Mathematics and the World 683and so on in place. Indeed, Sober's account cannot rule out any cleverly formulatedsceptical hypothesis. Furthermore, Sober is reluctant to appeal to simplicity orparsimony a s non-observational signs of truth, and so such sceptical problems aretaken to be scientifically insoluble. This is one important way in which contrastiveempiricism departs from standard scientific realism (and, arguably, standard sci-entific methodology). Although according to contrastive empiricism \"science attempts to solve dis-crimination problems\" [Sober, 1993, p. 391 and the burden of solving these prob-lems is placed firmly on the observational data, there is no restriction to hypothe-ses about observables, as in van Fraassen's constructive empiricism (emphasis inoriginal) : Contrastive empiricism differs from constructive empiricism in that the former does not limit science to the task of assigning truth values to hypotheses that are strictly about observables. What the hypotheses are about is irrelevant; what matters is that the competing hypothe- ses make different claims about what we can observe. Put elliptically, the difference between the two empiricisms is that constructive empiri- cism focuses on propositions, whereas contrastive empiricism focuses on problems. The former position says that science can assign truth values only to propositions of a particular sort; the latter says that science can solve problems only when they have a particular character. [Sober, 1993, p. 411 Much more could be said about contrastive empiricism, but we have seen enoughto motivate Sober's objection to the indispensability argument. Sober's main objection is that if mathematics is confirmed along with our bestempirical hypotheses, there must be mathematics-free competitors (or at leastalternative mathematical theories as competitors): Formulating the indispensability argument in the format specified by the Likelihood Principle shows how unrealistic that argument is. For example, do we really have alternative hypotheses t o the hypotheses of arithmetic? If we could make sense of such alternatives, could they be said t o confer probabilities on observations that differ from the probabilities entailed by the propositions of arithmetic themselves? I suggest that both these questions deserve negative answers. [Sober, 1993, pp. 45-46] It is important to be clear about what Sober is claiming. He is not claimingthat indispensability arguments are fatally flawed. He is not unfriendly to thegeneral idea of ontological commitment to the indispensable entities of our bestscientific theories. He simply denies that \"a mathematical statement inherits theobservational support that accrues to the empirically successful scientific theoriesin which it occurs\" [Sober, 1993, p. 531. This is enough, though, to place him atodds with the Quine-Putnam version of the indispensability argument.
684 Mark Colyvan In reply t o this objection, I wish to first point out that there are alternativesto number theory. F'rege showed us how t o express most numerical statementsrequired by empirical science without recourse to quantifying over numbers.52Furthermore, depending on how much analysis you think Hartry Field has suc-cessfully nominalised, there are alternatives to that also. (At the very least he hassuggested that there are nominalist alternatives to differential calculus.)53 I take the crux of Sober's objection then to be the second of his two questions,and I agree with him here that the answer to this question deserves a negativeanswer. I don't think that Field's version of Newtonian mechanics and standardNewtonian mechanics would confer different probabilities on any observationaldata, but so much the worse for contrastive empiricism. The question of whichis the better theory will be decided on the grounds of simplicity, elegance, and soon - grounds explicitly ruled out by contrastive empiricism. Supporters of theindispensability argument do not propose to settle all discrimination problems bypurely empirical means, so it should come as no surprise t o find that they run intotrouble when forced into the straight-jacket of contrastive empiricism. You might be inclined to think that since a mathematised theory such as Newto-nian mechanics and Field's nominalist counterpart have the same empirical conse-quences, it can't be said that the mathematics receives empirical support. Accord-ing to this view, the mathematised version is preferred on the a priori grounds ofsimplicity, elegance and so on, not on empirical grounds. In reply to this, I simplypoint out that there is nothing special about the mathematical content of theo-ries in this respect. As I've already mentioned, we prefer standard evolutionarytheory and geology over Gosse's version of creationism and we do so for the sameapparently a priori reasons. It would be a very odd view, however, that deniedevolutionary theory and geology received empirical support. Surely the right thingto say here is that evolutionary theory and geology receive both empirical supportand support from a priori considerations. I'm inclined to say the same for themathematical cases.54 Another objection to the whole contrastive empiricism approach t o theory choiceis raised by Geoffrey Hellman and considered by Sober [1993]. The objection isthat often a theory is preferred over alternatives, not because it makes certain(correct) predictions that the other theories assign very low probabilities to, butrather, because it is the only theory t o address such phenomena at all.55 Sober 52For example, 'There are two F s ' or 'the number of the F s is two' is written as: (3x)(3y)(((FxA Fy) A x # y) A (Vz)(Fz > (z = x V z = y))). 5 3 ~ h i sis only considering sensible alternatives. There are, presumably, many rather badtheories that d o without mathematics. Perhaps most pseudosciences such as astrology and palmreading d o without all but the most rudimentary mathematics. 5 4 ~ its perhaps best t o speak of the 'scientific justification of theories,' where this includesempirical support and support from a priori considerations. This is clearly the sort of supportthat our best scientific theories receive, so we see that Sober's concentration on purely empiricalsupport might be thought t o skew the whole debate. Thanks t o Bernard Linsky for a usefuldiscussion on this point. 55Hellman [1999]gives the example of relativistic physics correctly predicting the relationshipbetween total energy and relativistic mass. In pre-relativistic physics no such relationship is even
Mathematics and the World 685points out that the relevance of this t o the question of the indispensability ofmathematics is that presumably \"stronger mathematical assumptions facilitateempirical predictions that cannot be obtained from weaker mathematics\" [Sober,1993, p. 52].56 If this objection stands, then the central thesis of contrastiveempiricism is thrown into conflict with actual scientific practice. For a naturalistthis almost amounts to serious trouble. Indeed, Sober admits that \"[ilf this pointwere correct, it would provide a quite general refutation of contrastive empiricism\"[Sober, 1993, p. 521. I believe that Hellman's point is correct, but first let's considerSober's reply.Sober's first point is that when scientists are faced with a theory with no relevantcompetitors, they can contrast the theory in question with its own negation. Heconsiders the example of Newtonian physics correctly predicting the return ofHalley7scomet, something on which other theories were completely silent. Soberclaims, however, that \"alternatives to Newtonian theory can be constructed fromNewtonian laws themselves\" [Sober, 1993, p. 521. For example, Newton's law ofuniversal gravitation:57 Gm1mz r2 F = -competes with:and Gm1m2 r4 F = -and many others. There is no doubt that such alternatives can be constructedand contrasted with Newtonian theory, but surely we are not interested in whatscientists could do; we are interested in actual scientific practice.Sober takes this a step further and claims that this is standard scientific practicefor such cases [Sober, 1993, pp. 52-53]. He offers no evidence in support of this lastclaim, and without a thorough investigation of the history of relevant episodes inthe history of science it seems rather implausible. Were scientists really interestedin debating whether it should be r2, r3,or r4 in the law of universal gravitation?58The relevant debate would have surely been over retaining the existing theoryor adopting Newtonian theory. At the very least, Sober needs t o present someevidence to suggest that scientists are inclined to contrast a theory with its ownnegation when nothing better is on offer.postulated, indeed, questions about such a relationship cannot even be posed. 56For example, [Hellman, 19921 argues t h a t the weaker constructivist mathematics, such as thatof the intuitionists, will not allow t h e empirical predictions facilitated by the stronger methodsof standard analysis. 57Here F is t h e size of t h e gravitational force exerted on two particles of mass ml and mzseparated by a distance r , and G is t h e gravitational constant. 5 8 ~ otto mention r2.00000000o1r r1~99999999(9A.lthough it seems that cases such as these wereconsidered when the problems with Mercury's perihelion came t o light [Roseveare, 19831, theywere considered only in order t o save the essentials of Newtonian theory, which, by that stage,was already a highly confirmed theory.)
686 Mark Colyvan In his second point in response to Hellman's objection, Sober considers thepossibility of \"strong\" mathematics allowing empirical predictions that cannot bereplicated using weaker mathematics. Sober points out that strong mathematicsalso allows the formulation of theories that make false predictions, and that thisis ignored by the indispensability argument (emphasis in original): It is a striking fact that mathematics allows us to construct theories that make t r u e predictions and that we could not construct such pre- dictively successful theories without mathematics. It is less often no- ticed that mathematics allows us to construct theories that make false predictions and that we could not construct such predictively unsuc- cessful theories without mathematics. If the authority of mathematics depended on its empirical track record, both these patterns should mat- ter to us. The fact that we do not doubt the mathematical parts of empirically unsuccessful theories is something we should not forget. Empirical testing does not allow one to ignore the bad news and listen only to the good. [Sober, 1993, p. 531 It may be useful at this point to spell out the dialectic thus far. Hellman'spoint is that contrastive empiricism does not account for cases where a theory ispreferred because it makes predictions that no other theory is able to address oneway or another. If this is accepted, then contrastive empiricism as a representationof how theory choice is achieved seems at best only part of the story, and atworst completely misguided. Furthermore, if it is reasonable t o prefer some theorybecause it correctly predicts new phenomena that other theories are silent on, thenit is reasonable to accept strong mathematical hypotheses, since theories employingstrong mathematics are able t o predict just such phenomena. I take it that Sober's reply runs like this: Contrastive empiricism can accom-modate the Hellman examples of scientific theories that address new phenomena.This is done by contrasting such theories with their negations. Thus, a generalundermining of contrastive empiricism is avoided. This reply, however, seems toallow that strong mathematics is confirmed, because such theories correctly pre-dict empirical phenomena that theories employing weaker mathematics cannotaddress. So the cost of saving contrastive empiricism from the Hellman objectionis that Sober's original point against the empirical confirmation of mathematicsnow fails. Here is where the second part of Sober's reply is called upon. Thepoint here is simply that the case of strong mathematics is different from thatof bold new physical theories in that strong mathematics can also facilitate falsepredictions that competing theories are silent on. Thus, the mathematics cannotshare the credit for the successful empirical predictions, since it won't share theblame for unsuccessful empirical predictions. There are a couple of interesting issues raised by this rejoinder. First, therejoinder is in the context of a defence of contrastive empiricism and yet it isnot an argument for that thesis. Nor is it an argument depending on contrastiveempiricism. It seems like a new objection to the use of indispensability arguments
Mathematics and the World 687to gain conclusions about mathematical entities. What is more, this objectionappears t o be independent of contrastive empiricism and as such is the moresubstantial part of his objection to the indispensability argument. So far I've suggested that Sober is wrong about scientists contrasting bold newtheories with their negations. At the very least Sober needs to give some evidenceto support his claim that scientists do this.5g Indeed, it would be interesting t oinvestigate some candidate cases in detail to shed some light on this issue, butfortunately this is not necessary for our purposes, since even if I grant Sober hisfirst point (that contrastive empiricism can accommodate Hellman's examples ofbold new theories), the second part of Sober's reply also runs into trouble. Sober claims, in effect, that mathematical theories cannot enjoy the confirma-tion received by theories that make bold new true predictions because the math-ematics is not disconfirmed when it is employed by a theory that makes bold newfalse predictions. I've already noted that this point is stated independently ofcontrastive empiricism. Indeed, I take this to be a separate worry about the indis-pensability argument as applied to mathematical entities. Also bear in mind thatit is important t o Sober's case that there be a difference between mathematicalhypotheses and non-mathematical hypotheses in this respect. This last claim, though, is false. Many non-mathematical hypotheses can beemployed by false theories and not be held responsible for the disconfirmation.Hypotheses about electrons (notoriously) have been employed by many false theo-ries, and yet we are unwilling to blame electrons for the lack of empirical supportfor the theories in question. Astrologers refer to the orbits of the planets in grosslyfalse theories about human behaviour, and yet we do not blame the planets forthe lack of empirical support for astrology. It is surely one of the important tasksof scientists to decide which parts of a disconfirmed theory are in need of revisionand which are not. Sober would have us throw out the baby with the bathwater,it seems. Hellman [1999] points out that this partial asymmetry between confirmationand disconfirmation is a consequence of confimational holism. Whelf a theory isconfirmed, the whole theory is confirmed. When it is disconfirmed, it is rarely thefault of every part of the theory, and so the guilty part is to be found and dispensedwith. It's analogous to a sensitive computer program. If the program delivers thecorrect results, then every part of the program is believed t o be correct. However,if the program is not working, it is often because of only one small error. The jobof the computer programmer (in part) is to seek out the faulty part of the programand correct it. Furthermore, the programmer will resort to wholesale changes tothe program only if no other solution presents itself. This is especially evidentwhen one part of the program is working. In such a case the programmer seeksto make a small local change in the defective part of the program. Changing theprogramming language, for instance, is not such a change. 591t is worth pointing out that he must provide evidence that contrasting theories with theirnegations is a general phenomenon. Even if there are only one or two counterexamples, con-trastive empiricism is in trouble.
688 Mark Colyvan Now if we return to Sober's charge that mathematics cannot enjoy the creditfor confirmation of a theory if it cannot share the blame for disconfirmation, wesee that blaming mathematics for the failure of some theory is never going tobe a small local change, due to the simple fact that mathematics is used almosteverywhere in science. What is more, much of that science is working perfectly well.Blaming the mathematics is like a programmer blaming the computer language.And, similarly, claiming that mathematics cannot share the credit is like claimingthat the computer language cannot share the credit for the successful program.In some cases it may well be the fault of the mathematics or the programminglanguage, but it is not a good strategy to start with changes t o these. Furthermore, we see that mathematics is not alone in this respect. Many clearlyempirical hypotheses share this feature of apparent immunity from blame for dis-confirmation. Michael Resnik points out that conservation principles seem immunefrom liability for much the same reasons as mathematics. He goes even furtherto express doubts about whether such principles could be tested at all in the con-trastive empiricist framework and \"yet we do not want to be forced to deny themempirical content or to hold that the general theories containing them have notbeen tested experimentally\" [Resnik, 1995, p. 1681. Another untestable empiricalhypothesis is the hypothesis that space-time is continuous rather than discrete anddense. To sum up, then. I agree with Sober that there is a problem of reconcilingcontrastive empiricism with the indispensability argument, but for the most partthis is because of general problems with the former. In particular, contrastiveempiricism fails to give an adequate account of a theory being adopted because itcorrectly predicts phenomena that its competitors are unable t o speak to at all. Iagree with Hellman here that this looks like the kind of role mathematics plays intheory selection. Strong mathematics allows the formulation of theories that ad-dress phenomena on which other theories are completely silent. Sober's rejoinderis that mathematical hypotheses are different from other scientific hypotheses, inthat mathematical hypotheses allow false predictions just as readily as true ones,and yet mathematics remains blameless for the former. This rejoinder is in effecta new argument against the indispensability argument applied to mathematicalentities and, what is more, it is independent of the framework of contrastive em-piricism. Nevertheless, the rejoinder faces problems of its own. First, it seems tomisrepresent the type of holism at issue -the holism at issue has an asymmetrybetween confirmation and disconfirmation built into it. Second, it seems clearthat mathematics is not alone in its apparent immunity from blame in cases ofdisconfirmation. I should mention Sober's claim that the main point of his objection can be sep-arated to some extent from the contrastive empiricist epistemology. He does not,however, seem to have the residual worry that I discussed in mind. He is concernedthat you might think that contrastive empiricism can't be right because it ignoresnonempirical criteria such as simplicity. He then suggests that \"even proponentsof such nonempirical criteria should be able to agree that empirical considerations
Mathematics and the World 689must be mediated by likelihoods\" [Sober, 1993, p. 551. Sober is suggesting that atthe very least we discriminate between empirical hypotheses by appeal to likeli-hoods and that his objection goes through granting only this.'jO But why should weaccept that all discriminations between empirical hypotheses must be mediated bylikelihoods? After all, we have already seen that we cannot discriminate betweenthe hypothesis that space-time is continuous and the hypothesis that space-timeis discrete and dense on empirical grounds and yet these are surely both empiricalhypotheses. So Sober's objections t o the indispensability argument fail becausethey depend crucially on accepting the Likelihood Principle as the only arbiteron empirical matters. The independent residual point I identified fails because itdoesn't take account of the asymmetric character of confirmational holism. 6 THE UNREASONABLE EFFECTIVENESS OF MATHEMATICSIn this section I'll turn my attention to another important issue that arises in thecontext of philosophy of applied mathematics. This is the issue of how mathematicsmanages t o be so \"unreasonably\" suited to the business of science. The physicistEugene Wigner once remarked that [tlhe miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. [ ~ i ~ n e19r6,0, p. 141Steven Weinberg is another physicist who finds the applicability of mathematicspuzzling: It is very strange that mathematicians are led by their sense of math- ematical beauty to develop formal structures that physicists only later find useful, even where the mathematician had no such goal in mind. [ . .. ] Physicists generally find the ability of mathematicians toantici- pate the mathematics needed in the theories of physics quite uncanny. It is as if Neil Armstrong in 1969 when he first set foot on the surface of the moon had found in the lunar dust the footsteps of Jules Verne. [Weinberg, 1993, p. 1251And it's not only physicists who have waxed lyrical on the applicability of math-ematics. Charles Darwin remarked that: I have deeply regretted that I did not proceed far enough at least to understand something of the great leading principles of mathematics, for men thus endowed seem to have an extra sense. [Darwin, 19581 In each case the author seems to be suggesting something mysterious - evenmiraculous - about the applicability of mathematics. Indeed, this puzzle, which Gosince,according t o the indispensability argument, mathematics is empirical, and yet we can-not discriminate between mathematical and non-mathematical theories by appeal t o likelihoods.
690 Mark ColyvanWigner calls 'the unreasonable effectiveness of mathematics', is often remarkedupon by physicists and applied mathematicians61 but receives surprisingly littleattention in the philosophical l i t e r a t ~ r e .I~t i~s hard to say why this puzzle hasnot caught the imagination of the philosophical community. It is not becauseit's unknown in philosophical circles. On the contrary, it is very well known; itjust does not get discussed. This lack of philosophical attention, I believe, is due(in part) t o the fact that the way the problem is typically articulated seems topresuppose a formalist philosophy of m a t h e r n a t i ~ s . ~ ~ Given the decline of formalism as a credible philosophy of mathematics in thelatter half of the twentieth century, and given the rise of anti-realist philosophiesof mathematics that pay great respect to the applicability of mathematics in thephysical sciences (such as Hartry Field's fictionalism [Field, 1980]), it is worthreconsidering Wigner's puzzle to see to what extent, if any, it relies on a partic-ular philosophy of mathematics. The central task of this paper is to argue thatalthough Wigner set the puzzle up in language that suggested an anti-realist phi-losophy of mathematics, it appears that the puzzle is independent of any particularphilosophy of mathematics. At least, a version of the puzzle can be posed for twoof the most influential, contemporary philosophies of mathematics: one realist, theother anti-realist.6.1 What is the Puzzle?Mark Steiner is one of the few philosophers to take interest in Wigner7spuzzle[Steiner, 1989; Steiner, 1995; Steiner, 19981. Steiner has quite rightly suggestedthat Wigner's \"puzzle\" is in fact a whole family of puzzles that are not distin-guished by Wigner; it depends on what you mean by 'applicability' when talkingof the applications of mathematics. Steiner claims that it is important to distin-guish the different senses of 'applicability' because some of the associated puz-zles are easily solved while others are not. For example,.Steiner argues that theproblem of the (semantic) applicability of mathematical theorems64 was explained 6 1 ~ oerxample: Paul Davies [1992, pp. 146601; Freeman Dyson [1964];Richard Feynman [1965,p. 1711; R.W. Hamming [1980]; Steven Weinberg [1986] and many others in [Mickens, 19901. 6 2 ~ h o u g ht,hat may be starting t o change. See [Azzouni, 2000; Wilson, 20001 for some rela-tively recent discussion of this topic. 63Saunders Mac Lane, for example, explicitly takes the puzzle t o be a puzzle for formalistphilosophies of mathematics [Mac Lane, 19901. Others have taken the problem t o be a problemfor anti-realist philosophies of mathematics generally. See, for example, [ ~ a v i e s1,992, pp. 1 4 ~ 6 0 1and [penrose, 1989, pp. 556-71. One exception here is Philip Kitcher [Kitcher, 1984, pp. 104-51who presents it as a problem for Platonism. I will discuss, what is in essence, Kitcher's problemin section 6.2. 6 4 ~ h iis the problem of explaining the validity of mathematical reasoning in both pure andapplied contexts - t o explain, for instance, why the truth of (i) there are 11Lennon-McCartney+songs on the Beatles' 1966 album Revolver, (ii) there are 3 non-Lennon-McCartney songs onthat same album, and (iii) 11 3 = 14, implies that there are 14 songs on Revolver. (Theproblem is that in (i) and (ii) '11' and '3' seem t o act as names of predicates and yet in (iii) '11'and '3' apparently act as names of objects. What we require is a constant interpretation of t h emathematical vocabulary across such contexts.
Mathematics and the World 691adequately by Frege (19951. There is, according t o Steiner, however, a problemwhich Frege did not address. This is the problem of explaining the appropriatenessof mathematical concepts for the description of the physical world. Of particularinterest here are cases where the mathematics seems to be playing a crucial role inmaking predictions. Moreover, Steiner has argued for his own version of Wigner'sthesis. According to Steiner, the puzzle is not simply the extraordinary appropri-ateness of mathematics for the formulation of physical theories, but concerns therole mathematics plays in the very discovery of those theories. In particular, thisrequires an explanation that is in keeping with the methodology of mathematics- a methodology that does not seem to be guided at every turn by the needs ofphysics. The problem is epistemic: why is mathematics, which is developed primarilywith aesthetic considerations in mind, so crucial in both the discovery and thestatement of our best physical theories? Put this way the problem may seem likeone aspect of a more general problem in the philosophy of science -the problemof justifying the appeal t o aesthetic considerations such as simplicity, elegance, andso on. This is not the case though. Scientists and philosophers of science invokeaesthetic considerations to help decide between two theories that are empiricallyequivalent. Aesthetics play a much more puzzling role in the WignerISteinerproblem. Here aesthetic considerations are largely responsible for the developmentof mathematical theories. These, in turn, (as I will illustrate shortly) play acrucial role in the discovery of our best scientific theories. In particular, novelempirical phenomena are discovered via mathematical analogy. In short, aestheticconsiderations are not just being invoked to decide between empirically equivalenttheories; they seem to be an integral part of the process of scientific discovery. Steiner's statement of the puzzle is clearer and more compelling, so when Ispeak of Wigner's puzzle, I will have Steiner's version in mind. I will thus con-centrate on cases where the mathematics seems to be playing an active role inthe discovery of the correct theory -not just in providing the framework for thestatement of the theory. I'll illustrate this puzzle by presenting one rather classiccase and refer the interested reader to Steiner's article [1989] and book [1998]forfurther examples.65 In the case I'll consider here, we see how Maxwell's equationspredicted electromagnetic radiation. Maxwell found that the accepted laws for electromagnetic phenomena priort o about 1864, namely Gauss's law for electricity, Gauss's law for magnetism,Faraday's law, and Ampitre's law, jointly contravened the conservation of electriccharge. Maxwell thus modified Ampitre's law to include a displacement current,which was not an electric current in the usual sense (a so-called conduction cur-rent), but a rate of change (with respect to time) of an electric field. This modi-fication was made on the basis of formal mathematical analogy, not on the basis 65Steiner distinguishes between two quite different, but equally puzzling, ways in which math-ematics has facilitated the discovery of physical theories: Pythagorean analogy and formalistanalogy. Although this distinction is of considerable interest, it has little bearing on the mainthesis of this section, so I will set it aside. See [Steiner, 1998, pp. 2-11] for details.
692 Mark Colyvanof empirical evidence.66 The analogy was with Newtonian gravitational theory'sconservation of mass principle. The modified Ampbre law states that the curlof a magnetic field is proportiond to the sum of the conduction current and thedisplacement current. More specifically:Here E and B are the electric and magnetic field vectors respectively, J is thecurrent density, and c is the speed of light in a vacuum.67 When this law (knownas the Maxwell-Ampbre law) replaces the original Ampkre law in the above set ofequations, they are known as Maxwell's equations and they provide a wonderfulunity to the subject of electromagnetism. The interesting part of this story for the purposes of the present discussion,though, is that Maxwell's equations were formulated on the assumption that thecharges in question moved with a constant velocity, and yet such was Maxwell'sfaith in the equations, he assumed that they would hold for any arbitrary system ofelectric fields, currents, and magnetic fields. In particular, he assumed they wouldhold for charges with accelerated motion and for systems with zero conductioncurrent. An unexpected consequence of Maxwell's equations followed in this moregeneral setting: a changing magnetic field would produce a changing electric fieldand vice versa. Again from the equations, Maxwell found that the result of theinteractions between these changing fields on one another is a wave of electricand magnetic fields that can propagate through a vacuum. He thus predicted thephenomenon of electromagnetic radiation. Furthermore, he showed that the speedof propagation of this radiation is the speed of light. This was the first evidencethat light was an electromagnetic phenomenon.68 It seems that these predictions (which were eventually confirmed experimentallyby Heinrich Hertz in 1888) can be largely attributed t o the mathematics, since thepredictions were being made for circumstances beyond .the assumptions of theequations' formulation. Moreover, the formulation of the crucial equation (theMaxwell-Ampbre law) for these predictions was based on formal mathematicalanalogy. Cases such as this do seem puzzling, at least when presented a certainway. The question on which I wish to focus is whether the puzzlement is anartifact of the presentation (because some particular philosophy of mathematics is 66~ndeedt,here was very little (if any) empirical evidence a t t h e time for the displacementcurrent. 6 7 ~ h feirst term on t h e right of equation 3 is the conduction current and the second on theright is the displacement current. 6 8 ~ c t u a l l ythe story is a little more complicated than this. Maxwell originally had a me-chanical model of electromagnetism in which the displacement current was a physical effect. (For the details of the relevant history, see [Chalmers, 19731, [ ~ u n t 1,9711 and [Siegel, 19911.)This, however, does not change the fact that there was little (if any) empirical evidence for thedisplacement current and the reasoning that led t o the prediction of electromagnetic radiationwent beyond the assumptions on which either the equations or the mechanical model were based [Steiner, 1998, pp. 77-81.
Mathematics and the World 693explicitly or implicitly invoked), or whether these cases are puzzling simpliciter. Iwill argue that it is the latter.6.2 Is the Puzzle Due to a Particular Philosophy of Mathematics?Applicability has long been the Achilles' heel of anti-realist accounts of mathemat-ics. For example, if you believe that mathematics is some kind of formal game -as Hilbert did -then you need t o explain why mathematical theories are neededto such an extent in our descriptions of the world. After all, other games, likechess, do not find themselves in such demand. Or if you think that mathematicsis a series of conditionals - '2+2=4' is short for 'If the Peano-Dedekind axiomshold then 2+2=4' - the same challenge stands. In Wigner's article he seems t o be taking a distinctly anti-realist point of view(my italics): [Mlathematics is the science of skillful operations with concepts and rules invented just for that purpose. [Wigner, 1960, p. 21Others, such as Reuben Hersh, also adopt anti-realist language when stating theproblem (again, my italics)?' There is no way to deny the obvious fact that arithmetic was invented without any special regard for science, including physics; and that it turned out (unexpectedly) to be needed by every physicist. [Hersh, 1990, p. 671Some, such as Paul Davies [1992, pp. 140-601 and Roger Penrose [1989, pp. 556-71,have suggested that the unreasonable effectiveness of mathematics in the physicalsciences is evidence for realism about mathematics. That is, there is only a puzzlehere if you think we invent mathematics and then find that this inventibn is neededto describe the physical world. Things aren't that simple though. There arecontemporary anti-realist philosophies of mathematics that pay a great deal ofattention to applications, and it is not clear that these suffer the same difficultiesthat formalism faces. Furthermore, it is not clear that realist philosophies ofmathematics are home free. In what follows I will argue that there are puzzles forboth realist and anti-realist philosophies of mathematics with regard t o accountingfor the unreasonable effectiveness of mathematics. I will consider two philosophies of mathematics that we've already encoun-tered: one influential realist philosophy of mathematics -Quinean realism [Quine,1981b] and-and one equally influential anti-realist position -Hartry Field's fic-tionalism [Field, 19801. Both of these philosophical positions are motivated by,and pay careful attention to, the role mathematics plays in physical theories. It 69Alsorecall Weinberg's reference to Jules Verne in the passage I quoted earlier in this sectionand Steiner's remark (quoted at the beginning of this chapter) about the mathematician beingmore like an artist than an explorer.
694 Mark Colyvanis rather telling, then, that each suffers similar problems accounting for Wigner'spuzzle. Recall that the Quinean realist is committed to realism about mathematicalentities because of the indispensable role such entities play in our best scientifictheories. Now, granted this, it might be thought that the Quinean realist has aresponse to Wigner. The Quinean could follow the lead of scientific realists suchas J.J.C. Smart who put pressure on anti-realists by exposing their inability toexplain the applications of electron theory, say. It's no miracle, claim scientificrealists, that electron theory is remarkably effective in describing all sorts of phys-ical phenomena such as lightning, electromagnetism, the generation of x-rays inRoentgen tubes and so on. Why is it no miracle? Because electrons exist and areat least partially causally responsible for the phenomena in question. Furthermore,it's no surprise that electron theory is able to play an active role in novel discover-ies such as superconductors. Again this is explained by the existence of electronsand their causal powers. There is, however, a puzzle here for the anti-realist. AsSmart points out: Is it not odd that the phenomena of the world should be such as to make a purely instrumental theory true? On the other hand, if we interpret a theory in a realist way, then we have no need for such a cosmic coincidence: it is not surprising that galvanometers and cloud chambers behave in the sort of way they do, for if there really are electrons, etc., this is just what we should expect. A lot of surprising facts no longer seem surprising. [Smart, 1963, p. 391 There is an important disanalogy, however, between the case of electrons andthe case of sets. Electrons have causal powers -they can bring about changes inthe world. Mathematical entities such as sets are usually taken to be causally idle-they are Platonic in the sense that they do not exist in space-time nor do theyhave causal powers. So how is it that the positing of such Platonic entities reducesmystery?70 Colin Cheyne and Charles Pigden [1996] have suggested that in lightof this, the Quinean is committed to causally active mathematical entities. WhileI dispute the cogency of Cheyne's and Pigden's argument (see [Colyvan, 1998b]), Iagree that there is a puzzle here. The puzzle is this: on Quine's view, mathematicsis seen to be part of a true description of the world because of the indispensablerole mathematics plays in physical theories, but the Quinean account gives us noindication as to why mathematics is indispensable to physical science. That is,Quine does not explain why mathematics is required in the formulation of ourbest physical theories and, even more importantly, he does not explain why math-ematics is so often required for the discovery of these theories. Indispensability issimply taken as brute fact. It might be tempting to reply, on behalf of Quine, that mathematics is indispens-able because it's true. This, however, will not do. After all, there are presumably - 'OA few people have pointed t o this problem in Quine's position (see [Balaguer, 1998, pp. 110- 11, [Field, 1998, p. 4001, [Kitcher, 1984, pp. 104-51 and [Shapiro, 1997, p. 461).
Mathematics and the World 695many truths that are not indispensable to our best scientific theories. What is re-quired is an account of why mathematical truths, in particular, are indispensablet o science. Moreover, we require an account of why mathematical methods which,as Steiner points out [1995,p. 1541, are closer t o those of the artist's than those ofthe explorer's, are reliable means of finding the mathematics that science requires.It is these issues, lying at the heart of the WignerISteiner puzzle, that Quine doesnot address. The above statement of the problem for Quine can easily be extended to anyrealist philosophy of mathematics that takes mathematical entities to be causallyinert. This suggests that one way to solve the puzzle in question is to followCheyne's and Pigden's suggestion and posit causally active mathematical entities( a la early Maddy [1990a]or Bigelow [1988]). Now such physicalist strategies mayor may not solve Wigner's puzzle.71 But it is not my concern here t o decide whichrealist philosophies fall foul of Wigner's puzzle and which do not. My concern is t odemonstrate that realist philosophies of mathematics do not, in general, escape theproblem. In particular, I have shown that Quine's influential realist philosophy ofmathematics, at least if taken t o be about abstract objects, succumbs t o Wigner'spuzzle. Now consider Field's [1980]philosophy of mathematics in light of this problem.Recall that Field responds to Quine's argument by claiming that mathematics is,in fact, dispensable to our best physical theories. He adopts a fictional accountof mathematics in which all the usually accepted sentences of mathematics areliterally false, but true-in-the-story of accepted mathematics. There is no doubtthat Field's partial nominalisation of Newtonian gravitational theory sheds con-siderable light on the role of mathematics in that theory, and perhaps on appliedmathematics more generally. But it is interesting to note that despite Field'scareful attention to the applications of mathematics, he leaves himself open toWigner's puzzle. Field explains why we can use mathematics in physical theories- because mathematics is conservative. He also explains why mathematics oftenfinds its way into physical theories -because mathematics simplifies calculationsand the statement of these theories. What he fails to provide is an account ofwhy mathematics leads to simpler theories and simpler calculations. Moreover,Field gives us no reason to expect that mathematics will play an active role in theprediction of novel phenomena.72 If I'm correct that facilitating novel scientific predictions (via mathematicalanalogy) is at least partly why we consider mathematics indispensable to science,then Field has not fully accounted for the indispensability of mathematics until hehas provided an account of the active role mathematics plays in scientific discovery.So although Field did not set out t o provide a solution to this particular problemof applicability (i.e. the Steiner/Wigner problem), it seems that, nevertheless, heis obliged to. (Indeed, this was the basis of my criticism of Field in [Colyvan, 711t'~not clear to me that they do. 7 2 ~discuss this matter in more detail in [Colyvan, 1999b] and in [Colyvan, 2001a, chap. 41.John Burgess raises similar issues in [1983].
696 Mark Colyvan1999bI.) On the other hand, if this shortcoming of his project is seen (as I'm nowsuggesting) as part of the more general problem of applicability -a problem thatQuine too faces - Field's obligation in this regard is not so pressing. In short,it's a problem for everyone. Now the fact that Field does not provide a solution to Wigner's puzzle does notmean that he cannot do so. But whether he can provide a solution or not, thepuzzle needs to be discussed and that is all I am arguing for here. Still, let me putto rest one obvious response Field may be tempted He might appeal to thestructural similarities between the empirical domain under consideration and themathematical domain used to model it, t o explain the applicability of the latter.So, for example, the applicability of real analysis to flat space-time is explainedby the structural similarities between R4 (with the Minkowski metric) and flatspace-time. There is no denying that this is right, but this response does not givean account of why mathematics leads to novel predictions and facilitates simplertheories and calculations. Appealing to structural similarities between the two do-mains does not explain, for example, why mathematics played such a crucial rolein the prediction of electromagnetic radiation. Presumably certain mathematicalstructures in Maxwell's theory (which predict electromagnetic radiation) are simi-lar t o the various physical systems in which electromagnetic radiation is produced(and it would seem that there are no such structural similarities with the pre-Maxwell theory). But then Wigner's puzzle is t o explain the role mathematicalanalogy played in the development of Maxwell's theory. The fact that Maxwell'stheory is structurally similar to the physical system in question is simply irrelevantto this problem.To sum up this section then. I agree with Steiner that the applicability ofmathematics presents a general problem. What I hope to have shown is that theproblem exists for at least two major contemporary positions in the philosophy ofmathematics. Moreover, the two positions I discuss - Field's and Quine's - Itake to be the two that are the most sensitive to the applications of mathematicsin the physical sciences. The fact that these two influential positions do not seemt o be able to explain Wigner's puzzle, clearly does not mean that every philosophyof mathematics suffers the same fate. It does show, however, that Wigner's puzzleis not merely a difficulty for unfashionable formalist theories of mathematics.While the problems I've discussed in this paper for both Quine and Field are notnew, they can now be seen in a new light. Previously each problem was seen as adifficulty for the particular account in question (in the context of the realismlanti-realism debate). That is, whenever these problems were discussed (and I includemyself here [Colyvan, 1999b]),they were presented as reasons to reject one accountin favour of another. If what I'm suggesting now is correct, that is the wrong wayof looking at it. There are striking similarities between the problem that Burgess 7 3 ~ a r kBalaguer seems t o have something like this response in mind when he says that \"I d onot think it would be very difficult t o solve this general problem of applicability [of mathematics]\"[Balaguer, 1998, p. 1441. It should also be mentioned that if this response were successful, itwould also be available t o realist philosophies of mathematics.
Mathematics and the World 697and I have pointed out for Field and the problem that Balaguer and others havepointed out for Quine. I claim that these problems are best seen as manifestationsof the unreasonable effectivenessof mathematics. Moreover, these difficulties seemt o cut across the realismlanti-realism debate and thus deserve careful attentionfrom contemporary philosophers of all stripes - realists and anti-realists alike. 7 APPLIED MATHEMATICS: THE PHILOSOPHICAL LESSONS AND FUTURE DIRECTIONSLet me close with some general comments about the philosophy of applied math-ematics. Although much of the recent work on the applications of mathematicshas had a fairly narrow focus on the indispensability argument, there is much ofvalue to immerge from this work that transcends such a focus. For a start, bothMaddy's [1997] and Field's [1980; 19891 critique of the indispensability argument(and the subsequent discussion of these two) suggests that we need to pay carefulattention to the details of the way mathematics is used in various physical appli-cations; it is not sufficient t o simply note that mathematics is used in science. Weneed to consider whether the mathematics is merely providing a convenient modelof the system in question or is it doing more? For example, is the mathematicscontributing to the explanatory power of the theory? Is it helping to unify thetheory in question? What attitudes do scientists in the area in question take to-wards the mathematics they use? Indeed, what attitude do these same scientiststake towards the theory itself? All in all, the applications of mathematics to phys-ical science is a much more nuanced affair than perhaps was appreciated by someearlier writers. Also we should not forget that mathematics finds many and varied applicationsin areas of science other than physics. Although most discussions of applied math-ematics begin and end with physics, careful attention to other branches of sciencesuch as biology and chemistry are of considerable interest here. It is not clearthat mathematics plays the same kind of role in, say, the biological sciences7* Forinstance, it may be that the biological sciences are less satisfied with unification-style explanations (if they are explanations) - which mathematics is rather wellsuited to. Instead, there is some reason to suggest that biology is more interestedin causal explanations [Colyvan and Ginzburg, 20031. Furthermore, in the biologi-cal sciences there is the issue of abuse of mathematics and overmathernatici~in~.~~One rarely encounters such issues in physics, yet mathematical models in ecol-ogy, for instance, are treated with considerable suspicion by many ecologists. Oneconcern is that the mathematics is obscuring ecological detail or invoking simpli-fications that are not well supported by ecological theory. This again suggests 74See,for instance, [Ginzburg and Colyvan, 2004; May, 20041 for recent discussions of the roleof mathematics in the biological sciences. 75Some mathematical ecologists are even charged with \"physics envy\". (This is the \"crime\" ofinvoking sophisticated mathematical methods, that would be appropriate in physics but allegedlyinappropriate in ecology.)
698 Mark Colyvanthe possibility of a significant difference between the use of mathematics in thebiological sciences and its use in the physical sciences. Finally, the Wigner problem of the applicability always lurks in the background.It simply won't do to pass it off as a problem for Platonism, or formalism or anyother particular philosophy of mathematics. As I've argued above, it is a problemfor everyone. Moreover, a solution to this problem is likely to involve both carefulattention to the details of the scientific and mathematical theories in question,and also careful attention to the history of science. For instance, it might turn outthat my example in the previous section of Maxwell's positing of the displacementcurrent (and the consequent prediction of electro-magnetic radiation) rides rough-shod over historical or mathematical details - details that once brought to light,help us to understand why mathematics is apparently so unreasonably effectivehere. I should also add that Steiner's (19981 recent work on this topic suggeststhat the many and varied ways that mathematics is utilised in scientific theoriesmakes the prospects of a unified solution t o the problem of applied mathematicslook dim. It may be that we'll need to look a t the problem case by case.76 This brief overview of some of the issues in the philosophy of applied mathe-matics should give those interested in the topic considerable joy. There are somefascinating issues for future work - issues that cut deep into other fascinatingissues in theories of explanation, the nature of scientific analogies, philosophy ofbiology and, of course, the history of science and mathematics. And no doubtthere are many other issues I haven't addressed here that lead in equally interest-ing direction^.^^ BIBLIOGRAPHY [Armstrong, 19781 D.M. Armstrong, Universals and Scientific Realism, Cambridge University Press, Cambridge (1978). 7 6 ~ h e ries also t h e issue o f inconsistent mathematics and its applications. Inconsistent theo-ries, such as t h e early calculus, were remarkably successful in applications. This suggests thatconsistency is not as important as many classically-minded logicians and philosophers o f math-ematics would have us believe. There are substantial issues here in need o f further exploration.See [Mortensen,19951 for a nice treatment o f non-trivial inconsistent mathematical theories. Seealso Mortensen's chapter in this volume. 77Some o f t h e material in this chapter has been previously published. I gratefully acknowl-edge Oxford University Press for permission t o reproduce material from [Colyvan, 2001a], t h eeditors o f The Stanford Encyclopedia of Philosophy for permission t o reproduce material fromm y [Colyvan, 20041, Philosophia Mathematica for permission t o reproduce sections o f [Coly-van, 1998a],Mznd for permission t o reproduce a section o f [Colyvan, 20021, and Kluwer Aca-demic Publishers for permission t o reproduce sections o f [Colyvan, 1999a; Colyvan, 199913;Colyvan, 2001b] in Erkenntnis, Philosophical Studies, and Synthese respectively. T h e relevantcopyrights remain with t h e publishers in question. I'd also like t o thank t h e Center for Philos-ophy o f Science at t h e University o f Pittsburgh where I held a Visiting Research Fellowship int h e winter term o f 2004 and where some o f the work on this chapter was carried out. Thanksespecially t o m y colleagues there John Norton and Alan Chalmers for many interesting discus-sions. Work on this chapter was funded b y an Australian Research Council Discovery Grant (grant number DP0209896).
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Mathematics and the World 701add^, 19921 P. Maddy, Indispensability and practice, Journal of Philosophy, 89 (1992), 275-289.[Maddy, 19941 P. Maddy, T a h n g naturalism seriously, Logic, Methodology and Philosophy ofScience IX, D. Prawitz, B. Skyrms and D. Westersthl, eds., Elsevier, Amsterdam (1994),383-407.[Maddy, 19951 P. Maddy, Naturalism and ontology, Philosophia Mathematica (3), 3 (3) (1995), 248-270.ma add^, 19971 P. Maddy, Naturalism in Mathematics, Clarendon Press, Oxford (1997).[Maddy, 1998aI P. Maddy, Naturalizing mathematical methodology, Philosophy of MathematicsToday, M. Schirn, ed., Clarendon Press, Oxford (1998), 175-193.[Malament, 19821 D. Malament, Review of Field's Science Without Numbers, Journal of Phi-l o s o ~ h< v, .79 (,1982,),. 523-534. A[ ~ a20~041, R.M. May, Uses and abuses of mathematics i n biology, Science, 303 (6 February2004), 790-793.[Melia, 20001 J. Melia, Weaseling away the indispensability argument, Mind, 109 (2000), 455-479.[Mickens, 19901 R.E. Mickens, ed., Mathematics and Science, World Scientific Press, Singapore (1990).si or tens en, 19951 C. Mortensen, Inconsistent Mathematics, Kluwer, Dordrecht (1995).[Papineau, 19931 D. Papineau, Philosophical Naturalism, Blackwell Publishers, Oxford (1993).[Penrose, 19891 R. Penrose, T h e Emperor's New Mind: Concerning Computers, Minds and theLaws of Physics, Vintage Press, London (1990).[Putnam, 1971/1979] H. Putnam, Philosophy of logic, reprinted in Mathematics Matter andMethod: Philosophical Papers Vol. I , second edition, Cambridge University Press, Cambridge(1979) (first published in 1971), 323-357.[Putnam, 19791 H. Putnam, W h a t is mathematical truth?, Mathematics Matter and Method:Philosophical Papers Vol. 1 , second edition, Cambridge University Press, Cambridge (1979),60-78.[Quine, 1936/1983] W.V. Quine, R w t h by convention, reprinted in Philosophy of MathematicsSelected Readings, second edition, P. Benacerraf and H. Putnam, eds., Cambridge UniversityPress, Cambridge (1983) (first published in 1936), 329-354.[Quine, 1948/1980] W.V. Quine, O n what there i s , reprinted in From a Logical Point of View,second edition, Harvard University Press, Cambridge MA (1980) (first published 1948), 1-19.[Quine, 1951/1980] W.V. Quine, T w o dogmas of empiricism, reprinted in From a Logical Pointof View, second edition. Harvard University Press, Cambridge MA, 1980 (first published in1951), 2G46.[Quine, 1953/1976] W.V. Quine, O n mental entities, reprinted in T h e Ways of Paradox andOther Essays, revised edition, Harvard University Press, Cambridge, MA (1976) (first pub-lished in 1953), 221-227.[Quine, 19601 W.V. Quine, Word and Object, Massachusetts Institute of Technology Press andJohn Wiley and Sons, New York (1960).[Quine, 1963/1983] W.V. Quine, Carnap and logical truth, reprinted in Philosophy of Math-ematics Selected Readings, second edition, P. Benacerraf and H. Putnam, eds., CambridgeUniversity Press, Cambridge (1983) (first published in 1963), 355-376.[Quine, 1981al W.V. Quine, Five milestones of empiricism, Theories and Things, Harvard Uni-versity Press, Cambridge, MA (1981), 67-72.[Quine, 1981b] W.V. Quine, Success and limits of mathematization, Theories and Things, Har-vard University Press, Cambridge, MA (1981), 148-155.[Quine, 19861 W.V. Quine, Reply t o Charles Parsons, The Philosophy of W.V. Quine, L. Hahnand P. Schilpp, eds., Open Court, La Salle ILL (1986), 396-403.[Quine, 19921 W.V. Quine, Pursuit of %th, revised edition, Harvard University Press, Cam-bridge MA (1992).[Quine, 19951 W.V. Quine, From Stimulus t o Science, Harvard University Press, CambridgeMA (1995).[Resnik, 19831 M.D. Resnik, Review of Hartry Field's Science Without Numbers, No&, 17(1983), 514-519.[ ~ e s n i k1, 985al M.D. Resnik, How nominalist is Hartry Field's nominalism?, PhilosophicalStudies, 47 (1985), 163-181.
702 Mark Colyvan[Resnik, 1985bl M.D. Resnik, Ontology and logic: remarks o n Hartry Field's anti-platonist philosophy of mathematics, History and Philosophy of Logic, 6 (1985), 191-209.[Resnik, 19951 M.D. Resnik, Scientific vs. mathematical realism: the indispensability argument, Philosophia Mathematica (3),3 (2) (1995), 166-174.[ ~ e s n i k1,9971 M.D. Resnik, Mathematics as a Science of Patterns, Clarendon Press, Oxford (1997).[Roseveare, 19831 N.T. Roseveare, Mercury's Perihelion from Le V d e r t o Einstein, Clarendon Press, Oxford (1983).[Shapiro, 19831 S. Shapiro, Conservativeness and incompleteness, Journal of Philosophy, 80 (9) (1983), 521-531.[Shapiro, 19971 S. Shapiro, Philosophy of Mathematics: Structure and Ontology, Oxford Uni- versity Press, Oxford (1997).[Siegel, 19911 D.M. Siegel, Innovation i n Maxwell's Electromagnetic Theory, Cambridge Uni- versity Press, Cambridge (1991).[Smart, 19631 J.J.C. Smart, Philosophy and Scientific Realism, Routledge and Kegan Paul, London (1963).[Sober, 19931 E. Sober, Mathematics and indispensability, Philosophical Review, 102 (1) (1993), 3- 5- -57.[Steiner, 1989) M. Steiner, The application of mathematics t o natural science, Journal of Phi- losophy, 86 (9) (1989), 449-480. [Steiner, 19951 M. Steiner, T h e applicabilities of mathematics, Philosophia Mathematica (3), 3 (2) (1995), 129-156. [Steiner, 19981 M. Steiner, T h e Applicability of Mathematics as a Philosophical Problem, Har- vard University Press, Cambridge MA (1998). [Urquhart, 19901 A. Urquhart, The logic of physical theory Physicalism in Mathematics, A.D. Irvine, ed., Kluwer, Dordrecht (1990), 145-154. [van Fraassen, 19801 B.C. van Fraassen, The Scientzfic Image, Clarendon Press, Oxford (1980). [Weinberg, 19861 S. Weinberg, Lecture o n the applicability of mathematics, Notices of the Amer- ican Mathematical Society, 33 (1986), 725-728. [Weinberg, 19931 S. Weinberg, Dreams of a Final Theory, Vintage Press, London (1993). [Wigner, 19601 E.P. Wigner, The unreasonable effectiveness of mathematics in the natural sci- ences, Communications on Pure and Applied Mathematics, 13 (1960), 1-14. [Wilson, 20001 M. Wilson, T h e unreasonable uncooperativeness of mathematics i n the physical sciences, Monist, 83 (2000), 296-314.
INDEXa-field, 495 mathematical, 35-98 apartness relation, 325a posteriori knowledge, 3, 5, 6, 8, 157, Apostoli, P., x, 479, 481, 482, 484- 179-186, 199, 213-226 488a priori knowledge, 3, 5-7, 9, 18, 22, applicability of mathematics, 133 33 t o empirical science, 84-86 saying is believing, 18 approximation space, 479, 480Abelard, A., 234 Archimedes, 164aboutness Aristotelian realism (or Aristotelian- thick vs. thin, 46, 92 ism), viii, 103'absolute' rest, motion, simultaneity, Aristotle, 1-2, 53, 105, 131, 135, 138, 223 160-168,172,176,178, 192,absoluteness, 572-574, 576, 617 196, 203, 215-216, 225,239,abstract algebra, 112 356abstract objects, 94-98, 238 arithmetic, 239abstraction axioms, 470 Armstrong'sabstraction principle, 236 abstract objects, 664abstraction scheme, 463 naturalism, 664Ackermann function, 555, 559 Armstrong, D. M., 53, 132, 642Ackermann, W., 302, 537, 543, 556 Aronszajn tree, 435Aczel, P., 481 Australian school, 110additivity, 493, 496 axiom, 241agent-relative, 497 Axiom of Choice (AC), 314 405, 406aggregates, 256 Axiom of Constructibility, 431, 680aleatory, 497Aleksandrov, P., 418 Axiom of Convergence, 497algebraic theories, 353-354 Axiom of Dependent Choice, 324algorithm, 586 Axiom of Extensionality, 470, 476analysis, 214-216 Axiom of Foundation, 425analytic geometry, 241 Axiom of Independence, 499analytic sets, 419 Axiom of Infinity, 473analytic truths, 169-170 Axiom of Randomness, 498analyticity, 4, 5, 7, 8, 18, 20, 22, 24, Axiom of Reducibility, 336, 411 Axiom of Replacement, 423, 431 25, 33 axiomatic method, 2anti-foundation axioms, 466 axiomatic set theory, 279anti-platonism axiomatization, 119, 241 mathematical, 76-86 axioms, 242anti-realism, vii, 347 for arithmetic, 175, 178-179
Index for geometry, 171 boldness Frege-Hilbert dispute, 297, 298 of theories, 661Ayer, A. J., 44, 213Ayer-Hempel-Carnap, 49 Bolyai, J., 173Azzouni, J., 49, 85, 384, 676 Bonevac, D., ix, 352-353, 378 BonJour, L., 182, 184Baire Category Theorem, 413 Borel, E., 412, 413Baire property, 413 Bostock, D., viii, 245 boundedness, 538, 598Baire, R., 412, 413Balaguer, M., vii, 349, 367, 373-374, condition, 585,590,593,596,599 Bourbaki, 114, 132 381, 676 Bourbakism, 308Banach, S., 381 Boyer, C. B., 198Banach-Tarski Paradox, 415 Brady, R. T., 472, 635Bar Theorem, 326 bridges of Konigsberg, 111Bar-Hillel, Y., 479 Brittan, G., 263Barcan formula, 388 Brouwer, L. E. J., 39-40, 243, 320Basic Law V, 462, 464, 465 Brown, J., 243Bayes' theorem, 501 Burali-Forti Paradox, 410Bayesian, 141, 500 Burgess, J., 348, 350, 357, 367, 371,Bayesian conditionalisation, 501Bayesian net, 511 378, 389Bayesianism, x calculability, 535, 540, 555-577Beall, JC, 49 Campbell, K., 667Behmann, H., 543 Cantor's diagonal argument, 464belief, 345-349 Cantor's paradox, 18, 410belief function, 501 Cantor's theorem, vii, 402, 465Bell, J. L., 480, 481 Cantor, G., 14-16, 131, 198, 201, 246,Belnap, N., 641Benacerraf, P., 42, 61-64, 66, 112, 248, 300, 317, 359, 379, 381, 396, 461, 464, 478, 679 113, 157,199,206, 352-353, cardinal characteristics, 423 355, 365, 373, 377, 384-385 cardinal comparability, 403Bentham, J., 357-359 cardinal number, 244, 401Berkeley models, 384, 388 Carnap, R., 44, 148, 263, 350, 375,Bernays, P., 302, 424, 431, 536, 543, 668 552, 555, 563, 568, 575, 576 Cartesian dualism, 52Bernoulli, 504 Cartwright, N., 654Bernstein, F., 421 Cassirer, E., 231, 249, 250, 266, 267betting quotient, 501 Casullo, A., 184BHK interpretation, 329 categorical concept, 233Bigelow, J., 110, 642 categorical theories, 210-2 12Birkhoff, G., 480 Cauchy, A.-L., 114, 216, 364Bishop, E. A., 311, 332 causal inertness of abstract objects,Blamey, S., 485 52, 85, 93Boethius, 356 causal irrelevance principle, 513
Index 705Ceitin, G. S., 329 conservative extensions, 209-210 conservativeness, 366-373, 672, 671-ceteris paribus principles, 361-363chance, 499 673Chellas, B., 478 constraint graph, 525 constructed objects, 381Cheyne, C., 135, 676 constructible objects, 381, 383-385Chihara, C., 45-46, 80, 84, 159, 190, constructible universe, 431 construction, 232 191, 194-195,201, 212, 346, constructive empiricism, see empiri- 359, 669 cism, constructivechoice sequences, 318 constructive mathematics, 381-382,Church, A., 28, 538, 564, 568, 570- 685 572, 575, 576, 586, 593, 611, constructive proof, 311 622 constructivism, viii, 167, 200, 311 constructivist, 243Church's Thesis, 323, 537, 561, 564, contact with abstract objects, 51-54 569, 572, 573, 576, 586 contingency, 371Church-Turing Thesis, 314 of mathematics, 56-58, 93-94 continuity, 215-216, 242classical logic, 469 continuum, 131, 253, 260closed set logic, 634closure principle, 473, 475 hypothesis (CH), 17, 60, 68, 77- 78, 91, 249, 367, 399, 404,Cohen real, 442 679Cohen, P., 441coherence, 501 problem, 399, 433Coleman, E., 645 contrastive empiricism, seeempiricism,collective, 497Colyvan, M., xi, 75, 134, 371 contrastivecombinatory process, 538, 577, 580 contrivance, 359 control systems, 641Compactness Theorem, 428 convenience, 529Completeness Theorem, 428 conventionalism, 44, 45, 81compositionality, 376 Copi, I. M., 200comprehension, 473 countable additivity, 495 counterpart semantics, 484 axiom, 109, 471, 475 Cowan, T., 639 scheme, 463 Craig, W., 660 creative activity, 347computability, 535, 537, 539,576, 577, cross-entropy, 516 594, 610, 611, 617 cumulative hierarchy, 425 Curry's paradox, 635 theory, x Curry, H. B., 44computable function, 611 curvature of space, 173-175computation, 538 cylinder sets, 496computor, 584-587, 612, 618 Dalen, D. van, 318conceivability, 171-173, 217-218concept, 462conceptualism, 158conditional probability function, 494confirmation holism, 55-56conjectures, 142conjunction, 495
706 Indexdark matter, 654 Duhem, P., 667, 667, 668Darwin, C., 689 Dummett, M., 40, 167, 222, 389Davis, M., 547, 566, 568, 572, 573 Dunn, J. M., 635, 638De Finetti, 502 Dutch book, 501decidability, 535, 540-544 Easton, W., 443decision problem, 428, 536, 542, 543, Eculid, 211 549, 574 Edidin, A., 184 effective calculability, 570, 571, 575,Dedekind, R., 9, 12, 15, 26, 42, 178, 198,210, 215-216, 254,364, 587 396,408,412,535,537,545, effectively calculable function, 540 609 Einstein, A., 115, 119, 172, 234 Eklund, M., 350deductivism, 38, 45, 46, 78-80, 346- Eleatic principle, 134 348, 351, 353, 388 elegance, 661 elementary mathematics, 110definition, 175, 367 see analysisnege- eliminability, 367-368 Hilbert dispute, 297, 298 empirical, 504, 505 implicit, 297, 298 scrutability, 353, 365deflationary fictionalism, 349-350 strategy, 359Dehaene, S., 40 -based subjective probability, 506dependence of a variable, 277 empiricism, 104Descartes, R., 168,172,205, 240,246, constructive, 682 contrastive, 682-683 643 empiricists, 246descriptions, 368 Entscheidungsproblem, 539,582, 591,descriptive aid 609 mathematics, 86 epistemic, 529descriptive set theory, 414, 418 objectivity, 517 epistemological, 497Dever, J., 377Devlin, K., 680 argument against platonism, 50- 61diagram, 111, 139, 240dialethism, 631 epistemology, 136Dirac Delta Function, 640 equational calculus, 564-566, 573, 577,directed constraint graph, 525 611, 612discernibility of the disjoint, 486, 488 Equivocator, 516discovery vs. invention in mathemat- Erdijs, P., viii Erdos, P., 435 ics, 93 Erdmann, B., 39discrete, 467 Escher, M. C., 171 Esser, O., 473, 476disjunction, 496 Euclid, 1-2, 10-12, 19, 24, 112, 113, property, 316 138,164,171,173-175, 234,distributed computing, 599 239doctrine of the limitation of size, 478double extension set theory, 476, 477double set theory, 468, 476, 478du Bois-Reymond, D. P. G., 316dualism, 52duality, 644
Euclidean geometry, x Field, see Field, fictionalismEuler, L., 111, 144event space, 495 free-range, 363, 365, 373evolutionary theory, 682 Hermeneutic, 375-377ex contradictione quodlibet, 631 hermeneutic, 348exceptionalism, 346exchangeable, 502 instrumentalist, 358-360, 366existence mathematical, 35, 46-48, 76-81, mathematical - as consistency, 91-94, 98 298, 299 relative reflexive, 375-377 representational, 360-363, 366 non-spatiotemporal, 95-98 revolutionary, 348existence property, 316existential theories, 353 fictitious objects, 358-359experience, 159-160 Field, 495experiential equivalence, 350, 351experimental mathematics, 141 conservativeness, 671, 671-673explanation, 529 consistency, 672 critics of, 675-676 as unification, 661 entailment, 672 inference to the best, 654 fictionalism, 670 intrinsic, 671 indispensability, 653,669-670,679explanatory power, 661 motivation for nominalism, 670-extension of a concept, 465extension of an abstract set, 465 671extensionality, 471-473, 475 nominalisation, 671, 673-675 principle, 472external, 521-523 Platonistic methods, 671externalism (concerning knowledge), representation theorem, 675 Field, H., 46-47, 56, 57, 77, 78, 84, 182-185 92, 130, 158, 196, 202, 207-fabulous entities, 358-359 213, 345, 348, 353, 359, 363,facts 366-373, 378, 695 figuralism, 374-377 of the matter, 94-98 Fine, K., 478 physical, 85 finitary arithmetic, 239 purely nominalistic, 85 finitely additive, 496 purely platonisitc, 85Fan Theorem, 326 finitism, 303, 311, 337, 552, 557-559,Feferman, S., 201, 337, 443, 485 562Feigenbaum's bottleneck, 507Fermat's last theorem, 142, 652 finitist function, 557Fetzer, J., 125 finitist mathematics, 540,544-550, 562Feynman, R., 680 finitist proof, 558fictionalism, ix, 211-213, 226, 345- finitistically calculable functions, 553 Finsler, P., 319 389, 652 Fodor, J., 667 deflationary, 34!3350 forcing, 441 formal sciences, 123 formal system, 577, 610 formal theory, 538, 542, 548 formalism, ix, x, 36, 38, 44-45, 237
Index game, 44 Godel's first incompleteness theorem, 249, 306 metamathematical, 44Forrest, P., 642 Godel, K., viii, 26-27, 40, 41, 51, 52,Forti, M., 472 69, 157, 159, 191-193, 200,Fosen, G., 348 243,319, 564, 370-569, 572, 574,576,586,608-610,613-foundation, 426 618, 620, 622, 680foundationalism, 632 Gabbay, D., xiFraenkel, A., 421, 424, 479F'raenkel-Mostowski models, 421,443 Gaifman, 502 Galileo, 361Francis, G., 639 Gambling system, 498Franklin, J., viii game formalism, 45free-range fictionalism, 363, 365, 373 Gandy Machine, 538, 597, 599-601,freedom, 379 606Frege structure, 465, 481 Gandy's Thesis, 596Frege, G., 8-10, 14-17, 23-25, 39-41, Gandy, R., 537, 538, 572, 579, 584, 44, 45, 62, 76, 82, 83,89, 90, 109, 127, 129, 157, 161, 163, 586, 593-596, 608, 622 176-178, 188-189, 194-196, general recursive function, 561, 611 203-204, 207, 235,254, 294, general recursiveness, 575 410, 416, 462, 464, 481, 536, generic set, 442 548, 684 Gentzen, G., 307, 308 geometry, 104, 239, 367, 387, 639F'regean problem, 465, 467 Gilmore, P., 469-472, 478, 485 Goldbach's conjecture, 148frequency, 497 Goldman, A., 182 Goodstein, R. L., 339fruitfulness (of axioms), 192-193 Gosse, E., 682, 684full conception of the natural num- Gray, J., 265. Greek mathematics, 164-166,211-212 bers (FCNN), 63-68, 71-75 group, 132, 642full objectivity, 517 group theory, ix, 112full-blooded platonism (FBP),35,40- Hajnal, A., 438 41,49, 59-61, 68-75, 91-94, Hale, B., 52, 55, 57, 184, 236 98, 373-374 Hallett, M., 248 Halpern, 523function Hamel, G., 421 p recursive, 567 Hanf, W., 439 primitive, 537 Hankel, H., 293 calculable, 561 Hardy, G. H., viii effectively calculable, 588 Hart, W. D., vii finitist, 561 Hausdorff's paradox, 415 Hausdorff, F., 412, 414, 416, 417 finitistically calculable, 537, 561 Hawthorne, N., 346, 358 general recursive, 564, 565, 569, 572 primitive recursive, 555, 561 recursive, 537, 569 Turing computable, 579functionality, 636
Heath, T. L., 171 Husserl, E., 39 Hyper Frege, 475Heine, H. E., 293 hyper-continuous function, 485Hellman, E., 689 hyperuniverse, 468Hellman, G., 42, 45, 46, 79, 346, 378, idea, 168, 233 684 ideal, 233 ideal elements, 239Hempel, C., 44, 346 idealisation, 118, 162-163, 189-191,Henkin, L., 437Herbrand, J., 537, 538, 540, 543-544, 220, 360-363 idealism, 365 549, 551, 556, 559, 564, 566, idealist, 243 611 identity, 235, 237hereditarily finite sets, 192-196, 201 of indiscernibles, 235, 478hermeneutic fictionalism, 348, 375- if-thenism, see deductivism, 45 377 Ignorabimusstreit, 317 implication, 495Hersh, R., 40, 693 inaccessible numbers, 678, 678Heyting, A., 39, 40, 316, 327 incommensurability of the diagonal,Hilbert program, 637, 646Hilbert space, 481 113Hilbert, D., 16, 25, 41, 44, 45, 177, incompleteness, 556 Incompleteness Theorem, 428, 559, 211, 237, 239, 337, 367, 369, 610, 614 379, 381, 387, 535, 540, 544, Godel, K., 306 546-548, 551, 552,555,575, inconsistency, 363-365, 374 576, 608, 613, 615, 673-674 inconsistent mathematics, 631, 698 indescribable cardinals, 439 Grundlagen der Geometrie, 296 indeterminacy, 355-356, 373Hinnion, R., x, 468, 470-472, 474, indiscernibility, 466 of locations, 481 475 indispensability, 134, 197-202Hintikka, J., viii of mathematics to empirical sci-Hoare, C. A. R., 125Hodes, H., 375 ence, 84-86Hofstadter, D., 247 indispensability argumentHofweber, T., 347, 350 general, 653holism, 657, 667 pragmatic, 658-659 confirmational, 667 Quine-Putnam, 656 scientific, 654 moderate, see holism, semantic infant cognition, 137 semantic, 667 infinite, 232, 241, 245Holland, R. A., 263 infinitesimals, 29, 31, 32Holmes, M. R., 476, 477 infinity, 164-168, 190-191, 196-198,homeomorphism, 466homoiomerous, 109, 129 203-204,219,363,374,378,Honsell, F., 472 387Horgan, T., 367Howson, 510, 522Hume, D., 7, 25, 114, 168-169, 233, 235Hunter, 512
710 Index axiom, 473 Kleene's normal form theorem, 566,infintesimal nearness, 481 569, 572inner penumbra, 484insight, 138 Kleene, S. C., 324, 538, 566, 568, 576,instantiation, 274 582, 596 rules, 275 Kline, M., 198instrumentalism, 207-209 knowledge of mathematical objects,instrumentalist fictionalism, 358-360, 50-61 366 Kock, A., 481instrumentalist strategy, 359 Kolmogorov, A., 329, 592intended objects or structures, 67- Konig, D., 434 Kripke, S., 170, 378, 380, 381, 478 69, 71-75, 77 Kronecker, L., 314,319,535,544-548intensionality, 471 Kuratowski, K., 418, 421internal, 521-523 Kurepa tree, 435internal properties of mathematical Kurepa, R., 436 objects, 42-43 Loweinheim-Skolem Theorem, 425internalism (concerning knowledge), Lowenheim, L., 425, 542 Lakatos, I., 640, 668 182-185 languageintuition, 232, 243 definition of, 97 mathematical, 52-55 relativity, 508 (Godel's), 192-193 Laplace, 508intuitionism, x, 39-40, 167, 201, 311, lattice, 642 Law of extensions, 462, 463 633 Lebesgue measure, 413intuitionistic logic, 381-382, 389 Lebesgue, H., 412, 413, 679, 680invariance, 516 Leibniz, G. W., 172, 216, 234, 541-invention vs. discovery in mathemat- 542, 610 ics, 93 Lepore, E., 667Inwagen, P. van, 48 Levy collapse, 443 Levy, A., 438Jaynes, J., 148, 506 Lewis, D., 38, 52, 56-59, 132, 484,Juhl, C., 367 519, 520Kalderon, M., 347 Libert, T., x, 472, 474, 475Kamp, H., 379 Liebniz, G., 114Kanamori, A., x likelihood principle, 682 limitation of size doctrine, 465, 466,Kanda, A., x, 479, 481, 482, 484-488Kant, I., viii, 3-7, 135, 168-170, 176, 468 Lindenbaum, A., 421 213, 363-364, 374 Link, G., 346Kantianism, viii Linsky, B., 41, 70, 684Katz, J., 52, 56-59 Liouville, J., 398Keynes, J. M., 508, 513 Lobatchevsky, N. I., 173Kisielewicz, A., 476Kitcher, P., 37-38, 83-84, 158, 179- 191, 203, 225, 661Kladeron, M., 345
Indexlocal observations, 467 anti-platonism, 76-86locality condition, 538, 585, 590, 593, anti-realism, 35-98 fictionalism, 695 596, 599Locke, J., 39 intuition, 52-55logic, ix knowledge, vii, viii, x, xi, 50-61 physicalism, 36-38 alternative, 218-222 nature of, 213, 216-225 physics, 233 second order, 209 platonism, 40-44, 50-75logical, 504, 505logicism, viii, x, 40, 205, 271, 346, realism, 35-98 triviality, 635 347, 632 truth, 68logicists, 235Lowenheim, L., 544 mathematicslower approximation, 480 as descriptive aid, 86Luzin set, 418 as invention or discovery, 93Luzin, N., 418 in biology, 697Machover, M., 481 Maximum Entropy Principle, 505-510,Maddy's 512-515, 523 indispensability, 679 Maxwell, J. C., 691 mathematical fictions, 677-678 Mayberry, J., 244, 257 mathematical practice, 678-680 McCarty, C., viii problems with indispensability, 669, measurable cardinal, 427 measurement, 114, 133, 369 676 scientific fictions, 677 mechanical set theoretic realism, 653 computability, 610 V = L, 680Maddy, P., 37, 51, 53-54, 91, 158, procedure, 538, 574, 608, 611, 615, 617, 619 179,191-196, 203,225, 243, 352 process, 618Mahlo cardinals, 415, 439 Meinong, A., 37, 48-49, 81Mahlo, P., 415 Meinongianism, 37, 48-50, 81make-believe, 345, 375-377 Melia, J., 676Malament, D., 84, 367Malitz, R. J., 472 Mental, 497Mancosu, P., 198Manfredi, P. A., 184 Mental / Physical, 497manifold, 254, 272Marginal probability function, 494 mereology, 115Markov's Principle, 330 metamathematics, 282, 304Markov, A. A., 329 Meyer, R. K., 633, 635, 637Markovian constructivism, 311 Mill, J. S., viii, 37, 83, 87, 91, 158,marsupial constructions, 384-385mathematical 168-359 Minervan constructions, 383-384 minimum perceptibility, 481 Mirimanoff, D., 423 mixed, 521 modal fictionalism, 377 modal strategy, 359 modal structuralism, 346
Indexmodel theory, 242 nominalization of empirical science,models, 496 46, 84 of PFS, 484 non-deductive logic, 142Montague, R., 438 non-Euclidean geometry, ix, 234Moore's paradox, 658 non-spatiotemporal existence, 95-98Moore, A., 245 non-uniqueness objection to platon-Mortensen, C., x ism, 61-69Mostowski, A. M., 421, 437 non-uniqueness platonism (NUP), 67-multiple reductions objection to pla- 69, 73-75 tonism, 61-69 noncognitivism, 345 notions, 296nalve notion of set, 461 Nozik, R., 182naYve set theory, 463, 465, 468 null set, 194natural sciences, ix number, 113, 238naturalism, 657, 664 number theory, 462 numerals, 238 Quinean, see Quine, naturalism numerical ordinals, 214-215naturalized epistemology, 352 numerical quantifiers, 203-205, 209-naturalized platonism, 53-54NBG set theory, 205 210, 214necessary truth, 6 numerically definite comparisons, 206,necessity, 204, 362, 372-373 215-216 of mathematics, 56-58, 93-94negation, 495 object-platonism, 41-44neighbourhood, 467 objective, 497Nerlich, G., 251 Objective Bayesian net, 510, 512Neugebauer, O., 208 objective Bayesian semantics, 522Neumann, J. von, 423 objective Bayesianism, 501new colours, 171, 184-185 objective credal nets, 526new constructivism, 311 objectivity, 529new foundations, 472 obprogic, 524Newton, I., 114, 120, 172, 201, 211- Ockham's Razor, 87-90 Ockham, William of, 350, 357-358 212, 216, 223-685 ontological commitment, 346, 349-353,Newton-Smith, W. H., 201Newtonian mechanics, 234 357, 377-379, 387niminalization of empirical science, 77 ontological parsimony, 87-90no-class theory, 349, 350 open models, 386-387nominalism, 106, 130, 158, 202-207, operations (physical vs. mathemati- 335, 348, 350, 356, 366-373 cal), 176, 187-191 easy road to, 669 operations research, 124 hard road to, 669 ordinal number, 244, 404nominalistic content of empirical sci- ortholattice, 483, 484 ence, 85-86 of exact sets, 483nominalistic scientific realism, 85-86 ostensible commitment, 351, 358, 377 Ostwald, J., 677, 681
outcome space, 495 full-blooded (FBP), 35,40-41,49,outer penumbra, 484 68-75, 91-94, 98Papineau, D., 664 mathematical, 40-44, 50-75paraconsistent, 632paradox, 364, 631 naturalized, 53-54paradoxical case axioms, 475 non-uniqueness (NUP), 67-69,73-paradoxical set theory, 468, 472, 474, 75 478 object, 41-44parallel computation, 578, 599, 607 physicalistic, 37, 53-54Paris, 523 plenitudinous, 41, 49, 68-75parsimony, 661 plenitude, 488 ontological, 87-90 plenitudinous platonism, 41, 49, 68- Quine, see Quine, parsimonyParsons, C., 43, 52, 55, 184, 191, 192, 75 plenum, 488 198-199, 263, 478 pleonastic propositions, 346Parsons, T., 378 Poincark, H., 41, 200, 313, 334, 359,part-whole relation, 235partial set, 468, 469, 471, 478 379-381, 677, 681partition property, 434 Polya, G., 142, 151pattern recognition, 136 Popper, 498-500, 503patterns, 363 Porphyry, 356 positive comprehension, 472 mathematical, 42Peano arithmetic, 487 positive set, 468, 472, 478Peano, G., 178, 364, 388, 410, 416 Post worker, 580Peirce, C. S., 360, 366, 542, 646 Post, E., 28, 536, 538, 576, 578-582,Penrose, R., 639penumbral modality, 484 589-591perfect set property, 401, 418 Posterior, 507permutable models, 385personalist, 497 potential infinite, 246PFS, 484, 486, 488 powers of relations, 205-206physical, 497 pragmatism, 372physicalism predicament, 354-355 mathematical, 36-38 predicate calculus, 462physicalistic platonism, 37, 53-54Pigden, C., 135 predicative theories, 200-201, 212place selection, 498 predicativism, 311Plato, 1, 24, 40, 61, 157, 16&161, pretense, 367, 375 Priest, G., 49, 631 192, 213, 239, 356, 378 Prime Number Theorem, viiPlatonic realism (or Platonism), vii Principal Principle, 519, 520Platonism, viii, 106, 107, 127, 135, Principia Mathernatica, 312, 410, 416, 352-355, 369, 373-374, see 427, 645 realism, mathematical Principle of extensionality, 463 Principle of Indifference, 508-510 Principle of naive comprehension, 463 Prior, 507 probability function, 493, 495, 496 probability logic, 521
714 Indexprobability space, 495 new foundation, 464, 477probability theory, x ontic commitments, 665Progic, 521 -Putnam indispensability argu-progression, 112projective sets, 419 ment, 84-86proof, 139, 632 quantification, 659proof theory, 307 realism, 693proofs of correctness of computer pro- semantic holism, 667 unapplied mathematics, 678-679 grams, 125 V = L, 680propensity, 499 Quine, W. V. O., 35, 37, 40, 48, 52,propositional language, 495propositional variable, 495 55-56, 76-78,82,84-86,88,proximal Frege structure, 479, 481 90, 134, 158, 169, 191, 195-proximity space, 480 199, 201, 202, 218, 226, 353,proximity structure, 479 355, 359, 378, 381-383, 472psychologism, 36-50, 81-86Ptolemy, 208 Rado, R., 436Putnam Ramsey, F., 434 ranging-over idea, 277 goals of science, 656 ratio, 112 indispensability, 655-656, 658 rationalists, 246 intellectual dishonesty, 656 real number systemPutnam, H., 35,45,76-78,82,84-86, axioms, 299 88, 90, 134, 158, 191, 196- real numbers, 197-202, 211-213, 396 199,201,220-222, 226,346, realism, 158, 191-192 348Pythagorean Theorem, vii mathematical, 35-98, 652 metaphysical, 652quantification, 379 selective, 667quantifiers, 277 set theoretic, 653quantity, 104, 110 realist, 243quantum reckonable function, 538, 575-577 recollection (Plato's theory), 159, 161 field theory, 367 recursive, 498 logic, 21S222, 481 function, 559 mechanics, 641 reduction, 346, 347, 351, 357, 377, nominalization of, 84 388 theory, 172 reference class problem, 499, 502Quine(an) Reflection Principle for ZF, 438 confirmational holism, 667 regularity property, 414 continuity thesis, 666 Reichenbach, H., 218, 219 -Duhem Thesis, see holism, con- Reichenbach, R., 250, 263 relation, 108, 296 firmational relational structures, 243 first philosophy, 665 relative reflexive fictionalism, 375-377 indispensability, 655 repeatable, 496 naturalism, 657-658, 664, 664
repeatably instantiatable, 496 semantics, 352-353, 375-377replacement, 426 sequences of finite projections, see SFPrepresentational fictionalism, 360-363, series, 214-215 set theory, x, 243, 272, 366-367 366 set-theoretic indiscernibility, 483Resnik sets, 109, 132 indispensability, 659 settled models, 383, 386Resnik, M., 37, 39, 41-43, 52, 55-56, Shanin, N. A., 329 58, 59, 63, 64, 84, 105, 110, Shannon, C., 507 346, 367Restall, G., 41, 70-74 Shapiro, S., 41,42, 52, 58, 59, 64, 84,revisionism, 644 105, 110, 115, 210, 370-378revolutionary fictionalsim, 348Riemann hypothesis, 142, 145 Shepherdson, J., 437, 438Riemann, B., 173 Sieg, W., x, 597-610, 621-623Robertson, H. P., 265 Sierpiriski, W., 418, 421Robinson, A., 437, 638 Simons, P., ixRoscelin, 356 simple random variable, 495Rosen, G., 46, 85, 350, 356, 367, 376, simplicity, 661 378, 389Rosser, B., 538, 568 simply infinite system, 258Rothberger, F., 421 simulation, 140Rotman, B., 255rough set theory, 479 single-case / repeatable, 496Routley, R., 49, 472, 631, 635rules, 238 Singular Cardinals Hypothesis, 444Russell set, 468, 471, 472, 476, 477Russell's paradox, xi, 17, 24, 295, 300, Singular Cardinals Problem, 444 301, 410, 464Russell, B., 19-21, 109, 119, 193, 198, Skolem's Paradox, 425 203-204, 225, 250, 300, 312, Skolem, T., 337, 338, 424, 425, 551, 334, 346,349, 350, 357-358, 364, 368, 410,416, 462, 464, 553 536, 632, 643 Smart, J. C. C., 655, 694 Snir, M., 502Sainsbury, R. M., 200Salmon, N., 48 Sober, E., 367, 669, 682-689schema, 233 social challenges, 180-181, 183-187Schiffer, S., 34G347 Solovay, R. M., 443, 444; 446scholastics, 138 space and time, 232, 235, 369Schroder, E., 412, 542sciences of complexity, 124 Specker, E., 444, 477Scott, D. S., 439, 440, 446, 484, 485second-order logic, 462, 463 Spurr, J., xisecond-order set theory, 366-367,387 Stalnaker, R., 350 standard probabilistic semantics, 522 Stanley, J., 367, 375-377 Steiner, M., 42, 52, 55-56, 690-696 Steinitz, E., 421 Steps 1, 2, 3, and 4, 525 strong nets, 384 structural property, 109 structuralism, 41-44, 58-59, 64-66, 110, 114, 365, 653 structuralist models, 385-386, 388
716 Indexstructure, 110, 114 truth by convention, 668subitization, 137 Turing computor, 538, 597, 599, 601,subjective, 497 606subjective / Objective, 497 Turing machine, 313, 538, 579, 586,subjective Bayesianism, 501 606, 609, 612, 622substitutional quantification, 378-379 Turing's Thesis, 537, 564, 578, 587-success, 345Summerfield, D. M., 184 592sundials, x Turing, A., 28, 538, 580, 584-586,supervenience, 346, 347, 368Suppes, P., 367 588, 590, 596, 608, 609, 611,Suslin tree, 435 617-620, 622Suslin, M., 419 type theoretic, 464symbolic manipulation, 133, 145 type-level, 496symmetry, 112, 115, 117 type-neutrality, 205-206synthesis, 232synthetic a priori, 232, 263, 264, 266 Ulam, S., 427Szabo, Z., 348, 352, 378 ultimate belief, 503, 518 ultrafinitist, 133Tarski, A., 283, 352-354, 369, 379, ultraproduct construction, 440 421, 574 undecidability, 610 undecidable mathematical sentences,Tautology, 496tertium non datur, 312 see continuum hypothesis, 91-Thagard, P., xi 92The Maximum Entropy Principle, 504 undefinability theorem, 283theoretic virtues, 661-662 underdetermination, 522theories, 633 undermining, 520theories of inconsistent mathematics, understanding, 138 understudy properties, 385-386 X undogmatic, 502 unificatory power, 661Third Man Argument, 135 Uniform Continuity Theorem, 326Thomas, C. J., 293 uninstantiated universals, 106Thomasson, A., 48 unit-making properties, 109Tiles, M., viii universals, 105Token-level, 496 unreasonable effectiveness of mathe-topological set theory, 466 matics, 689-698topology, 112, 466, 468 upper approximation, 480transfinite arithmetic, 300 Urquhart, A., 366transfinite numbers, 399transparency, 635 vagueness, 360-361tree property, 435 Vaihinger, H., 363-365, 373Troelstra, A., 318 van Fraassen, B., 654, 665, 682, 683truth, 346-348, 352-353, 366, 377- Velleman, D., 355 vicious circle principle, 200, 313, 335 378 Vitali, G., 421 mathematical, 68 Von Mises, R., 498, 499
Indexvon Neumann, J., 306, 536, 550,556, 557, 653Walton, K., 345Wang, H., 550, 615, 616warrants (for knowledge), 179-180,182- 185weak counterexamples, 318weak nets, 385Weierstrass, K., 114, 216Weinberg, S., 689well-founded set, 425, 473well-ordering, 399Well-ordering Theorem, 405Weydert, E., 472Weyl, H., 336Whedon, J., 38&382Whitehead, A. N., 263, 312, 536Wiener, N., 417Wigner, E., 689William of Ockham, 350, 357-358Williamson, J., xWittgenstein, L., 45, 205, 237Woods, C., xiWoods, J., xiWright, C., 52, 55, 57, 236, 371Yablo, S., 46, 85, 348, 354-356, 371, 374-378, 384Yessenin-Volpin, A. S., 337, 339Zalta, E., 41, 48, 70, 75Zermelo's set theory, 407Zermelo, E., 21-22, 405, 407, 487, 653Zermelo-Fraenkel axiom, 479Zermelo-Fkaenkel set theory, see ZFZF, 193, 198, 199, 465, 466, 468, 471ZFC, 478ZFU, 366-373Zorn's Lemma, 415
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