One 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 physicalsciences that 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“intellectual dishonesty” (Putnam 1979b, p. 347).Moreover, mathematical entities are seen to be on an epistemic parwith the other theoretical entities of science, since belief in theexistence of the former is justified by the same evidence thatconfirms the theory as a whole (and hence belief in the latter). Thisargument is known as the Quine-Putnam indispensability argument formathematical realism. There are other indispensability arguments, butthis one is by far the most influential, and so in what follows,we’ll mostly focus on it.

In general, an indispensability argument is an argument that purportsto establish the truth of some claim based on the indispensability ofthe claim in question for certain purposes (to be specified by theparticular argument). For example, ifexplanation isspecified as the purpose, then we have an explanatory indispensabilityargument. Thus we see that inference to the best explanation is aspecial case of an indispensability argument. See the introduction ofField (1989, pp. 14–20) for a nice discussion ofindispensability arguments and inference to the best explanation. Seealso Maddy (1992) and Resnik (1995a) for variations on theQuine-Putnam version of the argument. We should add that although theversion of the argument presented here is generally attributed toQuine and Putnam, it differs in a number of ways from the argumentsadvanced by either Quine or Putnam.^[1]

• 1. Spelling Out the Quine-Putnam Indispensability Argument
• 2. What is it to be Indispensable?
• 3. Naturalism and Holism
• 4. Objections
• 5. Explanatory Versions of the Argument
• 6. Conclusion
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Spelling Out the Quine-Putnam Indispensability Argument

The 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.For typically 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 “doublestandard” with regard to ontology.)

For future reference, we’ll state the Quine-Putnamindispensability argument in the following explicit form:

(P1) We ought to have ontological commitment to all and only the entitiesthat are indispensable to our best 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. We 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.We’ll discuss its defense in the following section. We’llthen present some of the more important objections to the argument,before considering the Quine-Putnam argument’s role in thelarger scheme of things — where it stands in relation to otherinfluential arguments for and against mathematical realism.

2. What is it to be Indispensable?

The question of how we should understand‘indispensability’ in the present context is crucial tothe Quine-Putnam argument, and yet it has received surprisingly littleattention. Quine actually speaks in terms of the entities quantifiedover in the canonical form of our best scientific theories rather thanindispensability. Still, the debate continues in terms ofindispensability, so we would be well served to clarify 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 forit to be eliminableand that the theory resulting from theentity’s elimination be an attractive theory. (Perhaps, evenstronger, we require that the resulting theory bemoreattractive than the original.) We will need to spell out what countsas an attractive theory but for this we can appeal to the standarddesiderata for good scientific theories: empirical success;unificatory power; simplicity; explanatory power; fertility and so on.Of course there will be debate over what desiderata are appropriateand over their relative weightings, but such issues need to beaddressed and resolved independently of issues of indispensability.(See Burgess (1983) and Colyvan (1999) for more on these issues.)

These issues naturally prompt the question ofhow 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)since they 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 ontologicalcommitment by virtue of the fact that they have applicationsinother parts of mathematics. So long as the chain of applicationseventually “bottoms out” in physical science, we couldrightfully claim that the whole chain carries ontological commitment.Quine himself justifies some transfinite set theory along these lines(Quine 1984, p. 788), but he sees no reason to go beyond theconstructible sets (Quine 1986, p. 400). His reasons for thisrestriction, however, have little to do with the indispensabilityargument and so supporters of this argument need not side with Quineon this issue.

3. Naturalism and Holism

Although both premises of the Quine-Putnam indispensability argumenthave been questioned, it’s the first premise that is mostobviously in need of support. This support comes from the doctrines ofnaturalism and 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“unregenerate realism, the robust state of mind of the naturalscientist who has never felt any qualms beyond the negotiableuncertainties internal to science” (Quine 1981b, p. 72). For themetaphysician this means looking to our best scientific theories todetermine what exists, or, perhaps more accurately, what we ought tobelieve to exist. In short, naturalism rules out unscientific ways ofdetermining what exists. For example, naturalism rules out believingin the transmigration of souls for mystical reasons. Naturalism wouldnot, however, rule out the transmigration of souls if our bestscientific theories were to require the 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 inall the entities of your bestscientific theories. We take it that naturalism does give ussome reason to believe in all such entities, but that this isdefeasible. This is where holism comes to the fore: in particular,confirmational holism.

Confirmational holism is the view that theories are confirmed ordisconfirmed as wholes (Quine 1980b, p. 41). So, if a theory isconfirmed by empirical findings, thewhole 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 appealed to in justifying belief inthe mathematical components of the theory that is appealed to injustifying the empirical portion of the theory (if indeed theempirical can be separated from the mathematical at all). Naturalismand holism taken together then justifyP1. Roughly, naturalism gives us the “only” and holism givesus the “all” in P1.

It is worth noting that in Quine’s writings there are at leasttwo holist themes. The first is the confirmational holism discussedabove (often called the Quine-Duhem thesis). The other is semanticholism which is the view that the unit of meaning is not the singlesentence, but systems of sentences (and in some extreme cases thewhole of language). This latter holism is closely related toQuine’s well-known denial of the analytic-synthetic distinction(Quine 1980b) and his equally famous indeterminacy of translationthesis (Quine 1960). Although for Quine, semantic holism andconfirmational holism are closely related, there is good reason todistinguish them, since the former is generally thought to be highlycontroversial while the latter is considered relativelyuncontroversial.

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). Mostcommentators, however, are of the view that only confirmational holismis required to make the indispensability argument fly (see, forexample, Colyvan (1998a); Field (1989, pp. 14–20); Hellman(1999); Resnik (1995a; 1997); Maddy (1992)) and my presentation herefollows that accepted wisdom. It should be kept in mind, however, thatwhile the argument, thus construed, is Quinean in flavor it is not,strictly speaking, Quine’s argument.

4. Objections

There have been many objections to the indispensability argument,including Charles Parsons’ (1980) concern that the obviousnessof basic mathematical statements is left unaccounted for by theQuinean picture and Philip Kitcher’s (1984, pp. 104–105)worry that the indispensability argument doesn’t explainwhy mathematics is indispensable to science. The objectionsthat have received the most attention, however, are those due toHartry Field, Penelope Maddy and Elliott Sober. In particular,Field’s nominalisation program has dominated recent discussionsof the ontology of mathematics.

Field (2016) 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 mathematicaltheories don’t have to be true to be useful in applications,they need merely to beconservative. (This is, roughly, thatif a mathematical theory is added to a nominalist scientific theory,no nominalist consequences follow that wouldn’t follow from thenominalist scientific theory alone.) This explains why mathematicscan be used in science but it does not explain why itis used. The latter is due to the fact that mathematics makescalculation and statement of various theories much simpler. Thus, forField, the utility of mathematics is merely pragmatic —mathematics is not indispensable after all.

The second part of Field’s program is to demonstrate that ourbest scientific theories can be suitably nominalised. That is, heattempts to show that we could do without quantification overmathematical entities and that what we would be left with would bereasonably attractive theories. To this end he is content tonominalise a large fragment of Newtonian gravitational theory.Although this is a far cry from showing thatall our currentbest scientific theories can be nominalised, it is certainly nottrivial. The hope is that once one sees how the elimination ofreference to mathematical entities can be achieved for a typicalphysical theory, it will seem plausible that the project could becompleted 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 itssignificance. Recently, however, Penelope Maddy, has pointed out thatifP1 is false, Field’s project may turn out to be irrelevant to therealism/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 toallthe 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 thatthe actual 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 apparentlydo not believe inall the entities posited by our besttheories. We should thus rejectP1.

The next problem follows from the first. Once one rejects the pictureof scientific theories as homogeneous units, the question ariseswhether the mathematical portions of theories fall within the trueelements of the confirmed theories or within the idealized elements.Maddy suggests the latter. Her reason for this is that scientiststhemselves do not seem to take the indispensable application of amathematical 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 thatscientists will invoke whatever mathematics is required to get the jobdone, without regard to the truth of the mathematical theory inquestion (Maddy 1995, p. 255). Again it seems that confirmationalholism is in conflict with actual scientific practice, and hence withnaturalism. And again Maddy sides with naturalism. (See also Parsons(1983) for some related worries about Quinean holism.) The point hereis that if naturalism counsels us to side with the attitudes ofworking scientists on such matters, then it seems that we ought nottake the indispensability of some mathematical theory in a physicalapplication as an indication of the truth of the mathematical theory.Furthermore, since we have no reason to believe that the mathematicaltheory in question is true, we have no reason to believe that theentities posited by the (mathematical) theory are real. So once againwe ought to rejectP1.

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 to settle some of these questions, new axiom candidates havebeen proposed to supplement ZFC, and arguments have been advanced insupport of these candidates. The problem is that the argumentsadvanced seem to have nothing to do with applications in physicalscience: they are typically intra-mathematical arguments. According toindispensability theory, however, the new axioms should be assessed onhow well they cohere with our current best scientific theories. Thatis, set theorists should be assessing the new axiom candidates withone eye on the latest developments in physics. Given that settheorists do not do this, confirmational holism again seems to beadvocating a revision of standard mathematical practice, and this too,claims Maddy, is at odds with naturalism (Maddy 1992, pp.286–289).

Although Maddy does not formulate this objection in a way thatdirectly conflicts withP1 it certainly illustrates a tension between naturalism andconfirmational holism.^[7] And since both these are required to support P1, the objectionindirectly casts doubt on P1. Maddy, however, endorses naturalism andso takes the objection to demonstrate that confirmational holism isfalse. We’ll leave the discussion of the impact the rejection ofconfirmational holism would have on the indispensability argumentuntil after we outline Sober’s objection, because Sober arrivesat much the same conclusion.

Elliott Sober’s objection is closely related to Maddy’ssecond and third objections. Sober (1993) takes issue with the claimthat mathematical theories share the empirical support accrued by ourbest scientific theories. In essence, he argues that mathematicaltheories are not being tested in the same way as the clearly empiricaltheories of science. He points out that hypotheses are confirmedrelative to competing hypotheses. Thus if mathematics is confirmedalong with our best empirical hypotheses (as indispensability theoryclaims), there must be mathematics-free competitors. But Sober pointsout thatall scientific theories employ a common mathematicalcore. Thus, since there are no competing hypotheses, it is a mistaketo think that mathematics receives confirmational support fromempirical evidence in the way other scientific hypotheses do.

This in itself does not constitute an objection toP1 of the indispensability argument, as Sober is quick to point out(Sober 1993, p. 53), although it does constitute an objection toQuine’s overall view that mathematics is part of empiricalscience. As with Maddy’s third objection, it gives us some causeto reject 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 ofSober’s or Maddy’s objections is to hold the position thatit’s permissible at least to have ontological commitment toentities that receive no empirical support. This, if not outrightuntenable, is certainly not in the spirit of the original Quine-Putnamargument.

5. Explanatory Versions of the Argument

The arguments against holism from Maddy and Sober resulted in areevaluation of the indispensability argument. If, contra Quine,scientists do not accept all the entities of our best scientifictheories, where does this leave us? We need criteria for when to treatposits realistically. Here is where the debate over theindispensability argument took an interesting turn. Scientificrealists, at least, accept those posits of our best scientifictheories that contribute to scientific explanations. According to thisline of thought, we ought to believe in electrons, say, not becausethey are indispensable to our best scientific theories but becausethey are indispensable in a very specific way: they areexplanatorily indispensable. If mathematics could be shown tocontribute to scientific explanations in this way, mathematicalrealism would again be on par with scientific realism. Indeed, this isthe focus of most of the contemporary discussion on theindispensability argument. The central question is: does mathematicscontribute to scientific explanations and if so, does it do it in theright kind of way.

One example of how mathematics might be thought to be explanatory isfound in the periodic cicada case (Yoshimura 1997 and Baker 2005).North American Magicicadas are found to have life cycles of 13 or 17years. It is proposed by some biologists that there is an evolutionaryadvantage in having such prime-numbered life cycles. Prime-numberedlife cycles mean that the Magicicadas avoid competition, potentialpredators, and hybridisation. The idea is quite simple: because primenumbers have no non-trivial factors, there are very few other lifecycles that can be synchronised with a prime-numbered life cycle. TheMagicicadas thus have an effective avoidance strategy that, undercertain conditions, will be selected for. While the explanation beingadvanced involves biology (e.g. evolutionary theory, theories ofcompetition and predation), a crucial part of the explanation comesfrom number theory, namely, the fundamental fact about prime numbers.Baker (2005) argues that this is a genuinely mathematical explanationof a biological fact. There are other examples of alleged mathematicalexplanations in the literature but this remains the most widelydiscussed and is something of a poster child for mathematicalexplanation.

Questions about this case focus on whether the mathematics is reallycontributing to the explanation (or whether it is merely standing infor the biological facts and it is these that really do theexplaining), whether the alleged explanation is an explanation at all,and whether the mathematics in question is involved in the explanationin the right kind of way. Finally, it is worth mentioning thatalthough the recent interest in mathematical explanation arose out ofdebates over the indispensability argument, the status of mathematicalexplanations in the empirical sciences has also attracted interest inits own right. Moreover, such explanations (sometimes called“extra-mathematical explanations”) lead one very naturallyto think about explanations of mathematical facts by appeal to furthermathematical facts (sometimes called “intra-mathematicalexplanation”). These two kinds of mathematical explanation arerelated, of course. If, for example, some theorem of mathematics hasits explanation rest in an explanatory proof, then any applications ofthat theorem in the empirical realm would give rise to a prima faciecase that the full explanation of the empirical phenomenon in questioninvolves the intra-mathematical explanation of the theorem. For theseand other reasons, both kinds of mathematical explanation haveattracted a great deal of interest from philosophers of mathematicsand philosophers of science in recent years.

6. Conclusion

It is not clear how damaging the above criticisms are to theindispensability argument and whether the explanatory version of theargument survives. Indeed, the debate is very much alive, with manyrecent articles devoted to the topic. (See bibliography notes below.)Closely related to this debate is the question of whether there areany other decent arguments for platonism. If, as some believe, theindispensability argument is theonly argument for platonismworthy of consideration, then if it fails, platonism in the philosophyof mathematics seems bankrupt. Of relevance then is the status ofother arguments for and against mathematical realism. In any case, itis worth 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 beviewed in isolation.

The two most important argumentsagainst mathematical realismare the epistemological problem for platonism — how do we comeby knowledge of causally inert mathematical entities? (Benacerraf1983b) — and the indeterminacy problem for the reduction ofnumbers to sets — if numbers are sets, which sets are they(Benacerraf 1983a)? Apart from the indispensability argument, theother major argumentfor mathematical realism appeals to adesire for a uniform semantics forall discourse:mathematical and non-mathematical alike (Benacerraf 1983b).Mathematical realism, of course, meets this challenge easily, since itexplains the truth of mathematical statements in exactly the same wayas in other domains.^[8] It is not so clear, however, how nominalism can provide a uniformsemantics.

Finally, it is worth stressing that even if the indispensabilityargumentis 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.

Bibliography

Although the indispensability argument is to be found in many placesin Quine’s writings (including 1976; 1980a; 1980b; 1981a;1981c), thelocus classicus is Putnam’s short monographPhilosophy of Logic (included as a chapter of the secondedition of the third volume of his collected papers (Putnam, 1979b)).See also Putnam (1979a) and the introduction of Field (1989), whichhas an excellent outline of the argument. Colyvan (2001) presents asustained defence of the argument.

See Chihara (1973), and Field (1989; 2016) for attacks on the secondpremise and Colyvan (1999; 2001), Lyon and Colyvan (2008), Maddy(1990), Malament (1982), Resnik (1985), Shapiro (1983) and Urquhart(1990) for criticisms of Field’s program. See the preface to thesecond edition of Field 2016 for a good retrospective on thesedebates. For a fairly comprehensive look at nominalist strategies inthe philosophy of mathematics (including an excellent discussion ofField’s program), see Burgess and Rosen (1997), while Feferman(1993) questions the amount of mathematics required for empiricalscience. See Azzouni (1997; 2004; 2012), Balaguer (1996b; 1998), Bueno(2012), Leng (2002; 2010; 2012), Liggins (2012), Maddy (1992; 1995;1997), Melia (2000; 2002), Peressini (1997), Pincock (2004), Sober(1993), Vineberg (1996) and Yablo (1998; 2005; 2012) for attacks onthe first premise. Baker (2001; 2005; 2012), Bangu (2012), Colyvan(1998a; 2001; 2002; 2007; 2010; 2012), Hellman (1999) and Resnik(1995a; 1997) reply to some of these objections.

For variants of the Quinean indispensability argument see Maddy (1992)and Resnik (1995a).

There has been a great deal of recent literature on the explanatoryversion of the indispensability argument. Early presentations of suchan argument can be found in Colyvan (1998b; 2002), and most explicitlyin Baker (2005), although this work was anticipated by Steiner (1978a;1978b) on mathematical explanation and Smart on geometric explanation(1990). Some of the key articles on the explanatory version of theargument include Baker (2005; 2009; 2012; 2017; 2021), Bangu (2008;2013), Baron (2014), Batterman (2010), Bueno and French (2012),Colyvan (2002; 2010; 2012; 2018), Lyon (2012), Rizza (2011), Saatsi(2011; 2016) and Yablo (2012).

Arising out of this debate over the role of mathematical explanationin indispensability arguments, has been a renewed interest inmathematical explanation for its own sake. This includes work onreconciling mathematical explanations in science with other forms ofscientific explanation as well as investigating explanation withinmathematics itself. Some of this work includes: Baron (2016), Baron etal. (2017; 2020), Colyvan et al. (2018), Lange (2017), Mancosu (2008),and Pincock (2011).

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abduction |mathematical: explanation |meaning holism |naturalism |nominalism: in metaphysics |Platonism: in metaphysics |Quine, Willard Van Orman |realism

Acknowledgments

The author would like to thank Hilary Putnam, Helen Regan, AngelaRosier and Edward Zalta for comments on earlier versions of thisentry.