Tennant Naturalism

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Critical Studies / Book ReviewsWhat is Naturalism in Mathematics, Really?^

P E N E L O P E M A D D Y . Naturalism in Ma thematics. Oxford: ClarendonPress, 1997. Pp. x + 254. ISBN 0-19-823573-9.Reviewed by NEIL TENNANT*

Penelope Maddy's Naturalism in Mathem atics is perhaps the most defini-tive contemporary statement on its topic, lucidly written and technicallywell informed. The book's central theme, as Maddy states it on p. 28, ishow a 'final consensus' might be 'rationally achieved' on 'new axiom candi-dates', or, as she also pu ts it, 'experim ental hypotheses', for set theory. Herposition might best be characterized as qualifiedly Quinean. She lays stresson pragmatic factors in theory choice and on the testing of hypotheses bytheir consequences.The book is divided into three parts: The Problem; Realism; and Nat-uralism. It is the inadequacies she uncovers in her earlier Realism 1 thatprom pt Maddy 's move here to her version of Naturalism. In 1 below Ishall take each chapter in turn, summ arizing what it covers. 1 is intendedto be a fair exposition of the project in its own terms, with only an occa-sional critical comment. (Readers familiar with the book could thereforeproceed directly to 2.) In 2 my task is a more critical one, in both sensesof th e word. I shall offer some opposing thoughts on some of the mostimportant points on which one might take issue with Maddy, both as aninterpreter of the historical record and as a writer dealing with contempo-rary philosophical problems. I take up eight issues, not necessarily in theorder in which they first arise in the text: restrictiveness; naturalism; ex-trinsic versus intrinsic justifications; bivalence; unification; indispensability;Zermelo's 'pragmatism'; and the evolution of the notion of mathematicalfunction.

t I am grateful to Maddy for clarifying certain aspects of her position in personalcorrespondence, and saving me from some unp rodu ctive misunde rstandings. 1 also owethanks to Julian Cole, Robert Kraut, Stewart Shapiro and an anonymous referee forhelpful comments on an earlier draft.

* De partm ent of Philosophy, Ohio State University, Columbus, Ohio 43210-1365U. S. A. [email protected] P. Maddy, Realism in Mathematics. Oxford: Oxford University Press, 1990.

PHILOSOPH1A MATHEMATICA (3) Vol. 8 (2000), pp. 316-338.


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PHILOSOPHIA MATHEMATICA 3171. Exposition

Chapter I .I , 'The Origins of Set Theory 1, argues that set theory resultedfrom a confluence of Frege's philosophical investigations and Cantor's math-em atic al ones. M ad dy discusses Frege's ill-fated Axiom V (of Un limitedComprehension), Russell 's paradox, the vicious-circle principle and the the-ory of types (both ramified and simple). She endorses Quine's criticism ofth e Axiom of R edu cibility as 'self-effacing: if it is tr u e, th e ramifica tion itis meant to cope with was pointless to begin with. ' 2

Maddy explains how Cantor happened upon his notions of ordinal andcardinal numbers while trying to treat of 'derived sets ' of real numbers.She describes how the subsequent at tempt to provide for sets as mathe-matical objects, carried out mainly by Zermelo, led to the conception ofthe cumulative hierarchy of sets.

Chapter 1.2, 'Set Theory as a Foundation', rehearses how one can defi-nitionally reduce different branches of classical mathematics to set theory.The force of set theoretic foundations is to bring (surrogates for) all mathe-matical objects and (instantiations of) all mathematical structures into onearenathe universe of setswhich allows the relations and interactions be-tween them to be clearly displayed and investigated, (p. 26)Maddy stresses, correctly, that set- theoretic foundationalism lays no claimto giving the essence or intrinsic nature of the individual mathematicalobjects ( the 'posit ions within a structure ') for which set theory providessurrogatesprecisely because there are so many different set- theoreticstructures ensuring surrogacy (the famous Benacerraf point) .

Maddy discusses Quine's 'ontological reduction' briefly, but withoutmak ing i t clear whethe r she endorses th e Qu inean view. She rightly asks'if neither metaphysical insight nor ontological economy is forthcoming[from set-theoretic reduction or surrogacy of classical mathematics], what isgained by th e exerc ise?' (p. 26). W hile imp licitly conced ing th e antec eden t,she answers her own question in a positive spirit . The answer, she says,lies in mathematical rather than philosophical benefits... Mathematics is pro-foundly unified by this [set-theoretically reductive] approach; the interconnec-tions between its branches are highlighted; classical theorems are traced toa single source; effective methods can be transferred from one branch to an-other; the full power of the most basic set theoretic principles can be broughtto play on heretofore unsolvable problems; new conjectures can be evaluatedfor feasibility of proof; and ever stronger axiomatic systems hold the promiseof ever more fruitful consequences. As the desired m athe matica l payoffs can

be achieved by this modest version of set theoretic foundations, I will assumeno more than this in what follows, (pp. 26-28)The way that mathematics is 'profoundly unified' by taking set theory asi ts foundation is impressive despite two caveats to which Maddy draws at-

2 Quine, as quoted on p. 12.


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318 CRITICAL STUDIES / BOOK REVIEWStention. First, as we know from the work of Godeland as Zermelo didnot realize in 1908one could never prove, within one's chosen founda-tional theory, that that very theory was consistent. Secondly, although settheory is 'just one theory' with (perforce) its own methods of proof, oneought nevertheless to be mindful of the different casts of thought of combi-natorialists, analysts, algebraists and others, whose variety of approachesand resources of conceptualization might be unduly restricted were one toinsist on some uniformity of proof-methods in addition to the ontologicalunification afforded by set theory itself.

Chapter 1.3, 'The Standard Axioms', discusses the justificatory status ofthe various axioms of ZFC. Maddy deals with a loose distinction between'extrinsic' and 'intrinsic' justifications. A justification of an axiom is ex-trinsic when it adduces for consideration what follows from the axiom, oradverts to pragmatic considerations of simplicity, economy and so on. Bycontrast, a justification is intrinsic when it appeals to our intuitions, and/orgrasp of the concepts involved, without an eye to the axiom's consequences.The distinction between these two types of justification will be employed inher case for naturalism. The leading idea is that the choice of new axiomsfor set theory might admit only of extrinsic justifications.There is an interesting but brief discussion of the limitation-of-size doc-

trine (pp. 43-49), which ends with the widely accepted conclusion thatit is impossible to restrict Unlimited Comprehension in a principled waywithout adverting to 'measures of size' that themselves make sense onlyagainst the background of what sets happen to exist. Maddy maintainsthat the limitation-of-size doctrine is itself a consequence of the iterativeconception of set. It follows that any justification resting either directlyon the iterative conception, or appealing to limitation of size, will count asintrinsic. Maddy discusses each axiom of ZFC in turn, giving both intrinsicand extrinsic justifications.Chapter 1.4, 'Independent Questions', gives a nice history of the Contin-uum Hypothesis, and an explanation of the broad features of Godel's proofof its consistency relative to ZFC. By contrast, Maddy's explanation of thecentral idea behind Cohen's proof of the relative consistency of ->CH is veryterse (p. 66). She then moves on to a very informative discussion of Borelsets, projective sets, Lebesgue measurability, and important mathematicalconjectures involving these notions that are independent of ZFC.Chapter 1.5, 'New Axiom Candidates', begins with a brief examinationof the Axiom of Constructibility before moving on to large-cardinal axioms

and the axiom of determinacy. The reader is treated to a clear exposition ofinaccessibles, hyperinaccessibles and Mahlo cardinals (the so-called 'small'large card inals). The discussion then moves to measurable cardinals (whichcontradict V L) and supercompact cardinals. The aspirations of large-cardinal theorizing are capped off by inconsistency, as shown by Kunen,


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PHILOSOPHIA MATHEMATICA 31 9when one postu lates the existence of a cardinal th at is the critical point fora non-trivial elementary embedding of V into V. Just short of that, certain'extremely large, large cardinal axioms have been formulated by examiningKunen's proof and positing cardinals that seem as large as possible withoutallowing the argument to go thro ugh'. Bu t alas, 'soon after th e inventionof forcing, the hopes that large cardinal axioms would settle the full CHwere dashed by Levy and Solovay.'3

W hat, then , about determinacy? Full determinacy contradicts the axiomof choice, bu t the more limited axiom of Projective Determinacy (PD) leadsto 'an elegant and nearly-complete theory of the properties and behaviorof projective sets of reals' (p . 80). Bu t 'the trouble is that PD , in itself,seems too specialized, too opaque, to serve as a basic axiom for set theory'.Maddy quotes Martin and Steel:Because of the richness and coherence of its consequences, one would like toderive PD itself from more fundamental principles concerning sets in general,principles whose justification is more direct.

The 'culminating theorem' of Martin and Steel is that the existence of asupercompact cardinal (SC) implies PD. Hence PD cannot settle CH, sinceSC, by the result of Levy and Solovay, cannot. So CH remains elusive andintrac table, one whole century after Hilbert pronounced it the Most Wantedfugitive from mathematical law.Chapter 1.6, 'V = L', briefly surveys some of the things Godel wrote inarguing against accepting the so-called Axiom of Constructibility. It impliesCH, and Godel thought CH was false. V = L is a 'minimizing' principle'restrictive, limiting, m inimal, a n d .. . these things are antithetical to thegeneral notion of set'. (Thus Maddy at p . 84, in summarizing a samplingof set theorists' views on the matter. There is near unanimity, then, on anintrinsic argument.) Bu t 'there are also more clearly extrinsic argumentsagainst V = L... its numerous detractors clearly hold that its merits arefar outweighed by its demerits' (pp. 84-85). The rest of Maddy's book isdevoted to resolving in the negative the 'legitimate mathematical question'whether the universe of sets is constructible, 'on the grounds that V = L isrestrictive, in some sense or other'. And, although she does not note this,Maddy's explication of these grounds would count as extrinsic by her ownlights.

Chapter II.1, 'Godelian Realism', takes as its point of departure theproblem of assigning a truth-value to CH, rather than to V = L. Maddyshows how Godel saw an analogy between th e choice of scientific hypotheses(to explain our perceptual experiences) and our choice of mathematicalaxioms (to explain our mathematical intuitions).Chapter II.2, 'Quinean Realism', aims to find some justification for as-

3 The paper in question here is A. LeVy and R. Solovay, 'Measurable cardinals and thecontinuum hypothesis', Israel Journal of Mathematics 5 (1967), 234-248.


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320 CRITICAL STUD IES / BOOK REVIEW Ssuming a mind-independent world of mathematical objects. Quine's prag-matism sees only the constructible sets as needed for physical theorizing.Unlike Carnap, Quine does not see the existence of abstracta such assets and numbers as a 'pseudo-question', or a merely pragmatic policy-question as to whether one should adopt one linguistic framework ratherthan ano ther. Rather, com mitment to any kind of entitybe it physicalor mathematicalis incurred, for Quine, by the existential quantificationsmade within one's overall theory of the world. (This, indeed, is the tru e fontof Quinean naturalism see 2.2 below.) The choice of theory is constrainedonly by our empirical data (that is, our observational reports about observ-able objects), and by pragmatic desiderata such as simplicity, economy andso on. On Quine's view, mathematics is a fully-fledged pa rt of science, in-extricably bound up with physical hypotheses, and not merely part of thelinguistic framework within which scientific questions about what 'really'exists might be raised and answered. Quine is willing to apply Occam 'sRazor just as readily to the abstract realm as to the physicaleven if thisis contrary, in the case of set theory, to the desire to maximize that flowsfrom the iterative conception. Maddy is alive to th is point, but only muchlater in her exposition (p. 131):

crudely, the scientist posits only those entities without which she cannot ac-count for our observations, while the set theorist posits as many entities asshe can, short of inconsistency. . . . Quine counsels us to economize, like goodnatural scientists, and thus to prefer V L, while actual set theorists rejectV = L for its miserliness.Thus the ontologically wary pragmatist believes V = L, even if his set-theoretic colleagues have found intrinsic, conceptual reasons for thinkingthat there is more to V than is ever dreamed of in L.Chapter II.3, 'Set Theoretic Realism', amounts to a one-page statementof Maddy's strategy in her earlier book Realism in Mathematics (p. 108):4

The compromise goes like this. Take the indispensability arguments to providegood reasons to suppose that some mathematical things (e.g., the continuum)exist. Admit, however, that the history of the subject shows the best methodsfor pursuing the truth about these things are mathematical ones, not those ofphysical science.Chapte r II.4, 'A Realist's Case against V = L\ seeks to develop the casewithout appealing to the existence of measurable cardinals. Maddy clearlywishes to keep open the option of giving an extrinsic argument later on for

4 The 'compromise' Maddy is referring to here is between the Godelian and Quineanversions of set-theoretic realism, not a compromise struck by the naturalist co-optingcertain features of such realism. I am grateful t o Mad dy for clarifying her m ean ingin context here, and mention this lest any other reader fall prey to that misunderstan-ding. Maddy subsequently finds the compromise inadequate, in that the indispensabilityconsiderations, she argues, do not vouchsafe the ontological conclusions sought by theset-theoretic realist.


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PHILOSOPHIA MATHEMATICA 321the existence of measurables, by appealing to Scott's result that the exis-tence of a measurable cardinal contradicts V = L. She gives an interestingaccount of the principle of Mechanism in the history of physics, drawingon Einstein and Infeld's 1938 monograph The Evolution of Physics. Wesee how the Field Conception came to displace Mechanism, as anomaliescropped up for the latter and explanatory successes accrued to the former.Maddy's suggestion is that, analogously, mathematicians progressed froma principle of Definabilism to a 'much broader understanding of the natureof functions' (p. 122). It is an underdeveloped analogy at bes t; M addydemonstrates little organic connection, or even explanatory co-operation,between the two areas. The needs of physical theorizing impinged on thedevelopment of the notion of mathematical function on two occasions, butwith cancelling effects: in the mathematical treatm ent of a vibrating string;and in the solution of Fourier's heat equation. For the former case, Eulerliberalized the then dominant Definabilist conception; but his 'new' func-tions were to be subsumed under the Fourier expansions introduced forthe second case, thereby reviving Definabilism with a broader conceptionof what definitions would be available. Further developm ents away fromthe Definabilist conception, however, were prom pted by mathem atical, notphysical, considerations in the foundations of the calculus.

Maddy provides a detailed survey of these shifts in mathem atical atti tud etowards the nature of functions, beginning with Descartes and Fermat, andmoving on to Bernoulli pere, Leibniz, Euler, D'Alembert, Bernoulli 61s,Fourier, Dirichlet, Riemann, Darboux, Du Bois-Reymond, Baire, Borel,Lebesgue, Zermelo, Hadamard, Poincare, and Bernays. The fully liberal-ized conception of a function as an arbitrary single-valued correspondence,whether or not it is definable, led to mathematical advances that wouldnot be sacrificed once they had been accomplished. Zermelo's Axiom ofChoice in 1904 postu lated the existence of functions for which one need notbe able to specify any rule. In the ensuing dispute in 1905 over Zermelo'saxiom, Lebesgue posed th e crucial question: Can we prove the existence ofa mathematical object without defining it?

Lebesgue answered this question negatively, and both Zermelo and Ha-dam ard opposed him. Hadamard appealed to the lesson of history: 'theessential progress in mathematics has resulted from successively annexingnotions which... were "outside mathematics" because it was impossible todescribe them .' Th is challenge to Definabilism culminated in the Combi-natorialism of Bernays. In a lecture in 1934 Bernays explained Combina-torialism as exploiting 'an analogy of the infinite to the finite': 55 P. Bernays, 'On platonism in mathematics', in P. Benacerraf and H. Putnam, eds,Philosophy of Mathematics, 2nd edn. Cam bridge: C am bridge University Press, 1983,pp. 258-271; at pp. 259-260. Maddy's text had nn in place of n n , and I incorporate hercorrection from personal correspondence.


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322 CR ITICAL STUDIES / BOOK REV IEWSThere are \n n] functions [which assign to each member of the finite series1, 2 , . . . , n a number of the same series], and each of them is obtained by nindependent determinations. Passing to the infinite case, we imagine functionsengendered by an infinity of independent determinations which assign to eachinteger an integer, and we reason about the tota lity of these functions Se-quences of real num bers a nd sets of real num bers are envisaged in an analogousmanner.

Maddy summarizes thus (pp. 127-129):So, according to Combinatorialism, there is one function from reals to reals forevery way of making 2N independent assignments of a real to a real. Theseassignments are taken to exist, on analogy with the permutations [sic]6 of 1,2 , . . . , n, regardless of whether or not we have a rule to determine t h em ThusCombinatorialism deposed Definabilism, much as the Field Conception re-placed Mechanism. . . . Under these circum stances, the deep and widespreadresistance to adding V = L as a, new axiom seems perfectly rational.Chapter II.5, 'Hints of Trouble' , is a short segue to a consideration ofthe role of continuum mathematics in natural science.Chapter II.6, ' Indispensability and Scientific Practice' , challenges theQuinean indispensability argument for a mathematically realist view atleast towards those structures or entities involved in the mathematical the-orizing tha t is applied by nat ura l scientists. By me ans of th e interesting case

history of the gradual, and finally decisive, acceptance by the scientific com-mun ity of the real i ty of atoms, Maddy shows th at existent ial comm itmentshave to pass part icu larly dem anding muster . T he early explanatory suc-cesses of D alto n's a tom ic theory were not enou gh. T he Law s of Definite andof Multiple Proportions, Gay-Lussac's Law of Combining Volumes, Boyle'sLaw and C harle s's Law were all explained by atom ic theory. So were thephen om ena of isomerism, and Dumas's Law of Sub st i tut ion. W hen C an-nizzaro distinguished between molecule and atom in 1858, and Avogadro'shypothesis was confirmed in independent ways, the atomic theory's expla-nation of all these things was still not enough to convince the scientificcommunity at large of the reality of atoms. Something more was neededsom ething by way of ' indep end ent ' confirmation of their existence. Th iscame from the phenomenon of Brownian motion, where 'directly' observ-able mo vem ents of tiny pa rticles were 'directly' explained in ter m s of molec-ular collisions, and led to another determination of Avogadro's number, indecisive agreement with the resul ts of other methods. Maddy concludes

Perhaps the notion that all existence claims are on the same footingthe'univocality of "there is" ' as Quine calls itis inaccurate as a reflection of thefunction of scientific language, (p. 143)The challenge is to explain how certain experiments license a switch from

6 Maddy overlooks the fact that Bernays's nn functions from { 1 , . . . , n} to {1 , . . . , n}will not all be one-one. Permutations, by definition, are one-one.


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PHILOSOPHIA MATHEMATICA 32 3a fictional use of 'there are atoms' to a literal use.

[T]he case of atoms makes it clear that the indispensable appearance of anentity in our best scientific theory is not generally enough to convince scientiststhat it is real. If we still hope to draw conclusions about the existence ofmathematical things from the application of mathematics in science, we mustbe more attentive to the details of how mathematics appears in science andhow it functions there.So the question will be: does science make ineliminable and non-idealizinguse of the m athematical continuum in theorizing about n atural phenomena?Maddy notes that there are at least two kinds of idealization in scienceidealization for causal isolation (frictionless planes, and so on), and ideal-ization for simplification (continuous ideal fluids and so on) . Idealizationprevents one from drawing ontological conclusions from the applied math-ematical context in question (p. 146). Maddy also draws attention to theopinion of Feynman, who confessed bluntly:7 'I rather suspect that thesimple ideas of geometry, extended down into infinitely small space, arewrong.' Feynman also had misgivings over whether the fine structu re oftime is continuous. Maddy's cautious and 'sorry' conclusion is that

a space-time continuum is not something we can take as established... .giventhe state of current natural science, a responsible indispensability argument...seems unlikely to support the existence of more than a few (if any) mathemat-ical entities [fn] and these few cannot be expected to guarantee determinatetruth-values to the independent questions of set theory, (pp. 153-154)Chapter III. 1, 'Wittgensteinian Anti-Philosophy', is a chapter that Maddyencourages 'philosophers... with a technical bent, [who] tend to be unsym-pathetic to the style and content of the late [sic] W ittgenstein... to skipover' (p. 161). The aspect of the later W ittgenstein's philosophy tha t ismost relevant to Maddy's concerns (not that she endorses itsee pp. 169-170) is his hostility to the conception that pure mathematicians tend tohave of the kind of mathematics they pursuethe kind that might, forall they know, be forever devoid of any application. Wittgenstein insists8'It is the use outside mathematics, and so the meaning of the signs, thatmakes the sign-games into mathematics'. He offers no argument as to whythe signs used in pure mathematics should be beholden to applicationsin natural science in order to be meaningful. As Maddy observes, 'ThisWittgensteinian line of thought has damaging consequences for set theory'(p . 168). Indeed it does; the W ittgenste inian line of thought here is un-justified and irresponsible. It transgresses Wittgenstein's own stricture to'leave mathematics as it is'. No pure mathematician would tolerate havinghis or her discourse so easily divested of meaning, simply for want of an

7 The Character of Physical Law. Boston: MIT Press, 1967, at p. 166.8 Remarks on the Foundations of Mathematics. Revised edition, G. H. von Wright,R. Rhees, and G. E. M. Anscombe, eds., Boston: MIT Press, 1978; at V, 2.


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324 CRITICAL STUDIES / BOOK REVIEWSapplicat ion in natural science.

Th e lesson or 'clue' th at Maddy , curiously, detects from her excursus intothe later Wit tgenstein is t h a t we should ' [excise] all traditional philosophy,and in its place, [pay] careful attention to the details of the pract ice [ofhigher mathematics] itself. Would that Wit tgenstein himself had done so.In this crucial respect, his natura l i sm is the t rue ancestor of Quine'ssee2.2 below.Turning her at tent ion in Chapter III .2 to 'A Second Godelian Theme' ,M addy provides a reading of two of Godel 's important discussions (of Rus-sell 's vicious-circle principle, and of Cantor 's continuum hypothesis) ac-cording to which Godel is moved by entirely mathematical considerat ionsto draw the philosophical morals he does. Godel always stresses the impor-tance of being able to provide a foundation for all of classical mathematics .Foundational systems that cannot do th is are ruled out. Thus Maddy drawsthe conclusion (pp. 174-176) that. . . the support for the assumption of the existence of sets lies.. . in the re-quirements of ordinary mathematical practice.. . On this reading of Godel, hisargument for the legitimacy of [the question of the truth or falsity of] CHactually bypasses his philosophical realism altogether; its working parts areall drawn from mathematics itself. If a Wittgensteinian anti-philosopher wereto 'treat' Godel's discussion by excising the 'misguided' philosophy, nothingessential would be lost; the residue would be an inventory of ordinary mathe-matical considerations, jus t the sort of thing the anti-philosopher recommendsto our attention.In Chapter III .3, 'Quinean Natura l i sm ' , Maddy draws her characteriza-t ion of naturalism from the writ ings of Quine. Na turalism , according toQuine , is 'the recognit ion that it is within science itself, and not in someprior philosophy, that reality is to be identified and described. '9 Moreover,natural ismsees natural science as an inquiry into reality, fallible and corrigible but notanswerable to any supra-scientific tribunal, and not in need of any justificationbeyond observation and the hypothetico-deductive method.10Maddy herself is aware of the inadequacies of Quinean natura l i sm as faras mathemat ics is concerned. In Chapter I I I .4 , 'Mathemat ica l Natura l i sm ' ,she tries to effect a synthesis of Quine's posi t ion with 'the insights we'vegained from Wittgenstein and Godel ' (p. 182). Maddy's leading proposal(p . 184) is t h a t we should adopta mathematical naturalism that extends the same respect to mathematicalpractice that the Quinean naturalist extends to scientific practice. . .. mathe-

9 W. V. O. Quine, 'Things and their place in theories', in Theories and Things. Cam-bridge, Mass.: Harvard University Press, 1981, pp. 1-23; at p. 21.1 0 W. V. O. Quine, 'Five milestones of empiricism', reprinted in Theories and Things,pp. 67-72; at p. 72.


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PHILOSOPHIA MATHEMATICA 325matics is not answerable to any extra-mathematical tribunal and not in needof any justification beyond proof and the axiomatic method.

(We shal l return to this definitional template in 2.2 below.)Chap te r III.5, 'The Problem Revisi ted ' , introduces the two m axim s tha tMaddy sees as driving set-theorical endeavours. Maddy calls them UNIFYand MAXIMIZE. The first is expressed as follows: ' [I]f your aim is to providea single system in which all objects and s t ructures of m athem at i c s can bemodelled or ins tant ia ted , then you must aim for a single, fundamenta l the-ory of sets. ' (pp . 208-20 9; my emph ases.) T he max im UNIFY, th en , requiresone to set t le every set- theoret ical s tatement one way or the other : tha t is,aim for one complete theory. Although its inspiration is prima facie onto-logical, the ma xim 's express goal is merely theoretical completenessa con-di t ion tha t we know, by Godel's first incompleteness theorem, will foreverelude us, providing we obey the other overriding maxim of 'CONSISTENCYour legitimate preference for consistent theories ' (p. 230). It is left to thesecond maxim, M A X I M I Z E , to ensure a plenitude of objects and s t ructures :.. . the set theoretic arena in which mathematics is to be modelled should beas generous as possible, the set theoretic axioms from which mathematicaltheorems are to be proved should be as powerful and fruitful as possible.Thus the goal of founding mathematics without encumbering it generates themethodological admonition to MAXIMIZE, (pp. 210-211)The quest ion, of course, is: How does one tell, formally, when a theoryis maximizing? M add y's answer is at best part ial . In C h a p t e r III.6, 'ANatural is t ' s Case against V = L\ she a t t em pts to provide a formal criterionfor telling when a theory is (too) restrictive. Non-restrictiveness will be anecessary, but by no means sufficient condition for maximizing. Maddy isable at least to show that (on her formal explication) ZF C plus the Axiom ofConstruct ibi l i ty (V = L) is restrictive, and t h a t the latter axiom thereforeshould not be adopted .

2. Critical Discussion2.1. RestrictivenessMaddy's definition of the notion 'T is restrictive' is the heart of her originalcontribution in the book. Over pp. 220-224 she gives a sequence of defini-tions building up to it. The formal notion, however, does not accomplishmuch. For as Maddy concedes (p. 225) her definition of restrictiveness 'clas-sifies as restrictive certain theories that don't seem restrictive and... failsto classify as restrictive certain theories that do seem restrictive'. That is,her criterion admits false positives and false negatives, respectively. So itwill 'need supplementation by informal considerations of a broader char-acter', such as whether an inner model is 'optimal' (p. 227) or whether atheory (such as ZFC extended by the claim that QT does not appear in any


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32 6 CRITICAL STUDIES / BOOK REVIEWStransitive set model of ZFC) 'leans hard in [the] direction' of inconsistency(p . 229). Such considerations are so dependent on the refined intuitionsof expert practitioners that it is hard to see how any account that has toresort to them can count as fully naturalized (in the normal philosophicalsense of this termsee 2.2 below). And even if this shortcoming can beremedied, by providing a revised formal definition admitting (as far as onecan tell) no false positives or false negatives, still one can point out thatthe criterion of restrictiveness cannot be the whole story needed. Havinga formally specifiable negative filter on extensions of ZFC is not enough toconvince one that our eventual positive choices of new axioms to appendto ZFC can be explained by a genuinely naturalistic model, whether in theusual sense or in Maddy's sense.2.2. NaturalismI have three criticisms to offer of Maddy's natu ralism. They concern, re-spectively, her accommodation of the a priori; an internal tension in heraccount; and her quietism about classical methods.Maddy's naturalism, one could argue, is no such thing; it is rather, oncloser examination, an a priori investigation involving conceptual analysis,considerations of intended interpretation, and non-empirical striving forreflective equilibrium. In a way that would find agreement with Pla to,Descartes, Leibniz and Kant, Maddy emphasizes the a priori nature ofmathem atics as its defining trait . She divorces mathematics from natu ralscience in general, and insists that one seek to understand mathematics onits own terms. As a result, her naturalism (about mathematics) is leachedof the original philosophical significance that a Quinean naturalism wouldhave had for mathematics. How is this so?

As Steven Wagner and Richard Warner wrote in their editorial introduc-tion to a recent collection of essays on naturalism,11 'we take naturalismto be the view that only natural science deserves full and unqualified cre-dence'. This is echoed by the definition of naturalism offered in The OxfordComp anion to Philosophy, edited by Ted Honderich.12 Naturalism is

the view that everything is natural, i.e. that everything there is belongs to theworld of nature, and so can be studied by the methods appropriate for studyingthat world. . . . In metaphysics naturalism ... ins ists ... that the world of natureshould form a single sphere without incursions from outside by souls or spirits,divine or human, and without having to accommodate strange entities likenon-natural values or substantive abstract universals. (p. 604)Thus, prima facie, naturalism would incline one to a nominalistic view ofmathem atics. Quine, however, accorded mathem atical entities the statusI I S. Wagner and R. Warner, eds., Naturalism: A Critical Appraisal. Notre Dame, Ind.:University of Notre Dame Press, 1993, at p. 1.1 2 Oxford University Press, 1995.


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PHILOSOPHIA MATHEMATICA 32 7of scientific posits. Mathem atical entities were on an ontological par withtheoretical entities such as fundamental particles. This was because of theway hypotheses quantifying over them were integrated into an holistic webof mathematical-and-scientific belief. That web of belief faced the tribunalof experience as a whole. One could not factor out the special contributionof mathematics; nor did one have any intuitive intellectual access to suchthings as numbers and sets, independently of the role they would play inscientific theorizing about nature overall.So for Quine, nominalism is averted as a consequence of naturalism by in-tegrating mathematics into natural science, and not according mathematicsany separable status as an intellectual practice.Maddy's anti-Quinean apostasy is precisely to accord mathematics suchseparable status. Yet almost all other aspects of her philosophical temper-ament appear to remain Quinean. Her leading proposal from p. 182 wasquoted earlier. Her definitional template for mathematical naturalism ad-mits the following substitutions, yielding peculiar brands of 'naturalism':13a theological naturalism that extends the same respect to theological practicethat the Quinean naturalist extends to scientific practice theology is notanswerable to any extra-theological tribunal and not in need of any justificationbeyond prayer, religious indoctrination and the study of scripture.If that seems unbearable,14 the Quinean ought still to be troubled by

a semantic naturalism that extends the same respect to semantic practicethat the Quinean naturalist extends to scientific practice semantics is notanswerable to any extra-semantic tribunal and not in need of any justifica-tion beyond introspection on one's own grasp of meaning and the method ofmeaning-postulates.Maddy says her plan 'is to construct a naturalized model of [mathematical]prac tice', (p. 193) But this does no t mean a model adverting only to th epatterns of neurological firings in the brains of mathematicians, and thecausal story behind their jottings and their tappings on keyboardswhichis what one might expect, given the other characterizations of natural-ism quoted above. Ra ther, she is out to account for the 'actual justifica-tory structure of contemporary set theory' and to argue for the claim that'this justificatory structure is fully rational'. Any a priorist foundationalistwould be in her debt were she to succeed.

Maddy wants to identify the goals of set-theoretic practice, and '[elab-orate] means-ends defences or criticisms of particular methodological de-1 3 These substitutions are mine. Maddy anticipates only the objection that would issuefrom 'astrological' naturalism; she does not say anything about theological or semanticnaturalism.1 4 Por further interesting discussion of the difficulties, on Maddy's conception of natu-ralism, in distinguishing between theological and mathematical naturalism, see J. M. Di-eterle, 'Mathematical, astrological and theological naturalism', Philosophia Mathematica(3) 7 (1999), 129-135.


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32 8 CRITICAL STUDIES / BOOK REVIEWScisions' (p. 197). She aspires thu s to stradd le the boundary between th edescriptive and the normative. Moreover, she allows no room for any ra-tional critique of the goals that the community of mathematicians mightset for themselves. She provides no criterion by means of which one mightdistinguish between (i) the goals to be served by the practice as a whole(for example: boost the self-esteem of schoolchildren; or develop an appro-priate m athema tics for non-continuous spacetime, in case it is needed forthe theory of quantum gravity), and (ii) the intra-theoretic goals to be pur-sued within the practice (for example: solve the continuum hypothesis; orclassify the simple finite groups).

It is worth pointing out that mathematicians within the mathematicalcommunity will often take issue with the direction of their discipline, itsfundamental assumptions, its fundamental goals, and its own criteria forjudging excellence and importance of work in various areas of m athem atics.Just such a debate is under way at present, for example, over the issue oflarge-cardinal axioms, and how best to convince 'core mathematicians' oftheir importance for 'ordinary' mathematical problems.15 The mathemat-ical community itself should not be thought of as monolithic, producingself-regulatory norms commanding community-wide consensus within, andimpervious to criticism from w ithout. Some mathem aticians are very philo-sophically minded, and allow philosophical considerations to influence theirchoice of problems and methods. Maddy writes as though, from her natu-ralistic perspective, professional mathematicians would have to be allowedto be a law unto themselves. Bu t, first, th is exaggerates the degree ofunanimity on important issues even among professional mathematicians.Secondly, it downplays the legitimate interest, of those who are not mem-bers of that professional community, in the body of norms that govern themathematicians' own practice. For, as Maddy herself points out (pp. 204-205), mathematics is 'staggeringly useful, seemingly indispensable, to thepractice of natural science'. One might add also: to engineering and tech-nology; to medical diagnostics; to the financial markets; to actuarial science;and to a host of other areas of hum an activity in which everyone's interestsand concerns are engaged. So ramifying would be the consequences, boththeoretical and practical, of any lapse on the part of professional math-ematicians into bad practices resulting from erosion of their historicallyhard-won norms, that even those outside the community of professionalmathematicians have a permanent and legitimate concern in the natureof the norms governing the latte r's practice. In a word: mathem aticianscannot be allowed to be a law unto themselves. What they do is too im-1 5 This emerged clearly at the ASL panel discussion on the topic 'Does MathematicsNeed New Axioms?', at the University of Illinois at Urbana-Champaign on June 5, 2000.The panellists were Solomon Feferman, Harvey FYiedman, Penelope Maddy and JohnSteel.


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PHILOSOPHIA MATHEMATICA 32 9portant, and ought to be subject to outside constraints designed to protecteveryone's interests.The 'naturalistic' model that Maddy proposes is beset with an internaltension. On the one hand, the phenomena being modelled are intrinsicallysemantic and norm-governed. One is modelling 'practice', 'debates', 'argu-ments', 'discourse', 'utterances', 'testimony', 'reports' (p. 199). In doing so,one is descrying the theor ists' goals, as well as 'th e underlying justificatorystructure of the practice' in which 'sound and persuasive arguments' maybe implicit (p. 200). On the other hand,

. . . suppose mathematicians decided to reject the old maxim against incon-sistencyso tha t both '2 + 2 = 4' and '2 + 2 = 5' could be acceptedonthe grounds that this would have a sociological benefit for the self-esteemof school children. This would seem a blatant invasion of mathematics bynon-mathematical considerations, but if mathematicians themselves insistedthat this was not so, that they were pursuing a legitimate mathematical goal,that this goal overrides the various traditional goals, / find nothing in themathematical naturalism presented here that provides grounds for protest. [Myemphasis]16According to Maddy, 'our fundamental naturalistic impulse' is the 'convic-tion that a successful practice should be understood and evaluated on itsown terms' (p. 201). The footnoted concession emphasized in the last quo-tation reveals, however, that the uncritical quietism essential to Maddy's'naturalistic' methodology has reduced that methodology to absurdity. In-terestingly, it is not only the a priori foundationalist, shocked a t the thoughttha t m athematicians might propose to have 2 + 2 being equal bo th to 4 andto 5, who is rendered impotent; it is also the Quinean re-distributor oftruth-values! For Maddy tells us (p. 204) that 'the scientific natura list[has] reason to refrain from criticizing mathem atica l methods'. Astrology,she says, 'is subject to scientific correction in a way th at pure mathem aticsis not'. Implicit in these comments is a rejection of the Quinean view thatno statementnot even a mathematical oneis immune to revision as wetry to adjust the whole web of belief to the tribunal of experience. Maddy,by contrast, has conferred on mathematical statements an extraordinaryimmunity to revision, even when (as in the example of her footnote) wewould want to say that the whole community of mathematicians had takenleave of its collective senses.Such is the price of buying into Quineanism without its unrelenting,16 Fn. 9, p. 198. Maddy (personal correspondence) assures me that certain other offbeatma them atical ideas were slips of th e pen. In footnote 16 of p. 27 of the hardback edition,we are told that we can '[erect] an equilateral right triangle on the unit length of a line';and on p. 197 she tells us 'I want to prove that P is false, so I'll assume not-P andderive a con tradiction .' Bu t with what justice, on her own accou nt, would Maddy beable to claim that she was in error here, if she suddenly found herself surrounded bymathematicians unanimously agreeing on such things?


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33 0 CRITICAL STUDIES / BOOK REVIEW Snaturalizing holism. For Quine, it is the evidential holism in our theoryof nature that truly naturalizes other areas of thought, such as mathemat-ics. Natural science as a whole has to be understood in its own terms. Tobe anti-holistic, and separate mathematics off from science as a whole, asMaddy does, is to divert the springs of naturalism at their very source. Andto apply the Quinean dictum about science as a whole to a self-containedmathematics shorn thus of its connections with the theory of nature is tolapse into two intimately related kinds of original sin: a priorism in math-ematics; and the accompanying belief that mathematics will be immune torevision in the light of any human sensory experience.

Section III begins with a statement of M addy 's naturalism : '[I]f ourphilosophical account of mathematics comes into conflict with successfulmathematical practice, it is the philosophy that must give' (p. 161). But,it may be asked, is there such a thing as a philosophically neutral criterionof successful mathematical practice? How could one identify such practice(especially as successful) before engaging in any kind of philosophical re-flection on its role within the wider body of hum an thought? Why is thisuncritical quietism about the norms of a particular practice being dignifiedwith the label 'naturalism' ? 17Consider someone who is indisputably a natu ralis t, in that she is an athe-ist and materialist, committed to the methodological precepts of naturalscience in accounting for the workings of the perceivable world. Such a na t-uralist could believe that analysis of the meanings of logico-mathematicalexpressions is best pursued by inquiring after certain central patterns ofcorrect inferential use of those expressions. Our na tura list could believethat one outcome of such analysis is a reflective equilibrium in which notall of classical logic and mathematics is sacrosanct and beyond philosophi-cal criticism. Thus our naturalist, whose naturalism is genuine, might notlike to see the term 'naturalism' hijacked for what is really only quietismabout the canons of classical logic and mathematics.

2.3. Extrinsic versus Intrinsic JustificationsRecall that the leading idea behind the distinction between extrinsic andintrinsic justifications was to allow for the possibility that some new ax-1 7 In personal correspondence, Maddy claims 'there is nothing in my mathematicalnatura lism t ha t insists on (or is quietistic ab ou t) classical logic and m athem atics. Itleaves room for any and all arguments against these classical forms and for whatever elseyou like, as long as those argum ents are m athem atical, n ot philosophical.' And the re'sth e rub. The philosophical, meaning-theoretic argume nts of the D um m ettian anti-realistagain st classical logic and in favor of intuitionistic logic are hereby ruled o ut. Th e resultwill almost certainly be quietistic, since classical reasoning is so deeply entrenched inm athe m atics. Why forbid an external fulcrum of philosophical detac hm ent from thepractice, and the force of non-mathematical (such as meaning-theoretic) considerations,when seeking proper leverage on normative issues such as correct choice of underlyinglogic?


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PHILOSOPHIA MATHEMATICA 33 1ioms of set theory might admit only of extrinsic justifications, rather thanforce themselves upon us as true when we confine ourselves to reflecting ontheir conceptual ingredients. Intrinsic justifications are supposed to exploitonly the conceptual ingredients in the claims being justified; while extrinsicjustifications are supposed to advert to the consequences of those claims.But are the classes of intrinsic and extrinsic justifications really disjoint?An inferential-meaning theorist18 might wonder why Maddy calls extrin-sic (and by implication, non-intrinsic) those justifications that advert toderivable consequences. In so far as entailments help constitute meanings,considerations of what follows from what are at the heart of a properlyconducted meaning analysis. Thus what Maddy is inclined to call an ex-trinsic justification could be claimed to be nothing more than an exercise inconceptual grasp. Why should not deductive explorations be on an equalfooting with reflective intuition? In general, why should not any path toreflective equilibrium in the foundations of mathematics be claimed to bewholly concept-driven?

It transpires that each axiom (or axiom scheme) of ZFC can lay claimto an intrinsic justification, even though Maddy does not point this outby way of summ ary of her own discussion. Nor does she seem to noticethat the only argument (which she apparently endorses) against the Anti-Foundation Axiom19 is one that, by her lights, counts as intrinsic (p. 61).She appears to be m ore concerned to stress the use of extrinsic justificationswhenever they appear in the literature.Maddy's contrast between extrinsic and intrinsic justifications can beapplied to clarify her discussion of the axiom of constructibility. In herfinal chapter, 'A natu ralist's case against V L', Maddy does not happento m ention extrinsic or intrinsic justifications . Her criterion of restrictive-ness, however, involves definitions adverting to various consequences of thetheories under consideration. Among such consequences, for exam ple, are'everything has property '; 'everything below some inaccessible rank hasproperty '; 'if a set has property then so do all its members'; and 'thereare non- isomorphism types'. Thus the considerations marshalled in hercase against V = L are extrinsic, in her sense of this term , even though shedoes not explicitly draw attention to this fact.The case study may therefore be construed as a plausibility argum ent forthe claim that at least one axiom candidate can be decided upon (in thiscase, negatively) by bringing important extrinsic considerations to bear.Admittedly th is is only a case study. Given, however, tha t there would

1 8 In The Taming of The True (Oxford: Clarendo n P ress, 1997) I seek to provide aninferential meaning theory for the main logico-mathematical operators. Such an accountfocuses on introduction and elimination rules for expression-forming operators.1 9 This axiom has been used by Aczel and others in place of Foundation for their workin situation semantics.


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332 CRITICAL STUDIES / BOOK REVIEWSappear to be a sufficiently strong conceptual case against V = L, one mightwonder how such a case study (however adeptly pursued, and howeverstrong and subtle the extrinsic considerations for the same conclusion mightbe) can conduce to a naturalistic view about higher set theory. What thenatura list is offering us might be interesting; but the a priori foundationalistneeds something more before being moved to regard it as more than anafterthought, or codicil to the real conceptual work that already settlesthe status of the axiom to his or her satisfaction. The metaphilosophicalthesis of the naturalist needs to be something as strong as 'There is at leastone axiom candidate whose acceptance or rejection is motivated solely byextrinsic considerations.' Godel himself ventured the conjecture that thismight one day prove to be true; but offered no concrete example of such anaxiom. Maddy takes a careful look at one important axiom (V = L) butthe results of her study do not bear out Godel's surmise.2.4. BivalenceGodel's realism is very much a Platonic affair (concerning the existence ofabstract objects) and offers no argument for the move from the existence ofthose objects to the claim tha t all mathematical statements about them aredeterminately truth-valued. Let us call this latter claim Bivalence. Nowit is well-known that the intuitionist does not accept Bivalence. There isa defensible brand of intuitionism that nevertheless takes the objects ofmathematics to be mind-independent, and the provision of proofs to be arule-governed activity, not susceptible to the creative whims or idiosyncra-cies of individual mathematicians. That is to say, mathematical thought isstill constrained by 'what is out there', and the individual mathematicianis not free simply to 'make things up' as one goes along.

The brand of intuitionism in question was expressed as follows by CrispinWright:20... someone could hold both that it is correct to think of the natural num-bers as genuine objects... and that there are decisive objections to the real-ist's way of thinking about the truth or falsity of statements concerning suchobjects [SJomeone might be persuaded of the realitythe Selbststeindig-keit-of the natural numbers but. .. reject the realist conception of the meaningof statements about them.

The 'realist conception' at issue is that Bivalence holds.Elsewhere I have emphasized and developed further the essential pointherethat ontological realism can be combined with semantic anti-real-ism.21 Such an 'ontologically realist' intuitionist would be in full agreement2 0 Cf. C. Wright, Frege's Conception of Numbers as Objects. Aberdeen: AberdeenUniversity Press, 1983, pp. xviii-xx.2 1 Cf. N. Tennant, Anti-Realism and Logic. Oxford: Clarendon Press, 1987, in ch. 2,'Scientific v. semantic realism', pp. 7-12.


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PHILOSOPH IA MATHEMATICA 33 3with Godel's sentiment, quoted by Maddy: 'the objects and theorems ofmathematics are as objective and independent of our free choice and ourcreative acts as is the physical world.'22 Note how Godel here talks oftheorems, not of truth values of statements in general. He does not commithimself to anything as classically full-blooded as, say, 'the objects, and adeterminate truth-value for each and every statement, of mathematics areas objective and independent of our free choice and our creative acts as isthe physical world'. Yet it is this stronger reading that would be neededby anyone looking to Godelian realism for a route to the conviction thatCH has a determinate truth-value, possibly independent of our means ofcoming to know what that truth-value is.

As the reader will be able to check, every quo tation th at Maddy adducesfrom Godel's writings (with one exceptionsee the next footnote) expressesno more than what an 'objective intuitionist' would readily concede.The fundamental (and required) justificatory move is never made to de-terminacy of truth-value, whether for CH or across the board.23 Interest-ingly, the very paragraph that Maddy quotes from Godel in order to showhow 'it is the mathematical considerations, not the philosophical ones, thatare decisive', seems to embody decisively philosophical insights into thesource of mathematical truth and the insufficiency of ontological realism tosecure realism about truth-value. The paragraph in question isHowever, the question of the objective existence of the objects of mathemati-cal intuition... is not decisive for the problem under discussion here [i.e., themeaningfulness of the continuum problem]. The mere psychological fact of theexistence of an intuition which is sufficiently clear to produce the axioms ofset theory and an open series of extensions of them suffices to give meaningto the question of the truth or falsity of propositions like Cantor's continuumhypothesis.24

2 2 K . G o d e l , ' S o m e b a s i c t h e o r e m s o n t h e f o u n d a t i o n s o f m a t h e m a t i c s a n d t h e i r i m p l i c a -t i o n s ' (T he Gib bs L ec tu re , Bro wn U nive r s i ty , 1951) , i n S. Fe fe rman e t a i ., eds . , CollectedWorks. Vol . I I I . Oxfo rd U nive rs i ty Press , 1995, p p . 304 -32 3, a t p . 312, n . 17 . )2 3 In pe r so na l co r r e sp ond enc e , M ad dy d raw s a t t e n t io n to he r quo ta t i on (on p . 89 ) f romG o d e l : ' . . . if t h e mea n in gs o f t h e p r im i t ive t e r m s o f s e t t he o ry a s exp la ined on p ag e 262a n d foo tno te 14 a re acce p ted a s sou nd , i t f ol lows th a t t h e s e t - t h eo re t i ca l con cep t s andt h e o r e m s desc r ib e som e we l l -de t e rmined r ea l it y , i n wh ich Ca n to r ' s con jec tu re m us t b ee i t h e r t r u e o r f a ls e .' ( ' W h a t i s C a n t o r ' s c o n t i n u u m p r o b l e m ? ' ( 1 9 6 4 ) , i n S . F e fe r m a n eta/ . , eds. , Collected Works. Vol . I I . Oxfo rd: Oxford Un ivers i ty Pre ss , 1990, p p . 25 4-2 70 ;a t p . 260 . ) She com m ent s 'T h i s c l ea r ly mak es t h e move to de t e rm ina cy o f t r u t h va luefo r C H . ' B u t I m a i n t a i n t h a t i t d o e s n o t . R a t h e r , it s i m p l y states d e t e r m i n a c y ; i t d o e sn o t infer i t f rom c la im s of exis tenc e . A 'we l l -de te rm ined rea l i ty ' i s by def in i t ion one inwhich e v e r y p r o p o s i t i o n h a s a d e t e r m i n a t e t r u t h - v a l u e a fortiori one in wh ich CH hasa d e t e r m i n a t e t r u t h - v a l u e . W h a t i s s t il l m i s s in g i s a n a r g u m e n t f ro m t h e m e r e o b j e c t i v ee x i s t e n c e o f m a t h e m a t i c a l o b j e c t s , a n d p e r h a p s t h e o b t a i n i n g o f c e rt a i n p r i m i t i v e r e la -t i o n s a m o n g t h e m , t o t h e d e t e r m i n a c y of t r u t h - v a l u e o f every proposition about thoseobjects, r ega rd l e s s o f t hose p ropos i t i ons ' l og i ca l (quan t i f i ca t iona l ) complex i ty .2 4 ' W h a t i s C a n t o r ' s c o n t i n u u m p r o b l e m ? ' , he. cit . , p. 268.


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33 4 CRITICAL STUDIES / BOOK REVIEW SThus Godel, it would appear, grounds the meaningfulness of CH in th e fac-ulty of mathematical intuition, which is relied on to produce appropriateaxioms to settle the matter. Those intuitions might be only partly 'of ob-jects'; they might concern also the workings of our mathematical concepts,and the meanings of our mathem atical vocabulary. The brute indepen-dent existence of those m athematical objects, could it somehow be secured,would not go far enough to settle the continuum hypothesis. Somethingmore is needed: the contact of the mind, through its faculty of intuition,with th a t realm of objects. Such contact must issue in linguistically ex-pressed axioms, and they in turn will settle the question of the truth-valueof CH by means of appropriate proofs. One is tempted to ask: Where isthe 'naturalism' in all of this?2.5. UnificationThe unification of mathematics that is afforded by set theory amounts, onM addy's account, to the following: (i) set theory aspires to an ontologyso rich that every conceivable mathematical object or structure could beassigned at least one surrogate within that ontology; (ii) every statement (f>of classical mathematics can be translated (modulo appropriate definitionsof the primitives involved in ) into a set-theoretic statement (f>; and (iii)for every proof of a classical mathematical theorem (f> there exists a set-theoretic proof of


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PHILOSOPHIA MATHEMATICA 33 5could be homologously accommodated by the methods of proof availablein the foundational theory (in this case, set theory). One could imposethis sort of adequacy requirement, and as far as one can tell, set theoryactually meets it. This would enable one to stress further the significance ofset-theoretic foundations. On M addy 's construal, a piece of mathem aticalreasoning in some synthetic branch of classical mathematics, from a setof axioms A to a theorem , needs only to be matched by some proof orother, from the axioms of ZFC, of the set-theoretic translation e of (j>.But this is overly modest. Why not insist further that one should be ableto take the 'synthetic' proof of (f> from A, and 'homologously unfold it' intothe sought set-theoretic proof of


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336 CRITICAL STUDIES / BOOK REVIEWSof cognitive significance.27

The second suggestion is that on the mathematical side one should havethe intellectual courage to admit the stark and obvious fact that the disci-pline is a wholly a priori one, whose standards of commitment are intuitive,logical and conceptual. Even if spacetime is grainy, so what? As far asthe abstract mathematical continuum is concerned, the truth-value of thecontinuum hypothesis, if it is somehow fixed, will be so quite independentlyof what mathematics is needed for the empirical description of our contin-gent world of concrete objects. It is no wonder at all that set theoristswould not care whether Feynman's successors opt for non-continua to de-scribe spacetime. It is genuinely irrelevant, both conceptually and logically,to mathematical propositions and how we properly come to establish theirtruth-values.2.7. Zermelo's 'Pragmatism'Maddy contends that the ontology that results from the formation of ranksin the cumulative hierarchy of sets is, in effect, that of the cumulative theoryof types. The impression of seamless confluence, or conceptual convergence,between Fregean and Cantorian developments is underwritten further byher (unargued) claim that Zermelo's pioneering investigations of 190828were in pursuit of a purely pragmatic endthat of 'selecting a simple,efficient, and powerful set of axioms from among the jumble of controversialand mutually exclusive set theoretic principles being debated at the time'(p. 19).

Maddy does not remark, however, on the fact that the confluence oftype-theoretic and set-theoretic thinking is more of a transferencefromthe linguistic (i.e., the stratification of type theory) to the ontological (i.e.,the ranks of the cumulative hierarchy of sets). It could also be objected thatshe makes Zermelo out to be much more of a methodological naturalist thanhe might have been. In his 1908 paper Zermelo wrote (at reprint pp. 210-211)

The further, more philosophical, question about the origin of these principlesand the extent to which they are valid will not be discussed here. I have not yetbeen able to prove rigorously that my axioms are consistent But I hope tohave done at least some useful spadework hereby for subsequent investigationsin such deeper problems.

It is clear that Zermelo saw his task as that of formulating certain set-theoretic principles as formal axioms and then demonstrating their power2 7 Cf. The Taming of The True. Op. cit., chapter 11, 'Cognitive significance regained' ,pp. 355-402.2 8 E. Zermelo, 'Investigations in the foundations of set theory, I', reprinted in J. vanHeijenoort, From Frege to Godel: A Sourcebook in Mathematical Logic. Cambridge,Mass.: Harvard University Press, 1967, pp. 200-215.


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PHILOSOPHIA MATHEMATICA 33 7by deriving various important theorems. As Gregory Moore argues in hisbook on the Axiom of Choice,29 Zermelo's primary motivation in formu-lating the axiom was to formalize his informal proof of the well-orderingtheorem, so as to defend th e informal proof against various critics. Zer-melo's paper reveals very little of whatever sort of thinking might other-wise have resulted in his specific choice of axioms. His words quoted aboveamount to a 'limitation of space' apology. They invite the inference thathis method of discovery (of suitable axioms) might have had just as muchto do with philosophical introspection and analysis as it might have hadto do with logical manipulations within a body of sought theorems uponwhich a certain deductive organization had to be imposed. But even if thesole inspiration for Zermelo's choice of axioms lay in certain 'pragmatic'insights as to what would logically afford what, there is the further ques-tion as to how one explains the fact that in our informal reasoning aboutsets as mathematical objects we happen to proceed in such a way as is bestelucidated by that particular choice of first principles. Why do we choosethe starting points that we do?2.8. The Evolution of the Notion of a Mathematical Function.V L was formulated only after Bernays's statement of combinatorial-ism. One is led to wonder whether, had Godel's definition of L come muchsooner after Zermelo's formulation of AC, matters might have turned outdifferently in the annals of mathem atics. Th e trouble with appealing tohistory in seeking extrinsic justifications is th at one can lose sight of what-ever a priori or conceptual insights and reasons might have been drivingthe evolution of the mathematical thought that one chronicles.Maddy's own approach to this branch of intellectual history seems to bepremissed on the view that, if thinking has evolved this way, then theremust have been some good, justifying reason for it to have done so. Bu tto concede even that much is to allow there to be a deeper, a priori justifi-cation underlying what purports to be a merely pragmatic one (appealingto the history of mathem atical practice). One could, however, go evenfurther than Maddy's implicit view, and claim that clearer intuitions anda sharper grasp of abstract concepts will eventually get the upper handin mathem atics, because of its own internal norms. Un derstand it as thekind of a priori enterprise that it is, and one thereby has the essentials ofan explanation of why mathematicians' thinking about functions, say, hastaken the various turns that Maddy chronicles. Perhaps Combinatorialism,allowing both indefinable and impredicatively defined functions to exist, isa sort of conceptual attractor in the phase space of mathematical thought.2 9 G. Moore, Zermelo's Axiom of Cho ice: its Origins, Developm ent, and Influence. NewYork: Springer Verlag, 1982, chs. 2-3; esp. pp. 158-159.


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33 8 CRITICAL STUDIES / BOOK REVIEWSAny community of rational creatures engaged in mathematical investiga-tions might have to end up where we are now, with the Combinatorialconception firmly entrenched. Their starting points might be different, andthe history of their thou ght might be subjected to different pe rturba tion sof deviance (p redicativism, definabilism, finitism, and so on); bu t, perhaps ,as the debates are worked out, and the dust settles, it is always the combi-natorial conception that must shine through.

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