this is iike calling th e standard metre in Paris 'the metre'. 49 Compar e Benacerra f (1965) , 292: 'Question s o f identificatio n o f the referents of numbe r word s shoul d b e dismisse d a s misguide d i n jus t th e wa y tha t a questio n about th e referent s o f th e part s o f a rule r woul d b e see n a s misguided. ' Despit e Benacerraf's rejectio n o f th e propert y view , ther e ar e man y point s o f contac t between th e positio n advocate d her e an d tha t sketche d i n th e fina l page s o f hi s paper. I won't try to sort out our agreements an d disagreements .

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these words, an d th e temptation t o trea t numbe r propertie s a s sets is just one version of this misguided impulse. Still, Frege numbers ar e o f interest , even i f it i s wrong t o sa y tha t they ar e th e natura l numbers. 50 Th e extensio n o f 'equinumerou s with . . .' i s a collection; se t theory i s our theor y o f collections; it' s natural t o as k ho w thi s particula r collectio n fit s in . Askin g thi s question will tell us something about the nature of sets. One o f the stronges t argument s agains t the claim tha t Benacerraf' s Frege-style set s ar e a suitabl e se t theoreti c versio n o f th e Freg e numbers i s tha t the y don' t exist. ^ We'v e see n tha t thi s i s tru e because, fo r example , ne w three-elemen t sets ar e forme d a t ever y stage of th e iterativ e construction, s o there i s no stag e a t whic h th e set o f al l three-elemen t set s i s formed. Thi s collectio n i s too bi g t o be a set ; i n standar d terminology , i t i s a 'prope r class' . Othe r collections, lik e th e collectio n o f al l sets , ar e als o prope r classes . Thus th e common wisdo m i s that th e set of all three-membered sets would b e a goo d candidat e fo r th e rol e o f a Freg e numbe r were i t not fo r the unfortunate fact tha t i t is too bi g to exist. But 1 think this underestimates the drawbacks of the set of all three-membered sets. To se e this , consider : ordinar y arithmeti c i s use d t o coun t physical objects, perhap s set s o f thes e (lik e the tw o pair s of shoes) , perhaps occasionall y even sets of sets of these, and s o on, bu t neve r sets o f mor e tha n finit e rank . Thi s i s th e conten t o f m y insistence that arithmeti c i s par t o f th e theor y o f hereditaril y finit e sets . Eventually, of course, we go beyon d ordinar y arithmetic; we try t o number thing s lik e the set s o f equinumerou s sets o f rea l numbers, and the n w e are doing highe r set theory.52 Just fo r the moment, I' d like to suggest that this expansion o f our arithmetica l theory i s nontrivial, that—jus t a s th e mov e fro m ordinar y distances t o astro nomical one s require s a chang e i n ou r notio n o f spac e fro m Euclidean t o non-Euclidean—whe n w e deman d tha t ou r number s count mor e complicated , infinitar y things , we ar e askin g for mor e complicated numbers." 13 O n thi s picture , ther e ar e Frege-style set s 50 Reader s of Madd y (1981 ) an d (1983 ) wil l recogniz e a chang e of hear t here . I have Michae l Resnik t o thank for th e insigh t tha t the ver y complexit y of th e theor y of th e 198 3 article suggests it is not a theory of ordinary numbers , 51 Se e Benacerraf (1965) , 284. 52 Cantor' s continuu m hypothesis , t o h e discusse d i n the nex t chapter, says that there are exactly two set s of equinumerous infinite set s of reals. ' • Notice , thes e new number s are not mor e complicated in that they are infinite — I'm stil l talkin g abou t finit e numbers—the y ar e jus t mor e complicate d i n tha t th e finite sets they number can hav e infinite set s in their transitiv e closures.

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that correspon d t o th e Freg e number s o f ordinar y arithmetic . Th e trouble arise s only whe n we try t o expand ou r arithmeti c t o coun t objects o f highe r se t theory, bu t perhap s th e troubl e i s due t o th e vagaries o f highe r se t theor y an d no t t o th e simple r number s o f ordinary arithmetic , wit h which we began. Now suppos e this separatio n o f infinitar y versu s non-mfinitar y arithmetic ca n b e maintained. I f the onl y problem wit h Frege-styl e sets as surrogates fo r Freg e numbers were their size , we could the n identify th e 'smal l Freg e numbers' , th e Freg e number s of ordinary arithmetic, wit h th e correspondin g Frege-styl e sets ; tha t is , th e small Freg e numbe r thre e woul d b e th e se t o f al l three-membere d hereditarily finit e sets . Wha t I want t o clai m i s that eve n her e th e Frege-style sets are unsuitable. The smal l Freg e number s coun t physica l object s (an d set s o f these, etc.) , an d presumabl y th e physica l object s i n ou r worl d fluctuate: ne w animal s ar e born ; ol d star s explode ; tree s ar e converted int o tables an d chairs . Correlated wit h thes e shifts i n the population o f physica l object s ar e shift s i n th e populatio n o f sets : when th e cub is born, i t is a member of sets that didn't exist before ; when th e ol d sta r explodes , set s also vanish ; set s of trees give way to ne w set s o f table s an d chairs . Finally , these fluctuation s i n th e population o f set s brin g abou t fluctuation s i n th e populatio n o f three-membered, hereditaril y finite sets, s o the se t of al l these, ou r Frege-style candidat e fo r th e smal l Frege number three, i s now thi s set, no w that . This i s not t o sa y that th e Frege-styl e set changes its membership fro m tim e to time—set s ar e full y determine d b y their members, so they can't d o that—rather, the proposal tha t the small Frege number three be identified with th e Frege-style set of all threeelement, hereditarily finite sets i s the proposa l tha t i t b e identifie d with one set after another , wit h differen t set s at different times. 54 This is not satisfactory . I f we really thought Freg e numbers were the numbers , w e migh t wel l insis t tha t whateve r thre e is , it i s th e same thin g toda y a s i t wa s yesterday ; i f w e though t th e numbe r words ha d referents , they shoul d hav e the sam e referents toda y a s tomorrow. Bu t even if we rejec t (a s I have) th e these s that numbers are object s an d tha t numbe r word s hav e referents , I thin k w e should stil l object to thi s account o f the Frege numbers themselves. When w e aske d afte r th e collectio n o f all three-membere d sets , we didn't wan t no w thi s collection , no w that , dependin g o n th e 54 Thi s argumen t i s a n adaptatio n o f Hambourger' s (1977) , wit h tempora l considerations replacing modal ones.

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vagaries o f physica l existence . Wha t w e ha d i n min d wa s a collection whos e membershi p i s allowe d t o chang e fro m tim e t o time while it stays the same, tha t is, the collection whic h collects , at any given time, the three-element set s in existence a t that time . Such a collection, eve n if it is small enough, i s not a set . Frege numbers , then, ar e collections tha t ar e not sets . Larg e Frege numbers ar e to o larg e t o b e sets, s o the y ar e prope r classes . Smal l Frege numbers , thoug h smal l enoug h t o b e sets , ar e individuate d differently, s o the y ar e classes , too , withou t bein g proper . Thos e inclined t o insis t o n referent s fo r th e numbe r word s wil l nee d a theory o f set s an d classes . Eve n thos e immun e t o thi s temptatio n will fin d th e distinctio n betwee n thes e tw o type s o f collection s important t o th e clarit y o f thei r theor y o f sets . An d ther e remain s the possibilit y tha t classe s migh t do importan t wor k i n se t theor y itself.55 Le t me conclud e thi s section, then , wit h a brie f look a t th e distinction betwee n set s and classes . At th e beginnin g o f thi s chapter , I sketche d tw o foundationa l worries that contribute d t o th e development o f the moder n theor y of sets . Th e historica l pat h fro m th e trouble s wit h th e calculus , through Canto r an d Dedekind' s wor k o n th e rea l numbers , t o Zermelo's axiomatizatio n an d beyond , ha s bee n a mainl y math ematical developmen t whos e vicissitude s eventually led t o a well articulated pictur e o f collections formed sequentially , from previously given elements , i n a hierarch y of stages : th e iterativ e conception o f set. A t eac h stage , ne w collection s ar e forme d wit h complet e freedom, withou t concer n fo r an y metho d o f construction . Finit e combinatorics tel l us that there i s a uniqu e subcollection o f a finite collection fo r ever y wa y o f sayin g ye s o r n o t o eac h individua l element. Carryin g thi s notio n int o th e infinite , subcollection s ar e 'combinatorially' determined , on e fo r ever y possibl e wa y o f selecting elements , regardles s o f whethe r ther e i s a specifiabl e rule for thes e selections. This i s the mathematica l notio n of a collection : a collectio n forme d combinatorially , i n a series of stages tha t mak e up the iterative hierarchy.56 55 Fo r example , i n non-demonstrativ e arguments fo r ne w se t theoreti c hypo theses. Se e Maddy (1988# ) fo r some examples. 56 Thi s pictur e appear s i n Zermel o (1930) . The combinatoria l ide a i s explicit in Bernays (1935) . A s indicate d above , Enderto n (1977) , 7-9 , Boolo s (1971) , an d Shoenfield (1977 ) giv e moder n presentations . O f course , th e tempora l an d constructive imager y is only metaphorical ; sets ar e understoo d as objectiv e entities, existing i n their own right .

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The secon d historica l trai l i s th e on e tha t begin s wit h Frege' s logicism an d continue s throug h Russel l an d Whitehead' s effor t t o salvage som e aspect s o f tha t idea. 57 Her e th e origina l goa l wa s t o found arithmetic , and th e collections involved are extensions of things more o r les s like predicates. 58 This is a very different conception : th e entire universe is simply divided into two piles, depending on whether or no t eac h thin g satisfie s th e give n predicate . Thi s contrast s twic e with the iterative picture: we are fre e t o collec t absolutely any things , regardless of whether the y are al l available at som e stage , but w e ar e not fre e t o collec t combinatorially , without recours e to a rule of any kind. Th e Cantorian , mathematical , collectio n i s a set ; the Fregean , logical, collection i s a class.59 Collections of these two type s diffe r i n several other ways , one of which we've alread y noted: classes can be larger than sets. Thus, for example, ther e i s a clas s o f al l sets—the extension o f the predicat e 'is a set'—but it is not a set, because there is no stage at which it can be formed. Such a class is a proper class , because it is too larg e to be coextensive wit h an y set . On th e othe r hand , becaus e classe s can only b e forme d when ther e ar e suitabl e predicates, ther e ma y well be mor e set s o f rea l number s than ther e ar e classe s o f reals . Thi s depends, o f course, o n th e detail s of our theor y of predicates, but i t is hard t o imagin e how th e existenc e o f a predicate fo r ever y se t a t each stage could b e guaranteed without someho w presupposin g the combinatorial notion of set theory.60 We've als o see n tha t a class can chang e it s membership , whil e a set cannot. Th e collection of things with a certain property ca n vary in membershi p fro m tim e t o time , bu t a s long a s i t i s identified a s the extensio n o f th e appropriat e predicate , i t remain s th e sam e class. A set , o n th e othe r hand , i s completel y identifie d b y it s membership, s o the collectio n o f thing s wit h a certai n propert y i s now one set, now another . 57 I n fact , Russel l an d Whitehead' s positio n (1913 ) i s a hybri d between th e tw o notions considere d here : extensions , whic h depen d o n predicates , ar e forme d i n stages. A simila r amalga m i s propose d b y Keit h Devli n i n suppor t o f th e axio m candidate V = L (see ch. 4, sect . 4, below) . Godel (1944) , 464 , trace s the idea s behind this axiom to Russell. 58 Her e I' m includin g Frege' s concepts , Russell' s prepositiona l functions , m y scientific properties, etc, I won't distinguis h between these in what follows . 59 I discus s thi s contras t i n mor e detail , wit h mor e historica l considerations , in Maddy (1983) . Se e also Parson s (1974
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A nove l an d strikin g differenc e i s on e tha t show s u p i n th e structure o f th e membershi p relation . A clas s ca n b e a membe r of another class—th e clas s of vo n Neuman n ordinal s i s a membe r of the clas s o f infinit e collections—jus t a s set s ar e member s o f on e another. Bu t th e clas s o f vo n Neuman n ordinal s isn' t th e onl y member o f th e clas s o f infinit e collections—ther e ar e infinitel y many infinite collections—so the class of infinite collection s has th e distinction o f bein g self-membered . O f course , n o se t ca n b e self membered, because all its members must be formed at stage s before that at which it is formed. And finally, classes, unlik e sets, lead to paradox. We'v e see n that some collection s ar e self-membered—e.g . th e clas s o f al l infinit e collections—while other s ar e not—e.g . an y set . 'Bein g non-self membered' seems a perfectly unobjectionable predicate, satisfied b y some collections, not satisfie d b y others, so there ought to be a class that i s its extension. But this is Russell's paradox i n much the sam e form a s h e firs t presente d i t t o Frege: 61 i s th e clas s o f non-self membered collection s self-membere d or not ? An d again , o n th e iterative conception, al l sets are non-self-membered, s o the Russellian set, th e se t o f al l non-self-membere d sets, i s the se t o f al l sets . Bu t there i s no suc h set , becaus e there i s no stag e a t whic h i t coul d b e formed, and thus, no paradox . Most contemporar y se t theory i s done withou t explici t mentio n of classes . Theorie s tha t d o attemp t t o encompas s bot h sort s o f collections generall y separate set s from prope r classe s on th e basi s of size , withou t takin g int o accoun t th e fundamenta l difference s between set s an d smal l classes. Sometimes classes ar e no t allowe d as member s o f an y furthe r collections , whic h avoid s paradox , bu t seems restrictiv e an d artificial; 62 othe r time s the y ar e allowe d membership i n furthe r collections , but thes e collection s ar e forme d in stages , disallowin g self-membership , and ar e determine d com 61

Again , see his letter to Frege , Russell (1902). On e syste m lik e thi s i s vo n Neumann—Bernays—Gode l (se e vo n Neuman n (1925), Bernays (1937) , Gode l (1940)) , in which ther e is a class for ever y first-orde r formula wit h quantifier s rangin g onl y ove r sets . Thi s allow s th e familia r Zermelo Fraenkel axiom s t o b e condense d int o a finit e list , bu t ha s littl e effec t beyon d thi s metamathematical simplification . Morse—Kelle y (se e Kelle y (1955 ) an d Mors e (1965)) i s a stronge r system , allowin g quantificatio n ove r classe s i n formula s tha t determine classes , bu t i t stil l disallow s classe s a s member s an d ha s th e adde d difficulty tha t it s classe s loo k lik e littl e mor e tha n anothe r laye r o f set s tha t wa s somehow lef t ou t o f th e hierarchy . Se e Drake (1974) , 16-17 , Fraenkel , Bar-Hillel , and Lev y (1973), ch. 2, §7. 62

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binatorially, whic h raise s seriou s question s abou t ho w thes e additional layer s of non-set collections really differ fro m sets. 63 A parallel situation arise s in the theory o f truth, 64 with sentence s like 'Everythin g I've ever sai d i s false. ' I f everything else I'v e ever said i s false, the n thi s statemen t i s paradoxical. Russel l gives us a predicate that can't have an extension; here we have a sentence that can't have a truth value . Parallel to the class theoretic solutio n tha t disallows classe s a s members , we coul d insis t tha t th e questio n of truth no t b e raised for sentences involving the notion o f truth; thi s escapes paradox, but again it is too restrictive and artificial . Another solution , paralle l t o regimentin g classe s int o stages , requires tha t th e notio n o f trut h b e typed : truth x applie s only t o statements tha t don' t involve the notion o f truth a t all; truth2 only applies t o statement s involvin g truth 1? an d s o on . Thu s 'false * i n 'Everything I'v e ever sai d i s false ' i s not-true « fo r som e « , an d 'everything' only ranges over statements involving truth^-x. In both cases—class theory and truth theory—the diagnosis is that we can't survey som e entir e category—al l classes, al l statements—but only one level or type at a time, and i n both cases , the restrictions square poorly with the pre-theoretic notions. Thus the paradoxes o f truth an d clas s theory lead to unpalatable hierarchies in which statements can't refer to themselves and classes can't b e self-membered. A n alternative available in both case s is to allow gaps, that is, statements that are neither true nor false , classes of whic h som e item s ar e neithe r member s no r non-members . Kripke showed how suc h a system would go for truth theory, 65 and I have proposed an analogous theory of classes.66 To a certai n extent , thi s approac h produce s th e desire d results: the clas s o f infinit e collection s i s self-membered; the clas s of nonself-membered collection s i s neithe r self-membere d no r non-self membered. Freg e numbers , larg e an d small , ca n b e define d i n various ways , an d th e smal l ones a t leas t ar e fairl y wel l behaved . Still, th e contex t o f three-value d logic—true, false , an d neither — 63 Fo r example , Ackerman n (1956 ) seem s t o ad d severa l layer s of combinatori ally determine d 'classes' . (Se e Fraenkel, Bar-Hille! , an d Lev y (1973) , ch . 2, §7.7. ) Reinhardt (1974) , i n a descendan t o f Ackermann's system, reinforces thi s picture by explicitly assumin g th e axio m o f foundatio n fo r classes . (Foundatio n disallow s self membership, amon g othe r class-lik e pathologies. See Enderton (1977) , 206. ) 64 Fo r further discussion of the parallel, see Parsons (19741?) .

65 66

Se e Kripke (1975). InMaddy{1983) .

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makes th e syste m awkward , s o I thin k th e jur y mus t stil l b e considered ou t o n th e questio n o f a workable theor y o f classe s i n general and o f Frege numbers in particular.67 67 Madd y (1984c ) contain s some modification s and development s o f the system, as wel l a s answer s t o on e o f th e ope n question s o f Madd y (1983) : a fixe d poin t theorem o f William Tait implie s that th e constructio n does not reac h a fixed point. Robert Flag g ha s sinc e demonstrate d wha t Tai t conjectured , namely , tha t i f th e construction i s carried out ove r a standar d model of ZF, the n th e fixed point will be the first admissible ordinal greater than th e leas t upper bound o f the ordinal s of th e ground model . Madd y (1984c ) als o contain s som e soberin g informatio n o n th e prospects fo r axiomatization . Se e Feferma n (1984a ) fo r a compendiu m o f relate d systems.

4

AXIOMS 1. Real s and sets of reals The epistemolog y o f compromis e Platonis m follow s Godel' s i n being two-tiered : th e mos t primitiv e truth s ar e intuitivel y given , obvious; th e mor e theoretica l hypothese s ar e justified extrinsically , by thei r consequences , b y thei r abilit y t o systematiz e and explai n lower-level theory , an d s o on . I n Chapte r 2 , I sketche d th e se t theoretic realist' s versio n o f intuition , a neurologicall y base d phenomenon tha t produce s firml y held elementar y belief s an d provides the m som e preliminar y leve l o f justification . Th e re mainder o f ou r evidenc e fo r th e principle s w e choos e a s axiom s must come from theoretica l sources; the time has come to look int o the structur e o f thi s second typ e of mathematica l justificatio n an d t o confront the questions it raises. Because extrinsi c argument s involv e mor e advance d level s o f mathematical theorizing , the y brin g u s fac e t o fac e wit h mor e esoteric set theoretic matter s than hav e heretofore been relevant. As this i s obviousl y unavoidable , I be g th e indulgenc e o f m y non mathematical reader . Indeed , I hope sh e might com e awa y wit h a greater appreciatio n for why Hilbert and others refuse t o be budged from Cantor' s paradise. The story begins with th e mathematical concerns that led Cantor there in the first place.1 I'll turn to axiomatics in the next section. Cantor's dissertatio n was in number theory, bu t when he arrived at his first university teaching job in 1869, on e of his senior colleagues posed hi m a problem i n analysis: 2 conside r function s fro m real s t o reals tha t ca n b e represente d b y infinit e trigonometri c series ; ar e 1 I n ch . 3 , sect . 1 , above , I pu t greate r emphasi s o n Dedekind' s an d Frege' s foundationai motivations. Cantor's, as we'll see, were more strictly mathematical. 2 I n mathematics , 'analysis ' mean s th e stud y o f rea l an d comple x functions , which include s the calculu s an d it s foundations and extensions . In this discussion of Cantor's career and it s antecedents, I follow Dauben (1979) .

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such representation s unique ? Eduar d Heine , Cantor' s colleague , had prove d uniquenes s unde r certai n specia l circumstances , bu t a general solutio n elude d him , jus t a s i t ha d hi s fello w analyst s Dirichlet, Lipschitz , an d Riemann . Withi n months , Canto r ha d obtained th e desire d result : i f a functio n f(x) i s give n b y a trigonometric serie s tha t converge s fo r ever y value o f x, then tha t representation i s unique. But Canto r didn' t sto p there . Typica l o f relate d wor k i n th e theory o f function s wa s th e generalizatio n o f suc h theorem s b y allowing a numbe r o f 'exceptiona l points' , fo r exampl e point s a t which th e serie s is not require d to converge . Heine' s partial result s had allowe d finitel y man y exceptions , bu t Cantor , inspire d by th e work o f Hankel , hope d t o accommodat e infinitel y many . Th e method depende d cruciall y on th e distributio n o f th e exceptiona l points. Everyon e kne w tha t infinitel y man y point s containe d i n a bounded interva l wil l accumulat e aroun d a t leas t on e point. 3 Cantor coul d se e ho w t o dea l wit h infinitel y man y exceptiona l points i f the y ha d exactl y on e accumulatio n point ; wit h a littl e effort, h e coul d se e ho w t o allo w fo r finitel y man y accumulatio n points; indeed , hi s metho d woul d stil l wor k o n a n infinit e se t of accumulation point s a s long a s that se t had onl y on e accumulatio n point o f it s own , o r fo r tha t matter , finitel y man y accumulatio n points o f it s own; an d s o on. Th e outline s of th e generalizatio n he had i n mind wer e there , bu t Canto r neede d a way t o formulat e his most lenien t condition o n exceptional points wit h precision. He quickl y realize d ther e wa s n o hop e o f definin g suc h a complicated se t o f point s withou t a n accurat e theor y o f th e rea l numbers themselves; it was thi s problem that led him to his account of real s i n term s o f fundamenta l sequences. H e wa s the n abl e t o formulate th e requiremen t o n exceptiona l point s appropriat e fo r the generalizatio n of hi s uniquenes s theorem. Bu t what a n od d se t of point s i t was : infinite , an d quit e complex , ye t stil l someho w small enough , o r well-behave d enough , i n relationshi p t o al l th e reals, t o d o n o damage ! Thi s apparentl y got Canto r t o wonderin g how continuou s set s lik e th e real s relat e t o seemingl y smaller , discrete infinit e set s like the natural numbers. Meanwhile, thei r share d interes t i n th e theor y o f rea l number s had brough t Dedekin d an d Canto r int o correspondence. I n a lette r 3

A n accumulation point o f a set of points ha s point s of tha t se t arbitrarily clos e to it . fo r example , 1 is an accumulatio n point of • '/z , 2 /% -V^ 4 /5, . . . ; .

REALS AN D SET S O F REAL S 10

9

to hi s friend , Canto r raise d a naggin g question : ca n th e rea l numbers b e brough t int o one-to-on e correspondenc e wit h th e naturals? Admittedly , ther e see m t o b e mor e real s tha n naturals , but then , ther e see m to b e mor e rational s tha t naturals , too , an d Cantor had shown that there were not.4 He wrote:5 At firs t glanc e on e migh t sa y no , i t i s no t possible , fo r [th e se t o f natura l numbers] consist s o f discret e part s whil e [th e se t of real numbers ] build s a continuum; bu t nothin g i s wo n b y thi s objection , an d a s muc h a s I a m inclined t o the opinion that [the se t of naturals] an d [th e set of reals] permi t no such uniqu e correspondence, I cannot find the reason, an d whil e I attach great importance t o it, the reason ma y be a very simple one .

Dedekind had no easy answer. Cantor replied : I raised th e questio n becaus e I have considered i t for a number o f years an d have alway s foun d mysel f doubtin g whethe r th e difficult y i t gave m e wa s subjective o r whethe r i t was du e t o th e subjec t itself . Since yo u writ e tha t you are also in no position t o answer it , I may assume the latter.

Shortly thereafter, Cantor found his theorem: the correspondence is impossible; ther e are , i n thi s sense , mor e rea l number s than ther e are naturals.6 If ther e ar e s o ver y man y points o n th e line , ho w man y might there b e i n a plane , i n a three- , o r four- , o r w-dimensiona l space? This nex t questio n t o Dedekin d too k longe r t o answer , perhap s because Canto r s o firml y expecte d space s o f highe r dimension t o have mor e points . Whe n h e finally produced a one-to-on e correspondence betwee n the line and the plane, he was moved to remark: 'I see it, but I don't believ e it!'7 But there it was. Every infinite set he had considere d so fa r ha d eithe r the cardinalit y of the real s or th e cardinality of the naturals. This discover y serve d t o refocu s attentio n righ t wher e i t ha d started: o n sets of reals. Cantor wrote: 8 And no w tha t we have proved, fo r a very rich an d extensiv e field of [sets] , the propert y o f bein g capabl e o f correspondenc e wit h th e point s o f a 4

Se e Enderton (1977), 130. Thi s quotation and the next are drawn from Daube n (1979), 49-50. 6 Se e Enderto n (1977) , 132 . A se t i s 'countable ' i f i t i s n o large r tha n th e naturals; otherwise it is 'uncountable'. 7 Daube n (1979), 55. 8 Thi s an d th e next quotation fro m Canto r (1878 ) ar e translated by Jourdain in the introduction to his edition of Cantor (1895/7), 45. 5

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continuous straigh t lin e . . . the questio n arise s . . . Into ho w man y an d wha t classes (if we say that [sets ] of the sam e or differen t [size ] are grouped in th e same or differen t classes respectively) do [infinit e set s of reals] fall ?

Cantor had an opinion : By a process o f induction, into th e furthe r description o f which we will no t enter here , we are led to the theorem tha t the numbe r of classes i s two . . .

This conjecture, tha t every infinit e se t of real s is either countable or of the cardinalit y o f th e continuum , ha s com e t o b e calle d Cantor' s 'continuum hypothesis' (CH). Cantor's bes t effor t i n th e directio n o f a proo f o f thi s conjectur e involved th e notio n o f a perfect set. A closed se t of reals is one tha t contains al l it s accumulatio n points ; a perfec t se t i s closed , an d every one o f it s points i s an accumulatio n point.9 Perfection played a key role in Cantor's analysi s of continuity itself, an d eventuall y he was abl e to sho w tha t every non-empty perfec t set is equinumerous with th e continuum . Afte r a fals e star t pointe d ou t b y Iva r Bendixson, Canto r prove d tha t ever y close d se t o f real s ca n b e decomposed into a countable set and a {possibl y empty) perfect set, which implie s that th e continuu m hypothesis is true for closed sets . (From no w on , I'l l assume tha t perfec t set s ar e non-empt y b y stipulation.) A t thi s point , Canto r wa s optimisti c abou t th e possibility of generalizing this result, called 'the Cantor—Bendixso n theorem':10 'I n futur e paragraph s i t will b e proven tha t thi s remarkable theorem ha s a furthe r validit y eve n fo r [set s of reals] which ar e not closed . . .'. Of course, a complete generalizatio n would constitut e a proof of Cantor's continuum hypothesis in its entirety. Characterizing set s o f exception s o r singularitie s wasn't th e onl y problem tha t brough t analyst s of thi s period u p agains t question s about sets o f real numbers . From th e eighteenth centur y on, a wid e range o f considerations—fro m vibratin g strings an d hea t flo w t o the foundation s o f th e calculus—consistentl y pushe d mathema ticians from narrower to more inclusive notions of function.1 9 Fo r example , th e se t consistin g o f th e point s betwee n 0 an d 1 inclusiv e is perfect. Th e se t consistin g o f thos e point s plu s the singl e poin t 2 i s closed bu t no t perfect. 10 Canto r (1883) , 244, translated b y Dauben (1979) , 118 . Fo r other discussion s of th e histor y o f the Cantor—Bendixso n result , see Moore (1982) , 34-5 , an d Hallet t (1984), 90-2. 11 Thi s developmen t i s trace d b y Klin e (1972) , 335-40 , 403-6 , 505-7 , 677-9 , 949-54.

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1

What bega n i n Galileo's tim e wit h curve s an d continuou s motion s developed t o Euler's combinations of parts o f different curves , then to a serie s o f account s i n term s o f ever-widenin g clas s o f expressions tha t coul d legitimatel y defin e a function , unti l math ematicians wer e finall y face d wit h th e ide a o f a purel y arbitrar y function a s absolutel y an y correspondence betwee n real s and real s regardless o f ho w i t migh t o r migh t no t b e expresse d b y mathematical operations . Od d an d seemingl y pathologica l ex amples proliferated : function s tha t pas s throug h ever y valu e between a and b without bein g continuous,12 continuous function s that aren' t different } able,13 an d eve n Dirichlet's 'shotgun ' functio n (zero o n rationals , on e o n irrationals ) whic h wa s nowher e continuous, withou t eithe r derivative or integral. 14 Doubts circulate d about th e soundnes s o f this extremely general notion, an d b y 190 0 ther e was considerabl e controversy abou t th e proper exten t o f th e functio n concept. 15 I n a n effor t t o ge t a responsible handl e o n th e vas t range of non-continuous functions, the Frenc h analyst s Rene Baire , Emile Borel , an d Henr i Lebesgu e set out t o giv e a systemati c classification, Lebesgue' s version mad e use o f Borel's earlie r hierarch y of set s of reals. Th e simples t sets of reals ar e the closed sets , mentioned earlier , an d thei r complements , the ope n sets. 16 Th e unio n o f tw o close d set s i s closed , bu t th e union o f countabl y man y close d set s ma y b e open 17 o r worse. 18 This 'or worse' gives rise to the Borel hierarchy:19 S? = the open set s 111 — the close d sets ^2 = countable unions of closed sets 12

Du e to Darboux. Se e Kline (1972), 952 . B y Riemann , Cellerier , an d Weierstras s betwee n 185 4 an d 1875 . Se e Kline (1972), 955-6. 14 I n 1829. Se e Kline (1972), 950 . 15 Se e Monna( 1972). 16 Equivalently , a se t i s open i f it contain s o n ope n interva l ({ x a < x < b } ) around eac h of its points. 17 Fo r example , th e unio n o f th e close d interval s [l/n, («—!)/« ] i s th e ope n interval (0 , 1) . (B y notationa l convention , squar e bracket s indicat e tha t th e endpoints ar e included, roun d bracket s that they are not.) For example, the union o f the closed interval s [0 , (»—!}/«] is neither closed no r open. 19 Th e Bore l set s wer e introduce d b y Bore l (1898 ) a s set s resultin g fro m th e closed set s usin g complemen t an d countabl e intersection . Lebesgu e (1905 ) intro duced a hierarchy , thoug h no t thi s one , whic h i s du e t o Hausdorf f (1919) . I t continues t o generat e ne w set s of reals until the first uncountable ordinal. (Fo r limit ordinals \, ^= countable unions of sets in the FI^s for a a < \. ) For the basic theory of Borel sets, see Kuratowski (1966), §§5, 6, and 30 , or Moschovakis (1980), ch. 1. 13

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S 11" = complements o f S? sets . . .

2a + i — countable unions of D a set s Ila + i = complement s o f 2 « + i sets A« = sets that ar e both S " and 11" Borel sets = th e union of the £^s Lebesgue's hierarch y of 'Bore l functions 1 wa s define d i n term s o f these simpl e set s o f reals , an d h e prove d i t equivalen t to anothe r hierarchy given earlier by Baire,20 The Bore l sets, despit e thei r complexity, tur n ou t t o b e fairly wel l behaved. Consider , fo r example , th e perfec t subse t property : a collection o f sets of reals is said to hav e th e perfec t subse t propert y if ever y infinite se t in the collectio n i s either countabl e o r contain s a perfect subset. 21 Thus th e Cantor-Bendixson theore m say s that th e closed set s o f real s hav e th e perfec t subse t property . Pau l Alexandroff (1916 ) mad e goo d o n Cantor' s hunc h tha t thi s resul t could b e generalize d b y extendin g i t t o includ e al l Bore l sets . Another propert y o f interes t t o analyst s is separability: two set s A and B are separate d b y a set C if A is a subset o f C and B is disjoint from C . Waclaw Sierpirisk i (1924 ) prove d tha t disjoin t Borel sets in n° can b e separated b y a Bore l set i n A° , an d gav e application s o f this fact to the general theory of functions . Lebesgue wa s als o instrumenta l i n th e isolatio n o f a secon d important collectio n o f set s o f reals . Th e impetu s thi s tim e cam e from th e theor y o f integration, another field filled with perplexitie s at th e time . Amon g th e numerou s pathologica l function s unde r scrutiny wer e seemingl y unobjectionabl e examples tha t turne d ou t not t o b e integrable using the state-of-the-art Rieman n integral . T o extend th e concep t o f integration, Lebesgu e neede d a new gaug e of the siz e o f a se t o f reals , no t a generalizatio n o f number , lik e Cantor's cardinality, bu t a generalization o f lengt h tha t coul d tak e that concep t fro m interval s to more comple x sets . For this purpose , 20 T o se e th e connection , notic e tha t a functio n f i s continuou s i f i t ha s th e following property : wheneve r A is an ope n set , f~ l[K\ = (x f(x) € A } is also open. Lebesgue defined a function g to be S" iff g~ l[A] is £„ whenever A is S°. Thus the S? functions ar e th e continuou s ones . Baire's versio n o f thi s hierarch y appear s i n Bair e (1899). 21 Recal l I'm assumin g perfect set s are non-empty.

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he develope d th e notio n o f Lebesgu e measure, 22 an d eve n Dirichlet's functio n becam e integrable . Sinc e close d set s ar e Lebesgue measurable , an d th e collectio n o f Lebesgu e measurable sets i s close d unde r complemen t an d countabl e union , i t follow s that Bore l set s hav e anothe r nic e property : the y ar e al l Lebesgue measurable. The next step in this development, the last one I'll touch on here , was also precipitated by Lebesgue, but thi s time by an uncharacteristic error. One o f his analyses of Baire functions included a 'trivial' lemma tha t th e projectio n o f a Borel set is Borel.23 This isn' t true , but th e sli p went unobserve d fo r a decade , unti l Mikhail Suslin , a young student in Moscow, burst into his professor's offic e wit h th e news. Together , studen t an d professor , Nikola i Luzin , established the elementary properties o f these 'analytic' sets, 24 which led, some years later, to the introduction of a new hierarchy of sets of reals:25 SQ = ITo = S} = II} =

th e open sets th e complements of So sets th e projections of HQ sets th e complements of S} sets

. . .

S^+i = th e projections of II* sets + 1 = the complements of S^+i sets LI* . . .

A* = set s that are both S^ and II* projective sets = th e union of the S^ The relationshi p betwee n Bore l and projectiv e sets was cinche d by Suslin, who showe d that the Borel sets are the A} sets. Despite thei r adde d complexity , projectiv e set s inheri t som e o f 22

I n Lebesgue (1902). For an elementary exposition, see Williamson (1962). Instea d o f set s o f point s o n th e line , thin k o f set s o f point s i n th e plane ; Borelness and s o on ca n be defined fo r these sets just as easily. Then th e projection of a Borel set in the plane is, so to speak, the shadow i t casts on the x-axis. 24 Se e Suslin (1917) and Luzin (1917). 25 Introduce d b y Luzin (1925) an d Sierpirisk i (1925) . Fo r th e classica l theor y of projective sets , se e Kuratowsk i (1966) , §§38—9 , o r Moschovaki s (1980) , §§!E , ch. 2, and parts of ch. 3. 23

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the regularit y properties o f Bore l sets . Lebesgu e measurability and the perfect subset property fo r analytic or £j set s were immediately established b y Luzi n an d Susli n an d separabilit y came a fe w years later.26 Separabilit y wa s extende d t o H i b y P . Noviko v i n hi s (1935), but afte r that, th e best efforts stalled : separability remained open pas t Ilj, measurabilit y beyond S}an d H\, and fo r the perfect subset property, a s Luzin remarked: 27 There remains here only one important gap : w e d o no t kno w i f every uncountabl e complement of a n analyti c se t [i.e . ever y uncountabl e ll\ set ] ha s th e [cardinality] o f th e continuum. ' Ther e th e matte r stoo d fo r som e years. Thus, fo r al l the thoroughl y admirable foundational goals pursue d by Freg e an d Dedekind , th e deepes t contribution s t o th e moder n mathematical theor y o f sets , thos e o f Geor g Cantor , wer e inspire d almost exclusively by mathematical concerns, particularl y concerns arising fro m analysis . I hav e trie d i n thi s sectio n t o giv e a hin t o f how Cantoria n se t theor y gre w ou t o f th e stud y o f rea l functions and t o sketc h i n the sort s o f question s tha t aros e naturall y in tha t context. Le t me turn no w t o how an d wh y thi s naive mathematical theory came to be axiomatized.

2. Axiomatizatio n To understan d th e impuls e tha t le d t o th e axiomatizatio n o f se t theory, w e mus t retur n t o Canto r an d hi s continuu m problem . Around th e tim e h e wa s makin g hi s optimisti c prediction, quote d earlier, about generalizing the Cantor-Bendixson theorem , anothe r letter t o Dedekin d contain s th e first hint of what wa s t o becom e a corner-stone o f hi s theory of infinit e numbers , namely, the concep t of well-ordering. 28 Again the catalyst was hi s earlier work o n set s of singular points . In orde r t o describ e th e proces s o f takin g accumulatio n point s o f sets o f accumulation points o f sets o f accumulatio n points, etc. , h e had proceede d a s follows from a given point set A: 26 27 28

I n Luzin (1927) . Luzi n (1925), as translated b y Hallett (1984), 108 . Se e Moore (1982) , 40-1. Th e letter i n question is dated 5 Nov. 1882

.

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AO = A

A! = { x x i s an accumulation point of A0} . . .

A n+1 = {x | x is an accumulation point of A n} .

.

.

A^ — the intersectio n of the A Ms AUJ+I = {x x i s an accumulation point of A w} and s o on . Now , i n th e earl y 1880s , hi s interes t shifte d fro m th e derived set s themselve s t o th e subscripts . Wh y shouldn' t ther e b e to + c o afte r al l th e o > + m, and (t o + co ) + 1 afte r that ? Her e i s a sequence tha t carrie s int o th e transfinite , an d afte r an y batc h o f entries, there is always a next.29 Considered i n term s o f cardinality , thi s transfinit e sequenc e o f ordinals yield s a wonderful bonus . Th e collectio n o f al l countabl e ordinals is not itsel f countable. I n fact, a s Cantor ha d show n b y the late 1890s, 30 th e cardinalit y o f thi s se t i s th e ver y nex t infinit e cardinality afte r tha t o f th e natura l numbers . Thus , h e calle d th e latter K 0 and th e former Kj. An d the se t of ordinals of cardinality K! has cardinalit y K 2 , an d s o on , s o th e infinit e sequenc e o f ordinal s yields a n infinit e sequenc e o f cardina l number s a s well . Canto r extended the arithmetic operations—plus, times, exponentiation— from th e finit e number s int o th e infinite , an d showe d tha t th e cardinality o f th e continuu m i s 2 . B y thi s point , then , th e continuum hypothesi s ha d become : 2 ° = Kj . But , fo r al l Cantor' s efforts, i t remained unproved . Then, i n 1904 , a t th e Thir d Internationa l Congres s i n Heidel berg, cam e a shock ; Juliu s Koni g rea d a pape r tha t purportedl y showed th e continuu m hypothesis t o b e false . I n particular, Konig argued tha t th e continuu m coul d no t b e well-ordere d a t all , an d ipso facto, tha t i t coul d no t b e pu t i n one-to-on e correspondenc e 29 Technically , a well-ordering of a set A is an ordering in which every non-empty subset of A has a least member. This produces the ide a i n the text: as we run throug h the element s o f A , a t an y poin t ther e i s a leas t membe r o f th e element s w e haven' t listed yet. Se e Enderton (1977) , 172-3. 30 Se e his last major work , Cantor (1895/7).

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with th e well-orderin g X j . A contemporar y repor t describe s th e scene this way:31 For Canto r t o clai m tha t ever y set can b e well-ordered and , i n particular, that th e continuu m ha s th e secon d [infinite ] cardinalit y wa s a kin d o f dogma tha t was part an d parcel of what h e knew and believe d in set theory. Consequently Konig' s address , whic h culminate d in th e propositio n tha t the continuu m could no t b e a n alep h (henc e coul d no t b e well-ordere d either), ha d a stunnin g effect , especiall y sinc e it s presentatio n wa s extremely elaborate and precise.

Apparently Canto r wa s les s upse t b y th e proo f itself , of whic h h e was sceptical, tha n h e was b y what h e saw as his public humiliation before hi s colleagues.32 His scepticism, at least, was well taken; on e of Konig' s assumptions , a 'theorem ' o f Feli x Bernstein , turne d ou t to b e incorrect . Thi s wa s pointe d ou t t o th e member s o f th e Congress, o n th e da y directl y afte r th e presentatio n o f Konig' s proof, b y Ernst Zermelo . But a doub t remained . A s earl y a s 1883 , jus t afte r th e above described tette r t o Dedekin d i n whic h th e notio n i s introduced , Cantor wrote that: 33 The concep t o f a well-ordered se t turn s ou t t o b e essentia l t o th e entir e theory o f point-sets. I t is always possible to brin g any well-defined se t int o the form o f a well-ordered se t ... thi s la w of thought appear s t o m e to b e fundamental, ric h i n consequences , an d particularl y marvelou s fo r it s general validity . . .

By 1895 , h e realize d th e nee d fo r a proo f o f thi s fundamenta l principle, an d se t out t o fin d one . Hi s lette r t o Dedekin d o f 189 9 contains on e attempt. 34 Now , thoug h Konig' s attac k ha d bee n unsuccessful, i t brough t hom e th e possibilit y tha t no t onl y th e continuum hypothesis , bu t th e well-orderin g principl e itsel f migh t one day be overthrown . 31

Schoenflies , a s quoted i n van Heijenoorr (ed. ) (1967), 192 . Se e Dauben (1979) , 247-50, 283, fo r discussion of this episode . 33 Canto r (1883) , as translated b y Moore (1982) , 42. Fo r an analysi s of Cantor' s thinking on well-ordering, see Hallett (1984), § 3.5. 34 I n fact , Cantor' s argument is very clos e to th e on e usuall y give n toda y (se e e.g. Drake (1974) , 56); i t goes throug h wit h minor modifications onc e Zermelo' s axiom has bee n identified . Apparentl y it wa s th e (unnecessary ) use of prope r classes—hi s 'inconsistent multiplicities'—tha t disturbe d Cantor . I n an y case , neithe r h e no r others (includin g Hilbert ) who sa w th e argument at th e time were convinced by it. In 1903, whe n i t wa s rediscovere d b y Jourdain , Canto r refuse d hav e hi s versio n published. See Moore (1982) , §§1.6 and 2. 1 for details. 32

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7

Before 190 4 was over, a proof wa s presented, no t by Cantor, but again b y Zermelo,35 The surprisingl y short argumen t depends o n a novel assumption : . . . that even fo r an infinit e totality o f sets there are always mapping s tha t associate wit h every set one of its elements . . .

By wa y o f defendin g thi s principl e o f choice—th e functio n 'chooses' one element from each set—Zermelo admits that This logica l principl e cannot , t o b e sure , b e reduce d t o a stil l simple r one.. .

but i n its favour: it is applied without hesitation everywhere i n mathematical deduction . . .

As an illustration, he cites the seemingly obvious fact that a set can't be divided into more non-empty disjoin t parts tha n it has members, a fact whose proof als o depends on choice. In thi s unassumin g manner , Zermel o propose s tha t mathemat icians accep t a principl e tha t i s simple , tha t wa s s o obviou s t o previous theorist s tha t the y use d i t unconsciously , an d tha t i s necessary t o prov e variou s importan t an d natura l result s (lik e th e well-ordering principle , th e theore m tha t al l infinit e cardinalitie s are alephs, and th e partition principl e just noted). H e could hardly have bee n prepare d fo r th e stor m o f controvers y tha t ensued . Gregory Moore' s extraordinar y histor y o f thi s perio d trace s th e complex reactio n throug h its independent manifestations in France, Germany, Hungary , England , Italy , Holland , an d th e Unite d States.36 Choic e an d a welte r o f othe r se t theoreti c principles , including well-orderin g an d unlimite d comprehension , wer e suddenly u p fo r grabs , embrace d here, denie d there . I t was i n defence of hi s principl e o f choic e an d hi s proo f o f well-orderin g tha t Zermelo wa s driven to axiomatize the practice of set theory. Zermelo's defenc e o f the principle of choice an d hi s axiomatization of se t theor y appea r i n tw o paper s writte n withi n day s o f on e 35 Zermel o (1904) . This shor t pape r bega n a s a lette r t o Hilber t one mont h afte r the Congres s an d wa s published later that year . All the quotation s in this paragraph come from p . 141 . 36 Moor e (1982) , ch , 2. M y brie f accoun t o f th e controvers y ove r choic e draw s heavily on Moore's work.

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another i n th e summe r o f 1907. 37 Th e firs t o f thes e contain s a spirited respons e to critics of what was now hi s axiom of choice. He admits again , a s h e ha d i n 1904 , tha t th e principl e ha s no t bee n proved, bu t points out : 'eve n in mathematics unprovability . . . is in no way equivalent to nonvalidity, since , after all , not everything can be proved , bu t ever y proo f i n tur n presuppose s unprove d prin ciples' (Zermel o (1908#) , 187) . Thus , eve n his opponents must rel y on unprove d axioms . How , then , ar e thes e axiom s justified?—'b y pointing ou t tha t th e principles are intuitively evident and necessar y for science . . .' (Zermel o (1908#) , 187) . Fro m ou r se t theoreti c realist's perspectiv e on mathematica l evidence, Zermelo is recognizing both intrinsi c supports-—in term s o f 'intuitiv e evidence'—an d extrinsic supports—i n term s o f th e rol e o f th e axio m i n overal l scientific theorizing. He proposes to apply these criteria to choice . To confir m th e intuitivenes s o f hi s axiom , Zermel o cite s historical evidence : That thi s axiom , eve n thoug h i t wa s neve r formulate d i n textboo k style , has frequentl y bee n used , an d successfull y a t that , i n the most divers e fields of mathematic s . .. i s a n indisputabl e fact , . . Suc h a n extensiv e us e o f a principle can be explained only by its self-evidence . . . (Zermelo (1908a), 187)

Cantor, Dedekind, an d th e other early set theorists ha d passed ove r numerous use s o f choic e i n variou s form s withou t comment , usually withou t noticin g i t themselves. 38 Unconsciou s application s of th e principl e ca n als o b e foun d i n analysis , particularl y i n th e work o f Baire , Borel , an d Lebesgu e touche d o n i n th e previou s section.39 This surel y constitutes som e evidenc e for its obviousness , and hence , fo r it s intuitiveness , though th e initia l protes t agains t Zermelo's linguisti c formulatio n remain s t o b e explaine d (se e p. 12 3 below) . Zermel o anticipate s the objectio n tha t evidenc e for intuitiveness should not b e counted as evidence for truth: No matte r i f this self-evidence i s to a certain degre e subjective—i t i s surely a necessar y sourc e o f mathematica l principle s . . . an d Peano' s 4 0 assertio n that i t ha s nothin g t o d o wit h mathematic s fail s t o d o justic e t o manifes t facts. (Zermel o (1908a) , 187 ) 37

Thes e are Zermelo (I908d ) and (19086) . Fo r examples, see Moore (1982) , § 1.4 . Se e Moore (1982) , § § 1. 7 and 4.1 . Fo r th e rol e o f th e axio m i n th e wor k o f Suslin and Luzin , see §§ 3.6 and 4.1 . 40 Pean o was among the critics of Zermelo's axiom. See Moore (1982) , §2.8 . 38

39

AXIOMATIZATION 119 Here h e appeal s t o th e undeniabl e fac t tha t intuitivenes s is ofte n taken, i n practice , a s evidenc e fo r truth . Thi s isn' t enoug h (a s observed i n th e thir d sectio n o f Chapte r 2) , bu t th e additiona l considerations cite d there can be brought t o bear . Turning t o extrinsi c supports , Zermel o give s his versio n o f th e role of set theory i n our overal l theory: Set theor y i s tha t branc h o f mathematic s whos e tas k i s t o investigat e mathematically the fundamenta l notion s 'number', 'order', and 'function' , taking the m i n thei r pristine , simpl e form , an d t o develo p thereb y th e logical foundation s o f al l arithmeti c an d analysis ; thu s i t constitute s a n indispensable component of the science of mathematics. (Zermelo (19086J, 200) Set theor y i s essentia l t o mathematics , especiall y arithmeti c an d analysis. I f w e ad d t o thi s th e Quine/Putnam-styl e clai m tha t mathematics i s essentia l t o ou r theor y o f th e world , w e hav e a n indispensability argumen t fo r se t theory . Th e questio n the n becomes, what versio n of set theory i s essential to mathematics, and in particular, does that versio n include choice? Zermelo argue s that it does: the questio n tha t ca n b e objectivel y decided , whethe r th e principl e i s necessary fo r science [b y which Zermel o mean s the scienc e of mathemat ics], I should no w lik e to submi t t o judgemen t b y presenting a numbe r of elementary an d fundamenta l theorems an d problem s that , i n m y opinion, could no t b e deal t wit h a t al l withou t th e principl e of choice . (Zermel o (1908*), 187-8 ) He goes o n to lis t a series of theorems fro m se t theory an d analysis , from whic h he concludes : Now s o lon g a s th e relativel y simpl e problem s mentione d her e remai n inaccessible to Peano' s [choiceless ] expedients, and s o long as, on the other hand, th e principl e of choic e canno t b e definitel y refuted , no on e ha s th e right t o preven t th e representative s o f productive scienc e fro m continuin g to us e this 'hypothesis'—a s one ma y call i t fo r all I care—and developin g its consequence s t o th e greates t extent , especiall y sinc e an y possibl e contradiction inheren t i n a give n point o f vie w can b e discovere d only i n that wa y . . . principles must b e judged fro m th e point of view of science, and no t scienc e fro m th e poin t of view of principles fixed once and fo r all . (Zermelo (1908*), 189) If choic e produce s a better , mor e effectiv e theor y tha n choiceles s

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mathematics, Zermel o counsel s that we opt fo r choice and jettison any unscientific prejudice that stands in our way . Zermelo's contention tha t mathematics without choic e woul d b e 'an artificiall y mutilate d science' 41 wa s substantiall y confirmed in the year s following . Th e implici t use s o f choic e i n analysi s were uncovered gradually , an d i t i s no w clea r tha t th e theor y o f rea l functions an d Bore l an d projectiv e set s sketche d i n th e previou s section would change significantly withou t a weak versio n of choice called 'dependen t choice ' an d woul d collaps e completel y withou t the eve n weake r 'countabl e choice'. 42 Th e theor y o f transfinit e cardinal numbers, on th e other hand, coul d hardl y survive without the ful l axio m o f choice. 43 And, to cit e just one mor e example , th e importance of the principle in algebra is dramatically demonstrated in the history of Bartel van der Waerden's classic textbook. The first edition, publishe d i n 1930 , include d th e axio m an d it s alread y considerable stor e o f algebrai c consequences. 44 Th e boo k itsel f stimulated furthe r productiv e research alon g these lines , but Dutc h opponents convince d va n der Waerden to omit choice in his second edition an d retur n t o mor e familia r methods . Algebraist s wer e appalled b y thi s 'mutilated ' versio n o f thei r discipline , an d th e axiom an d it s consequences wer e reinstate d by popula r deman d i n the third editio n in 1950.45 In th e cas e o f th e axio m o f choice , then , w e hav e ou r firs t example o f a n extrinsi c defenc e o f a se t theoreti c hypothesis , beginning wit h a straightforwar d indispensabilit y argument: ou r best theor y o f th e worl d require s arithmetic and analysis , an d ou r best theor y o f arithmeti c an d analysi s requires se t theor y wit h a t least th e axio m o f dependen t choice . Beyon d thi s pur e Quine / Putnamism, th e compromis e Platonis t find s th e sor t o f intra mathematical argument s tha t Code ! anticipates. 46 First , a s 41

Zermel o (1908^), 189 . Th e principl e of countabl e choice say s tha t ther e i s a choic e functio n fo r an y countable collectio n o f non-empty sets; dependen t choice say s that these choice s can be mad e i n suc h a wa y tha t eac h depend s o n th e previou s one . Se e Moore (1982) , 103 an d 325 , an d Moschovaki s (1980) , 423 an d 445 , fo r discussion s o f the rol e of these principles. 43 Fo r example , th e principl e that fo r any tw o cardina l number s K and \ , eithe r K = \ o r K < X o r X < K i s equivalent to th e ful l axio m o f choice. Se e Moore (1982), §4.3, and 330-1, for more. 44 Fo r a discussion of these, see Moore (1982) , §3.5 . 45 Se e Moore (1982), §4-5. 46 I n Godel (1947/64) , 477 . Al l the quotation s i n this paragraph com e fro m tha t location. 42

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Sierpiriski noted, 47 i t ha s a numbe r o f 'verifiabl e consequences' , that is, 'consequences demonstrable withou t th e new axiom, whos e proofs wit h th e help o f the ne w axiom , however , ar e considerabl y simpler and easier to discover . . .' Second, it yields a 'powerful method' for solvin g pre-existin g ope n problems , fo r exampl e th e well ordering question . And finally, it systematizes and greatly simplifie s the entir e theor y o f transfinit e cardina l numbers , 'sheddin g s o much ligh t upo n a whol e field . . . that. . . [it] would hav e t o be accepted a t leas t i n the sam e sens e as any well-established physical theory'. These are not, however, the only arguments mathematicians have given i n favou r o f th e axio m o f choice . T o trac e th e sourc e o f th e other mai n lin e o f defence , le t m e retur n t o th e tw o concept s o f collection discusse d i n th e fina l sectio n o f Chapte r 3 . Th e mathematical notion, originating in Cantor's thinking about set s of pre-existing points , eventuall y develope d int o th e ful l iterativ e conception o f Zermel o (1930) . Th e logica l notion , beginnin g with Frege's extensio n o f a concept , no w take s a numbe r o f differen t forms dependin g o n exactl y wha t sor t o f entit y provide s th e principle o f selection , bu t al l thes e hav e i n commo n th e ide a o f dividing absolutel y everythin g into tw o group s accordin g to som e sort of rule. In th e earl y 1900s , thes e tw o notion s ha d no t ye t bee n distinguished, an d thi s ambiguit y i s wha t produce d th e deepes t division ove r Zermelo' s principle . On e o f th e grea t ironie s of thi s entire historical episode is that the strongest negativ e reaction to the axiom cam e from th e very group o f French analysts—Baire , Borel, and Lebesgue—wh o unwittingl y use d i t with grea t frequenc y an d whose wor k provide s par t o f th e basi c indispensabilit y argument for a t leas t dependen t choice. 48 Ye t i t i s not har d t o se e how thi s happened. Th e effort s o f thes e analyst s were originall y motivate d by thei r doubt s abou t th e extremel y genera l notio n o f functio n proposed b y Dirichlet an d Riemann ; instea d the y concentrate d o n developing thei r hierarchie s o f function s definabl e b y acceptabl e mathematical means . Th e conflic t her e i s betwee n th e notio n o f functions a s completel y arbitrar y correspondences , on e fo r eac h possible combinatio n o f pairs o f reals, an d th e notio n o f function s as transformations , determine d b y som e sor t o f definitio n or rule . 47 48

I n Sierpiriski (1918) . For discussion, see Moore (1982), §4.1. Se e Moore (1982), § § 1.7, 2.3 , an d 4.1 .

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Transferred int o th e real m of sets, this is just th e contras t betwee n the mathematical and the logical notions of collection . The tru e shap e o f thi s conflic t emerge d i n th e aftermat h o f Zermelo's first proof, in a series of letters between the three analysts and their opponent, Jacques Hadamard.49 Ther e Lebesgue writes: to defin e a se t M i s t o nam e a propert y P whic h i s possesse d b y certai n elements o f a previousl y define d se t N an d whic h characterizes , b y definition, th e elements o f M. . . . The question comes down t o this, whic h is hardl y new : Ca n on e prove the existence o f a mathematical object without defining it ? . .. I believe that w e can onl y buil d solidly b y granting that it is impossible to demonstrate the existence of an object without defining it . (Bair e et al. (1905), 314 )

Hadamard's position was the opposite: . . . Zermel o provide s n o metho d t o carr y ou t effectively th e operatio n which h e mentions , an d i t remain s doubtfu l tha t anyon e wil l b e abl e t o supply suc h a method i n the future . Undoubtedly , it would hav e bee n mor e interesting to resolv e the problem i n this manner. Bu t the question pose d in this wa y (th e effectiv e determinatio n o f th e desire d correspondence ) i s nonetheless completel y distinc t from th e on e tha t we ar e examinin g (doe s such a correspondenc e exist?) . , .. Ca n on e prov e th e existenc e of a mathe matical objec t withou t definin g it ? I answer.. . in th e affirmative . . . . the existence . . . is a fact like any other . . . (Baire etal. (1905), 312, 317)

For Lebesgue and th e rest , the existenc e of a choice function, or t o put i t mor e simply , a choic e set, 50 depend s o n ou r havin g a rul e with whic h t o determin e wha t i s i n th e se t an d wha t isn't . Fo r Hadamard, wha t rules we have is irrelevant, purely psychological; a set either exists or it doesn't. Lebesgue's poin t o f vie w i s obviousl y mos t plausibl e o n th e logical notio n o f collection , th e notio n i n fac t suggeste d i n th e above quotation . I n contrast , th e mathematica l notion, accordin g to which , fo r an y give n things , ther e i s a se t consistin g o f an y combination o f thos e things , align s wit h Hadamard' s thinking . Given a collectio n o f non-empty , disjoin t set s forme d fro m som e batch o f things , a choic e se t wil l b e amon g th e combination s o f those origina l elements. From thi s point of view, Zermelo's axio m becomes obvious . During th e debat e ove r choice , th e notio n o f 49

Bair e etal. (1905). Th e axio m ca n b e rephrased t o say : fo r an y collectio n o f non-empty , disjoin t sets, ther e is a set that contains exactly one element fro m eac h of them . 50

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collection i n questio n wa s stil l ambiguous , whic h i s wh y som e found the principle obvious and others foun d it preposterous. Indeed, th e ver y dept h o f th e convictio n o n bot h side s suggest s that bot h notion s o f collectio n enjo y som e intuitiv e backing. Thi s would als o explain why many opponents o f choice continued to use it unaware s eve n afte r th e principl e ha d bee n isolate d an d th e controversy wa s joined ; recal l tha t intuitions , properl y s o called , are commo n t o (nearly ) all . O n thi s theory , then , Zermelo' s historical evidenc e does suppor t th e intuitivenes s of th e principle , but hi s linguisti c formulation me t wit h protes t becaus e th e wor d 'set' therei n wasn't alway s connected u p with th e appropriate pre linguistic intuitive beliefs. Fleshing this idea out require s a few more speculative wrinkles t o the perceptual/intuitive story told in Chapter 2 , but le t me indicate how i t migh t go . W e lear n t o perceiv e 'somethin g wit h a numbe r property'. Thi s concep t develop s i n th e cours e o f extensiv e childhood experienc e with manipulatin g and rearrangin g mediumsized physica l objects , s o i t i s inextricabl y linke d t o finit e combinatorial notions, fo r example the idea that in any order, ther e are stil l ten pennies , or tha t absolutel y an y proper subcollectio n of the pennie s wil l numbe r fewe r tha n ten . Thi s underlie s th e mathematical, combinatoria l notion . O n th e othe r hand , o n mos t occasions o f counting, the child counts things collected unde r some umbrella: th e boy s i n th e garden , th e pennie s i n m y pocket. Even seemingly rando m collection s ar e fo r th e mos t par t spatiall y circumscribed: th e thing s i n thi s box . Thi s aspec t o f numerica l experience underlie s th e logica l notion . Thu s th e intuitiv e concept of 'that which has a number' contains elements of both notions . Only late r d o we realize that thes e tw o notion s ca n com e apart . In finite cases, i t might be argued that an y combinatorial possibility is also determine d b y a property—the property 'being this thing or that thin g or . . . ' through a finite list of the members—though this approach wil l be problematic, fo r example o n account s o f scientifi c properties a s 'natural ' collections . Bu t infinit e case s rais e mor e pressing doubts. I s there suc h a rule for determining membership in each an d ever y combinatoriall y determine d subse t o f th e natura l numbers? I t isn't obviou s tha t ther e is . Yet the combinatorial ideal suggests tha t eac h an d ever y suc h subse t ca n b e counted , jus t a s finite collections ca n be counted eve n when n o non-trivial membership rul e i s available . Thi s clas h o f intuitions—betwee n 'ever y

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collection i s collected b y some property' and 'an y collection, wit h a common property o r not, ca n be counted'—leads to confusion. Eventually, theoretical considerations take over . Set s rather tha n classes make th e mos t workabl e and fruitfu l mathematica l entities, and function s understoo d a s arbitrar y mapping s provid e a n important flexibilit y tha t narrowe r notion s o f functio n canno t equal. B y thes e means , w e ar e le d t o th e conclusio n tha t bein g collected unde r on e umbrell a wa s a n accidenta l featur e o f th e 'things wit h numbe r properties' o f ou r childhoo d experience , an d not a n essentia l feature of all 'things with numbe r properties'. No t surprisingly, then , i t i s t o suc h theoretica l fact s tha t Hadamar d ultimately appeals: From th e inventio n of th e infinitesima l calculu s to th e present , i t seems to me, th e essentia l progress i n mathematic s ha s resulte d fro m successivel y annexing notion s which , fo r th e Greek s o r th e Renaissanc e geometers o r the predecessor s o f Riemann , were 'outsid e mathematics ' becaus e it wa s impossible to describe them. (Baireetal. (1905) , 318)

In th e year s sinc e thi s controvers y raged , th e mathematica l notion ha s bee n develope d an d accepted , becaus e o f it s effective ness, an d the Hadamard positio n has prevailed. Thus D . A. Martin , one of our leading contemporary se t theorists, writes : much o f th e traditiona l concern abou t th e axio m o f choic e i s probably based o n a confusio n between sets and definabl e properties . I n man y cases it appear s unlikel y that on e ca n define a choic e functio n fo r a particula r collection o f sets. But this is entirely unrelated to th e questio n of whether a choice function exists. Once this kind o f confusion i s avoided, the axio m of choice appear s a s one o f th e leas t problematic of th e se t theoretic axioms . (Martin, 'Sets versus classes' (unpublished), 1-2 )

This las t pro-choice argument , then, i s that objections to choice are based o n th e wron g notio n o f collection . I t depend s o n bot h th e intuitive evidence for choice assuming the mathematical notio n an d the theoretical evidence that th e mathematical notion i s the correc t one. I have concentrate d o n th e axio m o f choic e becaus e it s fascinatin g history provide s th e cleares t illustratio n o f th e interpla y betwee n intuitive and theoretica l supports fo r set theoretic hypotheses, bu t a similar analysi s ca n b e give n fo r eac h o f th e currentl y accepte d

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axioms o f Zermelo-Fraenke l se t theory . I won' t d o thi s here, 51 because my goal is illustrative rather tha n exhaustive , bu t le t me a t least indicate the extreme variet y of that list. Pairing—for an y tw o things there is a set with exactly those members—and Union—any sets ca n b e combine d int o a se t wit h al l thei r member s a s members—have bee n cite d earlier 52 a s example s o f nearl y un adorned intuitions . I n contrast , th e axio m o f infinity—ther e i s an infinite set—proclaim s the bol d an d revolutionar y hypothesi s tha t led Canto r int o hi s paradise an d th e res t o f u s wit h him . There i s nothing obvious about it, but it launched modern mathematics, and the succes s an d fruitfulnes s o f tha t endeavou r provide s it s purely theoretical justification . 3. Ope n problems By th e mid-1930s , th e fundamenta l assumption s underlyin g se t theoretic practice had bee n codified int o a simple axiomatic system , ZFC, whic h wa s stron g enoug h t o impl y th e know n theorem s o f classical number theory an d analysi s and pre-axiomatic set theory . The well-orderabilit y o f the real s was provabl e within thi s theory , leaving the continuum hypothesis i n Cantor's preferre d form: Kj = 2 ° . Bu t th e questio n remaine d open , a s di d thos e raise d b y Susli n and Luzin : i s there a n uncountabl e 11 } set wit h n o perfec t subset? Are all 2l and H\ sets Lebesgue measurable? Can disjoint 2] or II] sets be separated? Developments i n logi c durin g th e twentie s an d earl y thirties , especially Godel' s completenes s an d incompletenes s theorem s o f 1930 an d 1931, 53 raised th e possibility o f a new sort of proof— a proof o f unprovability—an d i t wa s i n this directio n tha t progres s on these open problems was first made. The most dramatic result of Godel's wor k i n th e lat e thirtie s wa s hi s demonstratio n tha t th e 51

Madd y (1988a ) contains a summary of many of these arguments. I n Ch. 2, sect. 3, above. 53 Gode l (1930 ; 1931) . The completeness theore m establishe s that every logically valid formul a i s provable . (Se e Enderton (1972) , ch . 2 , fo r a now-standar d proo f using th e alternativ e method s o f Henki n (1949). ) Th e firs t incompletenes s theorem shows tha t ther e ar e sentence s expressibl e i n th e languag e o f ZF C tha t aren' t provable o r disprovabl e fro m ZFC . Th e secon d incompletenes s theore m give s a n example of such a sentence, namely, the on e expressing the consistenc y of ZFC. (Se e Enderton (1972) , ch. 3.) 52

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continuum hypothesi s ca n no t b e disprove d fro m th e axiom s o f ZFC.54 A t th e sam e time , Gode l note d som e consequence s fo r analysis whic h wer e finall y prove d som e tim e late r b y Joh n Addison:55 in ZFC, it cannot b e proved that al l uncountable Fl{ sets have perfect subsets or that al l A] sets are Lebesgue measurable. O n the questio n o f separation , Addiso n use d simila r methods t o sho w that ZF C doe s no t impl y eithe r th e separabilit y of X ] set s o r th e non-separability of FI * sets. 56 The possibilit y remaine d tha t th e axiom s o f ZF C woul d b e enough t o establish th e continuum hypothesis a s true, bu t Godel fo r one di d no t expec t this . Thi s conjectur e wa s base d partl y o n evidence tha t ZF C i s too wea k t o decid e th e questio n a t all , and partly o n hi s strong convictio n tha t the continuu m hypothesi s is in fact false : 'certai n fact s (no t known a t Cantor' s time ) . . . seem t o indicate tha t Cantor' s conjectur e wil l tur n ou t t o b e wron g . . .' (1947/64), p . 479). Thes e fact s consis t o f 'highl y implausibl e consequences o f th e continuu m hypothesis' , o f whic h Gode l list s several. Thus we find Godel arguin g against Cantor's conjectur e on extrinsic grounds , i n term s o f it s purportedl y undesirabl e consequences. The questio n is , wha t make s thes e particula r consequence s undesirable? Man y o f the m depen d o n th e ide a tha t set s o f real s which ar e larg e i n numbe r shoul d no t als o b e smal l i n measure theoretic terms . Ther e migh t wel l b e som e intuitiv e belie f lurkin g behind thi s vague principle, but i f so, it is extremely undependable : an elementar y theore m o f measur e theor y show s tha t ther e ar e uncountable set s o f Lebesgu e measure zero ; th e standar d exampl e goes bac k t o Cantor . Give n tha t th e suggeste d principl e ha s bee n discredited, i t might b e argued tha t Gode l i s relying on som e other , more dependabl e intuition . I f so, w e shoul d expec t mos t othe r se t theorists t o shar e hi s views , bu t the y don't . Martin , fo r example , 54 Unles s ZF C i s inconsistent; anythin g can b e bot h prove d an d disprove d i n a n inconsistent system . (Terminology : t o disprov e is to prov e th e negation , s o a proo f that C H canno t b e disprove d i s a proo f tha t it s negatio n canno t b e proved. ) Se e Godel (1940) . 55 I n Addiso n (1959) . Othe r partia l result s b y Mostowsl a an d Kuratowsk i wer e destroyed durin g th e war . Se e Addison's pape r fo r th e complicate d histor y o f these theorems. Moschovakis (1980), § 5A, presents Addison's results. 56 I n Addison (1958) .

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writes:57 'While Godel's intuition s should neve r b e taken lightly , it is ... har d fo r some o f us to se e why the example s Godel cite s are implausible a t all ' (Marti n (1976) , 87) . Th e final , mos t likel y possibility i s tha t Gode l i s relying , no t o n intuition , bu t o n hi s mathematical experience , exercisin g th e sor t o f theoretica l judge ment tha t produce s th e natura l scientist' s hunc h tha t a theor y of this sor t rathe r tha n tha t i s th e kin d tha t ough t t o work . Unfortunately, Godel' s effort s t o pi n dow n hi s idea s hav e sinc e proved unsuccessful. 58 But whateve r hi s reasons , Gode l wa s correc t i n hi s predictio n that th e continuum hypothesis would b e shown no t to follow from ZFC. Thi s resul t was finally established by Paul Cohen i n 1963. 59 Solovay the n extende d Cohen' s metho d t o trea t question s fro m analysis: ZF C canno t disprov e tha t al l uncountabl e Fl ] set s have perfect subset s o r tha t al l £] an d H ] set s ar e Lebesgu e measurable.60 Thi s inconclusiv e pictur e wa s complete d b y Le o Harring ton,61 who used Cohen' s metho d t o sho w tha t ZF C doe s not yield the separation property fo r X] or Fl] . Thus thes e various problems are ope n i n a n entirel y ne w sens e o f th e word ; provably , the y cannot b e decide d o n th e basi s of the standar d assumption s of set theory. Many hav e bee n provoke d t o philosophica l extreme s b y th e thought o f question s this open. Th e if-thenist , for example , simply declares al l such problems t o b e solved; what w e wanted to know , after all , was whether o r not th e continuum hypothesis and the rest follow fro m ZFC . Bu t whatever machinations might be available to her opponents, th e Platonist's position is clear. In Godel's words : It i s t o b e noted , however , tha t o n th e basi s o f th e poin t o f vie w her e 57 Se e also Marti n an d Solova y (1970) , 177 . Moor e (forthcoming ) attributes a similar lac k o f agreemen t t o Cohen . A n exceptio n t o thi s rul e i s Nyiko s (forthcoming), wh o agree s wit h Gode l tha t a t leas t on e o f thes e example s i s extremely implausible. 58 Fo r an account of these efforts, se e Moore (forthcoming), § 6. 59 Th e published version is Cohen (1966) . 60 Se e Solovay (1970) . Solovay' s result actuall y depend s o n th e relativel y weak additional assumptio n tha t th e existenc e o f a n inaccessibl e cardina l (se e th e nex t section) cannot be refuted i n ZFC. Moschovaki s writes , 'I n the present context this is surely a reasonable assumption ' (1980) , p. 284), an d I know o f no dissent fro m this view. 61 Se e Moschovakis (1980), 284 .

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adopted, a proo f o f th e undecidabilit y o f Cantor' s conjecture fro m th e accepted axiom s o f se t theor y . . . would b y n o mean s solv e th e problem . For i f th e meaning s o f th e primitiv e term s o f se t theor y [base d o n th e iterative conception ] ar e accepte d a s sound , i t follow s tha t th e set theoretical concepts an d theorem s describ e som e well-determine d reality , in whic h Cantor' s conjectur e mus t b e eithe r tru e o r false . Henc e it s undecidability fro m th e axiom s bein g assume d toda y ca n onl y mea n tha t these axiom s d o no t contai n a complete descriptio n of that reality . (Godel (1947/64), 476)

For th e se t theoreti c realist , the worl d consist s o f physica l objects , sets o f these , set s o f physica l object s an d sets , an d s o on , throug h the transfinit e level s of the iterativ e hierarchy. There i s a fac t o f th e matter abou t th e cardinalit y o f th e se t o f Dedekin d cuts— a se t of pairs of sets of sets of pairs of von Neumann ordinals—an d it is the set theorist's jo b to discover it. 62 And ho w migh t thi s elusive fac t b e ascertained? Godel's answer , and th e se t theoretic realist's , i s that w e nee d t o fin d ne w axioms , axioms we ca n justif y jus t a s Zermelo's axiom s wer e justified , b y a combination o f intrinsi c an d extrinsi c considerations. 63 Thi s approach i s seconde d b y ou r curren t se t theorists . Fo r example , Martin writes : 'Althoug h th e ZF C axiom s ar e insufficien t t o settl e CH, ther e i s nothin g sacre d abou t thes e axioms , an d on e migh t hope to find further axioms . . .' (Marti n (1976) , 84). Considerable work ha s bee n don e o n thi s project i n recent years , som e o f which will be described i n the next section . In anothe r attac k o n Godel' s ideas , Chihar a take s thi s situation as evidence against Platonism. 64 Godel' s an d Cohen' s result s sho w that bot h ZF C + CH an d ZF C + not-C H ar e consisten t theories , perhaps equally worthy o f investigation. 65 Chihara concludes tha t For Godel, . , . the proliferation o f set theories poses the thorny problem of 62 O f course , se t theor y i s as fallibl e a s an y othe r science , an d i t coul d tur n ou t that th e continuu m questio n i s base d o n fault y presuppositions , but ther e i s n o conclusive reason t o believe this now . 63 Se e G6del{ 1947/64), 476-7. 64 Se e Chihar a (1973) , 63-5 . Al l quotation s i n thi s paragrap h an d th e nex t come fro m thi s location. 65 I n fact , neithe r o f thes e theorie s i s much studied, a s such , because neithe r C H nor not-C H is considered a viable axiom candidate. Neither is intuitive and neithe r is sufficiently fruitfu l t o meri t acceptance o n extrinsi c grounds, though C H i s better off than not-C H i n thi s regard : Sierpirisk i (1934 ) derive s eighty-tw o proposition s fro m CH, non e o f whic h i s know n t o b e settle d b y not-C H (se e Marti n an d Solova y (1970), 143).

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determining whic h o f th e man y se t theorie s [on e fo r eac h possibl e cardinality o f the continuum ] i s the one that most truly describe s th e real world of sets. (Chihara (1973), 65)

Now n o Platonist would den y that th e continuum hypothesis poses a 'thorn y problem'—it ha s engage d many of the bes t se t theoretic minds since Cantor's—but it isn't immediatel y clear why this casts doubt on Platonism. After all , scientific realism leaves us with many thorny problems, fro m th e shape of the universe to the existence of a fre e quark , an d n o on e count s thi s a s evidenc e that ther e i s n o objective physical world. On th e Platonist' s view , ther e i s a rea l an d extremel y difficul t problem abou t th e cardinality of the continuum. Chihar a seem s to hold that this fact counts in favour o f alternative philosophies of set theory fo r whic h th e continuu m proble m present s n o suc h challenge. Fo r example , h e suggest s a s a 'reasonabl e option ' hi s 'mythological Platonism', which takes the continuum hypothesis to be analogou s t o 'Hamlet' s nose i s three inche s long', that is , to b e neither tru e no r false . Bu t ho w reasonabl e woul d i t b e fo r th e physicist t o solv e th e questio n o f th e fre e quar k b y adoptin g a philosophy o f physics according to which it is no longer a problem? In fact, thorn y problems are the life-blood of science, its motivator, and set theory is no different fro m th e rest. If w e ar e t o loo k toward s ne w axiom s fo r a solutio n t o th e continuum problem , i t i s wort h askin g i n whic h directio n tha t solution migh t be expected. Opinio n o n thi s matter i s divided, but opinion ther e is . N o on e pretend s t o hav e anythin g resemblin g conclusive evidence for any alternative, but a brief look a t the range of considerations offered wil l give the flavour of the debate. Cantor, o f course, held the continuum hypothesis to be true, and occasionally, durin g hi s man y efforts , eve n believe d tha t h e ha d proved it. 66 On e o f th e force s behin d hi s stron g convictio n ma y have been his confidence that th e partial solutio n containe d i n th e Cantor-Bendixson theore m coul d b e generalized. 67 T o a certai n extent, we'v e see n tha t thi s confidenc e wa s wel l placed : Cantor — Bendixson wa s extended , mos t dramaticall y b y Susli n t o th e S J sets. Thus, despite the fact that there are far fewer Si sets than there 66

Fo r examples, see Moore (1982) , 42-5, an d Hallett (1984), 92. Canto r had other reasons, too. Se e Maddy (1988a) , 490—2, and the reference s cited there. 67

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are arbitrar y set s o f reals, 68 thi s partia l resul t affirmin g th e continuum hypothesi s fo r a wid e rang e o f sampl e set s migh t b e taken as confirmation for the hypothesis in general. Unfortunately, whateve r confirmin g evidence Suslin's result ma y have promised, i t is severely undercut by the details of the argument itself. What h e actually shows i s that every uncountable Xj se t has a perfect subset . The trouble is that some uncountable sets don't have perfect subsets , s o there i s actually no hop e o f generalizing Suslin's result on 2} set s to all sets of reals. In Martin's words : 'Thus , while our simpl e [£{] sets have the cardinalities required b y CH, thi s is so because the y hav e a n atypical property , th e perfec t subse t prop erty' (Marti n (1976) , 88) . I n Cantor' s defence , i t shoul d b e note d that mos t o f th e set s h e wa s familia r wit h wer e S { a t worst , an d Bernstein's theore m o n uncountabl e set s withou t perfec t subset s didn't appear until 1908 . If Cantor' s reason s fo r believin g the continuu m hypothesi s ar e ultimately unpersuasive , GodeP s reason s fo r disbelievin g i t hav e also draw n fe w converts . B y contrast , th e sentiment s o f anothe r major playe r i n thi s dram a procee d alon g line s that man y see m t o find mor e plausible . Cohen' s thinkin g depend s o n a contras t between tw o way s in which large r cardinal s can b e generated fro m smaller ones . One metho d build s up fro m below . Cantor' s origina l procedur e for buildin g ever large r ordinal s depende d o n thre e principle s of generation.69 Th e firs t allow s th e passag e fro m on e numbe r t o it s successor, fro m 2 t o 3 , fro m 3 t o 4 . Ther e i s no larges t numbe r in this series , s o Cantor' s secon d principl e generate s thei r 'limit' , th e next numbe r after the m all , that is , co. Fro m here, th e first principle yields co -f 1 , co + 2 , etc. , an d th e second , t o + to . And s o on. But al l ordinals generate d b y these processe s ar e still countable. Th e thir d principle tells us that after al l ordinals o f a certain cardinality, there is a next , i n thi s case , CO] . Thi s ordina l i s uncountable; its cardinal number i s NI. These method s produc e a sequenc e o f infinit e cardina l num bers—K0, K! , K 2, N 3, etc.—bu t the y canno t tak e u s beyon d thi s point. T o buil d u p furthe r fro m below , w e nee d th e axio m o f replacement: give n an y set , i f eac h o f it s element s i s replace d b y something else , the resul t is still a set . Thus i f (0, 1 , 2, 3 , . . . } is a 69

Ther e are 2*° 2,} sets of reals and 2 2 " sets of reals altogether. Canto r (1883) . Se e Daube n (1979) , 96-9 , o r Hallet t (1984) , §2.1 , fo r discussion. 69

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set, a s the axio m o f infinity guarantees , and 0 is replaced b y the set of finit e ordinals , 1 by the se t of countable ordinals, 2 by the se t of ordinals of size Kl 5 and s o on, we have the set whose elements have the cardinalitie s K0, K l 5 K 2, an d s o on . I f we tak e th e unio n o f al l these sets, a s the union axiom say s we can, the result is a set whose cardinality i s th e nex t larges t afte r K 0, X l 5 N 2, etc . Thi s i s K w . Obviously, this process can be continued. The secon d wa y o f generatin g large r cardinalitie s i s ver y different. Beginnin g again with the set {0, 1, 2, 3, . . .}, this time we take the se t of its subsets, as allowed b y the powe r se t axiom. On e of Cantor' s mos t beautifu l theorem s show s tha t th e powe r se t of any set has a larger cardinality than the set itself;70 in this case, that larger cardinalit y i s 2 , als o th e cardina l o f th e se t o f reals. 71 Taking th e powe r se t o f thi s power se t yield s a se t o f cardinality 22 °, and s o on. Cohen's ide a i s simpl y tha t powe r se t i s stronge r tha n an y principle for building up fro m below. 72 In his book establishin g the unprovability of the continuum hypothesis, he writes: A point o f view which the autho r feel s ma y eventually come to b e accepte d is that C H i s obviously false . . . . Nj i s the se t of countable ordinal s an d thi s is merel y a specia l an d th e simples t wa y o f generatin g a highe r cardinal . The se t [o f subsets o f th e natura l numbers ] is , in contrast , generate d b y a totally new and more powerful principle, namel y the Power Set Axiom. I t is unreasonable t o expec t tha t an y descriptio n o f a large r cardina l whic h attempts t o buil d u p tha t cardina l fro m idea s derivin g fro m th e Replacement Axio m ca n eve n reac h [ a set o f siz e 2 ] . Thus [ 2 °] is greate r than K n, X w, X a, wher e a. = X M, etc . Thi s poin t o f vie w regard s [th e power se t of the set of natural numbers ] as an incredibly rich set given to us by one bol d ne w axiom , whic h ca n neve r b e approached b y any piecemea l process o f construction. (Cohe n (1966) , 151)

This lin e of thought harmonize s with variou s others insistin g that the continuu m hypothesi s place s a n artificia l an d unwarrante d restriction o n th e numbe r o f reals. 73 Mos t suc h thinkers—the y include Martin74 —feel that the continuum is likely to be quite large compared wit h K t . Thi s set s the m apar t fro m Godel , who , while 70

Canto r (1891) . See Enderton (1977) , 132-3. Se e Enderton (1977) , 149 . 72 Daube n (1979), 269, trace s this way of thinking to Baire. 73 Se e Maddy (1988a) , § n.3.4. This paper give s a more complete and detailed list of the various arguments for and against Cantor's hypothesis. 74 Se e e.g. hi s Trojectiv e set s an d cardina l numbers ' (unpublished) , 2. Som e of Martin's views are reported i n Maddy (1988a) , § v.4. 71

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rejecting Cantor' s hypothesis , seeme d t o lea n toward s a relatively small continuum of size X2.75 So the axiomatizatio n o f se t theory produce d a rang e o f problem s more ope n tha n ha d previousl y bee n possible , tha t is , problem s neither provable no r disprovabl e fro m th e accepted assumption s in the field . Th e mos t famou s o f thes e i s Cantor' s continuu m hypothesis, bu t variou s other s appeare d amon g th e natura l questions aske d b y analyst s in th e twentie s and thirties . From th e Platonist's perspective , ther e i s good reaso n t o believ e that thes e questions nevertheles s hav e unambiguou s answers ; som e eve n proffer opinion s abou t wha t thos e answer s migh t be . The hop e is for new , strongl y supporte d axiom s tha t wil l resolve these difficul t issues. 4. Competin g theories In th e curren t se t theoreti c landscape , tw o opposin g theoretica l approaches dominat e effort s t o solv e th e profoundl y ope n prob lems described above. My goa l in this section is to give an overview of each , wit h specia l attentio n t o perceive d strength s an d weak nesses. I n th e fina l sectio n o f thi s chapter , I'l l tur n t o th e philosophical questions raised by this controversy. The mos t complet e an d concis e o f thes e tw o theorie s stems fro m Godel's proo f tha t th e continuu m hypothesis cannot b e disprove d from th e axiom s o f ZFC . T o sho w this , Gode l describe d a simplified worl d o f set s i n whic h al l th e axioms , an d henc e al l consequences o f th e axioms , ar e true . Bu t i n thi s world , K j = 2 ', so the negation o f the continuum hypothesis cannot be proved fro m ZFC.76 This simplifie d worl d o f Godel' s i s arrange d i n a hierarch y o f stages, jus t lik e th e standar d iterativ e hierarchy, and ther e i s on e stage fo r ever y ordinal number, again i n imitatio n of th e standar d picture. The differenc e i s that a t any given stage, instead of forming all possible subsets of what ha s bee n given so far, in Godel's world See Moore (forthcoming), esp. § 6. 76 Se e Drake (1974), ch. 5, or jech (1978) , §§ 12-13, fo r textbook presentation s of this argument.

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one add s onl y thos e subset s explicitl y definabl e b y predicative 77 formulas. This procedure yields the constructible universe , called L, and th e clai m tha t th e constructibl e univers e is the rea l universe , written V = L , is the axiom o f constructibility. Godel's work show s that addin g th e axio m o f constructibilit y t o ZF C can' t introduc e any contradictions that weren't already present in ZFC alone. So on e liv e theoretica l optio n i s jus t that : ad d V = L t o th e assumptions of ZFC. Thi s move produces a theory so powerful tha t the axio m o f choic e i s no longe r needed ; i t can b e proved. I n fact , not onl y i s ever y se t well-orderable— a conditio n equivalen t t o choice78—but th e entir e univers e can b e arranged i n a gian t wellordering—a conditio n calle d 'globa l choice' . Thi s hold s i n L because th e relevan t formula s ca n b e well-ordere d a t eac h stage , which produce s i n tur n a well-orderin g o f th e set s introduce d a t that stage. The well-orderings of the stages are then strung together to produce a well-ordering of the entire constructible universe. The axiom als o ha s strikin g effects i n the are a o f cardina l arithmetic; it implies no t onl y th e continuu m hypothesis , bu t th e generalize d continuum hypothesi s a s well , tha t is , tha t N a + i = 2 Kot, fo r al l ordinals a. The open question s from analysi s are also decided in ZFC + V = L. Addison's unprovability results, like Godel's, ar e established by showing tha t th e negation s o f th e proposition s i n questio n follo w from th e axio m o f constructibility . I n th e constructibl e universe, there i s a n uncountabl e Fl j se t wit h n o perfec t subset , a nonLebesgue-measurable A 2 set , and separatio n hold s fo r FI^ , Fl^ , Us, an d s o on . Al l thes e follo w fro m th e existenc e o f a particularly simpl e well-orderin g o f th e reals ; a s a subse t o f th e plane, it is A£. So V = L answers all our outstandin g questions . Indeed , furthe r elaborations, largel y du e t o Ronal d Jensen , settl e nearl y al l important se t theoretic questions, and som e from othe r branches of mathematics a s well. 79 Her e i s a n axio m tha t clearl y provide s 77

Tha t is, formulas that only refe r t o sets formed a t previous stages . SeeEnderto n (1977), 151-4, 196-7, 199. Devli n (1977 ) make s th e cas e fo r V = L' s efficac y bot h insid e an d outsid e set theory. Th e importance of Jensen's contribution t o constructibilit y theory comes ou t in th e historica l note s t o Devlin' s compendiu m (1984 ) an d i n th e introduction : 'without hi s wor k ther e woul d hav e bee n practicall y nothin g t o writ e about! ' (p . viii). 78

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'powerful method s fo r solvin g problems'. 80 I t i s als o a saf e assumption; a s remarked above , i t engenders no contradictions tha t wouldn't alread y follow from ZF C alone. There ar e even those wh o find i t a 'natural ' assumption , beginnin g wit h Godel , wh o intro duced it this way: The propositio n [ V = L ] adde d a s a ne w axio m seem s t o giv e a natura l completion o f the axiom s of se t theory, in so fa r a s it determines the vagu e notion o f an arbitrar y infinite set in a definite way. (Godel (1938), 557)

Keith Devli n goe s eve n further , claimin g constructibilit y t o b e 'closely boun d u p wit h wha t w e mea n b y "set " ' , bu t h e defend s this b y identifyin g set s wit h extension s o f propertie s rathe r tha n combinatorially defined collections. 81 Despite th e remar k jus t quoted, Gode l soo n cam e t o rejec t V = L, an d despit e it s strengths , constructibilit y toda y ha s mor e detractors tha n supporters . Th e mos t fundamenta l reaso n fo r resistance t o th e axio m i s implici t in m y crud e sketch : instea d o f forming al l possible subsets of what has bee n given so far, one add s only thos e subsets explicitl y definabl e by predicative formulas. Thi s requirement o n subset s clearly violates the combinatoria l ide a tha t every possible collection b e formed, regardles s of whether ther e is a rule for determining which previously given items are members an d which ar e not . Devlin, t o bolste r hi s case , concoct s a cleve r compromise betwee n th e logica l an d th e mathematica l notion s o f collection—he proposes tha t extension s of properties b e formed in stages,82 thu s avoidin g inconsistency—but th e combinatoria l idea he reject s stand s a t th e en d o f a clea r historica l tren d i n mathematics, fro m function s an d collection s determine d b y rules towards function s an d collection s determined arbitrarily. Most set theorists hav e adopte d th e ful l iterativ e conceptio n an d thu s fin d the axio m o f constructibilit y an artificia l restriction . For example , Moschovakis writes : The key argument against accepting V = L ... i s that the axiom of constructibility appear s t o restric t undul y th e notio n o f arbitrary s e t . . . there i s n o 80

Gode l (1947/64) , 477. Se e ch . 3 , sect . 3 , abov e fo r thi s distinction . Th e quotatio n i s fro m Devli n (1977), p . iv . Th e relevan t notio n o f se t i s sketche d o n pp . 13-18 . Fraenkel , Bar-Hillel, an d Lev y (1973) , 108-9 , also cite a rang e o f consideration s i n favou r o f V = L. 82 Se e Devlin (1977) , 27-8. 81

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a prior i reaso n wh y every subset . .. should b e definabl e . .. (Moschovakis (1980), 610) And Godel : '[ V = L ] states a minimum property. Not e that onl y a maximum propert y woul d see m t o harmoniz e wit h th e concep t o f set. . .' (Gode l (1947/64) , 479) . Man y other s expres s opinion s alon g these lines. 83 Further argument s agains t th e axio m o f constructibility focu s o n its consequences . O f course , anythin g that migh t coun t a s a reaso n for disbelievin g th e continuu m hypothesi s woul d likewis e coun t against V = L ; it may be that Godel' s change o f mind was partially motivated i n thi s way . Thoug h anti-constructibilit y argument s o f this for m ar e sometime s offered , mor e commo n an d concret e objections involv e the consequence s o f V = L for the occurrenc e of so-called pathologie s lo w dow n i n the projective hierarchy , amon g the fairl y simpl e set s o f reals . Fo r example , th e axio m o f choic e implies that the real numbers can be well-ordered, bu t there is every reason t o suppos e tha t th e simples t such well-ordering (a s a subset of th e plane ) i s an extremel y complex set . Provably , it cannot b e as simple as £{, but i n the presence o f the axio m o f constructibility, it is A|. This means that, beginning from a set too simpl e to be a wellordering o f the reals , suc h a well-orderin g ca n b e obtained b y on e application of complementation and on e applicatio n o f projection , a prospec t whic h seem s highly unlikely to many . Indee d man y se t theorists fee l tha t such a choice-generated oddit y shoul d no t appea r anywhere amon g th e projectiv e sets . Th e sam e sor t o f thinkin g applies t o the uncountable I I j set with n o perfect subset an d the Al non-Lebesgue-measurable set ; thoug h choic e guarantee s tha t suc h sets exist, they should no t b e so simple.84 The secon d liv e theor y begin s fro m a styl e o f axio m fo r whic h Godel had high hopes, namel y the large cardinal axiom. Th e first of these, the axiom of inaccessible cardinals, roughly speaking , mean s nothin g els e bu t tha t th e totalit y o f set s obtainable b y us e o f th e procedure s o f formatio n o f set s expresse d i n th e other axiom s form s agai n a se t (and , therefore, a ne w basi s fo r furthe r applications of these procedures) . . . {Godel (1947/64), 476) 83

Se e e.g. Drake (1974), 131, Scot t (1977), p. xii, and Wang (I974a), 547 . Opinion s o f thi s sor t ca n b e foun d i n Marti n (1977) , 811 , (1976) , 88 , an d 'Projective set s an d cardina l members', p. 2; Moschovaki s (1980) , 276; an d Wang (1974a),547. 84

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The ide a i s tha t th e tw o mai n operation s fo r generatin g ne w set s from ol d postulated b y ZFC—replacement and power set—are not enough t o exhaus t al l the ordinals. 85 Obviously thi s thinking is ripe for generalization . Suc h axioms , accordin g t o Godel , sho w tha t ZFC 'ca n b e supplemente d withou t arbitrarines s b y ne w axiom s which onl y unfol d th e conten t o f th e [iterativ e conceptio n o f set] . . .' (Gode l (1947/64) , 477) . H e propose d larg e cardinal axiom s as a cure for the sort of open problem s we've been considering: It is not impossibl e that. . . some completenes s theore m woul d hol d whic h would say that ever y proposition expressibl e in set theory i s decidable from the presen t axiom s plu s som e tru e assertio n abou t th e largenes s o f th e universe of all sets. (Gode l (1946), 85)

By th e sixties , larg e cardinal s o f ever-increasin g siz e were a boo m industry. The mos t strikin g applicatio n o f inaccessible s themselve s i s Solovay's theorem , mentione d above , applyin g Cohen's metho d t o questions i n analysis . Thi s importan t wor k presuppose s th e irrefutability, i f not th e existence, of an inaccessibl e cardinal. Other small larg e cardinal s hav e consequence s fo r Bore l sets, 86 bu t th e most dramaticall y effectiv e larg e cardina l axio m postulate s th e existence o f a muc h large r measurabl e cardinal. 87 T o giv e jus t a modest hin t o f it s size , ther e ar e inaccessibl y man y inaccessibl e cardinals smaller than th e first measurable. The stronges t argument s fo r th e assumptio n o f measurabl e cardinals ar e extrinsi c ones, mos t notabl y Dan a Scott' s discover y that i t implie s V ^L. 88 Thu s th e stron g consideration s agains t th e axiom o f constructibilit y al l retur n a s extrinsi c support s fo r th e existence o f a measurabl e cardinal . And , onc e again , Solova y coaxed ou t result s i n analysis . H e showe d tha t i n additio n t o refuting constructibiiity , a measurabl e cardinal yields the preferre d results for projective sets: every uncountable ^\ set nas a perfect subset, ever y Si and FI ^ se t is Lebesgue measurable, and ther e is no 85 Inaccessible s were introduce d i n Zermelo (1930 ) an d i n Sierpirisk i an d Tarsk i (1930). Fo r th e sor t o f argumen t i n thei r favou r considere d here , se e als o Drak e (1974), 267 , an d Wan g (1974a) , 554 , Fo r argument s o f othe r types , se e Madd y (1988a), 501-4. 86 Se e th e detaile d an d innovativ e wor k o f Harve y Friedman , describe d i n Harrington e t al. (1985). 87 Measurabl e cardinal s were first proposed b y Ula m (1930) . See Drake (1974) , chs. 6 and 8, or Jech (1978), ch. 5, for textbook discussions. 88 Scot t (1961).

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A] well-ordering o f th e reals. 89 Bu t fo r al l the goo d news , ther e is also bad. The most conspicuou s open problem remain s so; Solovay and Levy' s application o f Cohen' s metho d show s tha t measurabl e cardinals cannot decide the continuum hypothesis.90 Then, in the late sixties, the appearance of a short paper by David Blackwell91 produce d a surg e o f interes t i n hypothese s o f a completely different sort , hypotheses given in game-theoretic terms. The notio n o f a n infinit e gam e wa s firs t introduce d b y th e Polis h school i n the thirties. Such a game can b e based on any set A of real numbers betwee n 0 an d 1 . Imagine two tireles s players who take s turns choosin g digits . Whe n the y ar e don e (!) , they wil l hav e constructed a n infinit e sequence , whic h ca n b e taken a s a decima l expansion, whic h represent s a rea l number , r . If r is in A , th e firs t player wins ; otherwis e th e secon d playe r wins . Th e gam e A i s determined if there is a winning strategy for one player or the other . Blackwell use d th e know n fac t tha t al l ope n games 92 ar e deter mined t o giv e a ne w an d elegan t proof o f Luzin' s theorem o n th e separability of S} sets. This cam e as a welcome surpris e to Moschovakis , Addison , an d Martin, al l of whom wer e engaged in efforts t o go beyond Novikov's theorem an d exten d th e separatio n propert y t o highe r level s of th e projective hierarchy . Fro m th e researche s o f th e earl y analysts , i t was known tha t separation holds for sets at the circled levels:

In th e constructibl e universe , Addison ha d show n tha t th e patter n extends o n the IT-side:

89

Solova y (1969). Lev y and Solovay (1967). 91 Blackwel l (1967). 92 Tha t is, games whose set of wins for the first player is an open subset of [0,1]. 90

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Even fo r disbeliever s in V = L , this meant tha t ZF C can' t disprov e separation fo r IL j sets , bu t fo r al l tha t wa s know n a t th e time , i t remained possibl e tha t ZF C coul d prov e Fl ] set s separable . Still , few expecte d this . A new hypothesis was needed . In thi s context , then , BlackwelP s proo f focuse d considerabl e attention o n determinac y assumptions . £ 3 set s wer e know n t o b e determined, a result which Marti n late r extended t o al l Borel sets.93 On th e othe r hand , th e axio m o f choic e implie s the existenc e o f a non-determined set, 94 But again, it seems natural to insis t that suc h a choice-generate d oddit y no t appea r amon g th e relativel y simpl e sets, fo r exampl e amon g th e projectiv e sets . Thu s protectiv e determinacy—the assumptio n tha t al l projectiv e set s o f real s ar e determined—was proposed b y a wide range of researchers.95 Projective determinacy, like the axio m o f measurable cardinals, is supported b y many of the consideration s though t t o coun t agains t V = L . Fo r example , i t extend s th e result s obtainabl e fro m a measurable cardina l by guaranteeing not onl y tha t ever y uncount able S | se t ha s a perfec t subset , bu t tha t ever y uncountabl e projective set has a perfect subset; not only that all 2j and FI^ setsa are Lebesgu e measurable , but tha t al l projectiv e set s ar e Lebesgu e measurable; not onl y tha t ther e i s no A ] well-ordering of the reals , but tha t ther e i s n o projectiv e well-orderin g o f th e reals. 96 Man y agree wit h Marti n tha t thes e ar e 'pleasin g consequences abou t th e behavior o f projectiv e sets'. 97 An d th e separatio n question , th e problem tha t inspire d thi s renewe d interes t i n determinacy , wa s solved b y Addiso n an d Moschovaki s an d independentl y b y Martin;98 th e initia l zigza g pattern continue s fo r th e lengt h o f th e projective hierarchy: 93 Th e resul t fo r £ 3 i s i n Davi s (1964) , Martin' s i n (1975) . Se e Moschovaki s (1980), §6r , fo r a proo f o f Martin' s theore m fo r th e finit e level s o f th e Bore l hierarchy. Martin (1985 ) give s a simplifie d proo f fo r al l Bore l sets. 94 Gal e and Stewart (1953) . See Moschovakis (1980) , 293. 95 Firs t Solova y an d Takeuti , independently , the n Addison , Martin , an d Moschovakis. (Se e Addison and Moschovaki s (1968), 708-9, Moschovakis (1970) , 31, an d (1980) , 422 , 605, and 610-11 , Marti n (1976) , 90 , (1977) , 814 , and 'Projective set s an d cardina l numbers' , p . 8. ) I n fact , mos t o f thes e propos e a stronger axio m candidate , quasi-projectiv e determinacy , bu t I won' t g o int o th e exact definitio n here . Se e Madd y (1988#) , 737 . Determinac y assumption s wer e introduced b y Mycielski and Steinhau s (1962). 96 Lebesgu e measurability appear s i n Mycielsk i an d Swierczkowsk i (1964) , an d the perfec t subset property i n Davis (1964) . The non-existenc e of a projectiv e wellordering follows because a well-ordering is not Lebesgu e measurable. 97 Marti n (1976), p. 90. 98 Addiso n and Moschovakis (1968) and Martin (1968) .

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This pattern i s considered mor e natural tha n the on e generated by V = L , if only becaus e AAAAAAAAAA A i s a more natural continuation of A tha n A " Other extrinsic evidence for projective determinacy is found i n its strong intertheoreti c connections wit h th e axio m o f measurable cardinals,100 and in the naturalness of the new game-theoretic proofs: One [reaso n fo r believin g projective determinacy] is the naturalness o f th e proofs fro m determinacy—i n eac h instanc e where w e prove a property of Il3 (sa y from [th e determinacy o f A ] sets]), the sam e argumen t give s a ne w proof o f the same (known ) property fo r H\ . . . Thus the new results appea r to b e natura l generalization s o f know n result s an d thei r proof s she d ne w light o n classica l [analysis] , (This is not th e case wit h th e proofs from V = L which al l depend o n th e [A^> ] well-ordering of [the reals] and she d no light on nj.) (Moschovaki s (1980) , 610 )

Moschovakis's thic k boo k contain s variou s beautifu l an d persuasive examples.101 But perhaps the most striking feature of determinacy hypotheses, what makes this a particularly fascinating case for the philosopher, is tha t al l argument s give n i n it s favou r fro m th e mid-sixtie s until the mid-eightie s ar e extrinsic . Determinacy supporter s were quit e explicit on this point: No on e claim s direc t intuition s . . . either fo r o r agains t determinac y hypotheses . . . (Moschovakis (1980) , 610 ) There i s n o a priori evidenc e fo r [projectiv e determinacy ] . . . (Marti n (1976), 90 ) Is [projectiv e determinacy ] true ? I t i s certainl y no t self-evident . (Martin (1977), 813 )

For twent y years, whil e extrinsic arguments o f th e sor t outlined here developed rapidly, there was n o change in the lack of intrinsic support. An d ye t projectiv e determinac y wa s stil l considere d a viable axiom candidate. "9 Se e Moschovakis (1970) , 33-4; Martin (1977) , 806, 811, an d 'Projectic e set s and cardina l numbers' , p . 8; Wan g (1974«) , 547 , 553^ . Fo r othe r reasons , see Maddy(1988
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The bes t hope for something mor e than purel y extrinsi c evidenc e lay i n th e possibilit y o f derivin g determinac y hypothese s fro m suitable large cardinal axioms. 102 Martin showed , early on, that the determinacy o f S| set s i s implied by th e existenc e o f a measurabl e cardinal: Some set theorists conside r larg e cardinal axioms self-evident , o r a t leas t as following fro m a priori principle s implie d b y th e concep t o f set . [Th e determinacy o f 2 } sets ] follow s fro m larg e cardina l axioms. I t i s possible that [projectiv e determinacy] itself follow s from larg e cardinal axioms, bu t this remains unproved. (Martin (1977), 813) One wa y t o increas e the evidenc e for [projectiv e determinacy] would b e to prove i t from larg e cardinal axiom s . . . (Martin, 'Projective set s an d cardina l numbers' (unpublished) , 8)

Unfortunately, attempt s t o exten d Martin' s resul t mad e us e o f cardinals s o larg e tha t eve n th e mos t enthusiasti c larg e cardina l theorists were concerned fo r their consistency. 103 Before turnin g t o th e development s o f th e mid-eighties , w e should paus e t o as k wh y derivin g determinac y hypothese s fro m large cardina l assumption s i s viewe d a s providin g non-extrinsi c support. Obviousl y n o suc h theore m ca n provid e direc t intrinsi c support fo r a determinac y hypothesis ; thi s a hypothesi s eithe r ha s or lack s o n it s own . Wha t happen s i n suc h a cas e i s tha t th e determinacy hypothesi s prove d inherit s th e existin g support s fo r the larg e cardina l assumptio n fro m whic h i t i s proved . Thu s Martin's theore m place s th e powe r o f argument s fo r measurabl e cardinals squarel y behin d th e determinac y o f 2 ] sets . Bu t th e arguments give n abov e fo r measurabl e cardinal s wer e al l ex trinsic!104 Where is the intrinsic evidence supposed to come from? The answe r i s tha t ther e ar e variou s othe r argument s fo r measurable cardinal s that don' t depen d o n consequences . I n fact , however, I think thes e argument s rest o n idea s that ar e no t happily classified a s eithe r intuitiv e o r extrinsic , idea s Marti n referre d t o 102 Moschovaki s (1970) , 31, note s Solovay' s conjectur e tha t thi s migh t he possible, and cite s Martin's resul t as confirming evidence. 103 Martin' s theore m o n measurabl e cardinal s an d determinac y appear s i n Martin (1970) . Marti n (1980 ) extends th e result to ii sets , an d Woodin too k thi s line eve n further . Se e Madd y (1988ii) , §vi , an d Marti n an d Stee l (1989 ) fo r discussion of these development s and th e very large large cardinals involved. 104 I n fact , it s implicatio n of th e well-supporte d X ] determinac y is also counte d as extrinsic evidence for measurabl e cardinals.

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above a s 'a priori principle s implie d b y th e concep t o f set' . On e example ha s alread y bee n given in favou r o f inaccessibles: th e ide a that the iterative hierarchy is too large and complex to be exhausted by th e simpl e operation s o f replacemen t an d powe r set . The mor e general ide a tha t motivate s al l larg e cardinal s i s simpl y tha t th e universe goe s o n throug h a s man y transfinit e stage s a s i t can. Various larg e larg e cardinal s ar e als o defended o n grounds arisin g from th e ide a tha t th e hierarch y o f stage s i s s o comple x an d ric h that i t mus t contai n stage s tha t resembl e on e anothe r i n certai n ways.105 For want o f a better term, I call these 'rules of thumb'. The rule s o f thumb underlying large cardina l axioms ar e clearly rooted i n the iterativ e conception, whic h i s drawn, I'v e suggested, from intuition . Why , then, d o I avoi d attributin g intuitiv e statu s also t o thes e rule s of thumb? The answe r i s simple: becaus e I think they exten d beyon d anythin g tha t coul d plausibl y b e trace d t o a n underlying perceptual , neurologica l foundation . Tha t set s ar e formed fro m previousl y give n things , tha t the y ar e forme d combinatorially, wit h n o concer n fo r rule s o f formation—thes e ideas might wel l hav e intuitive backing, an d ther e i s no doub t the y figure centrally i n the iterativ e conception. Bu t if , as I've suggested, the suppor t fo r th e assumptio n o f a n infinit e stag e i s purel y extrinsic, followin g from th e immens e success of modern infinitar y mathematics, the n par t o f th e standar d iterativ e conception , th e part that drive s the hierarchy into th e infinit e an d insist s that i t go on a s lon g a s possible , tha t par t i s base d o n th e developin g methodology o f set theory itself, not on simple intuition. On th e other extreme , why shouldn't thes e principles be counted as extrinsic ? Natura l scienc e ha s it s ow n principle s o f simila r generality, fo r exampl e Maxwell' s principl e that a la w o f natur e should b e vali d a t al l points i n spac e an d time. 106 Suc h principles are indirectl y subjec t t o extrinsi c support—if the y consistentl y led to ineffectiv e theories , they would eventuall y be dropped—but this proves nothing ; eve n intuition s mus t b e confirme d b y conse quences. An unavoidably central aspec t of the appea l of these rules of thumb—b e the y scientifi c o r mathematical—i s tha t the y 'see m 105 Se e Maddy (l9SSa), § vi.2. Though the arguments fo r large large cardinals are similar is some structural and philosophica l respects t o the arguments for small large cardinals, the y canno t b e sai d t o b e equall y convincing . Again , Madd y (1988a ) provides details. 106 Wilso n (1979 ) discusse s thi s example.

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right', eve n i f thi s seemin g i s unlikel y to enjo y a strictl y intuitiv e basis. I f we confine ourselves to a n unbiase d description o f practice , I think w e must admi t tha t rules of thumb fal l somewher e betwee n the intuitive and the extrinsic. Leaving aside fo r no w th e problem o f elucidatin g an d defendin g the efficac y o f this new categor y o f purported evidence , let me pick up th e threa d o f th e stor y o f determinacy . Larg e cardinal s ar e thought t o enjo y certai n non-extrinsi c justification s i n term s o f various rule s o f thumb . Thus , i f projectiv e determinac y coul d b e proved fro m suc h a n axiom , it s purely extrinsi c defenc e woul d b e enriched b y suppor t fro m thi s othe r source . This , then , wa s th e goal. What happened i n the mid-eightie s was a fulfilment o f this hope . Work o f Martin, John Steel , an d Hug h Woodin 107 established tha t projective determinacy 108 ca n b e prove d fro m th e existenc e o f a supercompact cardinal . Thoug h supercompact s ar e muc h large r than measurables—ther e ar e measurabl y man y measurabl e cardi nals belo w th e firs t supercompact—the y ca n b e viewe d a s a generalization o f that notion. 109 Thus th e theory ZF C + S C (there is a supercompac t cardinal ) enjoy s al l th e extrinsi c support s o f projective determinac y plu s an y non-extrinsic , rule-of-thum b evi dence available for large cardinals in general and fo r supercompact s in particular. 110 I t solve s al l th e outstandin g ope n problem s fro m analysis, an d i t does s o i n way s tha t man y fin d natural . In man y respects, then , ZF C + S C present s a n attractiv e alternativ e t o ZFC + V = L . But wha t abou t th e continuu m hypothesis ? I f i t i s tru e i n L , a minimal environment , perhap s i t i s fals e i n a maximize d world o f large cardinals . There i s some evidenc e i n tha t directio n i n result s proved fro m th e ful l (false ) axio m o f determinacy : i f ever y se t i s determined, then th e real s can b e mapped ont o N %2, K w + i , Nj"* 0) + l 5 and beyond . I n th e presenc e o f th e axio m o f choice , thi s woul d imply th e existenc e o f set s o f real s o f thes e cardinalities , badl y falsifying th e continuu m hypothesis , bu t ful l determinac y als o 107 Se e Martin an d Stee l (1988) , Woodin (1988) , an d Marti n an d Stee l (1989). These paper s also contain usefu l historica l information. 108 Indee d more . Th e ful l quasi-projectiv e determinacy , mentione d i n n . 9 5 above, i s provable from thi s large cardinal assumption. 109 Se e e.g . Solovay , Reinhardt , an d Kanamor i (1978) , §2 . Fo r a textboo k discussion of supercompactness, seejech (1978) , 407—13. 110 Forasurvevo f the latter, see Maddv (L988j) , §§vi.l an d vi.2.

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implies tha t choic e i s false . Indeed , i n tha t strang e worl d o f ful l determinacy, ever y uncountable se t of reals has a perfect subset, s o the continuu m hypothesi s i s tru e i n Cantor' s origina l form : al l infinite set s o f real s hav e th e cardinalit y o f th e natural s o r th e cardinality o f the reals themselves. But in Cantor's favoured form — Kj = 2 °—it remain s false ; th e continuu m isn' t o f siz e K a fo r an y a becaus e it can't be well-ordered. It isn' t clea r wha t al l thi s madnes s mean s fo r th e rea l worl d i f there i s a supercompac t cardina l an d th e axio m o f choic e i s true . Supporters o f ZF C + S C disagre e ove r thei r expectation s fo r th e size o f th e continuum, 111 no t becaus e som e suppor t restrictiv e principle like V = L, but because they disagree over whether Kj = 2 or it s opposit e i s th e leas t restrictive. 112 (Notice , i t i s possibl e t o read th e equatio n a s restrictin g th e numbe r o f reals , bu t als o a s maximizing the number of countable ordinals. ) Efforts ar e under way to extend th e theory to decide the question one way or the other. I hav e described tw o theories , tw o extension s o f ZFC, tha t canno t both b e true . Eac h theor y answer s a t leas t th e ope n question s of Luzin an d Suslin , and on e eve n decides the siz e of th e continuum . Each enjoys an array of extrinsic supports, supplemente d to varying degrees b y intuitive and rule-of-thum b evidence , a small portion of which ha s been describe d i n this summary. The philosophical ope n question is : on what rationa l grounds can one choose betwee n thes e two theories?

5. Th e challeng e We've see n tha t se t theory aros e i n response t o bot h foundationa l and purely mathematical concerns , tha t i t developed a s a branch of analysis an d a s a stud y i n it s ow n right , an d tha t bot h pursuit s produced natura l ope n questions . I n the confusion surrounding the paradoxes an d Zermelo' s controversia l proo f o f th e well-orderin g theorem, th e informa l practic e wa s axiomatized , whic h develop ment ha d tw o consequence s o f concer n t o u s here : th e rol e o f 111 Fo r example , Foreman (1986 ) proposes 'generi c larg e cardinal' axiom s that imply CH . Martin , o n th e othe r hand , ha s conjecture s abou t th e relationshi p between th e cardinals in the world of full determinac y and th e real world that impl y the CH is badly false. See Maddy (1988a) , § v.4. 112 See Maddy (\988a), §§ n.3.4 and n.3.11.

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extrinsic argument in mathematics was crystallize d as never before, and i t became possible t o sho w tha t various natural ope n problem s in se t theor y an d analysi s were ope n i n a ne w an d stronge r sense . The hop e o f answerin g thes e extremel y ope n question s le d t o th e emergence o f tw o competin g theorie s tha t answe r al l o r mos t o f them an d d o s o in very different ways . The difficul t proble m facin g the set theorists of our da y i s how t o adjudicat e between these , tha t is, how t o determine , withi n th e limit s of ou r cognitiv e capacities, which i s more likely to b e true. Proof i s obviousl y th e mos t commo n sourc e o f evidenc e i n mathematics, bu t eve n proo f mus t begi n fro m axiom s tha t ar e no t themselves proved . I n man y circles , th e preferre d accoun t o f ou r knowledge o f axiom s i s that the y ar e someho w self-evident ; in th e words o f Roderic k Chisholm , followin g Leibniz , 'once yo u understand it , yo u se e tha t i t i s true'. 113 Bu t ou r brie f surve y i n thi s chapter show s tha t eve n the accepte d axiom s o f ZFC d o no t enjo y this status , le t alon e th e mor e controversia l axio m candidate s lik e projective determinacy. A new account of our knowledg e o f axioms and o f th e evidentia l rol e o f non-demonstrativ e mathematica l arguments in general is clearly needed. For th e se t theoretic realist , non-demonstrative argument s com e in two varieties—intuitive and extrinsic—and examination of cases reveals what i s probably at least one more—those based on rules of thumb. I n Chapte r 2 , I argued tha t intuitiv e support i s prima-facie evidence fo r truth . Perhap s i t i s no t surprisin g tha t mos t o f th e available intuitive evidence is marshalled in support o f the accepte d axioms o f ZFC . I n th e disput e betwee n V = L an d SC , I'v e suggested tha t V = L labour s a t a modes t intuitiv e disadvantage, because i t reject s th e combinatoria l componen t o f se t theoreti c intuition. But, given that intuitive evidence is never conclusive, tha t it need s supplementatio n b y extrinsi c supports an d ca n b e (indeed has been ) overthrow n b y theoretica l counter-evidence , thi s alon e can hardly settle the question i n favour o f SC. What's neede d i s a dependabl e metho d fo r comparin g th e strength o f th e non-intuitive , non-demonstrative argument s rele vant t o thi s case , som e o f whic h wer e touche d o n i n th e previou s section. Bu t befor e w e ca n answe r th e questio n o f whic h axio m candidate i s supported b y bette r suc h arguments, w e mus t fac e th e prior questio n of whether thes e argument s carry an y weight a t all , "•' Se e Chisholm (1977) , 40.

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and i f so, why. We need t o explai n how , why , an d t o wha t exten t such arguments count a s evidence for the truth o f their conclusions . Only the n ca n w e determin e whic h amon g the m constitut e th e better evidence . A prerequisit e fo r thi s inquir y i s a n appreciatio n fo r th e ric h variety of extrinsic supports offere d b y set theorists. Th e discussion in this chapter alon e reveals a wide range of types: 1. Verifiabl e consequences . Fo r example , Sierpifiski' s demon stration tha t man y theorems prove d fro m th e axio m o f choice can also be proved, thoug h th e proofs are more complicated, without it. Another case, to which attention was not explicitly drawn, provides support fo r measurabl e cardinals . Marti n (1970 ) derive d th e determinacy o f Bore l set s (indee d 2} sets) fro m th e existenc e of a measurable cardinal. Five years later, i n his (1975) , this result wa s 'verified', that is, proved fro m ZF C alone. 2. Powerful ne w methods for solving pre-existing open problems. The axiom of choice settled the open question of whether o r not th e reals coul d b e well-ordered. V = L and S C both provid e method s for solvin g the long-standing open problems in analysis, and V = L even decides the continuum hypothesis. 3. Simplifyin g an d systematizin g theory. Th e axio m o f choic e brings order into the chaos of transfinite arithmetic . 4. Implyin g previous conjectures. The existenc e o f a measurable cardinal implie s tha t V = £L , an d a supercompac t rule s ou t projective well-ordering s o f th e reals . Bot h thes e ha d bee n previously conjectured. 5. Implyin g 'natural ' results . Th e zigza g patter n o f separatio n properties i n th e projectiv e hierarch y generate d b y projectiv e determinacy (an d hence by SC) is considered mor e natura l than th e Fl-side pattern o f V = L . 6. Stron g intertheoreti c connections . Th e detaile d intertheoreti c connections betwee n determinac y an d measurabl e cardinals , be yond Martin' s resul t o n 2 } determinacy , ar e to o complicate d t o summarize here , bu t a simple r sor t o f exampl e involve s th e extension o f know n pattern s fro m on e theor y t o th e next . Fo r example, i n ZFC, ever y uncountable 2 } set has a perfect subset; in the presence of a measurable cardinal , thi s extend s t o Si sets , an d with projective determinacy, to £3 sets and beyond . Thus th e three theories seem to be pulling in the same direction .

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7. Providin g ne w insigh t int o ol d theorems . Projectiv e deter minacy allow s man y classica l properties o f 11 } set s t o b e extende d to II 1. Alon g th e way , thi s procedur e ofte n provide s a ne w an d simpler proo f from ZF C of the classical theorem fo r Il{. The rang e o f rule s o f thum b is hardly les s bewilderin g tha n thi s array o f extrinsi c justifications . I'v e mentione d a handfu l mar shalled i n favou r o f larg e cardinals , bu t ther e ar e man y more. 114 And eve n thi s rough-and-read y classificatio n neglect s th e rol e o f conjectures—like th e non-existenc e of projectiv e well-orderings of the reals—an d othe r judgement s of plausibility—lik e tha t agains t choice-generated odditie s a t lo w level s of th e projectiv e hierarchy. So even as description, leaving aside explanation, my account i s far from complete . I' m sorr y t o sa y that I won't complete it here; fillin g in th e detail s o f structur e o f non-demonstrative , non-intuitiv e arguments an d evaluatin g thei r cogenc y i s a subjec t fo r anothe r book, a boo k I unfortunatel y don' t kno w ho w t o write . Afte r drawing attentio n t o the problem, fo r now I can do littl e more than highlight it s importanc e an d encourag e a concerte d investigation . Let me conclud e with a fe w words abou t ho w thi s las t migh t bes t proceed. There i s an obviou s similarity betwee n thi s project an d th e centra l business o f philosopher s o f science : givin g a n accoun t o f th e confirmation o f scientifi c theories . Indee d th e ver y description s of th e style s o f extrinsi c justification—verifiabl e consequences , simplifying an d systematizin g theory , stron g intertheoreti c connections—suggest that th e analog y i s a powerful one. I t woul d seem tha t th e compromise Platonist' s science/mathematic s analogy stands to gain furthe r detai l from thes e distinctive parallels between scientific and mathematica l modes of theoretical justification. To thi s lin e o f thought , som e wil l repl y tha t th e analog y i s superficial, tha t natura l scientists , no t mathematicians , tes t thei r theories b y experiment . Th e immediat e respons e i s that mathema ticians do use experiments. Le t me quote from Martin: 115 I thin k tha t ther e ha s bee n mor e o f wha t I migh t cal l subjectin g determinacy hypothese s [to ] experiment tha n i s suggeste d b y wha t I an d other writer s hav e sai d i n print . A n example : th e firs t theore m i n 114 m

Se e Maddy( 1988*). Persona l communication, Sept. 1984 .

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determinacy I proved was tha t [th e full axio m o f determinacy] implies that every se t o f degree s o f unsolvabilit y contain s o r i s disjoin t fro m wha t i s now calle d a 'cone'. 116 When I discovered the two-line proof o f this, I was very excited. I was sur e that, with a few minutes' thought, I would b e able to fin d a se t of degree s which wa s a counterexample . Thu s I would refut e [the ful l axio m o f determinacy ] an d surel y even [projectiv e determinacy] , and probabl y eve n Bore l determinacy . I starte d goin g throug h variou s simple sets of degrees I knew about , checking each one out. I was surprised to discove r tha t I coul d alway s find—b y elementar y recursion-theoreti c means—the con e whos e existenc e determinac y predicte d . . . the effec t o n me was muc h a s that o n a physicist when a theory predict s a new kind of particle and such particles ar e then observed.

Here a mathematica l experimen t i s undertake n quit e explicitly . Other case s ca n b e culled fro m materia l earlier i n this chapter ; for example, investigatio n o f th e consequence s o f V = L an d S C fo r low-level projective sets can be viewed as tests. But, opponent s o f th e analog y migh t continue , thes e ar e experiments of a very different sor t from thos e found in physics; no accelerators ar e involved , n o observation s o f instrument s o r computer outputs , n o clou d chambers , etc . O f course , thi s ca n hardly b e denied , bu t wha t need s t o b e appreciate d her e i s th e extent to which scientific methodology varies even from on e natural science t o another. Martin's experiments ma y use paper and pencil and depen d o n previou s result s i n recursio n theor y rathe r tha n using an electro n microscop e and dependin g o n previou s result s in subatomic physics , bu t th e botanist' s experiment s ar e differen t from both : i n anothe r era , sh e took a field trip and brough t bac k hand drawing s fo r compariso n wit h previousl y gathere d an d classified samples . W e don' t expec t a stud y o f th e methodolog y appropriate t o physic s to tel l u s al l we wan t t o kno w abou t ho w botanists, biologists , psychologists , astronomers , o r geologist s formulate an d tes t thei r theories , s o wh y shoul d w e expec t mathematical science to conform to confirmation techniques drawn from som e other science ? The answe r is that we shouldn't. If Martin's experiments ar e different fro m th e physicist's, thi s should come as no surprise and shouldn' t (b y itself) coun t against their efficacy . Respect fo r the variation between th e sciences also undercuts the opposite, overl y quick , reactio n t o th e analog y betwee n mathem atics an d natura l science. Some, citing the similaritie s noted above , 116

Thi s theorem i s the crucial lemma in Martin (1968).

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might b e incline d t o conclud e tha t th e epistemologica l issue s fo r this aspec t o f mathematica l knowledg e ca n b e identifie d wit h th e corresponding problem s i n th e philosoph y o f scienc e i n general . The motive for such a move would b e much like that of the logicists decades earlier ; th e proble m o f mathematica l epistemolog y i s reduced t o another—th e epistemology o f logic or the epistemolog y of scienc e i n general—whic h i s presume d t o b e a n easie r target . What I'v e bee n suggestin g i s that thi s won' t wor k either , tha t th e idiosyncrasies o f mathematica l theorizin g requir e individua l atten tion. A complete theor y o f the methods o f physics (or psychology o r biology), even if there were such a thing, would no t b e enough. So th e theor y o f mathematica l theor y formatio n an d confir mation we'r e afte r wil l exploi t parallel s wit h th e variou s natura l sciences whil e attendin g t o th e uniqu e aspect s o f mathematica l methodology. And , i f it i s to d o th e ambitiou s job se t fo r i t here, i t cannot res t conten t wit h pur e description . Whe n scienc e i s functioning smoothly , i t ma y b e enoug h t o describ e it s methods , but i n th e cas e o f contemporar y se t theory , eve n th e practitioner s aren't sur e ho w th e variou s non-demonstrativ e argument s shoul d be evaluated . Thu s Gode l write s tha t th e recalcitranc e o f th e continuum problem may b e due to purel y mathematica l difficulties ; i t seems, howeve r . . . tha t there ar e also deepe r reasons involve d and tha t a complete solutio n . . . can be obtaine d onl y b y a mor e profoun d analysi s (tha n mathematic s i s accustomed t o giving ) o f th e meaning s o f the term s . . . and o f the axiom s underlying their use . (Gode l (1947/64) , 473 )

I sugges t tha t th e 'mor e profoun d analysis ' require d i s th e ver y investigation I' m urgin g here . What' s neede d i s no t jus t a description o f non-demonstrativ e arguments , bu t a n accoun t o f why an d whe n the y ar e reliable , a n accoun t tha t shoul d hel p se t theorists mak e a rationa l choic e betwee n competin g axio m candi dates. Finally, thi s fundamenta l problem , th e proble m o f describing , explaining, an d evaluatin g non-demonstrative argument s i n math ematics, i s a centra l challeng e for th e se t theoreti c realist , bu t i t is not hers alone. Of course, any compromise Platonist will face it , but so will others. Anyon e who hold s tha t there is (most likely) a fact of the matter abou t the size of the continuum (or the open question s of Luzin and Suslin ) must admit that one or the other o f V = L and SC

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is false. This leave s the proble m o f adjudicating between thes e tw o theories, an d unles s som e alterativ e mean s i s found , thi s i n tur n requires confrontin g an d evaluatin g th e non-demonstrativ e argu ments fo r an d agains t them . Actually , th e rang e o f Platonis t an d anti-Platonist philosophica l position s fo r whic h thi s challeng e remains a real one is quite broad, a s will become cleare r in my final chapter, but for all that, it has been almost universall y ignored. Th e goal o f thi s chapte r wil l hav e bee n serve d i f m y portrai t o f th e challenge itself is vivid an d compellin g enough tha t th e reade r no w sees it as her own .

5

MONISM AN D BEYOND The mai n outline s o f se t theoreti c realis m ar e no w i n place . It s epistemology divide s loosel y int o tw o parts , a s befit s a versio n o f compromise Platonism: the intuitive (Chapte r 2 ) and the theoretical (Chapter 4). Benacerraf's ontological puzzl e is met by agreeing that numbers aren' t objects , bu t insistin g non e th e les s o n a clos e connection betwee n number s an d sets , namely , tha t number s ar e properties o f sets (Chapte r 3) . Finally , though i t i s far fro m solved , the mos t seriou s ope n proble m ha s a t leas t bee n formulate d wit h some care (Chapter 4). Only a fe w bit s o f tidying-u p remain fo r thi s closin g chapter . First comes a final look a t the ontolog y o f set theoretic realism , this time wit h close r attentio n t o th e relationshi p betwee n th e math ematical an d th e physical . Thi s i s followe d b y tw o section s comparing an d contrastin g se t theoreti c realis m wit h othe r con temporary positions , wit h a n ey e to illuminatin g some surprisin g convergences. I conclude, i n section 4, wit h a summary and a loo k to the future .

1. Monis m Let m e begi n wit h a forcefu l objectio n Chihar a onc e aime d a t th e very hear t o f set theoretic realism , that is , at m y clai m that 'set s of physical object s . . . have locatio n i n tim e an d spac e an d ca n b e literally perceive d b y the senses'. 1 I n the cours e o f lampooning thi s position, h e writes: Imagine tha t I a m sittin g a t m y desk . It s surfac e ha s bee n cleare d o f everything excep t a n apple . No w accordin g t o Maddy , w e ca n literall y perceive o n th e desk , i n addition t o the apple , the se t of apples o n m y des k (which happen s t o b e a uni t set) . It is claimed that thi s set has a location i n space, th e exac t spo t wher e th e appl e is . Supposedly , i t als o cam e int o 1

Chihar a (1982) , 223.

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existence a t a particula r tim e (whe n th e appl e did) , an d wil l g o ou t o f existence a t a particula r tim e (whe n th e appl e does) . Obviously , Madd y thinks this set can be moved abou t i n space. Now i f we can perceive this set with th e sens e o f sight , the n wha t doe s i t loo k like ? Evidently , i t look s exactly lik e th e appl e itself . Afte r all , I don' t se e anythin g a t tha t exac t region i n space that look s differen t fro m th e apple . One wonders ho w thi s object i s t o b e distinguishe d (perceptually ) fro m th e apple , sinc e i t ha s exactly th e sam e shap e an d color . Perhaps i t feel s different . Let' s touc h it . But I can't fee l anythin g there other tha n th e apple . Evidently, this strange entity feel s n o differen t fro m th e apple . Ho w abou t it s smel l o r taste ? Again, it would seem that th e set must be identical in smell and tast e to the apple. S o i t looks , feels , smells , an d taste s exactl y lik e th e appl e an d i s located i n exactl y th e sam e spot an d a t exactl y th e sam e time—ye t i t i s a distinct entity! One would thin k that a n entity with these properties would be o f interes t t o th e physicist . Furthermore , essentiall y the sam e reason s Maddy give s for maintainin g that thes e sets can b e perceived by the senses can also be given for claiming that a set of such sets can be perceived. Thus, we should be able to see the set whose only member is the unit set described above; w e shoul d b e abl e to perceiv e the unordere d pai r consisting of the apple an d th e abov e uni t set , an d s o on indefinitely . Presumably , all these different entitie s would look , feel , smell , and tast e exactl y alike. (Chihara (1982), 223^4)

Chihara's onl y explici t conclusio n i s th e undoubtedl y soun d on e that m y view differs fro m Godel's, 2 but th e sens e that thi s passage raises a n importan t questio n fo r the set theoretic realis t can hardl y be avoided. On th e fac e o f it , th e questio n i s one alread y considere d i n th e second section of Chapter 2: how is it, on a given occasion, that we see a physical mass rather than a set, or one set rather than another , when al l produce th e sam e retina l stimulation ? The answe r ther e was tha t difference s i n training , o r interests , o r attention , coul d produce differen t cell-assemblie s i n differen t individuals , or facili tate the activation of one cell-assembly rather than another withi n a single individual . An d th e activatio n o f differen t cell-assemblies , even given the same retinal stimulation, produces a difference in the purely phenomenological look of the scene. In Chihara's case , then , th e differenc e betwee n th e physical mass that make s u p th e appl e an d th e singleto n containin g th e appl e i s that th e latter has an unambiguous number property—one—while the forme r i s on e apple , man y cells , mor e molecules , eve n mor e 2

Se e ch. 2, sect, 4, above.

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atoms, an d so on. What makes the example unsettlin g is that in this case, the singleto n i s so conspicuous tha t w e rarely see the physical mass o r an y o f th e othe r sets . A topologist, heroicall y immersed i n her work , migh t se e lef t an d righ t apple-halves , bu t i t seem s unlikely tha t eve n a n infan t woul d se e a physica l mass undifferen tiated a s a uni t fro m it s background—tha t is , something wit h n o number propert y a t all—an d th e norma l adul t almos t invariably sees th e singl e apple. Indee d th e ver y question o f ho w th e physica l mass differs fro m the singleton ca n be asked i n such a way a s to beg it: what' s th e differenc e betwee n a singl e object and it s unit set ? A 'single object' alread y has a n unambiguou s number property ! This doesn't mea n tha t th e physica l mas s doesn' t diffe r fro m th e singleton—it does—bu t onc e th e physica l mas s i s individuated , separated fro m it s surroundings and see n a s an isolate d thing , tha t difference seem s to evaporate. So, whil e th e se t theoreti c realis t ha s a read y answe r t o on e question—what distinguishe s a physica l mas s fro m a set? — Chihara i s askin g another—wha t distinguishe s a n individuate d physical objec t from it s unit set ? Th e answe r t o thi s ne w questio n cannot b e that on e ha s a n unambiguou s number property an d th e other doesn't , becaus e both th e single object and th e singleton have the sam e numbe r property: one . I f there i s a difference , i t mus t lie somewhere else , and Chihara' s remark s pointedly sugges t tha t i t is not perceptual. The se t theoreti c realist' s firs t option , i n respons e t o thi s situation, i s t o insis t tha t ther e i s a n unperceivabl e difference . I t wouldn't b e the first such difference; we aren't ver y good a t seeing the difference betwee n gol d and fool' s gold, an d tha t between wate r and heav y wate r i s completel y invisible . I n science , unperceivabl e differences ar e detected b y more sensitive instruments or implied by well-supported theory . Perhap s theoretica l argument s coul d b e found i n mathematics , o r mor e likely , i n ou r theor y o f mathemat ics, in th e philosoph y o f mathematics , fo r distinguishin g individual things fro m thei r unit sets . These migh t hav e to d o wit h inviolable differences betwee n concret e an d abstract , o r betwee n mathemati cal and physical. The set theoretic realist's other optio n i s simply to deny that ther e is an y suc h differenc e a t all , perceivabl e or otherwise , tha t is , t o identify individual s with thei r singletons . Thi s i s no t t o sa y tha t every singleto n i s identica l with it s sol e member ; ther e i s every

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reason t o distinguis h betwee n {{0 , 1 , 2 , 3, . . .}} an d {0 , 1 , 2, 3,. . .}, startin g wit h th e fac t tha t on e i s finit e an d th e othe r infinite. Rather , w e tak e i t tha t th e physica l objects , x , th e individuals fro m whic h th e generatio n o f th e iterativ e hierarchy begins, ar e suc h tha t x = {x}. Afte r that , th e axio m o f extension ality3 guarantee s tha t set s forme d a t late r stage s wil l b e distinc t from thei r singletons. And , again, this option doe s not sugges t that the physical mass of apple-stuff i s identical with the singleton apple. Here ther e i s a rea l differenc e i n th e determinac y o f numbe r property. Al l that' s bein g denie d i s tha t th e individua l apple i s distinct from it s unit set. I thin k bot h thes e option s ar e ope n t o th e se t theoreti c realist , that i s t o say , bot h ar e full y consisten t wit h th e tenet s o f tha t position a s described in previous chapters. An d neithe r of them, as far a s I can see , doe s an y damag e o f the sor t implie d by Chihara' s rhetoric. I n particular , neithe r option give s up th e clai m t o a real , perceptual difference betwee n a set and a n undifferentiated physica l mass, between three apples and the unindividuated stuff tha t makes them up. Godel, considering an argument from Russell, remarks: Russell adduce s . . . against the extensional view of [sets ] . . . the existence o f . . . th e uni t [sets] , whic h woul d hav e t o b e identica l with thei r single elements. Bu t i t seem s t o m e tha t thes e argument s could , i f anything , at most prov e that. . . the unit [sets ] (as distinct from thei r only element) ar e fictions . . . not that all [sets] are fictions. (Godel (1944), 459)

The same reply could be given to Chihara . But even if both option s ar e open, eve n if a set theoretic realist is free t o follo w either, my own preferenc e is for the second. Th e only motivation I see for insistin g on a n unperceivabl e difference i s th e desire t o maintai n a stric t dualis m betwee n th e mathematica l an d the physical, and I feel no suc h desire. The remainder of this section will thus presuppose th e identification of individual with singleton , but le t me insist, one last time, that a se t theoretic realis t reluctant to take this turn should fee l no obligation to do so.4 3 Tw o set s ar e th e sam e i f an d onl y i f the y hav e th e sam e members . Se e Enderton(1977),2, 17. 4 Quin e (1969a) , §4 ) advocate s th e identificatio n o f objec t wit h singleto n fo r the purpos e o f simplifyin g hi s forma l theory . S o doin g require s a ver y mino r modification o f Zermelo-Fraenkel set theory and th e additional assumptio n that the individuals, th e element s fro m whic h th e iterativ e hierarch y begins , for m a se t with at leas t tw o members . Se e my Thysicalisti c Platonism' (forthcoming) , especiall y th e appendix.

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While Chihara' s exampl e i s aime d a t on e specifi c aspec t o f se t theoretic realism , othe r anti-Platonist s ma y b e bothere d b y a mor e general worr y arisin g fro m version s o f physicalism. 5 Physicalis m began i n th e hand s o f th e positivist s a s a ver y stron g thesi s abou t the reducibilit y o f al l science s t o th e vocabular y o f fundamenta l physics.6 I n thi s form , i t foundere d o n th e methodologica l independence o f th e specia l sciences / bu t th e crud e ide a tha t physics is somehow basi c retains its appeal. A sophisticated versio n appears i n th e writing s o f th e contemporar y physicalis t Hartr y Field. Field describe s th e doctrin e a s a well-supporte d methodologica l principle: . . . physicalism [is] the doctrin e tha t chemica l facts, biologica l facts , psycho logical facts , an d semantica l facts , ar e al l explicabl e (i n principle ) i n terms o f physica l facts . Th e doctrin e o f physicalis m function s as a high level empirica l hypothesis , a hypothesi s tha t n o smal l numbe r o f experiments ca n forc e u s to giv e up. I t functions , in other words, in much the sam e wa y a s the doctrin e o f mechanis m (tha t all facts ar e explicabl e in terms of mechanical facts ) onc e functioned . . . (Field (1972) , 357) Mechanism wa s rejecte d when Maxwell' s theor y o f electromagnet ism coul d no t b e explaine d i n mechanica l terms . Fiel d conclude s that Mechanism ha s bee n empiricall y refuted ; it s hei r i s physicalism , whic h allows a s 'basic ' no t onl y fact s abou t mechanics , bu t fact s abou t othe r branches o f physic s a s well . I believ e tha t physicist s a hundre d year s ag o were justifie d i n acceptin g mechanism , an d that , similarly , physicalism should b e accepted unti l we have convincing evidence that there i s a real m of phenomena i t leaves out o f account. (Fiel d (1972) , 357) He gives this example of how th e physicalistic principle functions: Suppose, fo r instance , tha t a certain woma n ha s tw o sons , on e hemophili c and on e not. Then, accordin g t o standar d geneti c account s o f hemophilia , the ovum fro m whic h one of these son s was produced mus t have containe d a gen e fo r hemophilia , an d th e ovu m fro m whic h th e othe r so n wa s produced mus t no t hav e containe d suc h a gene . Bu t no w th e doctrin e o f physicalism tell s us that ther e must have been a physical differenc e betwee n 5 I mentione d thi s possibilit y in ch . 2 , sect . 1 , above bu t postpone d discussio n until now. 6 Se e Carnap (1934) . 7 Se e e.g. Fodor (1975), 9-26 .

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the tw o ov a tha t explain s why th e first son had hemophili a an d th e secon d one didn't... We should no t rest conten t with a special biologica l predicate 'has-a-hemophilic-gene'—rather w e shoul d loo k fo r non-biologica l fact s (chemical facts ; an d ultimately , physica l facts ) tha t underli e th e correc t application of this predicate, (Field (1972), 358) Field doe s no t requir e tha t th e biologica l predicat e 'ha s a haemophilic gene' be translated int o the basic vocabulary of physics or eve n chemistry, or tha t al l biological laws be derivable from th e laws o f physic s o r chemistry , o r tha t an y rationa l biologica l methodology b e identical with that of physics or chemistry. He only insists that ther e be a chemical and ultimatel y physical explanation of wh y thi s particula r ovu m ha s a haemophili a gen e an d tha t on e doesn't.8 Now presumabl y a physical explanation i s one that involves only physical things, physica l facts. I f physical things an d physica l fact s are jus t thos e spoke n abou t i n physics , the n th e indispensability arguments make it hard t o se e why physicalism presents a proble m for an y versio n o f Platonism ; accordin g t o Quin e an d Putnam , physics speaks constantly and essentiall y of things mathematical. If being part o f physic s were al l there wer e t o bein g physicalistically acceptable, mathematica l thing s woul d ge t int o th e physicalist' s ontology o n th e groun d floor , eve n ahea d o f th e chemical , th e geological, th e astronomical, th e biological, and so on. Obviously, fo r those wh o tak e physicalism to raise a problem fo r Platonism, 'being physical' comes to more than 'being mentioned in physics'. T o se e what thi s something more might be , consider onc e again Field's version of the epistemological problem for Platonism: what raise s th e reall y seriou s epistemologica l problem s i s not merel y th e postulation o f causall y inaccessibl e entities; rather, i t is the postulatio n of entities that are causally inaccessible and can't fall withi n our field of vision and d o no t bea r an y othe r physica l relatio n t o u s tha t coul d possibl y explain ho w w e ca n hav e reliabl e informatio n abou t them . (Fiel d (1982) , 69) In contrastin g mathematica l entitie s wit h hi s ow n space-tim e regions, Field clarifies th e 'physical relations' h e has in mind: there ar e quit e unproblemati c physica l relations , viz. , spatia l relations , between ourselves and space-time regions, and this gives us epistemological 8 I n hi s 'Physicalism ' (forthcoming) , Fiel d differentiate s hi s vie w fro m variou s stronger and weaker versions of physicalism.

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access to space-tim e regions . For instance, because of their spatial relations to us , certai n space-tim e region s ca n fal l withi n ou r fiel d o f vision . (Fiel d (1982), 68)

To b e epistemologicall y accessible , t o b e acceptabl e o n Field' s world-view, an entity need not b e causally efficacious, bu t it must at least b e spatio-temporall y located . Assumin g tha t acceptabl e entities, fo r Field, are physical, this suggests that wha t i t takes t o be physical, ove r an d abov e bein g talke d abou t i n physics, is locatio n in space and time, and preferably in the causal nexus.9 On thi s reading , physicalis m create s obviou s problem s fo r traditional Platonism , wit h it s acausal , non-spatio-tempora l ob jects. Th e se t theoreti c realist , b y contrast , begin s i n a muc h stronger position : he r set s o f medium-size d physical object s ca n 'fall withi n ou r fiel d o f vision' . O f course , no t al l set s actuall y d o fall withi n ou r fiel d o f vision , but , t o paraphras e Field , 'thi s raises no mor e epistemologica l problem s for [sets ] than i t raise s for , say, tigers' (p . 68). If all that' s require d for physicalisti c acceptability is spatio-temporal location , th e se t theoretic realist' s impur e sets ar e unimpeachable. But wha t abou t pur e sets ? What abou t th e empt y set , th e se t of von Neuman n ordinals , an d it s powe r set ? Thos e unafflicte d b y physicalistic scruples are fre e t o respond tha t w e gain knowledge of the pur e set s b y theoretica l inferenc e fro m ou r elementar y perceptual an d intuitiv e knowledge o f impur e sets, bu t physicalist s will complai n tha t thi s reply doe s nothin g to solv e th e proble m of spatio-temporal location . Doe s thi s mea n tha t suc h a physicalis t must rejec t se t theoreti c realis m afte r all ? Thi s follow s onl y i f se t theoretic realis m i s irrevocabl y committe d t o pur e sets , an d I contend that it is not. In fact, the pure sets aren' t reall y needed. The set theoretic realist who woul d simultaneousl y embrac e physicalis m ca n tak e th e subject matte r o f se t theoreti c scienc e t o b e th e radicall y impure hierarchy generate d fro m th e se t o f physica l individual s b y th e usual power se t operation , excep t tha t th e empt y se t is omitted a t each stage . O n this picture, each set, no matter how exalted i n rank, is located wher e the physica l stuf f i n its transitive closure is located. The theor y o f thi s structur e differ s onl y triviall y fro m tha t o f th e 9

Cf . Armstrong (1977).

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usual hierarch y wit h individuals. 10 I t ca n serv e al l th e sam e purposes. So the set theoretic realist can locate all the sets she needs in space and time . Still , a mor e physicalisticall y satisfyin g ontology woul d be no t onl y spatio-temporall y located , bu t causall y efficaciou s a s well. Bu t notice: i f sets ar e indee d perceivable , a s I've argued, the n they mus t pla y th e sam e rol e i n th e generatio n o f m y perceptua l beliefs abou t the m as , say , m y han d play s i n th e generatio n o f my perceptual belie f tha t ther e i s a hand before me when I look at i t in good light , a rol e whic h is , presumably, causal. Or , t o us e a non psychological example , suppos e yo u deposi t thre e quarter s i n a soft-drink machin e and a soda drop s out . Whic h propertie s o f tha t which yo u deposite d ar e causall y responsible fo r th e emergenc e of the Pepsi ? Well, the weight o f the physical mass of metal, its shape , and als o th e numbe r property : three . (Th e machin e count s somehow.) Fro m th e se t theoretic realist' s perspective , tha t whic h has a number property, tha t i s to say, a set, is causally efficacious. I conclud e tha t se t theoreti c realis m i s consistent wit h physical ism. Onc e again , thi s i s no t t o sa y tha t ever y se t theoreti c realis t must b e a physicalist ; non-physicalist s ma y prefe r t o retai n th e standard versio n o f se t theoreti c realis m wit h it s pur e sets . M y point i s tha t th e positio n ca n b e physicalize d without significan t trauma. In thi s section , I'v e suggeste d tw o mino r alteration s i n th e se t theoretic realist' s ontology : th e identificatio n o f physica l object s with thei r singleton s an d th e eliminatio n o f pur e sets . Together , these move s produc e a powerfull y symbioti c pictur e o f th e relationship betwee n th e mathematica l an d th e physical : ever y physical thin g i s alread y mathematical , an d ever y mathematica l thing i s based i n the physical . In place of the customar y dualis m of 10 Wit h tw o individuals , x an d y , a versio n o f th e ordinal s ca n b e constructe d without pure sets—x, {x, y}, {x, y, {x, y] } , and so on—and the various axioms and theorems ca n b e tinkered with. Practicall y speaking, however, it is probably best t o keep the empty set as a notational convenience . Godel, for example, suggests it could be treate d a s a fictio n 'introduce d t o simplif y th e calculu s like points a t infinit y i n geometry' (1944 , p . 459). Fraenkel , Bar-Hillel , an d Lev y sa y th e empt y se t i s introduced fo r 'reasons of convenience and simplicity , and ca n be regarded as a mere notational convention ' (1973 , p . 24) , an d eve n Zermel o call s i t 'fictitious ' (19086 , p. 202) . See my 'Physicalistic Platonism' (forthcoming), appendix .

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mathematical an d physical , thi s pared-dow n se t theoreti c realis m offers a version of monism. 11 To appreciat e jus t ho w closel y th e tw o ar e intertwine d o n thi s view, tr y t o separat e them . A purely mathematical worl d woul d b e empty. Wha t woul d a purel y physica l worl d b e like ? A s soo n a s there ar e numbe r properties , ther e ar e set s tha t bea r them , s o a world withou t mathematica l thing s woul d hav e t o b e a worl d without an y things , a completel y amorphou s mass : th e Blob . T o add eve n th e structuring into individual physical objects is to admi t singletons, t o broac h th e mathematical . Th e onl y wa y t o confin e ourselves t o th e purel y physica l i s t o refrai n fro m an y differen tiation whatsoever . Perhaps suc h a worl d i s possible, bu t i t clearl y isn' t ou r world , with it s objects, kinds , patterns, an d structure s o f s o many, widel y varied sorts . I n place o f th e ol d picture—physica l realit y here an d now, mathematica l realit y nowher e an d nowhen—se t theoreti c monism offer s a spatio-tempora l realit y inseparabl y physica l an d mathematical. Physic s an d mathematics , o n thi s ne w picture , ar e two sciences , alon g wit h chemistry , biology , psychology , an d th e rest, tha t stud y aspect s o f thi s reality . Eac h scienc e ha s it s ow n vocabulary an d laws , it s ow n technique s an d methods , bu t thi s doesn't mea n tha t th e worl d itsel f i s divided int o th e physical , th e mathematical, th e chemical , th e biological , th e psychological , an d so on . Rather , everythin g i s ultimatel y physico-mathematica l o r mathematico-physical. Some wil l not e that , strictl y speaking , thi s vie w i s mor e Aristotelian tha n Platonistic . The y ar e right , i n th e sens e tha t Aristotle's form s depen d o n physica l instantiations , whil e Plato' s are transcendent . I retai n th e ter m 'Platonism ' here , no t fo r it s allusion t o Plato , bu t becaus e i t ha s becom e standar d i n th e philosophy o f mathematic s fo r an y positio n tha t include s th e objective existenc e o f mathematica l entities. Other s wil l point ou t that singletons don' t deserve the special status m y presentation ha s awarded them ; fo r th e monist , tw o object s (a s oppose d t o th e undifferentiated mas s o f physica l stuf f tha t make s the m up ) ar e already a doubleton, a s well. This i s also correct. Th e singleto n cas e is uniqu e onl y i n it s psychologica l impact : i t make s u s realiz e just how littl e it takes fo r mathematics to intrude. But for all this, I hope 11 Fo r thos e wh o thin k o f Platonis m as a for m o f theology , thi s i s a versio n o f pantheism.

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the genera l feature s o f se t theoreti c monis m ar e clea r enough . I leave i t as a n optio n fo r an y se t theoretic realis t t o who m i t migh t appeal. 2. Field' s nominalism Despite m y efforts t o tur n awa y objection s t o Platonis m base d o n epistemology, o n th e possibilit y o f multipl e reductions , an d o n purely physicalistic concerns, Fiel d remains unconvinced. Instea d of seeking ou t ne w objection s t o counter , a mor e fruitfu l strateg y a t this stage migh t be to have a look at Field's nominalisti c alternative to Platonism . Fiel d is not th e onl y nominalis t o n th e contemporar y scene, but his version ha s th e distinction o f being non-revisionist:12 he sets ou t t o show how w e can go on usin g classical mathematic s in scienc e exactl y a s w e hav e been , bu t withou t admittin g th e existence o f mathematica l things , an d h e attempt s thi s withou t short-changing th e Quine/Putnam indispensability arguments. Her e is a form of nominalism tha t should giv e the Platonist pause . Field an d I agree that th e indispensabilit y arguments provid e th e best evidenc e fo r mathematic s a s a whole. Movin g beyon d Quine / Putnamism int o Godelia n territory , w e als o agre e tha t ther e ar e other possible forms of mathematical evidence: if we assume that there i s at least one body of pure mathematical assertion s that include s existentia l claim s an d tha t i s true . . . then w e ar e assumin g that ther e ar e mathematical entities . From thi s w e can conclude tha t ther e must be some bod y of facts abou t these entities, and that no t al l facts abou t these entities are likely to be relevant to known application s to the physical world; i t i s the n plausibl e t o argu e tha t consideration s othe r tha n applications t o th e physica l world , fo r example , consideration s o f simplicity an d coherenc e withi n mathematics , ar e ground s fo r acceptin g some propose d mathematic s axiom s a s tru e an d rejectin g other s a s false . (Field (1980), 4)

Finally, we agree tha t the Benacerraf-styl e worr y i s a real one , tha t the Platonis t owe s a descriptiv e an d explanator y accoun t o f ou r knowledge o f (o r reliabilit y about ) mathematica l facts , an d I ca n 12 Recal l from ch . 1 , sect. 4, that the nominalist claims there ar e no mathematica l entities. Chihar a (1973 , ch. 5 ) is also a nominalist , but h e proposes a n abbreviated and reinterprete d mathematic s tha t ma y o r ma y no t b e adequat e fo r scientifi c purposes.

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extend thi s accord b y accepting fo r now th e further stipulation tha t this account mus t satisfy th e physicalist. Against thi s share d backdrop , tw o possibl e strategie s stan d out : take mathematica l statement s t o b e (mostly ) tru e an d mee t th e epistemological challeng e hea d on , o r tak e mathematica l state ments (a t leas t th e existentia l ones ) t o b e fals e an d explai n wh y these falsehood s ar e s o usefu l i n applications . Fiel d choose s th e second, whic h require s him t o someho w circumven t the indispensability arguments . What I want t o sugges t here i s that thi s arduou s undertaking doe s no t wi n hi m th e advantage s h e hopes for , indee d that i t doesn't exempt hi m fro m wha t I take to b e the mos t seriou s challenge to Platonism. The bes t approac h t o Field' s method s i s to return t o the thinking of the traditiona l Platonist : mathematics , i f true , i s necessaril y true , that is , it i s true regardles s o f the contingen t detail s o f th e physical world, tru e i n al l possibl e worlds . Le t M b e ou r necessaril y tru e mathematical theory . No w suppos e tha t N i s a consistent theor y o f what th e physica l world migh t b e like, and furthe r suppos e tha t N is nominalistic, tha t it makes no referenc e to mathematica l entities . Then N 4 - M mus t als o b e consistent ; otherwise , th e trut h o f N would impl y th e falsehoo d o f M , an d M couldn' t b e tru e i n a possible worl d wher e N i s true, contradictin g th e necessit y o f M . So, if M i s our tru e mathematical theory , it must be consistent wit h any consistent nominalisti c theory N. Now suppos e tha t A is a nominalisti c statement implie d b y N + M. The n N + not- A 4 - M i s inconsistent . But we'v e jus t finishe d arguing tha t M mus t b e consisten t wit h an y nominalisti c theor y that i s itsel f consistent , s o i t follow s tha t N + not- A i s als o inconsistent. B y elementary logic , thi s mean s tha t N implie s A . So we've shown , fro m th e perspective of the traditional Platonist , tha t if a nominalisti c statement follow s fro m a nominalistic theor y plu s mathematics, the n tha t sam e nominalisti c statemen t follow s fro m the nominalisti c theor y alone . I n technica l terms , mathematic s i s conservative over nominalistic physical science.13 The traditional Platonist's argument doesn't work for the set theoretic realist because she doesn't tak e mathematics to b e necessary (se e ch. 2, sect. 2 , above). The set theoreti c monis t als o denie s tha t ther e i s suc h a thin g a s nominalize d physics , because physica l object s ar e alread y mathematical , whic h severel y undercut s th e potential significance o f any conservativenes s claim.

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Of course , a s a nominalist , Fiel d reject s th e assumptio n tha t mathematics i s true a t all , le t alon e necessaril y true, bu t h e agree s with th e conclusio n tha t goo d mathematica l theorie s shoul d b e conservative: it would b e extremel y surprisin g i f it were t o b e discovered tha t standar d mathematics implie d that ther e ar e a t leas t 10 6 non-mathematica l object s in the universe , or tha t the Pari s Commune wa s defeated ; and wer e suc h a discovery to b e made, al l but th e most unregenerat e rationalist would tak e this a s showin g tha t standar d mathematic s neede d revision . Good mathematics i s conservative; a discover y tha t accepte d mathematic s isn' t conservative would b e a discovery that it isn't good. (Field (1980), 13 )

This conservativeness is a boon t o the nominalist : even someon e wh o doesn' t believ e i n mathematica l entitie s i s fre e t o us e mathematical existence-assertion s i n a certai n limite d context : h e ca n us e them freel y i n deducin g nominalistically-state d consequence s fro m nominal istically-stated premises . An d h e ca n d o thi s no t becaus e h e think s thos e intervening premise s ar e true , bu t becaus e h e know s tha t the y preserv e truth amon g nominalistically-stated claims. (Field (1980), 14)

So thi s i s th e beginnin g o f Field' s answe r t o th e indispensabilit y argument: h e admit s tha t mathematic s i s used i n scienc e t o deriv e physical claim s fro m othe r physica l claims, bu t insist s that we ca n believe these results without believin g the mathematics employed t o be true . W e nee d onl y believ e i t i s conservative , tha t whateve r i t implies is already implied by the physical theory itself . But thi s i s only par t o f th e story , fo r th e rol e o f mathematic s i n deriving one physical statement fro m another i s only one of its roles in physica l science . A s Putna m ha s argued , man y o f th e physica l statements themselve s mak e essentia l appea l t o mathematica l entities. I n other words , th e nominalisticall y stated physica l claims discussed s o fa r don' t cove r mos t o f physica l science . Field i s full y sensitive t o thi s point . H e conclude s onl y tha t 'once such a nominalistic axiom system i s available, the nominalis t is free t o use any mathematic s h e likes for deducin g consequences, a s long as the mathematics h e use s [i s conservative] ' (Fiel d (1980) , 14) . S o th e answer t o th e indispensabilit y argument s come s i n two parts . Firs t it mus t b e show n tha t physica l theorie s ca n b e stated withou t th e use o f mathematics , the n classica l mathematic s mus t b e show n t o be conservativ e ove r thos e restate d physica l theories . Tha t ac complished, th e scientist can use whatever mathematic s she likes in

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deriving nominalisti c consequence s fro m nominalisti c theory , because an y such consequence derive d usin g mathematics i s already implied by the nominalistic theory alone . For concreteness , le t m e rehears e a simplifie d bu t I hop e illustrative example o f this strategy. We ordinarily use real numbers to measur e distance s i n ou r theor y o f space . Thus , b y a standar d indispensability argument, an y confirmatio n of our theor y o f spac e also confirm s th e existenc e o f rea l numbers . But , Field argues , th e use o f th e real s i n thi s contex t i s actually dispensabl e afte r all . To show this , h e mus t firs t reformulat e ou r standar d theor y o f spac e without tal k o f distances . Ther e i s a wa y t o d o this , codifie d b y Hilbert; 14 rather tha n assignin g locations an d distances to points, i t makes us e o f comparativ e predicate s lik e 'between ' an d 'con gruent'. This theory i s nominalistic because it deals only with point s and regions of space and not with numbers. Call it H. Now suppos e we' d lik e t o establis h som e nominalisti c clai m A about space. To appl y rea l number theory, we first move to a larger theory, S , that combines H wit h som e set theory. I n S, we can prov e that ther e i s a functio n fro m pair s o f points t o rea l numbers 15 that does al l the righ t things : for example, the segmen t betwee n x an d y is congruent t o th e segment betwee n x' an d y' i n H i f and onl y if the real numbe r assigne d t o (x , y ) i s th e sam e a s th e rea l numbe r assigned t o ( x ' , y ' ) . I n fact , w e ca n sho w tha t th e spac e itsel f i s isomorphic t o th e se t of ordered triple s of real numbers. In this rich context, w e translat e A int o a n equivalen t statement A ' tha t talk s about distanc e an d henc e abou t rea l numbers , an d w e procee d t o prove A'. Because A and A ' are equivalent, this also establishes A in the theor y S . Bu t S , bein g a goo d mathematica l theory , i s conservative over H , s o A is also implie d by H alone . An d tha t wa s what w e wanted t o sho w i n the first place. But we needn' t assum e the truth of S to do it, only its conservativeness.16 Field extend s thi s techniqu e t o cove r application s o f classica l 14

Se e Hilbert (1899) . Whicheve r version of the reals we select for this measuring job. 16 Student s of Field' s theory wil l realiz e tha t I' m confinin g m y attentio n t o th e second-order versio n o f hi s view . (Se e Shapiro (1983fc ) an d Fiel d (1985 ) fo r discussion o f thi s distinction. ) I d o thi s becaus e it i s th e second-orde r version that offers th e ful l us e o f classica l mathematic s without a n ontolog y o f mathematica l entities—the first-orde r versio n offer s somethin g less—an d becaus e I thin k ou r understanding of space (and other mathematica l notions) is essentially second-order. (See Shapir o (1985 ) o n thi s las t point. ) Fo r a n introductor y accoun t o f th e differences betwee n first- and second-order logic, see Enderton (197 2), ch. 4. 15

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mathematics in Newton's gravitational theory, but som e commentators17 doub t tha t i t ca n b e adapte d t o othe r part s o f physics , i n particular to quantum mechanics . Because his efforts t o date are (at best) onl y partial , Fiel d admit s that th e indispensability argument s retain some force: At presen t o f cours e w e d o no t kno w i n detai l ho w t o eliminat e mathematical entitie s fro m ever y scientifi c explanatio n w e accept ; con sequently, I thin k tha t ou r inductiv e methodolog y doe s a t presen t giv e u s some justificatio n fo r believin g i n mathematica l entities . But . . . justification is not a n all or nothin g affair . . . . what we must do is make a bet on how bes t t o achiev e a satisfactory overal l view o f the place of mathematic s in the world, . . . my tentative be t is that we would d o better to try to show that th e explanator y rol e o f mathematica l entitie s i s no t wha t i t superficially appear s to be; and th e most convincing wa y to d o that would be t o sho w tha t ther e ar e som e fairl y genera l strategie s tha t ca n b e employed t o purg e theorie s o f al l reference to mathematica l entities . (Field (1989), 17-18)

Weighing wha t h e see s a s th e epistemologica l an d ontologica l drawbacks o f Platonis m agains t th e indispensabilit y arguments , Field wager s tha t mathematic s ca n b e show n t o b e dispensable , after all . This is his project. When thi s versio n o f nominalis m i s compare d wit h traditiona l Platonism, som e observers 18 argu e tha t it s space-tim e points an d regions ar e abstract , an d thu s a s susceptibl e a s number s t o epistemological challenge . Fro m th e physicalisti c poin t o f vie w sketched i n th e las t section , thi s ca n hardl y b e true . Space-tim e points an d regions have location, an d some such regions 'fall within our fiel d o f vision' . A s I'v e indicated , the sam e i s true o f th e se t theoretic monist' s impur e sets , s o a t thi s crud e level , nominalism and monis m ar e o n a par , an d bot h ar e preferabl e to traditiona l Platonism. Like th e traditiona l Platonist , Fiel d i s als o face d wit h th e accusation tha t hi s entitie s are causall y inert . Th e argumen t runs that i t i s th e object s i n space-time , no t space-tim e itself , that ar e causally efficacious . Fiel d respond s t o thi s charg e alon g tw o different lines . First, h e suggest s tha t physica l objects b e identifie d 17 18

e.g . Malament (1982). e.g . Resnik(1985a) ,

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with th e space-tim e region s the y occupy. 19 I n this way, a t leas t th e occupied area s of space-time enter int o the causal nexus. A stronger argument involve s the claim that a fiel d theor y i s most naturally construe d a s a theor y tha t ascribe s causal properties . . . to space-time points. (Fiel d (1982) , 70) In electromagneti c theor y fo r instance , th e behavio r o f matte r i s causally explained b y th e electromagneti c fiel d value s a t unoccupie d region s o f space-time . . . (Fiel d (1980) , 114)

Given th e omnipresenc e o f fields , thi s observatio n bring s causa l powers t o al l point s o f space-time . S o once again , th e nominalis t and the monist are on equal footing. To furthe r th e comparison , w e mus t conside r th e respectiv e ontologies mor e closely . Th e nominalist' s worl d consist s o f space time regions ; ordinar y an d theoretica l physica l object s ar e identified wit h th e space-tim e region s the y occupy , an d point s ca n b e identified wit h partles s regions. The monist's worl d consist s of sets; physical object s ar e singleton s amon g these . The contras t betwee n the tw o view s become s smalle r whe n w e realiz e tha t a thoroug h account o f th e monist' s discret e physica l object s wil l involv e th e study of perceptual continu a a s well: object boundaries, trajectories of movement , etc. 20 Thu s spatia l poin t set s joi n th e monist' s ontology. S o bot h nominalis m an d monis m embrac e discret e objects an d continua : fo r the former , both ar e species of space-time regions; fo r the latter, both are species of sets. Finally, what o f tha t ol d poin t o f contention , th e numbers ? The nominalist, o f course , eschew s them . Fo r th e monist , a s we'v e seen,21 the questio n o f th e existenc e of numbers i s a special case of the age-ol d proble m o f the existenc e of universals. Now Fiel d sides with old-fashione d nominalis m agains t universal s a s wel l a s hi s modern variet y against mathematical entities,22 and I see no reaso n why th e monist' s commitmen t t o number s nee d b e an y stronge r than Field' s commitment t o th e propertie s of hi s regions . I n so fa r as old-fashione d nominalis m is tenable, th e monis t ca n agre e wit h Field that numbers don't exist. So far, then , th e nominalis t and th e monis t ar e no t a s fa r apar t a s 19

Fiel d (1982) , 70. Se e ch. 2, sect. 3, above. I n ch. 3, sect. 2, above. 22 Se e Field (1980) , 35, 55-6, an d (1982), 70. 20

21

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rhetoric woul d suggest—wher e th e nominalis t see s a space-tim e region containin g th e stuf f o f three apples , th e monis t see s a set of three apples—but a dramatic difference soo n emerges. These spacetime regions are the end of the ontological stor y fo r the nominalist, but th e monist' s world , in addition t o set s of apples, als o contain s sets o f set s o f apples , set s o f set s o f set s o f apples , an d s o on . Physicalistically speaking , thes e set s o f highe r ran k ar e n o clea r liability; the y hav e locatio n an d the y ca n (a t leas t i n principle ) b e causally efficacious. I f physicalism doesn't rule them out, we should ask wha t motivate s th e monis t t o includ e the m an d wha t th e nominalist hopes to gain by abstaining. Part of the motivation for an escalation of ranks lies in arithmetic itself. Two pair s of shoes are naturally viewed as a set of two sets ; a series o f set s o f ever-increasin g rank , analogou s t o th e vo n Neumann ordinals, does good service as a measuring device for th e number properties o f finite sets; this set theoretic contex t allow s us to prove the Peano axioms and to provide a simple and explanator y theory tha t encompasse s an d explain s variou s well-entrenche d generalizations, fo r exampl e tha t th e unio n o f tw o disjoin t twomembered set s ha s fou r members . B y contrast , th e nominalist' s position her e is much lik e that of the aggregat e theorist considere d back i n Chapte r 2 : a statemen t o f numbe r concern s a mas s o f physical stuf f togethe r wit h a predicate . O n th e on e hand , i t isn' t clear tha t a smoot h an d flexibl e arithmeti c can b e establishe d o n this basis , but, on th e other , I doubt tha t a truly persuasive case of the postulation o f infinite ranks can be based on arithmetic alone.23 Let m e turn , then , t o th e continu a whic h inhabi t bot h th e nominalist's an d th e monist's universes . Both theorists hypothesize that thes e satisf y th e nominalist' s Hilbert-styl e axioms , and , give n that they are physical entities, the nominalist should be as interested as th e monis t i n answerin g furthe r question s abou t them. 24 A number of deep questions can be asked about physical structures of this complexity, 25 an d i t i s the consequence s of askin g them tha t I now want to explore . 23 I conside r thi s possibilit y a t greate r length , bu t n o mor e conclusively , i n my 'Physicalistic Platonism' (forthcoming) , § 7. 24 Field' s remark s abou t mathematic s quote d a t th e beginnin g o f thi s sectio n strongly sugges t tha t a s soon a s entities ar e admitted int o the nominalist's ontology, all facts about them ar e worthy o f investigation. 25 Man y observer s hav e remarke d tha t thes e question s includ e th e continuu m hypothesis. Se e Resnik (19856) , 198 . I'l l conside r othe r question s here, fo r reason s that will become obvious.

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Recall (fro m sectio n 4 o f Chapte r 4 ) tha t bein g determine d i s a property o f set s o f reals . Thi s propert y ca n b e define d mathemat ically i n fairl y simpl e terms ; it s nominalistic counterpart shoul d b e expressible i n Field's nominalisti c theory.26 Now suppos e tha t ou r nominalist observe s tha t a grea t numbe r o f th e simpl e region s h e comes acros s ar e i n fac t determined . A s lon g a s h e consider s nothing mor e complicate d tha n th e regions correspondin g t o wha t the monis t woul d thin k o f a s countabl e union s an d intersection s generated from open sets,27 he will encounter no set that is not determined. A s a scientificall y minde d inquirer , h e wil l wan t a n explanation o f this fact . The monis t ha s a n explanatio n fo r the correspondin g fac t abou t the world of sets in the for m of Martin's theorem tha t al l Borel sets are determined. 28 This explanatio n involves , however, a n inescap able commitmen t t o infinit e ranks ; Harve y Friedma n ha s show n that th e theore m require s them.29 S o we imagin e ou r scientificall y minded monis t hypothesizin g th e axiom s o f infinit y an d replace ment, expandin g he r ontolog y accordingly , i n orde r t o gai n a n explanatory theor y of the behaviour of Borel sets.30 Where doe s this leave our nominalist ? Presumably, he'd als o like to explai n th e behaviou r o f hi s Bore l regions ; surel y h e want s a theory o f hi s continuu m tha t explain s a s muc h a s th e monist's . I t might b e though t tha t th e nominalis t will have to brea k dow yn an d postulate som e aggregate-theoreti c analogu e of the monist's highe r ranks i n orde r t o gai n a counterpar t t o th e theore m o n Bore l regions. Bu t t o thin k thi s i s to forge t th e rol e o f conservativeness . Recall that if ZFC, whic h includes replacement, is conservative over the nominalist's theor y H , the n whatever ZF C ca n prove i s already true i n the nominalist' s world. In particular, i f ZFC i s conservative over H , the n al l Borel regions are determined, regardless of whether or not ther e really are higher ranks of any kind. So, i n orde r t o sho w tha t Bore l region s ar e determined , th e 26 Sayin g a set i s determined involve s quantification over rea l numbers. Methods of Shapiro (1983fc) an d Resni k (19856) sho w how t o simulat e such quantification i n Field's system using space-time points. 27 Fiel d (1980 , p . 63 ) describe s the nominalisti c versio n o f ope n sets . Countable unions and intersection s generate the Borel sets. 28 Marti n (1975 ; 1985). 29 Friedma n (1971). 30 O f course , thi s isn' t th e onl y reaso n fo r assumin g infinity o r replacement , bu t I'm simplifyin g here. For more, see Maddy (1988a) , § 1.8.

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nominalist need only establish the conservativeness of ZFC over the theory H. I n his book, Field gives a set theoretic proof o f this fact in a theor y slightl y stronger tha n ZF C itself , bu t thi s i s a metamath ematical argumen t i n terms o f models tha t i s obviously unavailable to th e nominalist . I n fact , conservativenes s itself i s usually define d metamathematically,31 s o it is unclear that there is even a legitimate nominalistic version of the bare claim that ZFC is conservative. Field's reply is that the conservativeness of ZFC, nominalistically stated, come s t o a clai m abou t logica l possibility , wher e logica l possibility i s a primitiv e notion no t defined , a s i t i s classically , i n metalogical terms . Wha t th e nominalis t need s t o know , then , i s (slightly mor e than) 32 tha t ZF C i s logically possibl e i n thi s sense . How doe s h e kno w this ? Well , presumabl y th e monis t als o claim s to know it, or something very like it, and Field suggests tha t it is no more problematic fo r a nominalist to clai m that a belief that [ZF C is logically possible] is reasonable tha n i t is for a platonist t o make this claim. (Field (1985), 140)

Thus, fo r example, bot h th e nominalist an d th e monist ca n cite the fact tha t n o on e ha s ye t derive d a contradictio n fro m thes e axioms.33 This situatio n ca n b e clarifie d i f w e conside r on e mor e hypo thetical example, returnin g this time to the concerns of the classical analysts. Let' s se e wha t happen s i f th e nominalis t examine s hi s regions an d ask s hi s nominalisti c version o f th e question : i s thi s region measurable? 34 Once again , the simples t region s he consider s will b e measurable . Indeed , thi s tim e h e ca n g o beyon d th e Bore l regions t o th e £ j region s an d stil l no t encounte r anythin g nonmeasurable. Bu t wha t wil l happe n i f h e goe s further ? Th e complements o f th e 2 } regions , th e FI { regions , wil l stil l b e measurable, bu t wha t abou t thei r projections, the 21 regions , an d beyond? Of course , th e monis t ha s considere d thes e question s i n Platonistic terms , an d we'v e see n tha t he r theor y ca n answe r the m 31 M i s conservative over N i f and onl y if, for an y nominalistic assertion A, if A is true in all models of M, then A is true in all models o f N . 32 Se e Field (1985), 139^0 . 33

Se e Field (1984) , 88,124. Fiel d mentions the possibility of non-measurable region s (1980, p. 144 , n . 26). And again , measurability i s a propert y tha t ca n b e stated mathematicall y using only quantification over reals , and thus, shoul d be expressible in Field's system. 34

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definitively.35 Wha t make s thi s cas e differen t fro m th e las t i s that this tim e th e monis t ha s a n embarrassmen t o f riches : tw o competing theorie s tha t answe r thes e question s i n differen t ways ; V = L answer s tha t ther e i s a A j non-measurabl e set , whil e S C implies tha t al l projectiv e set s ar e measurable . In othe r words , a s we've seen , th e monis t i s placed i n a positio n analogou s t o tha t of the natura l scientis t wh o mus t chos e betwee n tw o competin g theories, an d th e reaction of the set theoretic community is much as this analogy would suggest: theorists disagree, they offer competin g evidence, an d furthe r evidenc e i s sought t o decid e th e matter . I've suggested tha t describing and explainin g the justificator y powe r of this sor t o f evidenc e is th e mai n ope n proble m fo r contemporar y Platonism. Where does this leave the nominalist? In order t o se e which of his regions i s measurable , h e needn' t worr y ove r whic h o f th e platonist's tw o opposin g theories is true; he need only know whic h one i s conservative. 36 I n thi s connection, Fiel d says much what h e did before: 37 any reaso n tha t a platonist offer s fo r believin g that i t is [ZFC + SC] rather than [ZF C + V = L ] tha t i s tru e [o r vic e versa ] ca n b e take n ove r b y a nominalist t o argu e wit h jus t a s muc h forc e tha t i t i s [ZF C + SC ] rathe r than [ZF C + V = L ] that is possible. (Fiel d (1985), 140 )

In othe r words , th e nominalis t i s face d wit h exactl y th e sam e bewildering array of argument and counter-argument, evidence and counter-evidence, a s th e monist . Whic h mean s tha t i n orde r t o answer his physical question about the measurability of his regions, the nominalis t mus t fac e a carbo n cop y o f th e mos t difficul t epistemological ope n questio n tha t confront s th e monist : wha t makes these argument s good or bad ? Thus, despit e hi s noble effor t to refrain fro m committin g himself to higher ranks, or anything like them, th e nominalis t ha s no t thereb y save d himsel f fro m th e monist's most difficul t epistemologica l challenge. 35

See ch. 4, sect. 4, above. I say 'which one' because at most one can b e conservative. This is because part of conservativenes s is second-order semantic consistency, and al l standard models of set theor y agre e o n th e structur e o f th e real s an d thei r subsets . Of course , bot h theories might fail to b e conservative. 37 Fiel d actuall y speaks, not o f ZFC + V = L and ZF C + SC , but i n general, of two set theories M and M* which are related as these two are. 36

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Two conclusion s follow . Th e firs t involve s Field' s wage r tha t dispensing with mathematics in science will produce a better overall theory tha n Platonis m ca n provide . I'v e argue d tha t physicalism gives u s n o reaso n t o prefe r Field' s nominalis m t o th e monisti c version o f se t theoreti c realism . Further , th e elementar y epistemology of monism and this nominalism are comparable; a t the most fundamental level , sets o r region s 'fal l withi n ou r fiel d o f vision' . Finally, bot h th e monis t an d th e nominalis t ar e face d wit h a difficult epistemologica l question at the theoretical level. I conclude that Platonis m nee d be no mor e problemati c than Field' s nominal ism, and thus , that ther e is nothing t o balanc e the disadvantages of nominalism—e.g. th e nee d t o rewrit e science—whe n th e overal l merits of th e tw o theorie s ar e compared . O n thes e terms, monis m should prevail. Before turnin g to m y second conclusion , let me pause t o dra w a moral fro m thi s one . We'v e see n tha t th e mathematica l theor y of sets has it s metaphysical as well a s its historical roots in the theor y of the continuum, in the calculus and higher analysis. The Platonist takes the ontology o f this theory at fac e value. The nominalist tries to abstai n fro m thi s as a way o f avoiding some difficul t philosoph ical questions . I thin k th e mora l o f thi s discussio n is that anyone , including th e nominalist , wh o embrace s th e ful l continuu m an d analysis i n som e for m o r another , wil l en d up , soone r o r later , meeting thos e sam e difficul t philosophica l problems , perhap s lightly disguised , bu t stubbornl y undiminished . Thi s lesso n fall s neatly i n lin e wit h th e realisti c urging s o f Frege , Quine , an d Putnam. 38 My secon d conclusio n transcend s th e pro-Platonist propagand a that is the main theme of this book. Towards the end of Chapter 4 , I suggested that set theoretic realism in particular, and compromis e Platonism i n general , ar e no t th e onl y position s face d wit h th e difficult proble m o f assessin g and explainin g the rationalit y of th e non-demonstrative argument s fo r an d agains t V = L and SC . The discussion i n thi s sectio n shows , quit e surprisingly , tha t Field' s nominalism provide s a fres h example ; i n order t o answe r wha t h e counts a s physica l questions , question s abou t space-time-regions , he must deal with thi s sam e open problem. A problem commo n t o nominalist an d Platonis t i s likely to b e a n extremel y fundamental 38

Se e ch. 1, sect. 4, above.

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one, deservin g th e attentio n o f philosopher s fro m a wid e rang e o f persuasions. We'll meet with two more in the next section .

3. Structuralis m Before closing , I' d lik e t o touc h briefl y o n on e othe r conspicuou s position i n th e philosoph y o f mathematic s o f recen t years , namel y structuralism. While lip-servic e to th e general idea behin d this view is fairl y common—mathematic s i s about structures , no t objects — there i s n o complet e an d definitiv e statemen t o f structuralis t orthodoxy comparabl e t o Field' s writing s o n hi s ow n versio n o f nominalism. Michae l Resni k an d Stewar t Shapir o offe r th e mos t comprehensive contemporary statements, 39 bu t thi s work i s still in progress, s o a thoroug h compariso n betwee n structuralis m and se t theoretic realis m wil l hav e t o wai t til l anothe r day . M y mor e modest goa l her e i s simpl y t o sketc h th e mai n outline s o f a structural approac h t o mathematic s an d t o sugges t tha t i t differ s less fro m se t theoreti c realis m tha n partisa n rhetori c woul d indicate. Though structuralis t thinkin g goe s bac k a t leas t t o Dedekind, 40 modern version s ar e inspire d b y consideration s aki n t o thos e i n Chapter 3 above . Th e Platonis t claim s tha t mathematic s i s abou t objects, but , a s Benacerrafia n meditatio n o n multipl e reduction s indicates, w e seem to kno w nothin g abou t thes e objects other than that they are related to one another i n certain ways. If mathematical objects hav e distinguishin g features ove r an d abov e these , thos e properties ar e hidde n an d presumabl y unimportan t t o th e math ematician. How , fo r example , ar e w e t o sa y whic h particula r objects are the natural numbers? 39 Se e Resnik (1975; 1981 ; 1982) , Shapir o (1983
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The structuralis t replie s b y rejectin g the presupposition s o f th e question: In mathematic s . . . we do not hav e object s wit h a n 'internal' compositio n arranged i n structures, w e hav e onl y structures . Th e object s o f mathemat ics, tha t is , the entitie s whic h ou r mathematica l constant s an d quantifier s denote, ar e structureles s point s o r position s i n structures . A s positions i n structures, the y hav e no identit y o r feature s outside o f a structure. (Resni k (1981), 530. Se e also Shapiro (1983*), 534 )

Arithmetic, fo r example , i s no t th e stud y o f certai n objects , th e numbers, but th e study of the natural number structure , a n endless sequence of featureles s position s satisfyin g certai n conditions. On e instantiation o f tha t structur e i s th e vo n Neuman n ordinals , another i s th e Zermel o ordinals . Bu t set s themselve s ar e als o positions i n a structure , s o th e multipl e reduction s o f numbe r theory t o se t theor y jus t sho w tha t th e natura l numbe r structur e occurs many times within the set theoretic hierarchy structure. Some structure s ar e physicall y instantiated : fo r example , th e substructure o f th e natura l numbe r structure consistin g o f it s first three position s i s instantiate d b y th e apple s o n th e table . O n th e other hand , man y pattern s o f highe r mathematics—e.g . th e iterative hierarch y structure—presumabl y ar e no t physicall y re alized. Betwee n thes e extremes , som e application s of mathematics in scienc e com e t o th e postulatio n o f enoug h theoretica l physical entities to exemplify the relevant structure: the clai m tha t actua l spac e exemplifie s th e structur e o f Euclidea n geometr y involves an assertion tha t there is a continuum of space points. . . . science . . . proceed [s] b y discoverin g mathematica l structure s exemplifie d i n materia l reality, bu t th e discover y i s ofte n indirec t an d involve s th e postulatio n o f theoretical entities . (Shapir o (1983a), 540)

In sum: universals are the subject matte r o f mathematics; some bu t not al l of these universals are physically instantiated. Already, certain points o f contact betwee n structuralis m and se t theoretical realis m are obvious : bot h solv e the multipl e reductions problem b y exchangin g object s fo r universals. 41 Indee d muc h o f their tal k i s strikingl y similar . I n th e three-appl e case , th e se t 41 Th e structures themselves are universals (Shapir o (1983a), 536) or, in Resnik's terminology, 'patterns'. The positions in these structures count as 'objects' fo r both, but notic e that these objects ar e structure-dependent: the y have no propertie s apart from th e relations they bear to other positions in the same structure.

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theoretic realis t says there i s a set on th e table tha t i s equinumerous with {4> , {<)>} , {cj> , {4>}}} . Th e structuralis t says there i s a physica l arrangement o n th e tabl e tha t instantiate s the sam e patter n a s {cf> , {4>}, {(£> , {4>}} } under th e successo r relation . Bot h are claimin g that a physica l mas s ha s a certai n organization . On e call s tha t organization formin g a se t equinumerou s with {ct> , {4>} , {cj> , {4>}}} , the othe r call s i t instantiatin g the sam e patter n a s {4> , {(f>} , {4> , {cj>}}}. A t thi s point , I thin k it' s fai r t o wonde r i f an y rea l significance attache s t o thi s differenc e i n description, 42 bu t I won't undertake t o answer that questio n here. There ar e als o agreement s i n epistemologica l thinking . For th e structuralist, variou s claim s abou t a patter n o f dot s o n a piec e of paper are simpl y obvious t o anyon e wh o ha s sufficien t mathematica l experience to understand the m an d who attend s to the diagram. . . . they are in a sense read of f the drawing . S o long a s we ar e takin g our perceptua l faculties fo r granted, the y nee d n o furthe r justification . . . . [they] continu e to hol d whe n talk of dots is replaced by talk of a sequence of squares, stars, a row o f houses, a stac k o f coins , etc . . . . These additiona l assertions are a s evident or almos t as eviden t a s th e origina l ones . W e hav e thu s arrive d a t knowledg e o f a n abstract pattern or structure. (Resni k (1975) , 34)

In plac e o f th e non-spatio-temporal , causall y iner t mathematica l entities o f traditiona l Platonism , th e structuralis t substitute s per ceivable arrangements of things. 43 The existenc e of infinit e pattern s and facts about the m ar e then justifie d theoretically: 44 If [ou r theor y o f th e infinit e structure ] turn s out t o b e highl y coherent an d confirmed b y our knowledg e o f the finite patterns from which i t arose, the n our belie f in the existence of the pattern is justified. (Resni k (1975) , 36-7 ) 42 Th e paralle l i s jus t a s strikin g for rea l numbers : I sa y th e space-tim e point s have th e propert y o f continuity , which ca n b e detecte d usin g various set theoreti c constructions; Shapir o say s the y exemplif y th e 'structur e o f Euclidea n geometry' (Shapiro (1983a) , 540) , whic h i s als o exemplifie d b y variou s se t theoreti c constructions. 43 Ther e ha s bee n som e evolutio n i n Resnik' s thinkin g here . Th e accoun t i n Resnik (1975 ) suggest s tha t w e perceiv e the patter n itself ; i n Resnik (1982) , we see the physica l thing s an d abstrac t th e pattern . I n Resni k (forthcomin g a) , thi s abstractionist epistemolog y i s abandone d altogethe r i n favou r o f a yet-to-be developed 'postulational ' view . Shapir o (1983a , p . 535 ) stick s wit h th e abstrac tionist mode . 44 Shapiro' s accoun t o f ou r knowledg e o f infinit e structure s read s somewha t differently. Se e Shapiro (forthcoming).

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Thus th e structuralist' s epistemolog y parallel s th e two-tiere d account o f th e se t theoreti c realist . A t th e mos t elementar y level , both theorists tur n t o perceptual knowledge—of set s or patterns— and afte r that , to theoretica l knowledge , justifie d b y it s coherenc e and its consequences for lower-level theory. So far , then , th e structuralis t an d th e se t theoreti c realis t ar e i n broad ontologica l an d epistemologica l agreement : the y mee t th e problem o f multiple reductions o f number theory with a move fro m numbers as objects to number s a s universals and th e epistemologi cal proble m fo r traditiona l Platonis m wit h a two-tiere d epistem ology o f perceptua l an d theoretica l justification . Thi s onl y cover s the natura l numbers , bu t bot h advocat e th e sam e sor t o f ex change—objects fo r universals—fo r th e reals, and presumabl y for other traditiona l mathematica l object s tha t ca n b e though t o f a s universals, multipl y instantiate d i n th e iterativ e hierarchy. Wher e the tw o par t company , then , i s i n thei r vie w o f th e se t theoreti c universe itself . Fo r th e se t theoretic realist , thi s 'structure ' consist s of rea l objects , th e sets ; thes e ar e th e bedrock , th e thing s tha t instantiate th e various mathematical universals. For the structural ist, i t i s jus t on e mor e structure , mad e u p o f featureles s point s i n certain relations . Though thi s surel y sound s lik e a substantiv e disagreement , it s true significanc e i s difficul t t o assess . T o se e this , conside r th e structuralist's accoun t o f the interconnection s betwee n branche s of mathematics. I n order t o explain, for example, ho w the study of the natural numbe r structur e ca n b e advance d b y stud y o f th e rea l number structur e (i n analyti c numbe r theory ) o r th e iterativ e hierarchy structur e (i n reductions o f number theor y to se t theory), the structuralis t mus t spea k o f on e structur e bein g 'contained ' o r 'modelled' i n another . Fo r suc h purpose s an d others—e.g . fo r posing th e questio n o f whethe r o r no t V = L—the structuralis t must spea k o f severa l structure s a t onc e an d o f th e relation s between them . Thi s i n tur n require s a n overarchin g 'structur e theory'. Of course , se t theory ca n provide such a backgroun d theory ; al l structures ca n b e take n t o b e set s (o r prope r classe s o f th e leas t problematic kind), as can the functions and relations between them . This i s th e se t theoreti c realist' s position . Bu t th e thoroughgoin g structuralist woul d insis t on a yet-to-be-described structur e theor y

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strong enoug h t o encompas s al l structures, includin g th e iterative hierarchy structure . Th e se t theoretic partisan migh t wonder wha t such an all-encompassing structure theory would b e like, and wha t could mak e i t preferable to th e more familia r theor y o f sets, bu t i n fact, th e prio r questio n is : wha t woul d mak e thes e tw o theorie s different? Shapir o concludes that In a sense , th e theorie s [se t theor y an d a comprehensiv e structur e theory ] are notational variant s of each other. (Shapir o (forthcoming))

If thi s i s so, th e purporte d differenc e betwee n se t theory a s patter n and se t theor y a s bedroc k begin s t o elud e us , alon g wit h tha t between structuralism and set theoretic realism. Just fo r th e record , I' d lik e t o mentio n here tw o consideration s that incline me to resist, at least for now, th e characterization o f set theoretic realis m a s a for m o f structuralism . Th e firs t i s a n epistemological disanalog y betwee n arithmetic— a cas e fo r which even th e se t theoretic realist adopt s a structuralist approach—and set theory—th e cas e stil l ope n t o debate . Structuralis m fo r th e natural number s is so appealin g partly becaus e our understandin g of arithmeti c doesn't depen d o n whic h instantiatio n o f the number structure we choose t o study. For the purposes of simple perceptual access, a s Resnik notes, a pattern of dots wil l do, a s will a sequence of squares , stars, houses, coins , etc. The structuralis t migh t say the same fo r se t theory, tha t i t matter s not whethe r w e begi n fro m a n array o f dots , coins , o r whatever , a s lon g a s the y instantiat e th e initial stage s o f the iterativ e hierarchy pattern. Fo r example , Mar k Steiner, anothe r thinke r wit h stron g structuralis t tendencies, 45 writes: One imagine s o r look s a t materia l bodies , an d the n divert s one's attentio n from thei r concrete spatial arrangement. . . . This is how one might becom e familiar wit h th e standar d mode l o f Z F se t theory—b y abstractin g fro m dots o n a blackboar d arrange d i n a certai n way . Thu s on e arrive s a t a n intuition of the structure o f ZF sets. {Steiner (1975tf), 134-5)

But I think it is not, i n fact, th e properties o f such physical arrays that give us access to the simplest of set theoretic truths. Experience with an y endless row migh t lead us to think tha t every number ha s 45 Se e Steiner (1975#) , 134 . A s noted above (ch. 3, sect. 1) , Steiner's structuralis m is onl y epistemic : number s ar e objects , bu t th e onl y thing s wort h knowin g abou t them ar e their relations t o other numbers .

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a successor , bu t i t is experience wit h set s themselves that produce s the intuitive belief that an y two thing s can be collected int o a set or that a se t will hav e the sam e number o f elements eve n after i t ha s been rearranged . I n othe r words , thoug h an y instantiatio n o f th e natural numbe r structur e ca n giv e u s acces s to informatio n abou t that structure , ou r informatio n abou t th e se t theoreti c hierarch y structure come s fro m ou r experienc e wit h on e particula r instantiation.46 Thus on e motivatio n fo r th e mov e to structuralis m i n the case of number theory is undercut in the case of set theory. My secon d sourc e o f concer n abou t th e assimilatio n o f se t theoretic realis m to structuralis m arise s out o f the simpl e question of what se t theory is about. The set theoretic realist answers that set theory i s the study of the iterative hierarchy with physical objects as ur-elements; th e se t theoreti c monis t take s physica l object s themselves to b e sets an d eschews pure sets altogether. Th e trouble with these answers from th e structuralis t perspective is that som e o f the 'positions', i n particula r th e ur-elements , hav e propertie s beyon d those the y hav e solel y b y virtu e o f thei r relation s wit h othe r positions i n th e structure . Th e purel y relationa l structur e arisin g from th e iterativ e hierarch y wit h ur-element s woul d mak e n o distinction betwee n th e positio n occupie d b y thi s appl e an d th e position occupied b y this orange, betwee n the position occupied by the set of the apple and the orange and the position occupied by the set o f th e appl e an d thi s baseball , distinction s th e se t theoreti c realist will certainly want to preserve. Thus the structuralist bent on assimilating se t theoreti c realis m b y claimin g tha t th e iterativ e hierarchy with ur-element s is itself a purely relational structure will have to mov e to a larger, containin g pattern, fro m whos e poin t of view th e basebal l an d th e fruit s ar e jus t position s wit h onl y relational properties. 47 M y worr y i s how t o squar e thi s wit h th e naturalist's common-sense realism.48 But whatever the upshot of these inconclusive speculations abou t whether se t theoreti c realis m should o r shouldn' t b e considere d a version o f structuralism , m y mai n goa l her e i s to cal l attentio n t o yet anothe r poin t o f agreement . Notic e tha t i n th e pur e iterativ e 46

Parson s (forthcoming) , §9 , make s a relate d poin t abou t th e epistemologica l importance of recognizing non-relational feature s of sets. 47 Resni k (persona l communication ) has suggested thi s move, and ther e are hints of it in Shapiro (forthcoming) . 48 Se e ch.1, sect. 2, above.

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hierarchy structure, the continuum hypothesis is either true or false, the projectiv e set s either d o o r don' t includ e a non-measurable se t or a n uncountabl e se t without a perfec t subset . Thus , onl y on e of SC and V — L can b e true there. So again, as in the previous section, the non-partisa n conclusio n i s tha t structuralists , a s wel l a s compromise Platonist s an d Fieldia n nominalists , will hav e t o fac e the difficul t proble m o f assessin g th e rationalit y o f argument s fo r and agains t th e variou s theoretica l hypothese s tha t migh t answe r these open questions. There i s a variatio n o n structuralis m accordin g t o whic h math ematics i s th e stud y no t o f structure s bu t o f possibl e structures . Rather tha n investigatin g sets (compromis e Platonism ) or th e se t theoretic patter n (structuralism) , the modalis t investigate s wha t would b e the cas e if there were a se t theoretic hierarchy of the sor t the Platonis t describes. 49 ' 2 + 2 - 4' translate s t o 'i f ther e wer e a natural number structure, 2 plus 2 would equa l 4 in that structure'. This view has obviou s if-thenist elements , and i t suffers fro m man y of th e sam e difficulties. 50 Notic e als o tha t th e modalist' s actua l world i s purely physical ; all mathematica l things exist (i f at all ) in some other possible world. Eve n if the extreme monism of section 1 above is rejected, the pro-Platonist arguments of Quine and Putna m suggest tha t suc h a separatio n o f th e physica l fro m th e mathemat ical i s not feasible . Thu s th e modalist , like Field, must find a wa y t o defuse th e indispensability arguments. 51 Epistemologically, th e modalis t owe s a n accoun t o f moda l knowledge tha t ha s no t bee n forthcoming . On e migh t thin k al l that's needed is an explanation of the logica l implication from , say , the Pean o axiom s t o 2 + 2 = 4, bu t ther e i s more ; th e moda l translation wil l no t wor k properl y unles s th e Pean o axiom s ar e jointly possible. 52 I n se t theory , th e correspondin g requiremen t i s that th e iterativ e hierarch y b e possible , an d i n thi s possibl e 49

Thi s sor t o f translatio n i s suggeste d i n Putna m (\967a), thoug h h e doesn' t espouse modalism. Following Putnam's method, Hellman (1989 ) does. 50 Fo r a partia l lis t o f these , se e ch . 1 , sect . 4 , above . Fo r more , se e Madd y (forthcoming b ) or Resni k (1980) , ch. 3. 51 Se e Hellman (1989), ch. 3 , an d Fiel d (1988) , §§6-7 , fo r a n assessmen t o f th e modalist's prospects. 52 I f no such structur e is even possible , 2 + 2 = 4 an d 2 + 2 = 5 are bot h true , along wit h everything else. Again, the logi c involved mus t be second-order i f we ar e to speak of a unique natural number structure.

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structure, th e continuu m hypothesi s i s eithe r tru e o r false , th e projective set s eithe r d o o r don' t includ e a well-orderin g o f th e reals, an d s o on. Thus, the modalis t face s a questio n analogou s t o Field's—which o f V = L or S C i s conservative?—namely, which of these pattern s i s possible? I brin g u p modalis m her e primaril y t o point out tha t i t is among th e many positions, bot h Platonisti c an d non-Platonistic, tha t fac e no t jus t th e difficul t questio n o f whether or no t a supercompac t cardina l exists (her e or i n another possibl e world), bu t th e prio r an d perhap s mor e difficul t proble m o f ho w one might rationally answe r suc h a question . 4. Summar y Realism abou t a give n branch o f inquir y is the contentio n tha t it s subject matte r exist s objectively , that various effort s t o reinterpre t its claim s shoul d b e resisted , an d tha t mos t o f it s well-supporte d hypotheses ar e at least approximatel y true. I've endorsed common sense realis m abou t medium-size d physical object s on th e ground s that th e bes t explanatio n o f wh y i t seem s t o u s tha t ther e i s a n objective world of such objects is that there is an objective world of such object s tha t i s responsibl e fo r ou r beliefs . Thi s explanatio n takes place, not within a priori philosophy, but within our scientifi c theory o f th e worl d an d ourselve s as cognizers ; thi s i s naturalism. I've also adopte d scientifi c realis m about th e theoretica l entitie s o f natural science, because these unobservable things play a role in our best theor y o f the world . Simila r reasoning cite s the centra l role of classical mathematics in both th e statement and the development of natural science as evidence for mathematica l realism or Platonism . These ar e th e pro-Platonis t indispensabilit y argument s o f Quin e and Putnam. Quine/Putnam Platonism differ s fro m th e traditional variet y over the purporte d a priority , certainty , an d necessit y o f mathematica l truth. A s a complet e theor y o f mathematica l knowledge , i t als o differs fro m th e practice o f mathematics itself : it fails to account for unapplied mathematic s an d fo r th e obviousnes s o f elementar y mathematics; i t ignore s th e actua l justificator y practice s o f math ematicians. Godel' s version o f Platonism , b y contrast , present s a n appealing two-tiered accoun t of justification within mathematics— intuitive and theoretical—but fails to support the scientific status of

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mathematics a s a whol e an d rest s it s accoun t o f elementar y knowledge o n a n unpersuasiv e notio n o f mathematica l intuition . Nevertheless, Godelia n Platonism stands wit h Quine/Putnamis m in opposition t o the traditional variety. I've propose d a compromise between thes e tw o moder n version s of Platonism . Fro m Quine/Putnamism , it take s th e indispensabilit y arguments a s support s fo r th e (approximate ) trut h o f classica l mathematics. Fro m Godel , i t take s th e two-tiere d analysi s o f mathematical justification . Bu t t o provid e a complet e picture , compromise Platonis m owes a replacement for Godel's intuition; in deference t o naturalism , thi s replacemen t mus t b e scientificall y feasible. Th e leadin g theme o f this boo k ha s bee n the developmen t and defenc e o f se t theoreti c realism , a versio n o f compromis e Platonism designe d to fill in this outline. It has long bee n thought tha t Godel's intuition , his epistemologi cal bridg e between th e object s of mathematical knowledge an d th e mathematical knower , canno t b e develope d naturalistically . Benacerraf's classica l statemen t o f thi s worr y (1973 ) depend s o n the then-popula r causa l theorie s o f knowledg e an d reference , bu t I've argue d tha t neithe r thes e no r a particularl y robus t notio n o f truth ar e essentia l to posin g th e problem. What matter s i s that th e beliefs o f mathematician s ar e reliabl e indicator s o f fact s abou t mathematical things ; thi s fac t call s ou t fo r a naturalisti c explan ation. Fro m this point> various forces—among them the conviction that mathematic s i s a legitimat e science analogou s t o th e physica l sciences—lead t o th e convictio n tha t a t leas t par t o f thi s explan ation mus t involv e a perception-lik e connectio n betwee n objec t known an d knower . Ad d t o thi s th e traditiona l Platonist' s characterization o f mathematica l entitie s a s non-spatio-tempora l and acausal , and it' s eas y t o se e why a naturalistic account i s ofte n considered impossible. The se t theoreti c realis t meet s thi s proble m b y admitting set s o f physical objects to the physical world, giving them spatio-tempora l location wher e the physical stuff tha t make s up their member s (an d the member s o f thei r members , etc. ) i s located. Thes e impur e set s then prov e appealin g candidate s fo r th e subject s o f perceptua l numerical beliefs , an d psychologica l an d speculativ e neurologica l considerations giv e scientifi c suppor t t o th e vie w tha t the y ar e directly perceived . Thu s par t o f th e se t theoreti c realist' s per ception-like connectio n i s jus t perceptio n itself . A n accompanyin g

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neurological phenomeno n furnishe s a rudimentary intuitive faculty whose products—intuitive beliefs—provide fallible but prima-facie justifications fo r th e mos t elementar y genera l assumption s o f se t theory. This accoun t depend s essentiall y o n th e clos e relationshi p between numerica l belief s an d belief s abou t sets , whic h raise s th e familiar ontologica l questio n o f whethe r number s simpl y are sets . Part of the scientific support fo r set theory rests on the foundation it provides fo r numbe r theor y an d analysis , an d thi s foundationa l theory i s standardly expresse d b y identifyin g th e natura l an d rea l numbers with certai n sets . But, as Benacerraf has pointed out , thi s identification i s ultimately unsatisfying becaus e it can be done with equal ease in several different ways ; thi s is the problem of multiple reductions. I f there is nothing to decide between the von Neuman n and th e Zermel o ordinal s whe n identifyin g th e natura l number s with sets , ho w ca n eithe r sequence of sets clai m to actuall y be th e numbers? The set theoretic realist's answer , implici t in the accoun t of se t perception , i s tha t neithe r sequenc e i s th e numbers , tha t numbers ar e propertie s o f set s whic h eithe r sequenc e i s equall y well equipped t o measure . The sam e line of response works fo r th e real number s whe n the y ar e understoo d a s detector s fo r th e property of continuity. If th e first tier o f Godel's epistemological theor y can b e ascribe d to the set theoretic realist's perceptio n an d intuition , there remains the proble m o f describin g an d accountin g fo r th e rationalit y o f reasoning at the theoretical level . In set theory, despit e traces of the traditional Platonisti c view that axioms ar e obvious or self-evident, theoretical defence s fo r axio m candidate s ca n b e foun d eve n i n Zermelo's first axiomatization, an d the y figure prominently i n the search fo r new hypotheses tha t will decide natural analyti c and set theoretic question s lef t ope n b y th e currentl y accepte d axiom s o f ZFC. Th e proble m o f assessin g th e rationalit y o f variou s non demonstrative argument s fo r an d agains t ne w se t theoreti c hypo theses become s mor e acut e a s se t theorist s devis e alternative , conflicting, theories . Th e firs t ste p i n helpin g adjudicat e suc h disputes i s a descriptiv e catalogue o f th e evidenc e offered b y each side. A modes t contributio n t o tha t projec t i s al l tha t ha s bee n attempted here . The nex t step , th e evaluatio n of this evidence , is a daunting undertaking , but I'v e argued tha t th e se t theoretic realis t faces thi s challeng e i n th e distinguishe d compan y o f thinker s

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representing a wide range of competing mathematical philosophies, structuralism, modalism , and a version of nominalism among them . Finally, fo r th e benefi t o f thos e wit h physicalisti c leanings, I'v e sketched se t theoreti c monism , a mino r variatio n on se t theoreti c realism. For the monist, all sets have physical grounding and spatiotemporal location , an d al l physica l object s ar e sets . Thes e manoevres produc e a radica l 'one-worldism'— a realit y a t onc e mathematical an d physical—tha t should appea l t o philosopher s of this stripe. In sum , then , I certainl y d o no t clai m t o hav e show n tha t m y version o f Platonis m raise s n o difficul t philosophica l problems . A t best, a t best, I hav e show n ho w t o replac e th e tw o prominen t Benacerraf-style objection s t o traditiona l Platonis m wit h a ne w open questio n abou t th e justificatio n o f theoretica l hypothese s i n set theory . Bu t whateve r th e complexitie s o f thi s ne w problem , I think thi s trad e amount s t o progress . I n th e defenc e o f mathemat ical realism , th e ne w proble m enjoy s a clea r advantag e ove r it s predecessors: nothin g o n it s fac e i s likel y t o inspir e one o f thos e nagging a prior i argument s against th e ver y possibilit y of Platon ism. O n th e contrary , th e question s i t raises—question s o f rationality—are standar d i n th e philosoph y o f al l sciences , an d there i s no obviou s reason wh y the y should b e any les s tractable in mathematics than they are in physics or physiology . But there is more to b e said for this new problem tha n that it may lighten th e perceived burden on th e defende r of Platonism. I attach considerable importanc e t o th e fac t tha t i t arises also fo r adherent s of alternativ e philosophical positions; thi s suggests that i t taps into a fundamenta l issue insensitive to minor variations in philosophical fashion. An d beyon d this , ther e i s th e allurin g possibilit y tha t philosophical progres s o n question s o f mathematica l rationalit y could mak e a rea l contributio n t o mathematic s itself, especiall y to the curren t search for new axioms . Thus , once again , I recommend pursuit o f thi s ne w proble m eve n t o philosopher s blissfull y uninvolved in the debate ove r Platonism. Mathematicians ofte n thin k o f themselve s a s scientists , explorin g the intricacie s of mathematica l reality ; and , fo r goo d reason , the y are especiall y incline d toward s suc h view s i n th e absenc e o f philosophers, I hav e trie d t o sho w that , contrar y t o popula r

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philosophical opinion , somethin g clos e t o th e mathematician' s natural attitude is defensible. Theories of mathematical knowledge tend eithe r to trivializ e it as conventional o r purel y formal or eve n false, o r t o glamoriz e i t a s perfect , a priori , an d certain , bu t se t theoretic realism aims to treat it as no more nor less than the science it is, and t o be fair, al l at once, to the mathematician who produce s the knowledge, the scientist who use s it, and th e cognitiv e scientist who mus t explai n it . I propos e it , then , a s anothe r step—afte r Codel, Quine , an d Putnam—o n th e lon g roa d toward s math ematics naturalized.

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STEINER, M . (1978), 'Mathematical explanation', Philosophical Studies, 34, pp. 135-51 . STRAWSON, P. (1959), Individuals (London : Methuen). SUSLIN, M . (1917) , 'Su r une definitio n de s ensemble s mesurable s B san s nombres transfinis' , Comptes rendus d e I'Academie de s Sciences d e Pans, 164, pp. 88-91 . TARSKI, A . (1933) , 'The concept o f truth i n formalized languages', repr . i n his Logic, Semantics, an d Metamathematics, 2n d edn . (Indianapolis, Ind.: Hackett, 1983) , 152-278. TROELSTRA, A . S . (1969) , Principles o f Intuitionism (Berlin : Springer Verlag). ULAM, S . (1930) , 'Zu r Masstheori e i n de r allgemeine n Mengenlehre' , Fundamenta Mathematicae, 16 , pp. 140-50 . URMSON, J. O. (1956), Philosophical Analysis (Oxford: Oxford University Press). VAN HEIJENOORT , J . (ed. ) (1967), From Frege to Godel (Cambridge , Mass.: Harvard Universit y Press). VON NEUMANN , J . (1923) , 'O n th e introductio n o f transfinit e numbers' , repr. in van Heijenoort (ed.) (1967), 346-54. (1925), 'A n axiomatizatio n o f se t theory' , repr . i n va n Heijenoor t (ed.) (1967) , 393-413 . WANG, H . (1974a) , 'Th e concep t o f set' , ch . 6 o f hi s (19746) , repr . i n Benacerraf an d Putnam (eds. ) (1983) , 530-70. (1974fc), From Mathematics t o Philosophy (London : Routledg e an d Kegan Paul). WEDBERG, A . (1955) , Plato's Philosophy o f Mathematics (Stockholm : Almqvist and Wiksell). WILLIAMSON, J . H . (1962) , Lebesgue Integration (Ne w York: Holt , Rinehart, and Winston). WILSON, M. (1979) , 'Maxwell' s condition—Goodman's problem', British journal for the Philosophy of Science, 30, pp. 107—23 . (1985), 'What is this thing called "pain"?—th e philosophy of science behind th e contemporar y debate' , Pacific Philosophical Quarterly, 66 , pp. 227-67. • (forthcoming), 'Honorabl e intensions' , i n S . Wagner an d R . Warne r (eds.) Notes Against a Program. WITTGENSTEIN, L . (1953) , Philosophical Investigations (Ne w York: Macmillan). (1978), Remarks on the Foundations of Mathematics rev. edn., eds. G. H . vo n Wright , R . Rhees , an d G . E . M . Anscomb e (Cambridge , Mass.: MI T Press). WOODIN, H . (1988) , 'Supercompac t cardinals , set s o f reals , an d weakl y homogeneous trees' , Proceedings o f th e National Academy o f Sciences

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WRIGHT, C. (1983) , Frege's Conception o f Numbers a s Objects (Aberdeen : Aberdeen University Press). YOURGRAU, P . (1985) , 'Sets , aggregates , an d numbers' , Canadian Journal of Philosophy, 15 , pp. 581-92. ZERMELO, E. (1904), 'Proof that every set can be well-ordered', repr. in van Heijenoort (ed. ) (1967), 139-41. (1908(3), ' A ne w proo f o f th e possibilit y of a well-ordering' , repr . i n van Heijenoort (ed, ) (1967) , 183-98 . (1908b), 'Investigation s i n the foundation s of se t theory, I' , repr . i n van Heijenoort (ed. ) (1967), 199-215. (1930), 'fiber Grenzzahlen und Mengenbereiche', Fundamenta Mathematicae., 16, pp. 29-47.

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INDEX

abstract object s 2,21,36-7,40,59 , 152,163,172 accumulation points 108,114-1 5 Achinstein, P. 1 0 n. Ackermann, W. 10 5 n. Aczel, P. 4 0 n. Addison.J. 126,133,137- 8 Alexandroff, P. 11 2 analysis 10 7 n. analytic vs. synthetic 27- 8 Anderson, C . 6 2 n. a posteriori vs. a priori 9 , 30, 33, 41, 74, 177 a priori, see a posteriori vs. a priori Aristotle 12 , 158 Armstrong, D. 12-13 , 46, 51 n., 52 n., 156 n. Ayer, A. 6 n., 16 n. Ayers, M. 3 9 n. axiom of choice 117-24 , 133,135, 138,142-3,145,146 axiom of constructibility (V = L ) 103 n., 132-5,136,137-9,142-3, 144-5, 147-9, 168-9, 173, 176, 177 axiom of extensionality 15 3 axiom o f foundation 4 0 n., 105 n. axiom o f infinity 125,131,141,16 6 axiom of replacement 130—1 , 136, 141, 166 axioms 4,27,31-3,40,67,70,72-3 , 77-8,102,107,114,117-18,125, 128-9,132,136,143-4,148, 179-80 see also axiom of choice; axiom of constructibility ( V = L) ; axiom of extentionality; axiom of foundation; axiom o f infinity ; axiom of replacement; larg e cardinal axioms; pairing axiom; power se t axiom; projective determinacy; separation axiom ;

union axiom; unlimited comprehension; Zermelo—Fraenke l axioms (ZFC). Baire.R. 111-12,118 , 121,131 n. Bealer, G. 6 2 n. Benacerraf, P. 36 , 42, 43-4,45 n., 79, 81 n., 84-6, 88 n., 89-90,98-100, 178,179 Bendixson, I. 11 0 Berkeley, G. 6,16,2 2 Bernays, P. 10 2 n., 104 n. Blackwell,D. 137- 8 Bonevac, D. 3 6 n. Bonjour, L. 7 2 n. Boolos, G. 3 9 n., 102 n. Borel,E. 111,118,12 1 Borel sets 111-14 , 120,136, 138, 145, 166,167 Bower, T. 5 3 n., 54, 64 Boyer, C. 2 2 n. Bridgman, P. 1 0 n. Brouwer, L. 1 6 n., 22 Bruner, J. 6 6 n. Burali-Forti, C. 2 2 n. Burgess, J. 2 0 n., 43 n., 46-8, 49 n., 78 n. Cantor, G. 22,24 , 81, 86, 102,10710,114-17,118,121,125,126, 129-30,131 n. Cantor—Bendixson theorem 110,112 , 114,129 Cantor's theorem 109 , 131 cardinal numbers 115,120-1,130-1 , 145 Carnap, R . 1 0 n., 27-8,29,154 n. Casullo, A. 4 2 n. causal theory of knowledge 37 , 41-2 , 44,49, 72,178 causal theory of reference 38—41,42 , 48-9, 178

200

INDEX

cell-assemblies 56-8 , 66, 68-70, 73-4, 151 certain vs. fallible 21,30,33,71 , 128 n., 177, 179 Chihara, C. 33-4 , 78, 128-9, 150-4, 159 n. Chisholm, R. 14 4 classes 4 n., 61-2, 100 , 102-6, 11 6 n., 121-4, 134, 173 closed set 11 0 Cohen, P. 127 , 130-1 contingent vs. necessary 21 , 30, 41, 62-3,160-1,177 continuity 74-5 , 81 , 94-5, 110 , 164, 165,169,171,179 see also real number s continuum hypothesis (CH) 10 0 n., 110, 114-16, 125-32 , 133, 135, 142-3, 145,148 , 165 n., 176, 177 conventionalism 27—8 , 29 correspondence theory of truth 16—2 0 countable 10 9 n. countable choic e 12 0 DaubenJ. 8 1 n., 10 7 n., 10 9 n., 110 n., 116 n., 13 0 n., 131 n. Davis, M. 13 8 n. Davis, P, and Hersh, R. 2 n., 3 Dedekind, R . 22,75,81-2,86,94 , 102,108-9,114,118, 170 Dennett, D. 5 7 n. dependent choice 120,12 1 Descartes, R . 8-9 , 72 n. descriptive theory of reference 37— 8 detectors 53,57,58,65,8 0 see also cell-assemblies determinacy 137-40 , 142-3, 145-7, 166 see also projectiv e determinacy Detlefsen, M. 2 5 n. , 47 n. Devitt, M. 1 1 n., 1 3 n., 14 n., 1 5 n., 17 n., 38 n.,40n. Devlin, K. 10 3 n., 133 n., 134 Dieudonne, J. 2 disquotational theory o f truth 17—2 0 Drake, F. 8 2 n., 104 n., 116 n., 13 2 n., 135 n., 13 6 n. Dummett, M . 1 5 n., 16n.,23 Eklof, P., and Mekler, A. 4 n. Ellis, B. 8 7 n. Enderton, H . 4 n., 25 n., 27 n. , 39 n., 59 n.,62n., 71 n., 81 n., 82 n.,

83 n., 84 n., 86 n., 89 n., 95 n., 96 n., 102 n., 105 n., 109 n., 115 n., 125 n., 131 n., 133 n., 153 n., 162 n. epistemology 4 n. see also causal theory o f knowledge; epistemology naturalized ; justifie d true belief; reliabilism epistemology naturalize d 9,11 , 13,20,35,43,46-8,97 equinumerous 8 3 n. extensional vs. intensional 1 3 n., 61,

90-2 externalism vs. internalism 72— 3 extrinsic justifications 33 , 35, 71 n., 72-5, 7 6 n., 77-8, 80 , 82, 107, 118-21, 124-5, 126-7, 128 , 135 , 136-7, 138-«, 144-9, 150, 168, 172-3,176, 177,179-80

fallible, see certain vs. fallibl e Feferman, S. 2 5 n. , 29 n., 10 6 n. Field, H. 1 7 n., 18 n., 19 n., 26 n., 36n.,43^,45n.,47-8, 81 n., 154-6, 159-69, 176- 7 Flagg, R. 106n . Fodor,J. 5 n., 154 n. Foreman, M. 14 3 n. formalism 2-3 , 10 , 23-6, 28 see a/so Hilbert' s programme ; if themsm Fraenkel, A. 2 7 Fraenkel, A., Bar-Hillel, Y., and Levy , A. 10 4 n., 10 5 n., 13 4 n., 157 n. Frege, G. 2 2 n., 23-4, 25, 26, 37, 60, 67 n., 82-3, 86 n., 88, 89-92, 94, 103, 104,114 , 121, 169 Freidman, H. 13 6 n., 166 function 95-6,97,110-12,114,120 , 121-2, 124 Gale, D. 13 8 n . Gelman, R. 6 3 n. Gettier, E . 36-7 , 41 n., 51, 66 n., 72 Gibson, E. 5 4 n . Gibson, J. 5 0 n. Godel, K. 25 , 28-9, 31-5, 6 2 n., 71 n., 75-80, 10 3 n., 104 n., 120-1, 125-9, 130-2, 134-6 , 148 , 153, 157 n., 177-9,181 Goldman, A . 37 , 51 n., 71 n., 72 n., 73 Gottlieb, D. 3 6 n. Gregory, R . 4 6 n. , 50 n. , 51 n.

INDEX Grice.P. 4 6 n., 51 Grover, D . 1 7 n. Hadamard,J. 122,12 4 Hale, B. 4 1 n. Hallett,M. 11 0 n., 114 n., 116 n., 129 n., 130 n. Hambourger, R. 10 1 n. Harman, G. 3 7 n., 46 n. Harrington, L. 12 7 Hart,W. 4 2 Hausdorff, F . Il l n. Hebb, D. 4 9 n., 50 n., 52 n., 53-4, 55 8,61,65-7,68-70,73-4 Hellman, G . 3 6 n., 176 n. Hempel, C . 1 0 n., 67 n. Henkin, L. 12 5 n. Heyting, A. 1 6 n., 22 n. Hilbert, D. 24-5 , 11 6 n., 117 n., 162 Hilbert's programm e 24— 5 Hodes.H. 2 6 n., 9 I n. Hume, D . 7 idealism 6 , 7, 16, 22-3 if-thenism 25-6 , 127 , 170 n., 176 inaccessible cardinals 12 7 n., 135— 6 indispensability arguments for mathematical entities 29—31,34 — 5,47, 58-9, 62-3 , 7 6 n., 82,11920,121,155,159-63,176,177-8 infinity 22-4,78-9,81,82,115,17 2 instrumentalism 10—11,2 4 intrinsic justifications 33 , 72—5,118 — 19,128,139-43 see also extrinsic justifications; intuition intuition 31-5 , 70-5, 7 6 n., 77, 78, 107,118-19,123-5,126-7,13943,144,150,174-5,177-9 intuitionism 22-3 , 28 iterative conception o f set 39—40 , 61 n., 62 n., 84, 100,102^, 123 4,128,132-3,134,141,153,1567, 171, 173- 7 Jech,T. 13 2 n., 136 n., 142 n. Jensen, R . 13 3 Jourdain, P. 10 9 n., 116n. Jubien, M. 8 1 n. justified, true belief 36-7 , 4 1 n., 72 Katz,J. 5 9 n. Kaufman, E. 6 0 n.

201

Kelley,]. 10 4 n. Kim,J. 40n.,46n. , 59 n., 67 n. Kitcher, P. 3 0 n., 36 n., 64 n., 81 n., 84 n., 86 n. Kline, M. 2 2 n., 110 n., Ill n. K6nig,J. 115-1 6 Korner, S. 2 5 n . Kripke,S. 38,4 0 n., 79-80,105 Kuratowski, K. 11 1 n., 113 n. large cardinal axioms 135-7 , 140-2 , 143 n., 146 see also inaccessible cardinals; measurable cardinals; supercompact cardinal s (SC) Lear, J. 3 7 n. Lebesgue,H. 111-13,118,121- 2 Lebesgue measure 113,114,125—7 , 133,135,136,138,167-8,176 Leeds, S. 17n . Levy, A. 13 7 see also Fraenkel, A., Bar-Hillel, Y., and Levy , A. Lewis, D. 13 n., 14 n., 41 n., 80 n., 96 Locke, J. 12 logical notion o f collection, se e classes logical positivism 6 n., 9 n., 10 n., 16 n., 26-8,154 logicism 26-8 , 148 Luce,L. 9 1 n. Luzin,N. 113-14,11 8 n., 125,137 McCulloch,W. 52- 3 Machamer, P. 5 0 n . Malament, D. 4 7 n., 163 n . Martin, D. 10 3 n., 124, 126-7, 128, 130,131,135 n., 137-42, 143 n., 145,146-7,166 mathematical notio n o f collection, see iterative conception o f set maths/science analogy, se e science/ mathematics analogy Maxwell, G. 1 0 n. measurable cardinals 136—7 , 138,139, 140, 142, 145 Menzel, C. 8 8 n . Merrill, G. 8 0 n. metaphysics 4 n. Mill,]. 6n. , 16 n., 67 n. Miramanoff, D. 27 , 40 n. modalism 176-7 , 180 monism, set theoretic 158-9,163-9 , 175,180

202

INDEX

Monna, A. Il l n. Moore, G. 11 0 n., 114 n., 116 n., 117, 118 n., 120 n., 121 n., 127 n., 129 n., 132 n. Morse, A. 10 4 n. Moschovakis, Y. 2 , 3, 4 n., Ill n. , 113 n., 120 n., 126 n., 127 n., 134-5, 137-9, 14 0 n. MycielskiJ. 13 8 n. natural collections or kinds 14 , 39-40, 80, 96-8 natural numbers 35 , 82-94, 97-100, 164, 165 , 170-3, 174-5,176, 179 naturalism 9 , 13, 14, 17, 20, 21, 29, 35, 42-3, 46-8, 58, 78-80, 87, 88, 96,175,177-8,181 see also epistemology naturalize d necessary, see contingent vs. necessary Neisser, U . 50 n. nominalism 12 , 28-9, 43-4,46-8, 96, 159-69,176,180 Novikov, P . 114 , 13 7 Nyikos,P. 12 7 n. objectivity 6 , 14, 177 ontology 4 open set 11 1 operationalism 10 , 14 ordinal numbers 115 , 130, 157 n. pairing axiom 67,12 5 paradoxes 22 , 26 n., 62, 71 n., 83, 104, 105 Parsons, C . 31 , 34 n., 59 n., 84 n., 89 n., 103 n., 105 n., 170 n., 175 n. perception 31,46,49-67,74,76-7 , 123, 150-3,157 , 172-3,178-9 perfect set 11 0 perfect subset property 112 , 114, 125 — 7, 130,133,135, 136-7,138,143, 145,176 phenomenalism 6-7 , 8, 10, 14, 16, 32 Phillips,]. 54 , 63 n., 64 n., 74 n. physicalism 5 , 35, 46, 154-7,159, 160, 163,165,169,180 PiagetJ. 54 , 63,74 n., 89 Pitcher, G. 50n.,5 1 n. Plato 12,20-1,36,15 8 Platomsm 20-1,26,28-35,36-7,40 1,42,46-8,81,127, 129,132,

155, 158 , 159, 160, 163, 167-9, 170, 177 , 180 compromise 34-5 , 41, 48, 75, 76 n., 107, 120 , 146, 148, 150, 169, 176, 178 Fregean 2 6 Godelian 28-9 , 31-5, 75-9, 107, 128, 159,177- 8 mythological 12 9 Quine/Putnam 28-31,33,34-5,45 , 46,75,119-20,159,177-8 traditional 21,30,33,36-7,40-1 , 43-4,48, 59, 78,156, 160,163, 172, 177-8,179,18 0 see also realism, mathematical ; realism, set theoretic power set axiom 131,136,14 1 projective determinacy 138^0 , 142, 145-6 projective sets 113-14 , 120 , 135, 1369,145-6,147,168,176,177 proper classes, see classes properties 12-13 , 93-4, 96-8, 103 n., 123-4 see also universals pure sets 5 9 n., 156- 7 Putnam, H. 8 n., 10 n., 1 2 n., 13 n., 14 n., 16 n., 20 n., 25-6, 28-30, 34, 38 n., 47, 67, 80 n., 93 n., 96, 155, 161,169 , 17 6 n., 177, 181 see also Platonism, Quine/Putna m Quine,W. 7-9,11,13,27-8,28-31 , 34, 39 n., 41,93 n., 153 n., 155, 169,177,181 see also Platonism, Quine/Putnam realism common sense 6—9 , 14 , 31—2, 35, 175,177 mathematical 14-15 , 20-1, 177 ; see also Platonism; realism , set theoretic pre-theoretic 1-3,15-16,35,45-6 , 180-1 scientific 10-11,14,35,129,17 7 set theoretic 35,59 , 61, 67, 71 n., 75-8, 81 , 86-7, 96,107,118,128, 144, 150-4 , 156-9, 160 n., 169, 170—5, 178—81 ; see also monism , set theoretic about universals 11-14 , 15, 28, 96 8,164

INDEX

and truth 15-2 0 real numbers 22,23,35,62,81-2,86 , 94-5,97,108-13,162,166 n., 167 n., 168 n., 172 n., 173,179 reference 18-20,37-4 1 see also causal theory of reference; descriptive theory of reference Reinhardt,W. 105 n. see also Solovay, R., Reinhardt, W., and Kanamori, A. reliabilism 42— 3 reliability 18-20,43-5 , 73,148,159, 178 Resnik, M. 25 n., 36 n., 47 n., 81 n., 91 n., 100 n., 163 n., 165 n., 166 n., 170-3, 175 n., 176 n. rules of thumb 141-2,143,14 6 Russell, B. 22,25,26-7,32,76,103 , 104

Salmon, N. 38 n. Schoenflies, A. 11 6 science/mathematics analogy 2,15, 313,45-6,67-8,75,76-8, 87-8, 93, 98,128 n., 129,146-8,168,178, 181 Scott, D. 13 5 n., 136 Searle,J. 3 8 n. separability 112,114,125-7,133 , 137-9,145 separation axiom 7 1 n. set theory as a foundation for mathematics 4-5 , 81-2 , 89 , 1023,114,119,179 Shapiro, S. 4 7 n., 162 n., 166 n., 170 3,174,175 n. Shoenfield,]. 3 9 n., 102 n. Sierpiriski, W. 112,11 3 n., 121, 128 n., 136 n., 145 Simpson, S. 2 5 n . singletons 150-3 , 158 Skolem, T. 2 7 Skyrms, B. 3 7 n . Solovay, R. 127,12 8 n., 136,137, 138 n., 140 n. Solovay, R., Reinhardt, W., and Kanamori, A. 14 2 n. Steel, J. 14 0 n., 142 Steiner,M. 34,41-2 , 82 n., 86 n., 89 n., 174 Steinhaus, H. 13 8 n. Stewart, F. 13 8 n. Strawson, P . 3 8 n.

203

structuralism 2 1 n., 35,170-7,180 supercompact cardinals (SC ) 142-3 , 144,145,147,148-9,168,169, 176,177 Suslin, M. 113-14,11 8 n., 125,12930 Swierczkowski, S. 13 8 n . synthetic, see analytic vs. synthetic Tait,W. 10 6 n. Takeuti, G. 13 8 n. Tarski,A. 18 , 136 n. theoretical justifications, see extrinsic justifications Thomae, J. 2 3 transitive closure 5 9 Troelstra, A. 2 2 n. truth 15-20 , 43,105, 178 see also correspondence theory of truth; disquotational theory of truth; verificationism Ulam, S. 13 6 n . uncountable 10 9 n. union axiom 67,125,13 1 unit sets, see singletons universal* 11-14 , 21 n., 87, 96-8,164, 171,173 see also properties unlimited comprehension 7 1 n., 83, 117 Urmson, J. 7 n. van der Waerden, B. 12 0 van Heijenoort, J. 116n . verificationism 16-17,2 3 von Neumann,]. 27 , 84, 104 n. Wang, H. 2 9 n., 40 n., 135 n., 136 n., 139 n. Wedberg,A. 1 2 n. well-ordering 114-17,121 , 125, 133, 135,137,138,139,143,145,146, 177 Williamson,]. 11 3 n. Wilson, M. 1 2 n., 13 n., 141 n. Wittgenstein, L. 1 5 n., 79-80, 97 Woodin,H. 14 0 n., 142 Wright, C. 2 6 n., 41 n. Yourgrau, P. 8 8 n.

204 I N D E

X

Zermelo, E. 27 , 40, 84, 102, 116-23, 10 4 n., 125-8, 132-4, 136, 137-8, 136 n., 157 n., 179 142-3,144,145,146,15 3 n., Zermelo-Fraenkel axioms (ZFC) 2 7 n., 166-8 , 179