Indispensability Arguments in the Philosophy of Mathematics (Stanford Encyclopedia of Philosophy) Cite this entry Search the SEP • Advanced Search • Tools • RSS FeedTable of Contents• What's New• Archives• Projected ContentsEditorial Information• About the SEP• Editorial Board• How to Cite the SEP• Special CharactersSupport the SEPContact the SEP ©Metaphysics Research Lab,CSLI,Stanford University  Open access to the SEP is made possible by a world-wide funding initiative. Please Read How You Can Help Keep the Encyclopedia FreeIndispensability Arguments in the Philosophy of MathematicsFirst published Mon Dec 21, 1998; substantive revision Tue Jan 15, 2008One of the most intriguing features of mathematics is itsapplicability to empirical science. Every branch of science draws uponlarge and often diverse portions of mathematics, from the use ofHilbert spaces in quantum mechanics to the use of differentialgeometry in general relativity. It's not just the physical sciencesthat avail themselves of the services of mathematics either. Biology,for instance, makes extensive use of difference equations andstatistics. The roles mathematics plays in these theories is alsovaried. Not only does mathematics help with empirical predictions, itallows elegant and economical statement of many theories. Indeed, soimportant is the language of mathematics to science, that it is hardto imagine how theories such as quantum mechanics and generalrelativity could even be stated without employing a substantial amountof mathematics. From the rather remarkable but seemingly uncontroversial fact thatmathematics is indispensable to science, some philosophers have drawnserious metaphysical conclusions. In particular, Quine (1976; 1980a;1980b; 1981a; 1981c) and Putnam (1979a; 1979b) have argued that theindispensability of mathematics to empirical science gives us goodreason to believe in the existence of mathematical entities. Accordingto this line of argument, reference to (or quantification over)mathematical entities such as sets, numbers, functions and such isindispensable to our best scientific theories, and so we ought to becommitted to the existence of these mathematical entities. To dootherwise is to be guilty of what Putnam has called "intellectualdishonesty" (Putnam 1979b, p. 347). Moreover, mathematical entities areseen to be on an epistemic par with the other theoretical entities ofscience, since belief in the existence of the former is justified bythe same evidence that confirms the theory as a whole (and hence beliefin the latter). This argument is known as the Quine-Putnamindispensability argument for mathematical realism. There are otherindispensability arguments,[1] but this one is by far the mostinfluential, and so in what follows I'll concentrate on it.1. Spelling Out the Quine-Putnam Indispensability Argument2. What is it to be Indispensable?3. Naturalism and Holism4. Objections5. ConclusionBibliographyOther Internet ResourcesRelated Entries1. Spelling Out the Quine-Putnam Indispensability ArgumentThe Quine-Putnam indispensability argument has attracted a great dealof attention, in part because many see it as the best argument formathematical realism (or platonism). Thus anti-realists aboutmathematical entities (or nominalists) need to identify where theQuine-Putnam argument goes wrong. Many platonists, on the other hand,rely very heavily on this argument to justify their belief inmathematical entities. The argument places nominalists who wish to berealist about other theoretical entities of science (quarks, electrons,black holes and such) in a particularly difficult position. Fortypically they accept something quite like the Quine-Putnam argument[2])as justification for realism about quarks and black holes. (This iswhat Quine (1980b, p. 45) calls holding a "double standard" with regardto ontology.) For future reference I'll state the Quine-Putnam indispensabilityargument in the following explicit form:(P1) We ought to have ontologicalcommitment to all and only the entities that are indispensable to ourbest scientific theories. (P2) Mathematical entities are indispensable to our best scientifictheories.(C) We ought to have ontological commitment to mathematicalentities.Thus formulated, the argument is valid. This forces the focus onto thetwo premises. In particular, a couple of important questions naturallyarise. The first concerns how we are to understand the claim thatmathematics is indispensable. I address this in the next section. Thesecond question concerns the first premise. It is nowhere near asself-evident as the second and it clearly needs some defense. I'lldiscuss its defense in the following section. I'll then present some ofthe more important objections to the argument, before considering theQuine-Putnam argument's role in the larger scheme of things - where itstands in relation to other influential arguments for and againstmathematical realism. 2. What is it to be Indispensable?The question of how we should understand ‘indispensability’in the present context is crucial to the Quine-Putnam argument, and yetit has received surprisingly little attention. Quine actually speaks interms of the entities quantified over in the canonical form of our bestscientific theories rather than indispensability. Still, the debatecontinues in terms of indispensability, so we would be well served toclarify this term. The first thing to note is that ‘dispensability’ is notthe same as ‘eliminability’. If this were not so,every entity would be dispensable (due to a theorem of Craig).[3] What we require for an entity to be‘dispensable’ is for it to be eliminable and thatthe theory resulting from the entity's elimination be an attractivetheory. (Perhaps, even stronger, we require that the resulting theorybe more attractive than the original.) We will need to spellout what counts as an attractive theory but for this we can appeal tothe standard desiderata for good scientific theories: empiricalsuccess; unificatory power; simplicity; explanatory power; fertilityand so on. Of course there will be debate over what desiderata areappropriate and over their relative weightings, but such issues need tobe addressed and resolved independently of issues of indispensability.(See Burgess (1983) and Colyvan (1999b) for more on these issues.)These issues naturally prompt the question of how muchmathematics is indispensable (and hence how much mathematics carriesontological commitment). It seems that the indispensability argumentonly justifies belief in enough mathematics to serve the needs ofscience. Thus we find Putnam speaking of "the set theoretic‘needs’ of physics" (Putnam 1979b, p. 346) and Quineclaiming that the higher reaches of set theory are "mathematicalrecreation ... without ontological rights" (Quine 1986, p. 400) sincethey do not find physical applications. One could take a lessrestrictive line and claim that the higher reaches of set theory,although without physical applications, do carry ontological commitmentby virtue of the fact that they have applications in other parts ofmathematics. So long as the chain of applications eventually"bottoms out" in physical science, we could rightfully claim that thewhole chain carries ontological commitment. Quine himself justifiessome transfinite set theory along these lines (Quine 1984, p. 788), buthe sees no reason to go beyond the constructible sets (Quine 1986, p.400). His reasons for this restriction, however, have little to do withthe indispensability argument and so supporters of this argument neednot side with Quine on this issue.3. Naturalism and HolismAlthough both premises of the Quine-Putnam indispensability argumenthave been questioned, it's the first premise that is most obviously inneed of support. This support comes from the doctrines of naturalismand holism. Following Quine, naturalism is usually taken to be the philosophicaldoctrine that there is no first philosophy and that the philosophicalenterprise is continuous with the scientific enterprise (Quine 1981b).By this Quine means that philosophy is neither prior to nor privilegedover science. What is more, science, thus construed (i.e. withphilosophy as a continuous part) is taken to be the complete story ofthe world. This doctrine arises out of a deep respect for scientificmethodology and an acknowledgment of the undeniable success of thismethodology as a way of answering fundamental questions about allnature of things. As Quine suggests, its source lies in "unregeneraterealism, the robust state of mind of the natural scientist who hasnever felt any qualms beyond the negotiable uncertainties internal toscience" (Quine 1981b, p.72). For the metaphysician this means lookingto our best scientific theories to determine what exists, or, perhapsmore accurately, what we ought to believe to exist. In short,naturalism rules out unscientific ways of determining what exists. Forexample, naturalism rules out believing in the transmigration of soulsfor mystical reasons. Naturalism would not, however, rule out thetransmigration of souls if our best scientific theories were to requirethe truth of this doctrine.[4]Naturalism, then, gives us a reason for believing in the entities inour best scientific theories and no other entities. Depending onexactly how you conceive of naturalism, it may or may not tell youwhether to believe in all the entities of your best scientifictheories. I take it that naturalism does give us some reasonto believe in all such entities, but that this is defeasible. This iswhere holism comes to the fore: in particular, confirmationalholism.Confirmational holism is the view that theories are confirmed ordisconfirmed as wholes (Quine 1980b, p. 41). So, if a theory isconfirmed by empirical findings, the whole theory isconfirmed. In particular, whatever mathematics is made use of in thetheory is also confirmed (Quine 1976, pp. 120-122). Furthermore,it is the same evidence that is appealedto in justifying belief in the mathematical components of the theorythat is appealed to in justifying the empirical portion of the theory(if indeed the empirical can be separated from the mathematical atall). Naturalism and holism taken together then justify P1. Roughly, naturalism gives us the "only" and holismgives us the "all" in P1.It is worth noting that in Quine's writings there are at least twoholist themes. The first is the confirmational holism discussed above(often called the Quine-Duhem thesis). The other is semantic holismwhich is the view that the unit of meaning is not the single sentence,but systems of sentences (and in some extreme cases the whole oflanguage). This latter holism is closely related to Quine's well-knowndenial of the analytic-synthetic distinction (Quine 1980b) and hisequally famous indeterminacy of translation thesis (Quine 1960).Although for Quine, semantic holism and confirmational holism areclosely related, there is good reason to distinguish them, since theformer is generally thought to be highly controversial while the latteris considered relatively uncontroversial.Why this is important to the present debate is that Quine explicitlyinvokes the controversial semantic holism in support of theindispensability argument (Quine 1980b, pp. 45-46). Most commentators,however, are of the view that only confirmational holism is required tomake the indispensability argument fly (see, for example, Colyvan(1998); Field (1989, pp. 14-20); Hellman (199?); Resnik (1995a; 1997);Maddy (1992)) and my presentation here follows that accepted wisdom. Itshould be kept in mind, however, that while the argument, thusconstrued, is Quinean in flavor it is not, strictly speaking, Quine'sargument.4. ObjectionsThere have been many objections to the indispensability argument,including Charles Parsons' (1980) concern that the obviousness of basicmathematical statements is left unaccounted for by the Quinean pictureand Philip Kitcher's (1984, pp. 104-105) worry that theindispensability argument doesn't explain why mathematics isindispensable to science. The objections that have received the mostattention, however, are those due to Hartry Field, Penelope Maddy andElliott Sober. In particular, Field's nominalisation program hasdominated recent discussions of the ontology of mathematics. Field (1980) presents a case for denying the second premise of theQuine-Putnam argument. That is, he suggests that despite appearancesmathematics is not indispensable to science. There are two parts toField's project. The first is to argue that mathematical theories don'thave to be true to be useful in applications, they need merely to beconservative. (This is, roughly, that if a mathematical theoryis added to a nominalist scientific theory, no nominalist consequencesfollow that wouldn't follow from the nominalist scientific theoryalone.) This explains why mathematics can be used in sciencebut it does not explain why it is used. The latter is due tothe fact that mathematics makes calculation and statement of varioustheories much simpler. Thus, for Field, the utility of mathematics ismerely pragmatic - mathematics is not indispensable after all.The second part of Field's program is to demonstrate that our bestscientific theories can be suitably nominalised. That is, he attemptsto show that we could do without quantification over mathematicalentities and that what we would be left with would be reasonablyattractive theories. To this end he is content to nominalise a largefragment of Newtonian gravitational theory. Although this is a far cryfrom showing that all our current best scientific theories canbe nominalised, it is certainly not trivial. The hope is that once onesees how the elimination of reference to mathematical entities can beachieved for a typical physical theory, it will seem plausible that theproject could be completed for the rest of science.[5]There has been a great deal of debate over the likelihood of thesuccess of Field's program but few have doubted its significance.Recently, however, Penelope Maddy, has pointed out that if P1 is false, Field's project may turn out to beirrelevant to the realism/anti-realism debate in mathematics.Maddy presents some serious objections to the first premise of theindispensability argument (Maddy 1992; 1995; 1997). In particular, shesuggests that we ought not have ontological commitment to allthe entities indispensable to our best scientific theories. Herobjections draw attention to problems of reconciling naturalism withconfirmational holism. In particular, she points out how a holisticview of scientific theories has problems explaining the legitimacy ofcertain aspects of scientific and mathematical practices. Practiceswhich, presumably, ought to be legitimate given the high regard forscientific practice that naturalism recommends. It is important toappreciate that her objections, for the most part, are concerned withmethodological consequences of accepting the Quinean doctrines ofnaturalism and holism - the doctrines used to support the firstpremise. The first premise is thus called into question by underminingits support.Maddy's first objection to the indispensability argument is that theactual attitudes of working scientists towards the components ofwell-confirmed theories vary from belief, through tolerance, tooutright rejection (Maddy 1992, p. 280). The point is that naturalismcounsels us to respect the methods of working scientists, and yetholism is apparently telling us that working scientists ought not havesuch differential support to the entities in their theories. Maddysuggests that we should side with naturalism and not holism here. Thuswe should endorse the attitudes of working scientists who apparently donot believe in all the entities posited by our best theories.We should thus reject P1.The next problem follows from the first. Once one rejects thepicture of scientific theories as homogeneous units, the questionarises whether the mathematical portions of theories fall within thetrue elements of the confirmed theories or within the idealizedelements. Maddy suggests the latter. Her reason for this is thatscientists themselves do not seem to take the indispensable applicationof a mathematical theory to be an indication of the truth of themathematics in question. For example, the false assumption that wateris infinitely deep is often invoked in the analysis of water waves, orthe assumption that matter is continuous is commonly made in fluiddynamics (Maddy 1992, pp. 281-282). Such cases indicate that scientistswill invoke whatever mathematics is required to get the job done,without regard to the truth of the mathematical theory in question(Maddy 1995, p. 255). Again it seems that confirmational holism is inconflict with actual scientific practice, and hence with naturalism.And again Maddy sides with naturalism. (See also Parsons (1983) forsome related worries about Quinean holism.) The point here is that ifnaturalism counsels us to side with the attitudes of working scientistson such matters, then it seems that we ought not take theindispensability of some mathematical theory in a physical applicationas an indication of the truth of the mathematical theory. Furthermore,since we have no reason to believe that the mathematical theory inquestion is true, we have no reason to believe that the entitiesposited by the (mathematical) theory are real. So once again we oughtto reject P1.Maddy's third objection is that it is hard to make sense of whatworking mathematicians are doing when they try to settle independentquestions. These are questions, that are independent of the standardaxioms of set theory - the ZFC axioms.[6] In order tosettle some of these questions, new axiom candidates have been proposedto supplement ZFC, and arguments have been advanced in support of thesecandidates. The problem is that the arguments advanced seem to havenothing to do with applications in physical science: they are typicallyintra-mathematical arguments. According to indispensability theory,however, the new axioms should be assessed on how well they cohere withour current best scientific theories. That is, set theorists should beassessing the new axiom candidates with one eye on the latestdevelopments in physics. Given that set theorists do not do this,confirmational holism again seems to be advocating a revision ofstandard mathematical practice, and this too, claims Maddy, is at oddswith naturalism (Maddy 1992, pp. 286-289).Although Maddy does not formulate this objection in a way thatdirectly conflicts with P1 it certainly illustrates a tension between naturalism andconfirmational holism.[7] And since boththese are required to support P1, the objection indirectly casts doubton P1. Maddy, however, endorses naturalism and so takes the objectionto demonstrate that confirmational holism is false. I'll leave thediscussion of the impact the rejection of confirmational holism wouldhave on the indispensability argument until after I outline Sober'sobjection, because Sober arrives at much the same conclusion.Elliott Sober's objection is closely related to Maddy's second andthird objections. Sober (1993) takes issue with the claim thatmathematical theories share the empirical support accrued by our bestscientific theories. In essence, he argues that mathematical theoriesare not being tested in the same way as the clearly empirical theoriesof science. He points out that hypotheses are confirmed relative tocompeting hypotheses. Thus if mathematics is confirmed along with ourbest empirical hypotheses (as indispensability theory claims), theremust be mathematics-free competitors. But Sober points out thatall scientific theories employ a common mathematical core.Thus, since there are no competing hypotheses, it is a mistake to thinkthat mathematics receives confirmational support from empiricalevidence in the way other scientific hypotheses do.This in itself does not constitute an objection to P1 of the indispensability argument, as Sober is quickto point out (Sober 1993, p. 53), although it does constitute anobjection to Quine's overall view that mathematics is part of empiricalscience. As with Maddy's third objection, it gives us some cause toreject confirmational holism. The impact of these objections on P1depends on how crucial you think confirmational holism is to thatpremise. Certainly much of the intuitive appeal of P1 is eroded ifconfirmational holism is rejected. In any case, to subscribe to theconclusion of the indispensability argument in the face of Sober's orMaddy's objections is to hold the position that it's permissible atleast to have ontological commitment to entities that receive noempirical support. This, if not outright untenable, is certainly not inthe spirit of the original Quine-Putnam argument.5. ConclusionIt is not clear how damaging the above criticisms are to theindispensability argument. Indeed, the debate is very much alive, withmany recent articles devoted to the topic. (See bibliography notesbelow.) Closely related to this debate is the question of whether thereare any other decent arguments for platonism. If, as some believe, theindispensability argument is the only argument for platonismworthy of consideration, then if it fails, platonism in the philosophyof mathematics seems bankrupt. Of relevance then is the status of otherarguments for and against mathematical realism. In any case, it isworth noting that the indispensability argument is one of a smallnumber of arguments that have dominated discussions of the ontology ofmathematics. It is therefore important that this argument not be viewedin isolation. The two most important arguments against mathematicalrealism are the epistemological problem for platonism - how do we comeby knowledge of causally inert mathematical entities? (Benacerraf1983b) - and the indeterminacy problem for the reduction of numbers tosets - if numbers are sets, which sets are they (Benacerraf 1983a)?Apart from the indispensability argument, the other major argumentfor mathematical realism is that it is desirable to provide auniform semantics for all discourse: mathematical andnon-mathematical alike (Benacerraf 1983b). Mathematical realism, ofcourse, meets this challenge easily, since it explains the truth ofmathematical statements in exactly the same way as in other domains.[8] Itis not so clear, however, how nominalism can provide a uniformsemantics.Finally, it is worth stressing that even if the indispensabilityargument is the only good argument for platonism, the failureof this argument does not necessarily authorize nominalism, for thelatter too may be without support. It does seem fair to say, however,that if the objections to the indispensability argument are sustainedthen one of the most important arguments for platonism is undermined.This would leave platonism on rather shaky ground.BibliographyAlthough the indispensability argument is to be found in many places inQuine's writings (including 1976; 1980a; 1980b; 1981a; 1981c), thelocus classicus is Putnam's short monograph Philosophy ofLogic (included as a chapter of the second edition of the thirdvolume of his collected papers (Putnam, 1979b)). See also Putnam(1979a) and the introduction of Field (1989) which has an excellentoutline of the argument. Colyvan (2001) is a sustained defence of theargument. See Chihara (1973), and Field (1980; 1989) for attacks on the secondpremise and Colyvan (1999b; 2001), Maddy (1990), Malament (1982),Resnik (1985), Shapiro (1983) and Urquhart (1990) for criticisms ofField's program. For a fairly comprehensive look at nominaliststrategies in the philosophy of mathematics (including a gooddiscussion of Field's program), see Burgess and Rosen (1997), whileFeferman (1993) questions the amount of mathematics required forempirical science. See Azzouni (1997; 2004), Balaguer (1996b; 1998),Leng (2002), Maddy (1992; 1995; 1997), Melia (2000; 2002), Peressini(1997), Pincock (2004), Sober (1993) and Vineberg (1996) for attackson the first premise. Baker (2001; 2005), Colyvan (1998; 1999a; 2001;2002; 2007), Hellman (1999) and Resnik (1995a; 1997) reply to some ofthese objections.For variants of the Quinean indispensability argument see Maddy(1992) and Resnik (1995a).Azzouni, J., 1997, "Applied Mathematics, Existential Commitmentand the Quine-Putnam Indispensability Thesis", PhilosophiaMathematica, 5/3 (October): 193-209.Azzouni, J., 2004, Deflating Existential Consequence, NewYork: Oxford University Press.Baker, A., 2001, "Mathematics, Indispensability and ScientificProgress", Erkenntnis, 55/1 (July):85-116.Baker, A., 2005, "Are There Genuine Mathematical Explanations ofPhysical Phenomena?", Mind, 114/454 (April):223-238.Balaguer, M., 1996a, "Towards a Nominalization of QuantumMechanics", Mind, 105/418 (April):209-226.Balaguer, M., 1996b, "A Fictionalist Account of the IndispensableApplications of Mathematics", Philosophical Studies,83/3 (September): 291-314.Balaguer, M., 1998, Platonism and Anti-Platonism inMathematics, New York: Oxford University Press.Benacerraf, P., 1983a, "What Numbers Could Not Be", reprinted inBenacerraf and Putnam (1983), pp. 272-294.Benacerraf, P., 1983b, "Mathematical Truth", reprinted inBenacerraf and Putnam (1983), pp. 403-420 and in Hart (1996), pp.14-30.Benacerraf, P. and Putnam, H. (eds.), 1983, Philosophy ofMathematics: Selected Readings, 2nd edition, Cambridge: CambridgeUniversity Press.Bueno, O., 2003, "Is it Possible to Nominalize QuantumMechanics?", Philosophy of Science, 70/5(December): 1424-1436.Burgess, J., 1983, "Why I Am Not a Nominalist", Notre DameJournal of Formal Logic, 24/1 (January):93-105.Burgess, J. and Rosen, G., 1997, A Subject with No Object:Strategies for Nominalistic Interpretation of Mathematics, Oxford:Clarendon.Chihara, C., 1973, Ontology and the Vicious CirclePrinciple, Ithaca, NY: Cornell University Press.Colyvan, M., 1998, "In Defence of Indispensability",Philosophia Mathematica, 6/1 (February):39-62.Colyvan, M., 1999a, "Contrastive Empiricism and Indispensability",Erkenntnis, 51/2-3 (September):323-332.Colyvan, M., 1999b, "Confirmation Theory and Indispensability",Philosophical Studies, 96/1 (October):1-19.Colyvan, M., 2001, The Indispensability of Mathematics,New York: Oxford University Press.Colyvan, M., 2002, "Mathematics and Aesthetic Considerations inScience", Mind, 111/441 (January):69-74.Colyvan, M., 2007, "Mathematical Recreation Versus MathematicalKnowledge", in M. Leng, A. Paseau, and M. Potter (eds.),Mathematical Knowledge, Oxford: Oxford University Press,pp. 109-122.Feferman, S., 1993, "Why a Little Bit Goes a Long Way: LogicalFoundations of Scientifically Applicable Mathematics", Proceedingsof the Philosophy of Science Association, 2:442-455.Field, H.H., 1980, Science Without Numbers: A Defence ofNominalism, Oxford: Blackwell.Field, H.H., 1989, Realism, Mathematics and Modality,Oxford: Blackwell.Hart, W.D. (ed.), 1996, The Philosophy of Mathematics,Oxford: Oxford University Press.Hellman, G., 1999, "Some Ins and Outs of Indispensability: AModal-Structural Perspective", in A. Cantini, E. Casari and P. Minari(eds.), Logic and Foundations of Mathematics, Dordrecht:Kluwer, pp. 25-39.Irvine, A.D. (ed.), 1990, Physicalism in Mathematics,Dordrecht: Kluwer.Kitcher, P., 1984, The Nature of Mathematical Knowledge,New York: Oxford University Press.Leng, M., 2002, "What's Wrong with Indispensability? (Or, The Casefor Recreational Mathematics)", Synthese,131/3 (June): 395-417.Maddy, P., 1990, "Physicalistic Platonism", in A.D. Irvine (ed.),Physicalism in Mathematics, Dordrecht: Kluwer, pp.259-289.Maddy, P., 1992, "Indispensability and Practice", Journal ofPhilosophy, 89/6 (June): 275-289.Maddy, P., 1995, "Naturalism and Ontology", PhilosophiaMathematica, 3/3 (September): 248-270.Maddy, P., 1997, Naturalism in Mathematics, Oxford:Clarendon Press.Maddy, P., 1998, "How to be a Naturalist about Mathematics", inH.G. Dales and G. Oliveri (eds.), Truth in Mathematics,Oxford: Clarendon, pp. 161-180.Malament, D., 1982, "Review of Field's Science WithoutNumbers", Journal of Philosophy, 79/9(September): 523-534 and reprinted in Resnik (1995b), pp. 75-86.Melia, J., 2000, "Weaseling Away the IndispensabilityArgument", Mind, 109/435 (July):455-479Melia, J., 2002, "Response to Colyvan", Mind,111/441 (January): 75-80.Parsons, C., 1980, "Mathematical Intuition", Proceedings ofthe Aristotelian Society, 80 (1979-1980):145-168; reprinted in Resnik (1995b), pp. 589-612 and in Hart (1996),pp. 95-113.Parsons, C., 1983, "Quine on the Philosophy of Mathematics", inMathematics in Philosophy: Selected Essays, Ithaca, NY:Cornell University Press, pp. 176-205.Peressini, A., 1997, "Troubles with Indispensability: ApplyingPure Mathematics in Physical Theory", PhilosophiaMathematica, 5/3 (October): 210-227.Pincock, C., 2004, "A Revealing Flaw in Colyvan's IndispensabilityArgument", Philosophy of Science, 71/1(January): 61-79.Putnam, H., 1979a, "What is Mathematical Truth", in MathematicsMatter and Method: Philosophical Papers, Volume 1, 2nd edition,Cambridge: Cambridge University Press, pp. 60-78.Putnam, H., 1979b, "Philosophy of Logic", reprinted inMathematics Matter and Method: Philosophical Papers, Volume1, 2nd edition, Cambridge: Cambridge University Press,pp. 323-357.Quine, W.V., 1960, Word and Object, Cambridge, MA: MITPress.Quine, W.V., 1976, "Carnap and Logical Truth" reprinted in TheWays of Paradox and Other Essays, revised edition, Cambridge, MA:Harvard University Press, pp. 107-132 and in Benacerraf and Putnam(1983), pp. 355-376.Quine, W.V., 1980a, "On What There Is", reprinted in From aLogical Point of View, 2nd edition, Cambridge, MA: HarvardUniversity Press, pp. 1-19.Quine, W.V., 1980b, "Two Dogmas of Empiricism", reprinted inFrom a Logical Point of View, 2nd edition, Cambridge, MA:Harvard University Press, pp. 20-46; reprinted in Hart (1996),pp. 31-51 (Page references are to the first reprinting).Quine, W.V., 1981a, "Things and Their Place in Theories", inTheories and Things, Cambridge, MA: Harvard University Press,pp. 1-23.Quine, W.V., 1981b, "Five Milestones of Empiricism", inTheories and Things, Cambridge, MA: Harvard University Press,pp. 67-72.Quine, W.V., 1981c, "Success and Limits of Mathematization", inTheories and Things, Cambridge, MA: Harvard University Press,pp. 148-155.Quine, W.V., 1984, "Review of Parsons', Mathematics inPhilosophy," Journal of Philosophy,81/12 (December): 783-794.Quine, W.V., 1986, "Reply to Charles Parsons", in L. Hahn and P.Schilpp (eds.), The Philosophy of W.V. Quine, La Salle, ILL:Open Court, pp. 396-403.Resnik, M.D., 1985, "How Nominalist is Hartry Field's Nominalism",Philosophical Studies, 47 (March):163-181.Resnik, M.D., 1995a, "Scientific Vs Mathematical Realism: TheIndispensability Argument", Philosophia Mathematica,3/2 (May): 166-174.Resnik, M.D. (ed.), 1995b, Mathematical Objects andMathematical Knowledge, Aldershot (UK): Dartmouth.Resnik, M.D., 1997, Mathematics as a Science of Patterns,Oxford: Clarendon Press.Shapiro, S., 1983, "Conservativeness and Incompleteness",Journal of Philosophy, 80/9 (September):521-531; reprinted in Resnik (1995b), pp. 87-97 and in Hart (1996),pp. 225-234Sober, E., 1993, "Mathematics and Indispensability",Philosophical Review, 102/1 (January):35-57.Urquhart, A., 1990, "The Logic of Physical Theory", in A.D. Irvine(ed.), Physicalism in Mathematics, Dordrecht: Kluwer, pp.145-154.Vineberg, S., 1996, "Confirmation and the Indispensability ofMathematics to Science" PSA 1996 (Philosophy of Science,supplement to vol. 63), pp. 256-263.Other Internet Resources[Please contact the author with suggestions.] Related Entries abduction | meaning holism | naturalism | nominalism: in metaphysics | Platonism: in metaphysics | Quine, Willard van Orman | realismAcknowledgments The author would like to thank Hilary Putnam, Helen Regan, AngelaRosier and Edward Zalta for comments on earlier versions of thisentry. Copyright © 2008 byMark Colyvan<mcolyvan@usyd.edu.au> |
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