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

WHAT I S TH E Q U E S T I O N ? 3

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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

38 P E R C E P T I O

N AN D I N T U I T I O N

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

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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 + 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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1

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

79

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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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5

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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9

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 , {} , {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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1

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.

REFERENCES ACHINSTEIN, P . (1965) , Th e proble m of theoretica l terms' , repr . i n Brody (ed.) (1970) , 234-50. ACKERMANN, W . (1956) , 'Zu r Axiomati k de r Mengenlehre' , Mathematiscke Annalen, 131, pp. 336-45. ACZEL, P . (1988) , Non-Well-founded Sets, Cente r fo r th e Stud y o f Language and Information, Lecture Notes, no. 14. ADDISON, J. W. (1958) , 'Separation principles in the hierarchies of classical and effectiv e descriptiv e se t theory' , Fundamenta Mathematicae, 46 , pp. 123-35 . (1959), 'Some consequences of the axiom o f constructibility', Fundamenta Mathematicae, 46 , pp. 337—57 . and MOSCHOVAKIS , Y. N. (1968) , 'Some consequences o f the axio m of definabl e determinateness', Proceedings o f th e National Academy o f Sciences (U. S. A.), 59, pp. 708-12. ALEXANDROFF, P . (1916 ) 'Su r la puissanc e de s ensemble s mesurable s B' , Comptes rendus de I'Academie de s Sciences de Paris, 162, pp. 232—5 . ANDERSON, C . A . (1987) , 'Revie w o f Bealer' s Quality an d Concept', Journal o f Philosophical Logic, 16, pp. 115—64 . ARISTOTLE (1952) , The Works o f Aristotle Translated into English, 1 2 vols., ed. W. D. Ross (Oxford: Oxford University Press). Categories, in his (1952). Metaphysics, i n his (1952). Physics, in his (1952). ARMSTRONG, D . (1961) , Perception an d th e Physical World (London : Routledge and Kegan Paul). (1973), Belief, Truth an d Knowledge (Cambridge : Cambridg e University Press). (1977), 'Naturalism , materialis m an d firs t philosophy' , repr . i n hi s (1981), 149-65. (1978), Universals an d Scientific Realism (Cambridge : Cambridg e University Press). (1980), 'Agains t "ostric h nominalism".' , Pacific Philosophical Quarterly, 16 , pp. 440-9. (1981), The Nature o f Mind (Ithaca , NY: Cornel l University Press).

REFERENCES 18

3

AYER, A . J. (1946), Language, Truth, an d Logic, 2nd edn . (New York : Dover, 1952). AYERS, M. R . (1981) , 'Lock e versus Aristotle on natura l kinds', Journal of Philosophy, 78, pp. 247-72. BAIRE, R . (1899) . 'Su r le s fonction s d e variable s reelles' , Annali d i matematica pura ed applicata, 3, pp. 1-122. BOREL, E., HADAMARD, J. , and LEBESGUE , H. (1905) , 'Five letters o n set theory', repr. in Moore (1982), 311-20. BARWISE, J . (ed. ) (1977) , Th e Handbook o f Mathematical Logic (Amsterdam: North Holland). BEALER, G. (1982), Quality and Concept (Oxford: Oxford University Press). BENACERRAF, P. (1965) , 'What numbers could not be' , repr. i n Benacerraf and Putnam (eds.) (1983), 272-94. (1973), 'Mathematica l truth' , repr. i n Benacerraf and Putna m (eds. ) (1983), 403-20. (1985), 'Comment s o n Madd y an d Tymoczko' , i n Kitche r (ed. ) (1985), 476-85. and PUTNAM , H. (eds. ) (1983) , Philosophy o f Mathematics, 2n d edn . (Cambridge: Cambridge University Press). BERKELEY, G. (1710), The Principles of Human Knowledge, i n his (1957). (1713), Three Dialogues between Hylas an d Philonous, in his (1957). (1734), The Analyst, i n his (1957). (1957), The Works o f George Berkeley, Bishop ofCloyne, 9 vols., ed. A. Luce and T. Jessop (London: Thomas Nelson and Sons). BERNAYS, P . (1935) , 'On platonis m i n mathematics' , repr . i n Benacerraf and Putnam (eds.) (1983), 258-71. (1937), ' A syste m o f axiomati c se t theory , I' , Journal o f Symbolic Logic, 2, pp. 65-77. BLACKWELL, D . (1967) , 'Infinit e game s an d analyti c sets' , Proceedings of the National Academy o f Sciences (U.S.A.), 58, pp. 1836-7 . BONEVAC, D. A . (1982) , Reduction i n the Abstract Sciences (Indianapolis , Ind.: Hackett). BONJOUR, L . (1980) , 'Externalis t theorie s o f empirica l knowledge' , Midwest Studies i n Philosophy, 5 (Minneapolis : Universit y o f Minnesota Press), 53—73. BOOLOS, G . (1971) , Th e iterativ e conception o f set' , repr . i n Benacerraf and Putnam (eds.) (1983), 486-502. BOREL, E . (1898) , Lemons su r l a theorie de s fonctions (Paris : Gauthier Villars). BOWER, T . G . R. (1966), 'The visual world of infants', Scientific American, 215, no . 6, pp. 80-92. (1982), Development i n Infancy, 2n d edn . (San Francisco: W . H . Freeman and Company).

184 R E F E R E N C E

S

BOYER, C . B . (1949) , Th e History o f th e Calculus an d it s Conceptual Development, (Ne w York : Dover , 1959) . (Origina l title: The Concepts of the Calculus.) BRIDGMAN, P . W . (1927) , The Logic o f Modem Physics (Ne w York: Macmillan). BRODY, B. (ed.) (1970), Readings i n th e Philosophy o f Science (Englewoo d Cliffs, NJ : Prentice Hall). BROUWER, L. E. J. (1913) , 'Intuitionis m and formalism' , repr. in Benacerraf and Putnam (eds. ) (1983) , 77-89 . (1949), 'Consciousness , philosophy , an d mathematics' , repr . i n Benacerraf an d Putnam (eds. ) (1983) , 90-6. BRUNER, J . (1957) , 'O n perceptua l readiness' , repr . i n R . Harpe r e t al. (eds.), Th e Cognitive Processes (Englewoo d Cliffs , NJ : Prentic e Hall , 1964), 225-56. BuRALi-FoRTi, C. (1897) , 'A question o n transfinit e numbers' , repr . i n van Heijenoort (ed. ) (1967), 104-12. BURGESS, J. P. (1983). 'Why I am no t a nominalist', Notre Dame Journal o f Formal Logic, 24, pp. 93-105. (1984), 'Syntheti c mechanics' , Journal o f Philosophical Logic, 13 , pp. 379-95. (forthcoming a), 'Synthetic physics and nominalis t realism', to appea r in C . W . Savag e an d P . Erlic h (eds.) , The Nature an d Function o f Measurement. (forthcoming b) , 'Epistemolog y an d nominalism' , t o appea r i n A . Irvine (ed.), Physicalism in Mathematics. CANTOR, G. (1872) , 'Uber die Ausdehnung eines Satzes aus der Theorie de r trigonometrischen Reihen', Mathematische Annalen, 5, pp. 123—32 . (1878), 'Ein Beitrag zur Mannigfaltigkeitslehre' , Journal fu r di e reine und angewandte Mathematik, 84 , pp. 242—58 . (1883), Grundlagen einer allgemeinen Mannigfaltigkeitslehre (Leip zig: B . G. Teubner). (1891), 'Ube r ein e elementar e Frag e de r Mannigfaltigkeitslehre' , Jahresbericht de r Deutschen Mathematiker-Vereinigung, 1 , pp. 75—8 . (1895/7), Contributions to th e Founding o f th e Theory o f Transfinite Numbers, ed. P. E. B. Jourdain (Chicago: Open Court , 1915) . (1899), 'Lette r t o Dedekind' , repr . i n va n Heijenoor t (ed. ) (1967) , 113-17. CARNAP, R . (1934) , The Unity o f Science (London : Kega n Paul , Trench , Trubner an d Company). (1936/7), 'Testabilit y an d meaning' , Philosophy o f Science, 3 , pp . 428-68, and 4, pp. 1^0. (1937), Logical Syntax o f Language (London : Routledg e an d Kega n Paul).

REFERENCES 185 (1950), 'Empiricism , semantics , an d ontology' , repr . i n Benacerra f and Putnam (eds.) (1983), 241-57. CASULLO, A . (forthcoming ) 'Causality , reliabilism , an d mathematica l knowledge', to appear . CHIHARA, C . (1973) , Ontology an d th e Vicious-Circle Principle (Ithaca , NY: Cornell University Press). (1982), ' A Godelia n thesi s regarding mathematical objects : D o the y exist? And can we perceive them?', Philosophical Review, 91, pp. 21127. CHISHOLM, R . (1977) , Theory o f Knowledge, 2n d edn . (Englewoo d Cliffs , NJ: Prentice Hall). COHEN, P . J . (1966) , Se t Theory an d th e Continuum Hypothesis (Ne w York: W. A. Benjamin). DAUBEN, J . W . (1979) , Georg Cantor (Cambridge , Mass. : Harvar d University Press). DAVIS, M . (1964) , 'Infinit e game s o f perfec t information' , Annals o f Mathematics Studies, 52, pp. 85-101. DAVIS, P . J. , an d HERSH , R . (1981) , Th e Mathematical Experience (Boston: Birkhauser). DEDEKIND, R . (1872) , 'Continuit y an d irrationa l numbers' , i n Essays o n the Theory of Numbers (La Salle, III: Open Court, 1901), pp. 1-27. DENNETT, D. C. (1978), Brainstorms (Bradford Books). DESCARTES, R . (1641) , Meditations o n First Philosophy, 2n d edn. , i n hi s (1967). (1967), Philosophical Works o f Descartes, 2 vols., ed. E. S. Haldane and G. R. T. Ross (Cambridge : Cambridge University Press). DETLEFSEN, M. (1986) , Hilbert's Program (Dordrecht : Reidel). DEVITT, M . (1980) , '"Ostrich nominalism " o r "mirag e realism"?' , Pacific Philosophical Quarterly, 61 , pp. 433-9. (1981), Designation (Ne w York: Columbia University Press). (1984), Realism an d Truth (Princeton , NJ : Princeto n Universit y Press). DEVLIN, K . (1977) , Th e Axiom o f Constructibility (Berlin : Springer Verlag). (1984), Constructibility (Berlin : Springer-Verlag). DRAKE, F . (1974) , Se t Theory: A n Introduction t o Large Cardinals (Amsterdam: North Holland) . DUMMETT, M . (1975) , 'Th e philosophical basi s of intuitionist logic' , repr . in hi s (1978) , ch . 14 , and i n Benacerraf and Putna m (eds. ) (1983) , 97 129. (1977), Elements o f Intuitionism (Oxford : Oxford Universit y Press). (1978), Truth an d Other Enigmas (Cambridge , Mass. : Harvar d University Press).

186 R E F E R E N C E

S

EKLOF, P . C , an d MEKLER , A . H . (forthcoming], Almost Free Modules: Set-Theoretic Methods, forthcoming fro m Nort h Hollan d Publishers , Mathematical Library Series. ELLIS, B, (1966), Basic Concepts o f Measurement (Cambridge : Cambridg e University Press). ENDERTON, H. (1972) , A Mathematical Introduction t o Logic (Ne w York: Academic Press). (1977), Elements o f Set Theory (Ne w York: Academic Press). FEFERMAN, S . (1984tf) , 'Towar d usefu l type-fre e theories , P , Journal o f Symbolic Logic, 49, pp. 75-111. (1984&), 'Kurt Godel: Conviction an d caution' , repr. in S. G, Shanker (ed.), Godel's Theorem i n Focus (London : Croo m Helm , 1988) , 96-114. (1988), 'Hilbert's progra m relativized : Proof-theoretical and founda tional reductions', Journal o f Symbolic Logic, 53, pp. 364—84. FIELD, H. (1972) , 'Tarski's theory of truth', Journal o f Philosophy, 69 , pp . 347-75. (1980), Science without Numbers (Princeton , NJ : Princeto n University Press). (1982), 'Realis m an d anti-realis m about mathematics' , repr . i n hi s (1989), 53-78. (1984), 'Is mathematical knowledge just logical knowledge?', repr. in his (1989), 79-124. (1985), 'On conservativeness and incompleteness', repr. in his (1989), 125-46. (1986), The deflationar y conceptio n of truth', in G, MacDonald an d C. Wright (eds.) , Fact, Science and Value (Oxford: Basil Blackweli), 55117. (1988), 'Realism , mathematics , and modality' , repr . i n hi s (1989) , 227-81. (1989), Realism, Mathematics, an d Modality (Oxford : Basi l Blackwell). (forthcoming), 'Physicalism'. FODOR, J. A . (1975) , Th e Language o f Thought (Ne w York: Thoma s Y . Crowell). FOREMAN, M . (1986) , 'Poten t axioms' , Transactions o f th e American Mathematical Society, 294, pp. 1-28 . FRAENKEL, A . A . (1922) , 'Z u de n Grundlage n de r Cantor-Zermelosche n Mengenlehre', Mathematische Annalen, 86, pp. 230—7 . BAR-HILLEL, Y., and LEVY , A. (197'3), Foundations o f Set Theory, 2n d rev. edn . (Amsterdam: North Holland). FREGE, G. (1884), The Foundations o f Arithmetic, 2nd rev. edn. (Evanston , 111: Northwestern Universit y Press, 1968). (1892a), 'On concept and object', in his (1970), 42-55.

R E F E R E N C E S 18

7

(18926), 'On sense and reference', in his (1970), 56-78. (1903), Grundgesetze de r Arithmetik, vol. ii. Relevan t section s ar e reprinted in his (1970), 182-233. (1970), Translations from th e Philosophical Writings o f Gottlob Frege, ed. P. Geach and M. Black (Oxford: Basil Blackwell). (1979), Posthumous Writings, ed . H. Hermes , F . Kambartel, and F. Kaulbach (Chicago: University of Chicago Press). FRIEDMAN, H . (1971) , 'Highe r se t theor y an d mathematica l practice' , Annals o f Mathematical Logic, 2, pp. 325-57. GALE, D. , an d STEWART , F . M . (1953) , 'Infinit e game s wit h perfec t information', Annals o f Mathematics Studies, 28, pp. 245—66 . GELMAN, R. (1977), 'How young children reason about small numbers', in N. Castellan , D . Pisoni , an d G . Pott s (eds.) , Cognitive Theory, i i (Hillsdale, NJ: Lawrence Erlbaum Associates), 219-38. GETTIER, E . (1963) , 'Is justifie d tru e belie f knowledge?' , Analysis, 23 , pp . 121-3. GIBSON, E . (1969) , Principles o f Perceptual Learning an d Development (New York : Appleton-Century-Crofts). GIBSON, J . J . (1950) , Th e Perception o f th e Visual World (Boston : Houghton Mifflin) , GODEL, K . (1930) , 'Th e completenes s o f th e axiom s o f th e functiona l calculus of logic', repr. in van Heijenoort (ed.) (1967), 582—91. (1931), 'O n formall y undecidabl e proposition s o f Principia Mathematica an d relate d systems , I' , repr . i n va n Heijenoor t (ed. ) (1967), 596-616. (1938), 'The consistency of the axiom of choice and of the generalized continuum hypothesis' , Proceedings o f th e National Academy o f Sciences (U.S.A.), 24, pp. 556—7 . (1940), Th e Consistency o f th e Continuum Hypothesis (Princeton : Princeton University Press). (1944), 'Russell' s mathematica l logic' , repr . i n Benacerra f an d Putnam (eds.) (1983), 447-69. (1946), 'Remark s befor e th e Princeto n Bicentennia l Conference o n problems i n mathematics' , i n M . Davi s (ed.) , Th e Undecidable (Ne w York: Raven Press, 1965), 84-8. (1947/64), 'Wha t i s Cantor' s continuu m problem?' , repr . i n Benacerraf and Putnam (eds. ) (1983), 470-85. GOLDMAN, A . (1967) , ' A causa l theor y o f knowing' , Journal o f Philosophy, 64, pp. 357-72. (1975), 'Innat e knowledge', i n S. Stich (ed.) , Innate Ideas (Berkeley: University of California Press), 111—20. (1976), 'Discrimination and perceptual knowledge', Journal of Philosophy, 73, pp. 771-91. (1977), 'Perceptual objects', Synthese, 35, pp. 257-84.

188

REFERENCES

GOLDMAN, A . (1979) , 'Wha t i s justifie d belief?' , i n G . Pappa s (ed.) , Justification an d Knowledge (Amsterdam : Reidel), 1-23 . (1980), Th e internalis t conceptio n o f justification' , Midwest Studies i n Philosophy, 5 (Minneapolis : Universit y of Minnesot a Press) , 27-51. GOTTLIEB, D . (1980) , Ontological Economy (Oxford : Oxfor d University Press). GREGORY, R. L. (1970), The Intelligent Ey e (Ne w York: McGraw-Hill) . (1972), Eye and Brain, 2nd edn. (Ne w York: McGraw-Hill). GRICE, P . (1961) , 'Th e causa l theor y o f perception' , repr . i n R . Swart z (ed.), Perceiving, Sensing an d Knowing (Berkeley : Universit y o f California Press) , 438-72. GROVER, D. , CAMP , J. , an d BELNAP , N. (1975) , 'A prosentential theor y o f truth', Philosophical Studies, 27, pp. 73-125. HALE, B. (1987). Abstract Objects (Oxford : Basil Blackwell). HALLETT, M . (1984) , Cantonan Se t Theory an d Limitation o f Size (Oxford: Oxfor d Universit y Press). HAMBOURGER, R. (1977) , ' A difficult y wit h th e Frege—Russel l definition o f number', Journal of Philosophy, 74, pp. 409-14. HARMAN, G . (1973), Thought (Princeton : Princeton University Press). HARRINGTON, L . A. , MORLEY , M . D. , SCEDROV , A. , an d SIMPSON , S . G . (eds.) (1985) , Harvey Friedman's Research o n th e foundations o f Mathematics (Amsterdam: North Holland). HART, W . H . (1977) , 'Revie w o f Steiner' s Mathematical Knowledge', journal of Philosophy, 74, pp. 118-29. HAUSDORFF, F . (1919) , 'Ube r halbstetig e Funktione n un d dere n Verallgemeinerung, Mathematische Zeitschrift, 5 , pp. 292—309. HEBB, D . O . (1949) , Th e Organization o f Behavior (Ne w York: Joh n Wiley and Sons). (1980), Essay on Mind (Hillsdale , NJ: Lawrenc e Erlbaum Associates). HELLMAN, G . (1989) , Mathematics Without Numbers (Oxford : Oxfor d University Press). HEMPEL, C . G . (1945) , 'O n th e natur e o f mathematica l truth' , repr . i n Benacerraf and Putnam (eds.) (1983) , 377-93. (1954), ' A logica l appraisa l o f operationalism' , repr . i n hi s (1965) , 123-33. • (1965), Aspects o f Scientific Explanation (Ne w York: The Fre e Press). HENKIN, L . (1949) , 'Th e completenes s o f th e first-orde r functiona l calculus', Journal o f Symbolic Logic, 14, pp. 159—66 . HEYTING, A. (1931), 'The intuitionist foundations of mathematics', repr. in Benacerraf and Putnam (eds. ) (1983) , 52-61. (1966), Intuitionism: A n Introduction, 2n d rev . edn. (Amsterdam : North Holland).

R E F E R E N C E S 18

9

HILBERT, D. (1899) , Foundations o f Geometry (L a Salle, 111.: Open Court , 1971). (1926), 'O n th e infinite' , repr . i n Benacerra f an d Putna m (eds. ) (1983), 183-201, an d in van Heijenoort (ed.) (1967), 367-92. • (1928), 'Th e foundation s o f mathematics' , repr . i n va n Heijenoor t (ed.) (1967) , 464-79. HODES, H . (1984) , 'Logicis m an d th e ontologica l commitment s o f arithmetic', Journal of Philosophy, 81 , pp. 123—49 . HUME, D. (1739), A Treatise of Human Nature, vol. i, in his (1886). (1886), Philosophical Works, 4 vols. , ed . T . H . Gree n an d T . H . Grose (London). JECH, T. (1978) , Set Theory (Ne w York: Academic Press). JUBIEN, M. (1977), 'Ontology and mathematical truth', Nous, 11, pp. 133 50. KATZ, J . J . (1981) , Language an d Other Abstract Objects (Totowa , NJ : Rowman and Littlefield) . KAUFMAN, E . L, LORD , M. W. , REESE , T . W., and VOLKMANN , J. (1949) , 'The discrimination of visual number', American Journal o f Psychology, 62, pp. 498-525. KELLEY, J. L. (1955), General Topology (Princeton , NJ: van Nostrand). KIM, J . (1977) , 'Perceptio n an d referenc e withou t causality' , Journal o f Philosophy, 74 , pp. 606-20. (1981), 'The role of perception in a priori knowledge', Philosophical Studies, 40, pp. 339-54. KITCHER, P . (1978), 'The plight of the Platonist', No«s, 12, pp. 119-36 . (1983), The Nature o f Mathematical Knowledge (Ne w York: Oxford University Press). -(ed.) (1985) , PS A 1984 , i i (Eas t Lansing : Philosoph y o f Scienc e Association) KLINE, M . (1972) , Mathematical Thought from Ancient t o Modern Times (New York : Oxford University Press). KORNER, S . (1960), The Philosophy o f Mathematics (London : Hutchinson University Library). KRIPKE, S. (1972), 'Naming and necessity', in D. Davidson and G . Harman (eds.), Semantics o f Natural Language (Dordrecht : Reidel) , 253—355 , 763-9. (1975), 'Outlin e o f a theor y o f truth' , Journal o f Philosophy, 72 , pp. 690-716 . • (1982), Wittgenstein o n Rules an d Private Language (Cambridge , Mass.: Harvard Universit y Press). KURATOWSKI, K. (1966), Topology, i (New York: Academic Press). LEAR, J . (1977) , 'Set s an d semantics' , Journal o f Philosophy, 74 , pp . 86— 102.

190 R E F E R E N C E

S

LEBESGUE, H . (1902) , 'Integrate , longueur , aire' , Annali d i matematica pura ed applicata, 7, pp. 231-359. — (1905), 'Su r les fonctions representable s analytiquement, Journal d e mathematiques pures et appliquees, 60 , pp. 139—216 . LEEDS, S . (1978) , Theorie s o f referenc e an d truth' , Erkenntnis, 13 , pp . 111-29. LETTVIN, J. Y. , MATURANA , H . R. , MCCULLOCH , W. S., an d PITTS , W . H . (1959), 'Wha t th e frog' s ey e tell s th e frog' s brain' , repr . i n W . S . McCulloch, Embodiments of Mind (Cambridge, Mass.: MIT Press, 1965), 230-55. LEVY, A. , an d SOLOVAY , R . M . (1967) , 'Measurabl e cardinal s an d th e continuum hypothesis', Israel Journal o f Mathematics, 5, pp. 234—48 . LEWIS, D . (1983) , 'Ne w wor k fo r a theor y o f universals' , Australian Journal o f Philosophy, 61, pp. 343-77. (1984), 'Putnam' s paradox' , Australian Journal o f Philosophy, 62 , pp. 221-36. (1986), On th e Plurality of Worlds (Oxford : Basil Blackwell). LOCKE, J . (1690) , A n Essay Concerning Human Understanding (New York: Dover, 1959) . LUCE, L . (1988), 'Frege on cardinality' , Philosophy an d Phenomenological Research, 48, pp. 415-34. LUZIN, N . (1917) , 'Su r l a classificatio n d e M . Baire' , Comptes rendus d e I'Academie de s Sciences de Paris, 164, pp. 91—4 . (1925), 'Sur les ensembles projectifs d e M. Henr i Lebesgue', Comptes rendus de I'Academie de s Sciences de Paris, 180, pp. 1572-4 . (1927), 'Su r le s ensembles analytiques', Fundamenta Mathematicae, 10, pp. 1-95 . MACHAMER, P . (1970) , 'Recen t wor k o n perception' , American Philosophical Quarterly, 7, pp. 1-22 . MADDY, P. (1980), 'Perception an d mathematica l intuition', Philosophical Review, 89, pp. 163-96. (1981), 'Sets and numbers', Nous, 15, pp. 494-511. (1983), 'Proper classes', Journal o f Symbolic Logic, 48, pp. 113—39 . (1984a), 'Mathematical epistemology: what is the question?', Monist, 67, pp. 46-55. (19846), 'How the causal theorist follow s a rule', Midwest Studies i n Philosophy, 9 (Minneapolis: University of Minnesota Press) , 457-77. (1984c), 'Informal notes on proper classes' , unpublished notes. (1986), 'Mathematica l alchemy' , British Journal fo r th e Philosophy of Science, 37, pp. 279-314. (1988a), 'Believin g the axioms' , Journal o f Symbolic Logic, 53, pp . 481-511,736-64. (19886), 'Mathematical realism' , Midwest Studies i n Philosophy, 1 2 (Minneapolis: University of Minnesota Press), 275-85.

R E F E R E N C E S 19

1

(forthcoming a), 'Physicalistic Platonism', to appear i n A. Irvine (ed.), Physicalism in Mathematics. (forthcoming b), The root s of contemporary Platonism' , to appear in the Journal of Symbolic Logic. MALAMENT, D . (1982) , 'Revie w o f Field' s Science Without Numbers', Journal of Philosophy, 79 , pp. 523-34. MARTIN, D . A . (1968) , 'Th e axio m o f determinatenes s an d reductio n principles i n th e analytica l hierarchy' , Bulletin o f th e American Mathematical Society, 74 , pp. 687—9 . (1970), .'Measurabl e cardinal s an d analyti c games' , Fundamenta Mathematicae, 66, pp. 287-91. (1975), 'Borel determinacy', Annals of Mathematics, 102, pp. 363-71. (1976), 'Hilbert's first problem: The continuum hypothesis', Proceedings o f Symposia i n Pure Mathematics, 28 , (Providence , RI: American Mathematical Society) , 81-92. (1977), 'Descriptiv e se t theory : projectiv e sets' , i n Barwis e (ed. ) (1977), 783-815. (1980), 'Infinite games', Proceedings of th e International Congress of Mathematicians (Helsinki, 1978), pp. 269-73. (1985), 'A purely inductive proof o f Borel determinacy', Proceedings of Symposia in Pure Mathematics, 42 (Providence, RI: American Mathematical Society), 303-8. 'Projective sets and cardinal numbers', unpublished photocopy. 'Sets versus classes', unpublished photocopy. and SOLOVAY , R. M . (1970) , 'Interna l Cohe n extensions' , Annals o f Mathematical Logic, 2, pp. 143—78 . and STEEL , J . (1988) , 'Projectiv e determinacy' , Proceedings o f th e National Academy of Sciences (U.S.A.), 85, pp. 6582-6. (1989), ' A proo f o f projectiv e determinacy' , Journal o f th e American Mathematical Society, 2, pp. 71—125 . MAXWELL, G. (1962) , 'Th e ontological statu s o f theoretical entities' , repr . in Brody (ed.) (1970), 224-33. MENZEL, C . (1988) , 'Freg e number s an d th e relativit y argument' , Canadian Journal o f Philosophy, 18 , pp. 87-98 . MERRILL, G . H . (1980) , 'Th e model-theoreti c argumen t agains t realism' , Philosophy o f Science, 47, pp. 69—81 . MILL,]. S. (1843), A System o f Logic, in his (1963/88), vols. vii and viii. (1865), An Examination of Sir William Hamilton's Philosophy, in his (1963/88), vol. ix. (1963/88), Th e Collected Works o f John Stuart Mill, 2 9 vols. , ed . J. M. Robson and J. Stillinger (Toronto: Universit y of Toronto Press). MIRIMANOFF, D . (1917#) , 'Les Antinomies de Russell et de Burali-Forti et le problem e fondamenta l de l a theori e de s ensembles', L'Enseignement mathematique, 19, pp. 37—52 .

192

REFERENCES

MIRIMANOFF, D . (\9\7b), 'Remarque s su r l a theori e de s ensemble s e t le s antinomies Cantoriennes , I' , L'Enseignement mathematique, 19 , pp. 209-17. MONNA, A . F . (1972) , 'Th e concep t o f functio n i n th e 19t h an d 20t h centuries', Archive for History o f Exact Sciences, 9, pp. 57—84 . MOORE, G . H. (1982) , Zermelo's Axiom of Choice (Ne w York : SpringerVerlag). (forthcoming), 'Introductor y not e t o 194 7 and 1964' , Th e Collected Works o f Kurt Godel, vol. ii , forthcoming from Oxford Universit y Press. MORSE, A. (1965), A Theory o f Sets (New York: Academic Press). MOSCHOVAKIS, Y . N . (1970) , 'Determinac y an d prewellordering s o f th e continuum', i n Y. Bar-Hillel (ed.) , Mathematical Logic an d Foundations of Se t Theory (Amsterdam : North Holland) , 24-62. (1980), Descriptive Se t Theory (Amsterdam : North Holland). MYCIELSKI, J. , an d STEINHAUS , H . (1962) , ' A mathematica l axio m contradicting th e axio m o f choice', Bulletin d e I'Academie Polonaise de s Sciences, 10 , pp. 1-3. and SWIERCZKOWSKI , S . (1964) , 'O n th e Lebesgu e measurabilit y and th e axio m o f determinateness' , Fundamenta Mathematicae, 54 , pp. 67-71. NEISSER, U . (1976), Cognition and Reality (Sa n Francisco: W. H. Freema n and Company). NOVIKOV, P . (1935) , 'Su r l a separabilit e de s ensemble s projectif s d e seconde classe' , Fundamenta Mathematicae, 25, pp. 459—66 . NYIKOS, P . (forthcoming) , 'Testimony o n larg e cardinals and set-theoreti c consistency results'. PARSONS, C . (1965) , 'Frege's theory o f number', repr. i n his (1983a), 150 75. (1974d), 'Sets and classes', repr. in his (1983d), 209-20. (1974ft), Th e lia r paradox', repr. in his (1983d), 221-67. (1977), 'What is the iterativ e conception o f set?', repr. i n his (1983