Introduction to the Philosophy of Mathematics 0631115803, 9780631115809

Table of contents :
Introduction
Part I: Existence in Mathematics
Chapter 1: if-Thenism
Chapter 2: Postulationism
Chapter 3: Mathematical Principles as Analytic
Chapter 4: Carnap's Theories
Part II: Mathematical Knowledge
Chapter 5: Fictionalism, Proof, Gödel's View of Mathematical Knowledge
Chapter 6: Intuitionism
Chapter 7: Mathematical Knowledge as Empirical: J. S. Mill and D. Hilbert
Chapter 8: An Empiricist Theory of Knowledge
References
Name Index
Subject Index

Citation preview

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APQ !LIBRARY Of PHILOSOPHY

,

HUGH LEHMAN

I

.·· Oxford, 1979 PUBLISl'IED BY BASIL BLACXWELL

©

American•Philosophic.a.I Quart.erly/979 ISBN 06.!H 11580 3

British Library Catal9guinl in Publication Data Lehman, Hugh Introduction to the philosophy of matkematics. ('American philosophical quarterly' library of philosol'hy; 2). I. Mathematics - fhilosophy

I. Title S1!0'.I

11. Serles QA8.4'

.. PRINTF.D4N fN(Jl.AND•

by Billing & s~ns Limilcd, Guildford, London and Worcester

TABLE OF CONTENTS PREFACE lNTRODUCTlaN:

I. The aim of this work is to discus!f ontological and epistemological issues. 2. It will a'lloid assuming extensive knowledge of mathematics or logic. 3. Concermng the ontological and epiitemologieal issues a'1d of the present approach in sapporting these views;. 4. Statements of mathematical theories ca1"y ontologicat. implications. for e:immple, lhe axioms of real numbers. 5. Numbers are unobser~ble, neither physical nor mental and universals, i.e., numbers are queer enlities. 6. Consideration of tbe subMJlutional interpretation as a way :of accepting matflematicaJ truths without queer entities.

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PART I:: E.XISTENCE·IN MATHEMATICS CHAPTBR ONE: lf-17HENISM

7. Mathematical statcmenlts arc alleged to be corodilional. There arc two versions of \his view. 8. But mathematical statements arc net material impliica lions. 9. A second version of if-tienism is the view that mathematical statements are logical nnplieations. This view is open to two sorls of criticisms. We may asl. whtther acceptance of logieail trU1ths involves making 'Ontological commitments. v

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11 12

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H I L 0 S 0 P f-1 Y 0 F iM /\ T II E M A T I C S

10. We may als ask whether mathematical statements are logical implications. Definitions or "real number" must be considered. 11. Someone may object to my claim because he subscribes to a more inclusive '1otion of logicat truth. But, I ask, haw does he distinguish logical from non-logical truths. In my view a logical truth is one which is an tnstance of a formula which is .. trae in all possible worlds". 12. One further way of defining "real number" is considered. 13. Conclusion fl chapter one.

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CHAPTER TWO: POSTULATIONISM

14. The definition of real number considered in section 12 suggests a view of mathematics held (at one time) by Bertrand Russell and atso by Henri P~incare, namely the "ew that mathematical statement's assert only that theorems are consequences of certain as.-iumptions. 15. But, even if postulationism ~s true with respec:t to pure matheltlalics, in applications of mathetmttics categorical mathematical assertions are made. Thus, postulationism does riot enablle us to use mathe~atical knowlcdte while avoiding the implication that queer entities e1ist. t 6. But postulatironistn does not give a correct destription or the activity· or the "pun:" mathematician. Ex~ ample 11egardin1 convexity. ~ 7. Nor does ·it correctly desc:ribe the class of mathematical truths, as w'as pointed out by Quine. ~ 8~ Conclusion df chapter two.

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CHAPTER TFIREE: .MATHEMATICAL PRINCIPl.ES AS ANALYTIC

l 9. Some philosophers have claimed that mathematical propositions are analytic. In particular they have assertecd that mathematical truths are true by definition and lacking in factual, import.

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A PRELIMINARY STATEMENT OF THESES

vii

20. Is "3 + 2 ... 5" I rue by virtue of derinitions or 'J', '2', '+ ', 'S' and ' - '? We are not ct>mpl'lled to sa,:y thal it is. But evefl if it is true by definition, tht theory presupposed by these derinieions has ontological imP.lications. 21. Defenders cl tlie view that mathematical prorx>mtions are .analytic have claimed that considerations of the ways in which such propositions are learnee are irrelevant to epistemology. But this position is wrong. It lead!; to ~n absurd consequence as consideration of the notion learning shows. 22~ Defenders cl tie view that mathematical propositions are analytic bve claimed tbat arithmetic: truths cannot be refuted by observed counter-instances. But this claim is apparently untenalllle. 23. Some philosophers, notably t.I. Lewis and Ii. INagel have maintaiaed that ma\hematical and logical propositions are prescriptioni> and the~fore ~have no ontolosical ~mport. We argue that while nchi propositions hll'Ve a pre.i;criplive role it does not follow that they ha'Ve no ontological iniport. :24. Conc:lusion of chapter three.

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CHAPTER FOUR: CARNAP'S THEORIES

:is.

In this section we try to e~lain some of the basic ideas of Catnap's theorie.'i. In particular we e,.plain lhe notions er i;yntactical and i;emantical systems. 26. An incompltle example of a :syatactical and semantical system is given. A complete system or ithc sort given constilutes, according to Carnap, the rules of ordinary logiical reasoning. 27. Here we oonsider some truths of mathtmatics, • namely Peano's postulates. Carnap pre!lenled a syntactic system ror these postulates and also gave semantical tuloes. He also gave derinitions SQI that statements of reano's postulates could be obliained in the semantical system of ordinitry logic.

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viii

PHILOSOPHY OF MATHEMATIC~

23. While the tramlations of Peano's postulates into the semantical system developed b)' Carnap arc L•trae. we may ask whether Carnllp has shown that mathematical tirinciplcs make no ontological com.. itments. Fot enc thing wt may ast whtther Carnap's tranllations really t:xprcss Peano's postulates. Also. lthc axiom& of the semantical system stated by CarftaJ' appear to hbve ontological implications. Further, the axioms needed to. complete the semantical system, namely the axioms nec:d«I for derivation of p•indplcs of hlllthematics contain existential implications and are·. not L·true. This is shown through consideration Qr t•c axiOlll& of ~hoice and infinity. Thus, even though mathematical truths may be translatable into stateMenls of this sema1tical system, that does not show that lhe mathemati::al truths contain mo Ol'tological il'ftplications. 29. Consideration of predicative deiNlitions, predicative Sunctions. the vicious circle ptlinciple and the axiom of reducibility. 30. Carnap has argued that tlttre is a distinction between inle"1al and ea:temal questions and thal external questions re&11rdins e11istcnte arc meaningless. We ctitkiz.c the disthtclion and argue Ghat his conclusioh that external questions regarding existence are 1ncaninglCS11 is ratse.

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PART 11: MATHEMATICAL KNOWLEDGIE •

CHAPTER Fl~E: FICTIONALISM, PROOF. CiODEL'S VIEW OF Ml\filEMATICl\L KNWLEDGE

3~ :. Consideration

of the view of Hans Vaihingdr that mathematical concepts are fictions. His theory that mathematical sratements are all fictions or sc:mifictions seems mistaken since ~l is incompatible with the fact that !SOW1e people have mathematical knowledge. On our view a pragmatic theory of mathematical knowledge is•correct. J2. Comiideration of the nature of mathematical proof.

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ft. PR 13 LIM IN 11 RY ST AT F,~ INT 0 F TH ES ES

3.l. Some mathematics! princiJYles; muu be knmwn without proof, since there is nlhlhematical knowlctlge through proofs and the number cir premisi;es af such knowledge is finite. )lt ConsideralioA Gf the view or Kurt Godcl that mathematical knowledge rests on principles !known via intuition or mathematical objects. Objections lo this view. 35. Mathematical intuition and the causal theor)I of perception. Views or Mark Steiner. 3'5. Conclusion or chapter five.

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CHAPTER SIX:: ll'llTUITIONiSM

3'"1. Eitplanatitm or intuitioni11t \licws on mathetnatical 18.

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existence and knowledge. Consideration er Intuitionist and Wittgcnsleinian objections lo the law of e"cluded middle. Their objections are IMll sound. Consideratiollll of intuitionist il"e.'11 number theory (of intuitionist tlteory of the continttum). The Intuitionists cannot ;act;0unt for all o( our mathematical knowledge. Criticism of the intuitionist view that mathematical knowledge rests on self-evidcttl principles and of the intuitionist theory or the reference or malhc:tn11tical terms. Intuition and l~rning or rnalhematics. Conclusion.

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CllAPTER SF.VEN: MATHEMATICAL KNOWLEl'>Ge AS EMPIRICAL: U.S. MILL !\ND I>. lllL11P.RT

· i'3, J.S. Mill's view of mathematical knowledge. 44~ Criticisms of Mill's view by JlOsitivists. 451• Gottlob Frege's crilicism!i of Mill. 46. Hilbert's tha:iry of mathematical knowledge. '41. Criticisms ot Hilbert'!> theory. '48. Discussion and criticism of I laskell Curry's version of formalism.

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PH I L'O S 0 PH Y 0 F ~AT II EM AT IC 5

49. Critical discussion or the formalist theory of Abraham Robiuon. Robinslln avoids the objections directed agaiti!lt Hilbert's epiMemology. But his view is essentially ~ncomplete. 50. Conclusion. CHAPTER EtGttT: KNOWLEDGE

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EMPIRICIST

THEORY

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OF

51. Skepticism teYisitcd. Disc"ssion or the -Acw or Stephan Korftcr. 52. Discussion er the significance or non-euclidean geometrics and of alternative set theories with rc.'pect lo the CJtiSlCRCC or mathematical knowledge. 53. Brier c,;plartlltion of our dtcory or mathctnatical knowledge. Mathematical principles are confinned via hypothctico-deductivc inr""cnccs. S4. Consideration of an objection: It is alleged sometimes that mathematic:all ll:nowledgc is urtain and so cannot be empirically confirmed. SS. Could science dlspence \Vith real number lh~ry and make do with rational number theory instead? %. Does confirmation of.mathe11natical principles show that while we arc warranted in using such P.rincipfcs in natural sdence we are not warranted in believing that they.arctruc? 57. Consider a lion of the view that . knowledge or mathematical principles presupposes that (I) no impredicative definitions occur in such principles, (2) there is no reference to inllinite totalities in such principles, (3~ there is no rcfierence to the exii;tcnce or sets in such principles or (4) that such principles 1 be "constructive".

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. I SS 156

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RtWFRF.NCF-'i

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NAME INDEX

171

SUBJECT INDEX

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PREFACE lt giYcs me 11real pleasure to thank the editor of this series, Nicholas Rtscher. for his encouragement and a~sistance with the publication this. book, and Canada Council for assistance with funds for typing this manuscripl and to Mrs. Judy Martin and ~ss Sheila MacP•erson for lyping. I also wish t4:1 acknowledge my indebtedness to my parents and tcachen.

or

This book is Ac:dieated to my wife and children.

Introduction I. My aim in this book is lo discuss certain ontological and eriistc:mologicar issaes.;. In particular, ~ asi;umc that modern man is in possession of a gries which are, in our mind, s"mcicntly weighty lo require their rejection. 6. Contrary to t.rhat we have c~imed some philosophers would affirm that we can accept mathematical statements as true without

INTRODILJCTION

1

accepting the existence of queer e•lities such as numbers. Such philosophers would argue that stale1•ents such as (A) There is a number which when added lo 5 yields O.

tlo not imply l'1e existence or numberi>. Acoonling to these philosophers, in arguing that sitnce (A) is true Lhcre must bt •umbers, I am inte11prcting the expresllion "There is a number ..... as if it could not be true unlcs.1; there ill an object which ill a aumber and which has lhc property in question. Contrary lo this i'fltcrprctation or statements such as (A) thc.1;c philosophers would ravor a "substitutioaal" interpretation or such statements. Statements such as (A) have the form (I) ( :U)F.t

where the signs prior to the "Fx" are called an existential 1t.uantifier. Statements or this fMm are sometimei; expressed as ""T•erc are some fi" or "There :is al least one f." In referring l~ the "objcctual" interpretation or th~e statements, what is meant is that the.-ic slal~nenls are to bc inleqircled as satisfying tht folllwing rule regarding their truth ¢onditions: (Obj) Statements or the form "f 3 x) Fx"arc true: if and only if there is at leasl one object having the property F. The substitutionnl interpretation or these stateme,.ts rejects the rule (Obj) and replaces it with a rule such as the following: (Subs) Statements or the form "( 3 .~) F.t" are true if and only ff there is a.t least one name which satistics the predicate .. Fx."

J\ccording lo the substitutional interpretation or statements such a$ (A] we would not lie ontologically committed to :numbers. Rathet ..,e would only be ontologically committed to narmcR or numbers... that i!I. to numerals. On this int~rprctation (i\) would be true ff and only if there ill a name which satisfies the prcditale "-equal~ t.ei:o when added lo 5." There arc two quclllions to a!'lk with respect tio, the substitut-onal interpretation. First. is it a satisfactory iinlcrprelation or nd-ordcr quantificational logic.' 11. In our discussio11 or if-thenism so· far we have a&'lumed that if statements or mathematics are logical implications then they arc either instances or valid logical formulas or can be transformed into such instances through replacement or some terms by their definitions. Let us refer to statements which meet eitr.er or these conditions as logical truths. So far we have been arauing that the axioms or real number theory are not logical tr11ths. Someone might reject my conclusion beta115e: he subscribes to a more inclusive notion or logic than I do. Such a person might hold simply that mathematics is logic and thus that the u.ioms or set thebrJ (lo which I referred as substantive axioms of set theory) ar~ i• reality, principles or logic. He could then com:lude that the axioms or real number theory are also logical truths. In reply to this sort or criticism I would reply aloag the followiitg lines: tr one is to 11pcak meaningrully or logical truths then there must be some nol\-arbitrary distiAction between such truths and non-logical trulhs. tr someone includes the uioms or set theory· among the logical truths then I want lo know how he makes 5uch a distinction. As for me, I can see no more reason for including the a0tioms of set theory (no matter which one among the current sel theories with which I am acquainled) amongst logical truths than l can for including rurther statements such as tho.'ie from some axiomatized VC~ion or !IOme branch or a moderA scientific theory, e.g., biology or physics. The ir-thenisl, or course, has a perfect right to ask me lo explain or justiry my appeal to quantificational logic in my fo1mulation or a cri1erion ror determining whether mathematical p•inciplcs are logical implicationi;, that is, for my criterion or lo~ical truth. To this quei;tion I would reply in the following way: A logical truth,

IF-THRNISM

11

a«ording lo my view, i;hould be "true in all pos.~ible worlds" iR tt'hich anything exi:lll. Now, the e.xpres11ion "true in all possible 'ft'orlds" is metapherical. However. I believe that the semanticat theories developed by Tarski and others provide a reasonably literal interpretation for tltis expression and so I shall continue to use it.' Set theories which are strong enough to provide proofs of the Hioms of real number theory arc oot true in every possible world. For example, there are possible worlds in which there arc only a finite number or objects. Or there arc po.cisible worlds in which it is not the case that power set or every set of sets exists. 7 Since the uioms of such set theories would be false in such possible worlds lhey are not, in my view, logical truths. l 2. Let us, al this point, mention fine further type or definition of the expression "a and b are real numbers" (and of related expressions). The e:tpression .. a apd bare real numbers" might be •nderatood to mean that a and bare part or a set or objects which satisfy the axiom5 ef real number tfleory. That is to say that a and bare part of a set or objects which are closed under addition and multiplication, etc. Then indeed, tht l>latement "if a and b arc real numbers then a+b•b+a" would be an elementary logical implication. Have we at last found! a satisfactory way of showins that the truths oF mathematics have no ontolosical iRtrlications? In the next chapter we shall explain this view of mathematies in mo"' detail and then co111ider further criticisms. 13. Conclusion: In this chapter we have considered the view that mathematical statements have no ontolotticnl implications becau~ they are all statements of logic:al implicaHons. In criticiT.ing this view we have conlidercd statements of real nund>er theory. They arc not statements or logical implications in thermielive.'I and further they· are not transformed into such statements whe1 mathematical terms are replaced by their definitions.

NOTES I. View~ of lhi~ ~t'lrl hnve been exprc~ are these quantificational statcraents to be undeistood? In the second principle we noted that we have, in effect~ a quantifier whieh ranges over all properties. Thus, thiii statement is true only if there is a non~empty domain of properties.• This suggests that while the logical principle may be true in virtue of the meaning of identity (or of other logical npres.,ions). the statements explaining the meaning of idenUlJ have ontological implications. Ad•ocates of the. philosophical view we have been criticizing, e.g.,, Ayer, Hempel, Carnap and others, subscribe to the claim that if a statement is true iy virtue of the definitions of its terms, then the statement has no factual or atttolog:ical import. Bat.. as we have s • there is reason to question this claim. What we bave just argued is, in effect, that the definitions of the terms ol the logical theory presuppose another theory which in turn m.s ontological impfica lions. 21. We argued abo•e that we are not compelled to say that aritbmetic truths are true by definition of the te•ms of the stattments in which t•ey are ~1'prc:!JSed. further support for this vi~ is obtained when one considtfns the ways ini which these state1T1enl5 are learned. One does not come to know that 2 + l =S by ~king up the c:lefinitions of these terms in sQme sort of a dictionary. Clearly. the way that everyone first learns these st.atements involves counting appropriate sets of obj~ts. For example, one countt1 a set having two olljects and then a disjoint set havi.ng lhree objects and then counts the union of these two sets. Them who accepted views such as: tf10se of Hempel and Carnap would respond to t~ above argument by claiming that talk of counting in the above context is irrelevant. They would say that such: talk confuse!! pRychological er genetic matters, such as matters relevant to how one learns a truth, with e'istemological mallers. For example, Ayer has said \ ' It is obvious lhal malhemalics and legic have lo be learned in the same way as chemistry and history l\lave to be learned ... Whal we arc ffiscussing however, when we Siil! lhat logical an~ 11111lhematical lrulhs are known independently of eicperience, is not a hislorical que:;Jion concerning the way in which, these lrulhs were 01iginally discoveted, nor a ~ychological question concerning lhe way in which each of us comes IOI learn them, bul an epistemological question.'

Ml\THENl\TICl\L PRINCIPLES l\S l\NAl.YTIC

H

At thb point it. i~ natural lu a~ how we are to distin~uish epmtemological maUcrs from these t11thcr psycholot1ical or historical matters? And to this, Ayer Hggests the reply thal. the epiistemological question concerns lhe way in which such pro· positions a.re ..validated." Perhapi; this sue;gcslioo can be under.. ill the following way. To see how a prnpusitwn is validated is t.o see what reasons scientists offer in evidence for the proposition. With regard lo many proposilionll\, it is clear lh31 the wayi> in which we come lQ learn that they nrc true may not involve learning "1e reasoning through which they ane "validated." For example. we may learn that twelve limes twc~c: is one hundred and forty-fous because someone tells us that this i11 true and not becauioe we hav1 studiea the arithmetical proof. ln other cal'ies we may learn that some principle is true through a combination of facitlrs which ma)( ~vnlve (I) witnc.~ng examples iffVQMng the applicatioo of the principle and (2. being told or readintt that the principle is true, etc.. w~ the evidence considered as ncce.'\sary lo prove the J?roposition is coosidered too ex~nsivc or too comrlu for us lo grasp. Thus, we can grant Ayer and. I lentpel the claim that there is a legitimate distinction between the experience.~ through which an ifldividual learns a particulnr proj?otition and the evidence which i.11 considered as proving or validntiilg that proflOSilian. But. that dot$ 1101 ahow that con11iderations •elating lo the ways in which arithmetical propositions are learned are totally irrelevant lo the question or how they are proved. If Ayer and Hempel a.re correct in their view, Chen we should hue to say that until the work of Peano and Boole and others was completed in the nineteenth century, while 111\l\ny people had ftcairned arithmetic, nobody knew that arithmetical principlci; were· liruc since nobodl bad learned the appropriate def,inilinns. In other words, on this view we 11hould !iay that ma thcmaticians such a!t l~ibniz, Newton, li>escarle.11. Archimedes, elc., di,:1 nol know such lrut~s as that 2 + 3""' 5 since l~4ltn05itions were lrue flrior lo their writing advanced work!i in logic. Tltey did nol know these arithmetical propositions were true because, on Ayer'~ view 110 one could have learned them (through the first kind of learning). If no one could have learned them in Lhe first wny then no one would haveJnown them.· 22.- In defending their view thaL arithmetical 1n:opositions are analytic, Hempel and Ayer have ar1ued that such propositions are logically immune from refutation through observed counter· instances. Hempel gave the followini ctample: -'We place some microbes on 11 111idc, pnUin,; down lirsl tliree o( them and fhc:n another two. Afterwards we couinl all the microbes lo lesl whether in 1hi11 ini;lancc J and 2 actually ndrttd ur to 5. Suppo.r of another. I have ilalici:r:c0nd to truths or mathematics {in natural langaages) and that the truth or thae sentences is determined by the semantical rules or the semantik:al system. We shall attempt lo keep oor presentation sufficiently simple !io that it can be followed by people who are nol mathematicians or mathematical logicians. Only after we have rully preo;cnted Carnap's thoory will we engage in critical lfiscussion or it wiith rdcrcnce lo the basic questions or this book. 26. The syntactic system whrch we shall present consists or certain signs Ctlled logical signs. Tbe1e arc several kinds or sach signs. These are: I. 2. J. 4. S. 6.

Scntentia I variables: p,q,p1,q1 ••• lndil'idual variables: .1,y,.11,y1.r2.yz, ... Predicate variables: F,G,F1,Fj, ... Gro~ing indicators: (,), [,]. Con$lants: :J. -.., Quantifiers: 3, 'fl.

We dall rder to sequences of syntactical &)'Item only certs in These are given by the formation are called wen-formed formulas. lhe system are:

such signs as formulas. In lhis sequences of signs may occur. rules and the permitted formulas Some or the formation rotes:. of

I. Any predicate variable followed ~ any number of individual or predicate variabtes is a well-formed formula as is any sentential variable. 2. Ir Pand Qase well-formed formulas then so is (P-;, Q). J. lr P ii a well-formed formula then so is ( .-P). 4. Ir Pis a well-formed formula then so is (¥ )P. (W~re the blank Sf!RCC arter the •v• can be occupied by any variable.) E1am111les or some wcll-rormcd formulae arc~ F:11., FF, (¥ .r)FA', (¥ f) F:11., ( F.t :J Gx), etc. Ordinary rules or logic enable us lo deduce some sentences from other sentences;. The sentences in a dcd•ction'rorm a sequence. In a syntactical sysll!m which is to correspond to our logic there may be sequences or well-formed formulae. These sequences may be obtained by stipulating that certain formulae are lo lie given as

!-

CARNAP'S TllF.ORIF-S

43

primitive formulae and then staling lhc rufe.-; by which sequenc~ or formulae may be generated from the primitive formulae. The rullc9 which allow for lhc generation or sequences arc called lrllnsrormation rules. Some or Ihe primitive form11lac or the system arc

I. rn-p) :J Pl :JP 2. pl:J [ ( - pl :J q] 3. (V~)f'K:J Fy 4. (V G)(Y f)(y y)[(V x)(Gy :J Fx) :J (Gy ::> (V x) F.t)J 1

Some or ihe transformation rulC!i

ror lhc system are

I. From two rormulas or the form P and (P:J Q) one may obtain the formula Q. 2. Yrom a formula or the form (A :J Bi) in which the part arter the • ::J' contains the variable 'x' while '1.' doe..~ not orcur in A. the fo1mula ( /\ ::> (V xrBx) may be obtained.

The specification of lhe syntactical ~ystem which we have given serves primarily al this point as aa ex.ample or tht sort of; thing that Carnap had in mind in his reSercncc to a syntactical system. Jn order that the rormulac or f'Uth a system be made lo corrupond to valid principles or ordinary logic semantical rules must be given. In the absence of semantical r•les the above formulae are mtaningless. The addition of the semantical rules turns lhe syntactical system Into a semantical syi;tem. In Foundations llf l..ogic and Mathematic.' Carmtrp spoke or two sorts or sernantical rule.-;. Riles or the rirsl sort a11sign dcsigmta as the referents or tenn. In the syntactical system we have sJM:cilied above there may be; well-rormcd formulae soch as "F~·· and "GF." Se1nantical rules lor these formulae or the kind indicated by Carnap are; I. '11' de.11i1tnates Ihis book. 2. 'F' dcsignate5 the properly or being long. J. 'G' designates the property or being widely manirest.

GiYen thesc three semantical rule.~... Fx" become5 ..This book i11; long", and "GF.. becomes "The properly of being long is widely manifo.-;t." In addiliiJn to sernnnticaD rule11; which assign designata to lern" there are semantical rules which spcciry the eondilions under which slalcmenl11 are true. Thus we might have

4

PHILOSOPHY OF MATHEMATICS

I. A· sentence of lle form "Fx" is; true if and onliy if the object designated by 1 ](' has the properly designated by

'F'. 2. A sentence of the forrn .. _ Fx" is true if and only if the sentence .. F](" is not true• 3. A scnte~ of the form P:> Q is lrue if aad only if either Pis not Erue or Q is true. 4. A sentence whici is not true is false. S. A sentence or the form "(\f' x)Fi" is true if and! only if .. Fx" is frue for every individual, i.e., for every individual in the aniversc if 'lf' is assigned that individual as 8c:signat11m then .. Fx.. is true, (This wouW be true only if every individual had the J>roperly designated by 'F' .)'

In presenting the above example: of a semantic::;d system we have attempted to present a system of logic which is complete with peel to validity,. thus, tllere wm1 certainly be principles of logic f\ich are valid which do not corrt.\pond to formulas that can be educed from the axioms we have stated by the rults that we have lated. Those read~n interested ill formulation of complete systems ( nnt•order quantincational logic may coasult any, one of a range logic tetts. A distinction which Carnap believed was qµite important nvolves noting that it is pc11siblc that the truth of same sentences is ctermincd solely by the way lhaHihe sem11ntical rules apply to the 1tcnce. The semantical rules guarantee tllat the Slmtence is true. n other cases the semantiical rule5 guarantee that the senlence is dse. The semantical rules we have stated guarantee tbt the olowing sentence is true (no matter what predicale is designated y'F'): (V.t) (Fx:::> Fx) ~nd they guarantee: that the following sente11ce is fatse:

l

-(\f' ir)(F.t :::> F.ir)

or

~f the semantical rules a sematltical system ue sufficient to guarantee that some sentence P or the 5yslem is true, then Carnap ~lled the sentence l·truc. If the semantical rules guarantee that a ~ntence P is false then Carnap called the sentence l-False. Scntem;es which arc either L-True or L-False are called L· "1eterminare. Sentences which are not L-determinate:were called by parnap factual.

CARNAr'S Tlll!ORIES

4.5

27. Since we are concerned with questions or ontology in relation to mathematic11I theories we must consider now the lnnguagc of a mathematical theory. In Carnap's view, underlying ~uch a thCOfy there will be a syntactical systc1n (also called a calculus). Then, or course,. there will be semantical! rules irHcqirctiflg the signs of the calculus. If one were just making a calculus fo11 no JlUrpose at all then one could form it in aoy way he cho.~e. l lowe\'er. since we want our calcullis lo be the calculus of a mathematical theory, llfo leads h> the formation or certain sorts of calculi rather tbn others. It does not, However" determine a calculus uniquely. Carnap believed that for mathemillical theories the underlyif118 calculus includc:s as a part a calculus sUlitable for logic. However, it might appear that additional initial fermulae and additional semantical rules might be needed in order lo extend the logii::al calcalui:; lo one suitable for mathernatics. However, according, lo Catinap, suG:h additions lo the semantical systlcm or lf!gic arc not necessary. He believe-' that tlle slalcment'i of mathematic.o; C(l)Uld be translattd into statements or logic; that is. in lo sla lement,;, which forrespoad to formulae or the basic logicnl calculus as i,1lerpreted by lhc semantical rules or logic. Orcounc, Carnap thowght thal the truths or mathematics>: arc, when properly tmnslaled into statements or logic. L·true. He attempted, in lhe Foundations of Logic aad Mathematics, to show that 1kmentary aritfametic. based 0n Peano's postulate.~. consisti; or a 5ysten1 of L-true statements. Let us now consider h6w this may be done. 1 Carnap presented Peano's pontulales as follows:

Pl. bis aa N. P2. For every x, xis an N. then x' iii an N. PJ. For every x,j; ir (x is an ·N and y is an N aind x •I) then ,. ... y. N. For every x, ir xis an N, then it is nou the case that b-

i"

Jt.

PS. For e'i'ery F. if ( b is an F and, for every x (if x is an F then ~ is an f)) then {for every y, if y is an N then y is , an f) .. Carnap regarded thc,sell and Whitehead is the axiom of choice (which they called the multiplicative axiom). Im section 21 we stated this axiom a11 follows: In any set S of disjoint seti; not containing Ihe aull set,

there ellists a function fwhose domain is Sand such that for each member m of S~ /{ m) is a member of m.

Let us brteny explain the content or this axiom. To say< that a function exists is-10 say t!hal ihcre i11 a set of objects which arc arguments of the function end that there ifs a set of objects which afe values< of the function. In the case of the choice function, the set of arsument.'I are the sets which are members or sand the llel of values is Lhe set containing the individual members of members of S which are picked out by the choice f.nction. But thi~ axiom. Hke the a:ir.iom of infinity, is an cx.il\lcntial as:mmplion. This axiom C9Uld be exprei;sed in the symboli~m or the syntactical system we have given above. But I.he sc1uantical rules we have slated (essentially the flemantical rule:; used by Carnap) do not imply that the statcnrent is true. To convince ourselves (informally) that the axiom of intlinity is nl!)t true simply in virtue of senmntical rule.o; which give the

52

PHILOSOPHY

or

MATJllEMA"llCS

customacy meanings or the logical connectives and the univenml quantifier, it is only necessary lo consider the sU:rposition; that the •umber er entities in the universe (individuals aad properties and properties or properties) is finite. The supposition that the number 11>r entities in the universe is finite. is, or course, the supposition tha.t there is some c~unting number n such that each entit:r in the 1niverse could be paired with a 1umber Jess than or equlfl to n • lhat no .r1Jmber was pair~ wit II more thaa one entity and no entity was paired with more than one number. In other words, this is the supposition that there is a one·to·one eorrespondence from the entities i1 the universe onto the CGunting numbers up to n. We may suppose that this supposition is lrue. We are. not prevented Crom doing thiS by the semantical ruh?s stated above. Jr the axiom of infinity were true in vinue or semantical 1ulC11, then the semantical rules would rule oul (be iteonsistent with) the suppositioo1 that the number or entities in the universe is finite.! In similar maaner we can argue that the a:dora or choiCe is not lrue in virtue of the semantical rodes. If it were true in virtue or th1 semantical rules then the semantical rules would be inconsistent with describing a set or sets in which the uiom of choice is. raise. It is nol dirficult 10 describe: such a set or sets and I see no reason r°' thinking that such a set ii; inconsistent with the semantical rules. A set or sets which cannot exist if lhe axiom of choice is true is the rollowing: The members or the set S arc a set containing the numbef5 one and two and a set cvntalnina the nu11nbers J and 4. 5 has two !lets as members in this case and the members are disjoin1 and the null set is not a member or S. BUil the set or values or the choice runction i~ not a member or S either. Carnap, in trring lo show that mathematical principles were L-true was clearly trying to show that mathematical principles have no ontological import We bve criticized Carnap's position. We have suggested that his translations of PtaltO's postulates into the semantical s}lslem he sketches may not render the cll5lomar)' meaning of the rostulate.'i. Further, while Carnap does n then we sheuld l•a¥C lo say that certain cq11a1ions, which wt now think of as having roots, cnuld not be sol•ed. e.g., the equation Problems rcprci;enled by suc:h e lhal aJe not satisfied by· lhe .~ct of rational rmmbcrs due lrk general tcrmio for making statements about the propc!!I ies or relation.!! of the entities in question. These could be obtained through havin~ in the syntactical syi;tcm certain constant tcrrn!i which C(luld be inter·

60

P ti I L 0 S 0 P II Y 0 f

M I\ 1"11 E M I\ T I C S

preted as designating lhe required properties or relations. The semanlical rules supply the interpretation, i.e .. they oorrelate lhe terms with the properties, relalions or other objects. Carnap also suggests thal in a liaguislic framew4>rk suitable for discussing several kinds or entities there should be variables or di'itinctive types for referring to the kinds or enlitie5 in question. But. it is not clear to me why advantages would be gained by lhe adoption of different style.o; or variables and this does not seem to be essential to his view. Quine also has discussed this aspect of C8lrnap's view and finds the refere11ce to style or variables ines. may be accc11u:d or rejcclr•.L Prc.~umably then lherc may be criteria or standards of evaluation wilh reference to vrhich a decision among alternative frameworks can be made. Carnap, points out. for example. that as a con· sequence or obi;crvations we rind dc.r;cribing spatial relation11 by w~ing thrcc-dimcn!!ional coordi1111tc systems more i:;ntii:;foclory than either two· or fot1r·dimcnsional systems. He ahm suggest!\ lhal considerations having to do with mathematical llimrlicity incline us lo use a framework which is based on lhc real numbcr11 rather than one which is based on the rnlional 111u1nbers. even though, so rar as 011,servation and measurement is cnnccrncd, identifying ruints of space with respect lo rational numbers would be justified. This suggests two criteria for cv;ilur1ting. rramcworkll, nar11ely a~rccmcnl with observation arod mathcnrntical si11111lidly. I lowcver, 1hcrc arc other criteria too nnd connicl5 among criteria are possible. For

62

PH I L 0 S 0 PH Y 0 F M AT H EM AT IC S

eumple, if one is working on the solution of some problem in arclsaeology. let us say a problem inW>lving classification of fossil bones, one might have to decide between two systel'IJls of classi· ficatiorl, system one and system two. System one mitht be simpler to apply in most cases, but, for some cases, might not yield a decision on classification. System 1wo might be much more ditncwlt lo apply in general but mjght yield decisions in a wider range of cl!!lcs so that, using system two there would be fewer freab. I have two objections to the position Carnap has expressed in this paper. For one lhing, in dtscribin.1 the questions with which a seienti:!ll is concerned I very often do not know whether t"y should bcdeseribed as internal or eiternnt questions. Foraootlaer.1 do not see hvw Carnap's claim that the. decision to aectpt a linguistic framework does not involve onlological commilrn~nt can be justified. Consider first lhc internaJ.:external distinction. With respect to the archaeological eHmple~ is the archaeologisLwho is trying: to classify the bone specimens trying to answer an internal or an external question.· One can describe what lie is doins as trying to decide between two linguistic frameworks-which makes his question appear external to either framework. Or. oae could say that he is working within a larger framework accepted by archacologists withiR which his que.'llion is internal. Lei u1 look al this in more detail. The archaeologist is trying to claBSily some. bone specimens. He may have lo invent some species-names in order to carry out tht classification. Thus he is · mating up a language. The decisions ooncerning what names to use would appear, within Carnap's way of thinking, to be answer1 to e)lttcrnal questions. But, he wants to arrive at a classification which reOects a real distinction. In particular, if he decides that two bone specimens came from two different animals and he giwes different species-names to each animal it reOecl'i his belief that the animals belonged to distinct populations which were not interbreeding. The dec::isicin regarding species-names reflects a ractual belief and clearly the statements made using the new species-name, statements such as "This bone is the jaw of an ... •• are factual statements. Carnap's distinction between internal and external questions seems misgt1idcd. The archaeologi!lls original question, namdy, "What species-names should be used in classifying these bone spcdmens7" may be external to the linguistic framework at which the ar1:haeologist rinally arrives after having made his decil;ion. But the question is not merely practical as opposed to

CARNAP'STllP.ORIES

theoretical or cognilive? The answer lo the question (cx.prern:d in the language wliich the archacolC>t!isl finally adopts) is a fac1uaJ statement, and not simply an expression of a decision to choose one linguistic framework rather than another. That the distinclion between external and internal questions with reference to linguistic frameworks is misguided can be i.hown also with respect to a mathematical example. Suppose that a mathematician is wondering whet~er some theorem which is lrne for real numbers also holds true for more general topolo~ical spaces. In order to answer this quc.~lion, the niatlllematician may have to introduce some new mathematical termi110logy. Thus he has to adopt a linguistic framework. In particular, suppose he ha!i tQI adopt a linguistic framework containing a gencra.lized concept of continuity. The mathematician may raise the qu~tion "Mow should 'continuity' be defined?" This question is external to the new linguistic fra1tlework. But the aniiwer lo it is arrived al in light of cqgnitive considerations. If he chooses lo define continuity in a parlicular way lhis renects certain factual beliefs. In particular it rdlecl.s the belier that functions which were continuous under the original definition or "continuity" will remain continuous under the generali:r.ed definiition. Again, the question "How should •con· tinuity' be defined?" ill not practical as opposed tn cognitive. Carnap wanted lo claim that the question "Doe.111 the system Uld accept mathematical principlc..11 and still remain uncommitted to the ontological implications 0f such principles. NOTES I. lfl prc~cntin@ Ihis senantlcal system we hlWt nol folloWed Carnap's eumplc in detail in (10). 2. In the discunion which lollciws ~ have lried to simplify Carnap's presen· !Ilion. I do not bclicYC tfql the mroirrcations introduced arrcct the philosophical iuua. 3. Carnap, ( 10) p. 34. 4. The: use or variables dislinguished wit• rcspec:t to lcvelt i& normally accom· plished by adding numerical tubsc:ripta or npu5cripls to the predicate YBriablea.

c ARN AP' s T ll F.O R ms The' le\'cls are also referred In in 8omc wnrk~ as IYJ!C.'I· Th«" i11lrotluction nf level or tyre distinction• was inrended, of course, to iwuid contrBdiclionir. such as Ru~ll's parado'l.

S. Carnar. (10) p. 33. 6. For critical disca11sion or the veriricaliol'I thcl1ry cf meaning see Slion a bit further we might say that just as statements based on sensory evidence are not known with certainly lo be lrue SCI also slatementi; bai;ed on mathematical intuilioo are not absolulcly certain. Noncthclc5'.,, the derender or GOdel's view may hold. mathematical intuition ha~ some: epistemic valuie. That is, the existence or rna1hcrna1 i\';tl intuition is strong evidence lhat the statement ii; true. Me might even say that a statement affirmed on the basis of mathemntical intuition ought to be considered a~ I rue unless it is known to be seif·contradictory. (J1in as in physical science.~ one might hold that statements based on sen11ory evidence ought lo be accepted as true unle~s one has good reason for surposing them false.) A second objection lo the view that intuition provides evidence for the truth or mathematical statement is, ii might be said, lhal lo let mathematicians decide which uioms are true Ol1I the buis of their intuitions is nothing more th.an letting them Mcide whal is lrae on the basis of their taste. To u.-.c this method of discovering the truth is. in effect, lo use what C.S. Peirce called "the a priori metliod or• ri~ing beliefs!' Dul then we cannot expect, so this objection runs, tha1l the beliefs arrived al in lii5 way will correspond lo any objective realiti.cllo. Thal is to say, we have no reason lo regard a belief a!I true unleli.~ it is .. determined by nothing human, but by snme e~ternal permanency-by 110mething upon which our thinking lml'l no effec1:•io We may be caused to believe that the axioms of a set theory arc lrue by some social cir· cumstance.4l. That tley are regarded as true may be a result of some myth lo whicb we have been socially conditioned. Thus we sho•ld not let the facl that we reel forced to accept these axioms count as evidence for their truth.u The defender or Gooel'e; view mivy r~pond to lhis objection in the.following way. He may suggcitl chat it is quire incorrect to say thal the appeal to inluitions in matJicmatics is nothing more than settling on belicrs on the basis or iiflciination or ta1te. He might claim that intuitions regarding the truth of smne aixioms can be conrirmed by rcferc•cc to other matlicmatical truths. Indeed GOdel has taken this p!isibililies exists, we

96

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may not assert that Neither eitists or does not e:llist. In this sense we can say that the law or excluded middle may not be used here. 6

The intuitionist rejects the law or excluded middle as 'a valid principle or reasoning because, he argues, if it were accepted as a valid rUlle of proof, then one could give mathematical proofs of the existende or objects which had not been constructed, that is, which had not been generated from the basic intuition. But, at this point, someone might argue that the intuitionists have conflated the meaning of two different statements. To i;ay of an object that it exists is one thing; to say of an object that it can be comtructcd is another. Why should we not say of the Dedekind Cut C, for example, that this object exists but has not been constructed. We would say of the solar system that it exists even though it has not been constructed. To this sugsestion, I think, the intuitionists would reply that with respect to physical objects such as the solar system we have come to know threugh our experience of the existence of objects which no one has constructed. Thus · it is meaniniful to speak of objects of that !iort (physical objects) which exist but have not been constructed. However, we have no e1perie1ce of mathematical objects or that sort since our only experieace of mathematical objects has been with respect to objects that we have constructed. Thus, in the context of mathematics, the only meaning which .. to exist'_~ can have is "to be comt•uctcd." If you ai;11ert that some non-constructed object exists in mathematics, you are using "exists" in some metaphysical sense which has no place in mathematics.' Given that all mathematical objects must be constructed in the mind, one might think that intuitionilltS would eschew all statements concerning properties of infinitely large sets of objects. In fact, some intuitionists have taken this position. But, as we have indicated above, some intuitionists have not found this position necessary. Intuitionists or the latter sort can aC(:Cpt claims such as, there are an infinite number of prime ndlTlbcrs, providin9 that such claims arc properly understood. Such claims must not be understood as implying that the individual has constructed in his mind all of the prime numbers and has inspected the entire set and found that it is infinite in number. However, according to Heyting, it may be understood ait a statement concerning a hypothetical construction.' To say that there are an infinite number of primes may be undcrlltood as saying that if one has constructed a natural number n then he can construct a larger prime. To prove such a

INTUITION ISM

97

statement, according to Mcyling, one must show how the lar~cr prime can be conii;lrucled. This con1>lruclion (or the larger prime) when combined with the constructinn of the natural number in question would be a proof of the existence of the rrimc in question. (To construct the priime in question we first com;trucl n! + I and then factori1,e this number. Each or itti prime factors is azreatcr than the natural number n.) The process of construction must. or coune. be a process generated from the basic intuition of two-oneness. . We have argued above that in mathematjc.,. some principles are known as a re.oi;ult of a proof and some principles arc known without proof. According to the intuitionists 11ome principles are known immediately on inspecting the objects that the mind ha11 created. However, some eo11s£ructions are very comrlex and properties of such constructions may not be recognized immediately by someone who has not yet made the construction (or the malltematician may himself forget how he made the const1uction). In such cases iL may be helpful to have some written state1nents indicating how the coastruetion is lo be performed or which objects we ought to inspect (introspect) in order to see lllat the comple" construction has the properly in question. Thtts. nn the intuitionist view, a person may acquire some new mathematical knowl'edge by studying. a proof. However. for the intuitioni11t there i5 no fundamental difference between what is known through a proof and what is known without proof, since in both cases the individual acquire..'! knowledge rrom inspecting an object which he has mentally C()flslructed. Further, according to the intutionist the following scquen~ of statements would not be a proof. siace the nrst statement does not reflect any con11lruclion that can be performed: I. Either Fermat's last theorem is true or Fermat's last theorem is not true. 2. If Fermat'$ last theorem is true then 2 + 2-4. 3. If Fermat's last theorem lo; aot true then 2+2-5. 4. Thus, 2+2•4or2+2-S.

38. , The connection between the law of e"cluded middle and the ontological and epistemological views which we have been con· sidering should be thoroughly e11.pi0Ped. We shall undertake such exploration in the next two sections. In section 38 we shall conl'ider the law of excludcG middle itself and in 11eclion 39 we 11hall COfl!lider in more detail the consequences in respect to knowledge of

98

PH I LOS 0 PH V 0 F MATH E M A TI CS

applied mathematics of the intuitionist principles. \Ye !\hall take the following as oar statement or the law or excluded middle: (EM) For any statement S, either Sis true or the: negation or S is true.

Denial of (EM} is tantamount to asserting that (NEM) For some statement S. it is. not the case that Sis true and it is not the case that the negation of S is true.

h might be argued that the. intuitionist rejection of the law or excluded middle is raise since (NEM) is logic;ally im:onsistent. This argament might go as follows: To assert NEM is, .in eUect, to as5ert that for some statement S. both Sis true and Sis not true and! thii; is contradictory. However. the fo~uilionists cotdd (and would) avoid this conclusion in the following way. They would arsuc ·that deriving a contradictiOll. from the aceeptance of ·(NEM) entails.accepting as a valid prindpJc the prineiplelhl:ll (DN) It is not the case thatthc negation of Sis true is logically equivalent to Sis true. Since the intuilionisu have no wish IO embrace inco11$isleneies they willingly give up net only the law or excluded middle (EM) but also the law of double negation {DN), They would argue that (NE.Ml is not logically i11consistent. To say that it is not lhe case that S is true is only to say that the mathematician has not been able to construct the objects which would exist if S were true. To say that it is not the case that the negation or S is true is only to say 1hat the matflematician has not been able to deduce a logical contradiction from the Hsumption that S is true. Surely ii is entirely pos!lible both that a person cannot perl'.orm the construction11 required by some statement Sand al the same time thal he cannot deduce a conlradiction from the assumption of S. It; should be noted that for lhe intuitionist, lo deduce a contradiction rrom the assumption Ghal Sis true is lo show that if S •ere true then enc would be able to create an object which woald prove a statement that is koown lo be fah;e. For eumple. one would show the .negation or S by showing lhllll on assuming that Sis lrue one ceuld prove that one is the same as two. The law of contradiction, that is (LC) It i11 not the case that some statement S is true and thal the nega ti:m ors is true

INTUITION ISM

is not logically eqwivalcnt, in the intuitionist view. lo the law

99

or

excluded middle. To reject the law of contradiction ii; lo hold that there is some statement S sucli that one can perform the construction which proves S and that one cannot perform !his C!'i!'iintr the properly A where the property A is identified by the following dcfinitio1: 11

INTUITION ISM

IOI

"The natural number n has the property A if and only if there are ail least 100- n perfect numbers.'' However, the questien of how many perfect numbers there arc is as yet undecided. Thus, we cannot determine which natural number is the smaHest natural number thal poii;scs.~es A. Since the Cllistence of the smallest natural number possessing A was established by an argement by reduction to absurdity but since we cannot actually determine which natural number is the smallest natural number which possesses A, McCall thinks 1hat the validity of arguments by rcd11ctits or the numerals of the decimal expansion of the square root of two out to the 10 11'"1.h place. Probably no one has e~er completely co1nputed this series. Yet it is rinite. If Wittgenstein is imggcsting &hat it is correct to say that either' a~pears in this series or it dQC.'li not appear, then what grounds can he have far denying the applicability of this principle (the law of excluded middle) to the infinite series? ·Wittgenstein's statement qtmtcd above implies that the statement of the rnle give.i; you no information concering whether ~ appeari; in the series. Out thi!i is nm necessarily correct. ll may be that the statement or thr, rule in qnc~tion would inoded give 1'0"1~ inlortnation which could be used lo determine whether~ appears in

104

PHILOSOPHY OF MATHEMATICS

lhc: i;cries. It may be however that no human being has, as yet, recognized the information in questiln and been able to utilize that information in the development of a proof which shows either that ti apriears in the seric..~ or that il does not. And this fact, that no one has as yet been able to utilize the information contained in the statement of the rule to decide whelher ' appears in. lhe series does not imply that the continuation is net determined. The reference to a misleading picture in the la.st line of the abeve quotation picks up another strand of Witt1emtein'11 reasoning. Earlier in the above work he claimed When someone sets up the law of excluded middle. he is

as it were putting two pictures before us to choose from, and sayiag that one mw correspond to the fact. But what if it is questionable whether the pictures can be applied here?" Then he also argues

Jn an arilhmetic in which one does not count further than 5 the question what 4+ 3 makes doesn't yet make sense. . . . That is to say: the questioa makes no more sense than does the law of excluded middle in application to it. 1" Here. Wittgenstein was arguing perhaps that in an arithmetic in which does not count beyond five ii is not correct to say that either 4+3•7 or that 4+J+7. Wittaenstein may have been thinking that in some cultures the language used does not contain words for numbers grealer than 5. In such a language it might indeed make no sen11e to say that either 4 + 3 • 7 or 4+3+1 since, in such a laaguage the sign •7• might not be a meaningful eJtprcssion. However, docs the fact that such an expression is not meaningful at a particular time or in a particular culture show that a mathemali(;ian who uses the pri111Cipfe of excluded middle in a mathematical proof is making an error? I do not think so. It is true. that 4 + 3- 7 or that 4+3+1, regardless of lite fact that this may be meanin1Jess in the language or some culture. For many people in our culture the expression e may be meaningless. Nonetheless it is still true that e'"- - I or e 1"f -1. Of course, Wittgenstein may not have been thinkin[ about the limitations regarding numerical expressions 111 other languages. But if he was not thinking of this matter, then. what is he thinking when he speaks of an arithmetic in which ane does not count further than

INTUITIONISM

105

S? Conceivably he was thinking or arithmetic modulo five. Dul ir •e was then the expres.~ion 4 + 3 is meaningrul. In such an arithmetic pr~umably 4 + 3 == 2. More than likely Wittgenstein lnew about modulo arithmetic and was not rderring to it in this quotation. In any case I do nlll sec that he has established the claim· that the law or excluded middle is not aprlicablc in mathematical proors in the ways that most mathematicians wisl' to use it. In our discussion or the law or excluded middle, we ccrt::iinly have not established that it is a valid rule or inrcrencc. I do not lit"e how lo csiablish such a claim. However neither the lntuilionisls nor Wittgenstein has produced logically compd1ing reasons for rejecting the principle. Ir there were no alternative lo the Godclin view regarding mathemath:al epis;tcmology beside, intuitK>nism amt ir there were no compelling objections to intuitionism then we might be inclined to accept the intuitionist strictures with reganl 10 the non-el{istencc or complete i11finite sets and non--eonstructible sets. Such an 01tology might be quite rlausible if one regarded mathematical objects as having brcen created by the malhematiclan or al least by a number of mathematicians working cooperatively. However, we have not yet considucd some serious objections to the intuiiionisl theory and there arc alternative epistemologics lo both intuitionism and GOdelianism. Intuitionist Theory of the Continuum As noted above the intuiltonist accounl of eur knowledge of principles or natural numbers. integers and rational numberi; rcsls on our ability lo create and mentally inspccf mental objects. However, as noled in our di5e1111i;i obtained from earlier formulae by application of the rules of derivation and such that the forn1ula to be proved occurii in lhe sequence. Proof, as so conceived, has no cpiiilemic value. i.e., it does Ml C.'litablish thal a formula correspondi; to reality. The 1hcQry of proof and of 1ruth lo which formalism leads then is quite diffcrcnt from the ideas on these matters which were part or Hilbert's lhoughL The view that mathematical i;tatcmcnts are merely formulae in format systems is not an nccurnle rencction of the way that most working mathematicians think aoout Lhc statementi; of mathematics. Consider the fact that mathema1icians in their work orten provide diagrams lo represenl the content of the statements lhal they make. Sets are represented as closed regions on a plane surface. Geometrical figures are used to represent triangles, etc. If the statements were considered merely as meaningless objects in a formal system then how could any diagram be appropriate as a representation or the content or the l'tatcrnent. If lhc formalist view of the nature of mathematics is correct then n111ny mathematicians must be said lo have very misguided ideas ai:o ln the nature of what they are doing. But this suggestion is not very plausible.

49. A philosophical position close to th~ or Hilbert and Curry has also been eJi:prcssed by Abraham Robinson. Robinson affirms that statement,; in mathematics which would normally be inter· preted as affirming the e:itistencc or infinite lolnlitic.11 should be regarded as ideal or !'not literally meaningful." Robimmn dOC$ not go so far as Hilbert in holding lhat finitary mathematical stat~ments arc about physical objects. Robinson considered views such as Hilbert's to be nominalistic and he rejected the nominalist point of view. He said "I do not feel compelled lo follow the nominalisls who 11cem ta have little trouble in gr:u1ring the notion of Bl] individual but feel incapable or procecdin,r 10 the notion or a clas5."J 1 It appears then lhal Robinson i11 willing to allow lhc ellisiencc of such queer entities as classes. However, he is not willing lo allow that there are infinitely large claS.'iC5 all of whose members actually e11isL His view is like Hilbert's in lhat he holds that such slatemeAts arc "ideal" or like Curry's in that he allows that such statements are uninterpreted parts or a primitive frame.

I 38

P II I L 0 S 0

r

H Y 0 F M A T II E M A T I CS

He holds that even though such statements are not literally meaningful we arc justified in retaining them as part of our mathematical theories due to the fact that such theories prove to be extremely uscrul when they arc applied in natural sciences. Herc again he wanl.5 to rollow the view or Curry in holding that a mathematical theory can be acceptable even if it is not literally meaningful. He held that "A mathematical theory is acceptable if it can serve as a roundalion for the natural scicnces."n We have criticized Hilbert's version or formalism by arguing that it is a mistake to regard mathematical terms as referring lo phy11ical particulars. Curry's version or formalism avoids this objection but, as we argued above. il provides no satisfactory theory or mathematical truth or knowledge. Curry's view, as we noted. yields the consequence that mathematical truth is relative. I believe that there is a fair interpretation of Robinson's view which avoids both or these objections. . In order to present this interpretation let us consider a theory from some brartch of a natural science. fl might be a theory or gravitation from physics or a theory concerning lhe now or blood in a mammars circulatory system. Let .us call the theory T. As.'lotiatcd with T ts a mathematical theory M. Mis an essential part or Tin that logical inrcrcnces which are made by scientists using T would be invalid unless principles included in M are presurrosed. Let us suppose for the ~;akc or this example that M is the theory or real numbers to which we have referred in several plnce.o; in this work ..Now, let us suppose that Tis formulated in a language or nrst-order predicate logic. In this language there will be certain predicate constants, let us say P,Q,R, and S. Certain relationships involving these predicates will be determined by the axioms or T. Robinson's theory is that many of the statements or T may be literally true. Included among these statements which may be true arc even many statements rrom the mathematical part or T. Howc:ver, in certain statements or T. the predicate constants would, if interpreted literally, rcrer to innnite totalities. Robinson's view is that' such predicates cannot be interpreted literally. These statements which, ir true, would imply that there exists an infinite totality are to be regarded as ideal. Robinson is willinging lo allow that a theory as a whole can be acceptable even though some of its statements are ideal. This is equivalent lo saying that a theory can be acceptable even though some or the terms occurring within the theoretical lltatcmenti; have no literal interpretation. 1r one thinks about the above theory T which includes real

J.

MATHEMATICAL KNOWL~DGF, AS EMPIRICAL

139

number theory M as a part, some questions arise as to how Robinson's view is lo be understood. In particular, are we lo regard all or the statements or real number theory as ideal since taken as a whole these statements imply the eitislence or inftnile lotalities. This would be the simplest course to take. ll would imply that such terms as real number, addition, negative, uprer bound. etc., are not literally meaningrul. However, Robinson has indicated that he regards statements about finite totalities as bolh literally meaning ful and as not rererring to physical objects. Thus, it would appear that we should regard some or lhe axioms or real number theory as meaningrui. But this po!le.'11 problems. SupPQSC for "ample that we say that the field uioms are meaningful since I.hey do not imply the existence or an infinite totality but that Ute uioms asserting the existence or an ordered field are not literally meaningful since these axioms do imply the existence or an infinite totality. But now we find that rnosl of the predicates for the axioms of an Ofdercd field als.o occur in the axioms for a field. It appears that we rnui;t therefore hold that while these predicates are meaningful in the contcitt or the field uioms, they arc no longer meaningful (literally mcaningrul) in the conlul or the axioms for an ordered field. Now it might be argued that this complexity in Robinson',; theory is not, or course. an objection that need be taken r.eriously. The silualion in regard lo mathematical terminology it mig.hl be said, is entirely analogous lo other terms in physical theory. Some lcrms in the context or physical theory may be literally meaninllful but the same terms may also occur in the theories advanced in writings or science fiction. Very orten, in the..i;e contexts, the !lame terms do not make sense. Such statements cannot be understood a:; having !he reforents that they have in their scientific uses. However. I am nol sure that this remark conccrnin!? the fact thal the sRme term!I occur in bolh scienliric and fictional contcxt!i sufflce5 to answer the difficulty which we have raised in Robimmn's theory. When one thinks about the u~ or scientiric terminology in rictional contexts nnd asks why such ui:cs sometimes muke no sense sever~! an5wers come to mind. Sometimes the u11c of such Lerms in ficlio.nal conleJtl5 n1akcs no i;cnse ~cause the i;tatcments made in 11uch 'contexts contradict acccrtcd princirlcs of natural science. Other limes statements in fictional contexts 1nakc no seni;e because such statements arc, a11 it were, simply lhrown inlo the dialogue. No eITort is made t-0 develop a coherent I heoretica I framework for the sta1ement11. In the absence nr such a framework the statements 8

140

PlllLOSOPHY OF MATHEMATICS

have no literal meaning. For reasons such as these, terms and statements in fictional contexts may rail to be literally meaningful even though the same terms and statements are literally meaningful in other contexts. However, the.11c reasons do not apply to the cue or theory T. The mathematical principles or real number theory contradict no accepted statements of laws of nature and statements in which reference is made to infinite totalities are integral parts of coherent theoretical frameworks. We have been discussing a difficulty with Robinson's theory as follows: Scientists accept physical theories such as theory T which include real number theory. According to Robinson's theory we should regard the fteld axioms by themselves as meaningful and the predicates used within the statements of these axioms as having a literal interpretation. However, we cannot regard the axioms which assert the existence or an ordered field as meaningful and so the terms w.hich occur in these , statements have no literal interpretation. The problem is that the same terms occur in both axioms. Thus Robinson's view seems to commit him to saying that certain terms both have and do no• have a literal interpretation. We have considered one possible answer to lhis difficulty. This is to say that the terms can be meaningful in some contellts and meaningless in others because in some contexts the use or terms is analogous to the use of terms in f M:tional uggesl!\ that he thought that they were in some sense "circular definitions." But, as Quine has pointed out, impredicative definitionr; are not circular defanitioM. They do not smuggle the term to be defined implicitly into the definition.' Nor do such ellpres.-;ions as the least upper bound of a set of objects satisrying some condition C appear Lo be particularly ambiguous. It has also been argued by roincare that avoidance of impredicative specifications 'Of sets is justified as this is the correct way to avoid paradoxes such 111 Russell's paradoJt.' However, as many people have noted, the parado:ites can be avoided wilhout making this rei;triction. There seems to be no sound argument which establishes that this is the .. correct •• way avoiding the paradoxes. Restricting Frege's axiorn or abstraction seems lo be equally correct. Conditions (ii) and (iii} above are similar in that lhey both relit on claims lo the effect that statements of the sort that woulu violate these conditions are meaningless. Abraham Robinson said 0 1 must regard a theory which refers to an infinile totality as meaninsless in the sense that its term" and sentences cannot pos."iess the direct interpretation in an actual structure that we

or

or

or

160

PHILOSOPHY OF MATHEMATICS

should expect them to have by analogy with concrete (e.g., empirical) situations.' Nelson Goodman has claimed that rcfor· ence11 to clas..-.es ere "incomprehensible."' But Robinson's claim seems simply to be mistaken. Each term in a mathematical theory, such as the theory or real numbers can, so rar as I can see, "possess a direct interpretation." Terms like tr and .[ 2 can be interpreted as namc11 or numbers. Sentences containing these terms can be interpreted as asserting that these null)ben have certain propercie.'i or stand in certain relations, etc. Robinson's reference to actual structures and concrete situations suggests that he: is saying tliat terms or theories which refer to infinite totalities cannot be construed as names of objects and, al the same time, correlated "directly" to sense perceptions. In other words, he rnay be saying that there are no sensory criteria which constitute sufficient conditions of something's being w or .,{2. But, if our arguments above are correct, then the same would be true even if our mathematical theories were rutricted so as to imply only that there are a finite number of numbers. The point is lhat numbers are not direelly ob.111crvabte. They cannot be identined with speeif.c atrokes along the: lines suggested by Hilbert. The semantic criteria which Robinson may be suggesting rule out finite as well as infinite totalities so rar as mathematical enlitic~,111 are concerned. Further, many other scientific terms would be ruled out also as we are not in a position lo give sufficient conditions ror the application of these terms which would satisfy extreme empiricist semantical rules. Consider, for example, biological terms such as "genotype.. or "species." However, ir one is prepared to admit that theoretical terms such as "genotype" need only be "partially inlerpreted" by reference to sense perceptions in order to be rncaningrul then it appears that terms such as continuity", "Cauchy sequence" J' 2 or .,, from theories which rerer lo infinite totalities can also be partially intcrprcled and so meaningful. For example, empirical criteria can be gi\'cn ror approximating to .,, or J' 2. In any case, Robinson's claim seems lo be mistaken. H some mathematical terms are mc:aningrul in a direct or lilcral sense then there seem to be no reasons for claiming that terms of theories which rder to infinite totalities arc not meaningful also. Nelson Goodman explains hi!i objection more th9roughly. He concei\'cS or the universe as consisting or individuals. Many objects of our experience are to be conceived as heaps or "sums" of individuals. The only collections of which Goodman can conceive 0

EMPIRICISTTUEORY

161

arc such sums. Set theory is incomprehensible lo Goodman because it allribule.'i ei1istence to objects that are not mere sums of individuals in the following sense: Objects may be conceived as sums of individuals providing th;it no two objects consi!ll entirely of the ~me individuals. Set theory violates this condition. For example: in set theory the object {a.b} is not the same object as {{a,b}} even though both objects arc "built up" out of the same individuals, namely of a and b. The former object has two members namely a and b. The laller object lrns only one member namely, ( a,b}. Since sets are identical if and only if they have the same members the.~ two objects arc not identical. My response to Goodman is to 1tay lhat this fcalurc or set theory does not lead me to say that set lheory is incomprehensible um! lo argue for this in terms of ell.am pies. A mathcmalic.'i class in school. for example, may consist of exactly the same individuals as an cnglish class. But, if someone saic.l that Lhc ma1hc111atia claAA is not the same as the cnglish class this would not be incomprchcnsibk. Or consider the following possibility, n and b above might be different populations and the sel ( ;1.b} might be a species. On the other hand the set {{ 11,b}} whose member is Ihe set {11.b} wo1dd be a genus. ln the context of biological theory one might want to :my of the genus lhal al the present time it hm; only one member, namely the specie.c; {a.b} but lhal it is not identical with this species. Other example11 where it makes sense to speak of different objects constituted of the same individuals could be given alStJ. I do nol, of course, hope to persuade Goodman hy cxamplc.o; such ac; the..c;e lo modify his position. fie would, no doubt, try to provide translations of the statements that I used in givinr, these examples so as lo show that where there aJc different objects lhere are also different individual:c; constituting them. I le would thus hope lo account for the meaningfulness of the examples I have given wirhout having lo allow that there arc sets in the sense to which he objects. Mowever, as we have been convinced of the existence of l'iCls on other grounds (through the empirical confirmation of mathematical principles which imply th:ll sets cxii;t) we shall nol try-to refute him al this point by i;lmwinl!, that possible translations which he might provide nre unncccptnble to ui;. Our point in giving these examples was merely to show that, to us at any rate, statements implying the C1'i11tcncc of sets arc quite comprehensible. They would remain so even should a translation along the lines that Goodman would accept rrovc unavailable. So far a5 I can see the sort of consi74. lJ Hilbert, David. "On the lnfini1c". Pl1iln.~ophy of Mafl1em:ilia, Selec:led Readin11, Benaccrrar. l'aul and Pu1nnm, llilary (Englewood CIJff5, Prentice-Hall, Inc .• 1964). 14 lscmi.,ger, G;iry, togic and l'hi/01>op/1y, Sclcclrd Readings (New York, Appleton·Century·Crofts, I%&). lS James, William, fa.~ays in Pragmatism (New York. Harner Publii;hing Co., 1948). 16 Kalmar, Lan.lo, "Fo11ndation11 of Mathemnlic"-·Whither Now?'' Problems in the Philosophy of Matl1cmalic.~·: Lakato.~. lmre (Amsterdam, North·llolland Publishing Comp.'\ny. 1967). 37 Kilcher, Philip, "Hilbert's Epistemology", Pl1ilosophy of Science, Vol. 41, No. I, March 1976. JR Korner, Stephan. The Philosophy of Mathemalic..~ (New York, Harper Torc:hbooks, 1960). 19-. 0 An Empirici:;t Ju11lification of Mathcmalici;", Lo[lic. Methodology and Philosopl1y nf Science. Bar-llillcl, Yehoshua (Amslerdam, Norlh·Holland Publishing Comrany, 11/f\!i). 40 Lakatos, lmrc... Proofs and Rcru1a\iom1", The British Journal for t/Je PbilmophyofScience, Vol. XIV. No. 53, 196J. 41 - . "lnnnite Regress and Foundations of Mathematics", Arisroteli,rn SIJC;eiy. Supplementary Volume. XXXVI, 1962. 42 - - i .. A Renaissance or Empiricism in lhe Recent Philosophy of Mathematics", The Brililih .lournal for the Pl1i/osophy of Science. Vol. 27, No. J, Seplembcr 1976. 0 Leach, James, and Bulls, Robert. and Pearce. Gl~nn. Science, Decision and Value (Dordrechl. Reidel Publid1ing Co.. 1972). 44 Lehman, H., ''Queer ArithmetiC11", Au,,tralasian Journal of Phi~ophy, Vol. 411, No. I. Mny 1970. 45 Leibni7.. G.W., "Leiter lo Canon Ff111cher", l.cibniz Selections. Weiner, Philip P. (New York. Charle." Scribner~ irnd Som1. 1951 ). 46 Levy. Stephen H., "On the Nnturc or Arithmclic Truths... paper read at meeting11 of thi: J\ meriel\n Philo!iophical As!iodation, I 971. 47 Lewis, Clarence Irving; "A Prattmatic Conception (If the A rrinn",

Meaning and Km1wlt:dge. S15tc111,1tic; Re11Jin1(11 in f:pi.~lcmolnJ!~·. · Brandt, R.B. and Nagel, E. (New York, lfarcourl, Brace and Worid, 1965). 48 Locke, John, An Eed on The Theory or Types'', in BtNrand Russell, twiic .1ml Knowledse. Marsh. Robert C. (London, Geor(!.C Allen and U11win Ltd., 195f1). 82 -.., Myslicism and Logic (G11rdc11 City. Douhlcd:IJ' Anch«)r Books}. 83 -.., Introduction to Matl1ematical Philosophy I London, George Allen and Unwin Lid .. 1919). 84-.., The Prindplcs of Marhernatics, liecond edition (London, George Allen and Unwin Ltd., 1931). 85 Schefner, Israel. The Anatamy of ltlquiry {London, Routledge and Kegan Paul Lid .. 1964). 86 Simmons, Oeorge F., Topqlo1y HOO Maclcrn Analysis {New York, McGraw-Hill Book Company lncorporaLed, 196J). 87 Steiner, Mark, Marl1ematical Know/edger (Ithaca, Cornell University Pre1s, 1975). 88 Troelstn, A.S., Principles of ln111;1w11ism (New York, Springer· Verlag, 1969). 89 Vaihinger, Hana, Philosophy of the 1h·lf (London. Routledge and Kegan Paul, l9J~). 90 Wai5mann. Friedrich, ln!roduC'lion lo Mathematical Thinking (New York, Harper and Brother5, 19~9). 91 Willgenstein, Ludwig, Remarks on the Foundalian.s of Mathematics (Cambridge. ih~ M.l.T. Press, l9S6).

I 70

PH I L 0 S 0 P H Y 0 F M AT H E M AT I CS

I

L I

NAME INOll.X

171

NAME INDEX Archimedes~ JI Ariltotle, 100, 134 Ayer, A.J., 17n, 27, 28, JO, Jl, 32,33, 36, 38, 39n, 122, 123, 142n BcnaCCt'l'af, P., I 32 Bieth, E.W., 119 Boole. 0 .• 11 Bolr.a119, 8., I08 Brouwer, L.E.J., 9lff, I 19n, 130, 141, 144,152 Brown, S.I., 90n Cantor,G., 128, IJO Carnap, R.. 17n, 28, 30, 39, 40fT Cauchy, A.. 70, 93, 106. 107. 108 Chihara, C., 6Sn, 90n. 16ln Copi, I., 6Sn Curry,H., IJO, ll2, 133ff, 141 Darwin, c.. JSB Dedekind, R., 14ff, 93f Dacart~ R.. l I Drobicch. 67 Eenin-Volpin, A.S., 113 Euclid, 19, 22, 128 Euler, t... 70, 94 Fermat, P., 1111 Fraenkel, A., 26n, Mn. 1I9n Frcge.G.,Jl,67,110, 12:m.142n, 1SO, 159 Gaus.,, C.F.. 67, 711, 119 Geach, P., 142n, 143n G&lcl, K., ll&rf, 79ff, 9fln, 91, IOS, 114, I IB, 130, ll4, l41, 14~n. 144, 148

Goethc,66 Goodman, N., IS9f, 16ln GoUlieh, 0., 8, 9, IOn Ori5i, G.F.C .. 111 Uahn, H.• 17n Harvey, W., 156 II ate her, W.S., 6Sn Heine, E., I 27

Hempel. c .• 21, 211. 30. 3 t, JJ, 34, Jll. l9n, 122, 123. I 42n Heyting, A.. imr. II 9n, 130 Hilbert, D.• 12W. I 37ff, 141, 142n. 143n, 144 Kalmar, I.., 163 Kant, I., 91, 110

Kitchu, P.. I 42n Korner. S .. 111. 112, ll9n, IJOff, 14Jn, 14Sfr, 157, l