Abstract
We identify four different minimal versions of the indispensability argument, falling under four different varieties: an epistemic argument for semantic realism, an epistemic argument for platonism and a non-epistemic version of both. We argue that most current formulations of the argument can be reconstructed by building upon the suggested minimal versions. Part of our discussion relies on a clarification of the notion of (in)dispensability as relational in character. We then present some substantive consequences of our inquiry for the philosophical significance of the indispensability argument, the most relevant of which being that both naturalism and confirmational holism can be dispensed with, contrary to what is held by many.
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Notes
- 1.
- 2.
Cf. Liggins (2008) for a reconstruction.
- 3.
In fact, we believe that other possible (and possibly more plausible) forms of platonism could be fashioned, and that the thesis just mentioned should then be more correctly called ‘ontological platonism’. However, insofar as it is not part of our present aims to argue for such distinction, we avoid this specification and call it ‘platonism’ tout court. Nothing in this thesis mandates that mathematical objects are abstract, and indeed, though this is generally (or at least, often) admitted, IA can support the thesis without going into details about the nature of mathematical objects.
- 4.
Semantic realism as conceived here is distinct from what Michael Dummett called ‘realism’ (see, e.g. Dummett 1978): the latter is a thesis about statements possessing an objective, mind-independent truth value, whereas the former is the claim that the relevant statements are true (possibly in a mind-independent way, but not necessarily).
- 5.
- 6.
- 7.
Colyvan (2001, p. 12).
- 8.
Cf. Colyvan (2001, p. 12): ‘[…] I should point out that the first premise, as I’ve stated it, is a little stronger than required. In order to gain the given conclusion all that is really required in the first premise is the ‘all,’ not the ‘all and only, ’ I include the ‘all and only, ’ however, for the sake of completeness and also to help highlight the important role naturalism plays in questions about ontology, since it is naturalism that counsels us to look to science and nowhere else for answers to ontological questions’.
- 9.
Cf. §§ V to VIII of Putnam (1971).
- 10.
- 11.
The argument is presented in slightly different terms in the two occasions. We are here using the one in Resnik (1995).
- 12.
Resnik (1995, p. 171).
- 13.
Though Colyvan does not explicitly equate ‘best’ with ‘best justified’, the list of scientific virtues he considers in Colyvan (2001, pp. 78–79) for a scientific theory to count as good—among which are empirical adequacy, consistency, simplicity and parsimony, unificatory and explanatory power, boldness and fruitfulness and formal elegance—makes clear that he (like other supporters of IA) has much more in mind than simply currently accepted theories. Notice, in passing, that the ‘ought to’, as opposed to the ‘best justified’, has both a permissive and a prescriptive component. We will not put much weight on the latter.
- 14.
Cf. Colyvan (2001, pp. 22–24) for some qualifications. Briefly, [QC] states that the ontological commitment of a theory t is given by the objects that must be counted in the range of the objectual quantifiers in the existential theorems of (the canonical reformulation of) t. [QC] plays in Colyvan’s argument the same role that quantification plays in Putnam’s argument. Cf. Quine (1948).
- 15.
Notice that a weaker notion, like that of acceptance of a scientific theory, modelled, for example, on the lines suggested by Van Fraassen (1980), will not be strong enough to deliver the required mathematical realist or platonist conclusion. We will consider later the possibility of appealing merely to confirmation rather than justification.
- 16.
- 17.
Cf. Colyvan (2001, p. 11). Cf. also footnote 8 above.
- 18.
Those who believe the first premise to be too harsh can still accept a weaker formulation in which that premise is discharged and the conclusion is conditional in form, i.e. ‘If there are true theories, then …’.
- 19.
Also Azzouni’s blueprint reported in Sect. 13.2 above could be thought to be schematic. However, this is so in a different sense. Whereas our arguments are schematic in that they can be turned into strictly different versions of IA by further specifying some of the notions involved, Azzouni’s blueprint is rather a matrix from which explicit and logically valid versions of IA can be obtained through the addition of other assumptions.
- 20.
A different issue is whether an argument based on the indispensability of the use of theories can be retrieved from arguments based on the indispensability of theories tout court: we will come back to this in Sect. 13.6.
- 21.
Whether we should consider a mathematical theory m indispensable to a theory t when only some parts of m are as a matter of fact used for the formulation of t depends, among other things, on whether the employed part of m is such that it can be considered an independent (sub-)theory of m. This is what Peressini (1997) labels ‘the problem of the unit of indispensability’.
- 22.
According to Craig’s Theorem (1956), given a recursively enumerable theory t, and a partition of its vocabulary into an observational one, o, and a theoretical one, t, then there exists a recursively axiomatizable theory t′, whose only non-logical vocabulary is o, comprising all and only the consequences of t expressible in o. Craig himself warned against the philosophical import of his result, claiming that the theorems of t′ obtained by his re-axiomatization method are not more ‘psychologically or mathematically […] perspicuous’ than those of t, this being ‘basically due to the mechanical and artificial way in which they are produced’ (p. 49).
- 23.
Colyvan’s discussion of ‘the role of confirmation theory’ in his (2001, pp. 78–81) hints to the relational character of the notion of preferability. We take our clarification of (in)dispensability to improve on that suggestion.
- 24.
As pointed out to us by an anonymous referee, our schematic definition of (in)dispensability assigns no special role to the notion of applicability of a mathematical theory. It goes without saying that we acknowledge the greatest importance to the problem of the applicability of mathematics and to its role within the debate concerning IA, although it is impossible to discuss these issues here. We do believe, however, that, although the two notions will be certainly connected eventually, they can be beneficially treated separately at a general level of analysis as ours. Whether and how a particular conception of applicability affects a given version of IA—either by facilitating its conclusion or by preventing it—is, indeed, something that we believe will have to be considered case by case, according to versions of the argument appropriately specified so to involve, for instance, one’s preferred notion of applicability in the specification of either the equivalence relation ε or the criterion of virtuosity α.
- 25.
But cf. footnote 30, below.
- 26.
Cf. Colyvan (2001, p. 37): ‘As a matter of fact, I think that the argument can be made to stand without confirmational holism: it’s just that it is more secure with holism. The problem is that naturalism is somewhat vague about ontological commitment to the entities of our best scientific theories. It quite clearly rules out entities not in our best scientific theories, but there seems room for dispute about commitment to some of the entities that are in these theories. Holism helps to block such a move since, according to holism, it is the whole theory that is granted empirical support’. For discussion of this passage and other issues connected with holism in Colyvan’s framework, cf. Peressini (2003, pp. 220–222).
- 27.
See Sereni (2013) for a way in which Frege may be taken to have reasons—based on considerations on applicability—for endorsing premise (iii), despite being alien to a holist conception of confirmation and to the idea that confirmation is relevant for the justification of mathematical theories.
- 28.
Putnam would clearly endorse premise (iii). But he has recently dispelled any doubt that his endorsing it hinges on holism: ‘I have never claimed that mathematics is ‘confirmed’ by its applications in physics’ (cf. Putnam (2012, p. 188)).
- 29.
Cf. the observation made at the very beginning of the present Sect. 13.5.2.
- 30.
Hellman (1999) attempts at avoiding these and others unpalatable consequences by suggesting that even though confirmation is holistic and it is conceded that it is transferred from the testable hypotheses of a theory to its inner parts, it should not be taken to transfer equally, so that different parts of a theory, like those expressing idealized conditions or mathematical hypotheses, could be taken to be confirmed to different degrees. Even if Hellman is right, his suggestion does not affect our present points concerning the non-necessity of confirmational holism for endorsing IA.
- 31.
We shall come back in Sect. 13.5.2.2 to the sufficiency of [CH] and the conditions it requires in terms of the connections between justification and confirmation.
- 32.
One could claim, however, that some form of holism (presumably non-confirmational in nature) is somehow presupposed by any criterion of ontological commitment uniform across statements, namely, in order to ensure that such a criterion uniformly applies both to scientific and to mathematical theories. Against this latter supposition, one could argue, for example, that the notion of aboutness employed in premise (iv) cannot be given a content-neutral characterization and does not apply to mathematical objects. A case in point is Azzouni’s (1998, 2004) suggested alternative to [QC]. One could think, then, that an appropriate form of holism is required for rejecting this possibility. Notice, however, that this is far from necessary: even if some understanding of [QC] or, better, some specification of the schematic notion of aboutness employed in premise (iv) of [PE] and [PnE] (or of other schematic notions employed in some of our versions of IA) turned out somehow to presuppose some form of holism, this would leave untouched that an epistemic version of IA and, then, a fortiori, IA as such, do not necessarily require any appeal to it; at most this thesis would be involved in some particular instances of such an argument (just as it happens for IBE: cf. footnote 35 below).
- 33.
One could, however, question this condition in some quite particular cases, as those involving highly theoretical physical theories, for example, string theory. One could indeed maintain that in cases like these, the relevant scientific theories can be justified and are actually considered to be so, independently of any empirical confirmation they may receive or have received. It is more likely, however, that in the complete absence of empirical confirmation, we would not take ourselves to be justified, however weakly, in taking a scientific theory to be true; rather, such a scientific theory will be said to enjoy a number of virtues that will merely make it acceptable in the scientific community for many practical and theoretical purposes. Nonetheless, this form of acceptance, it goes without saying, will not be strong enough to support the conclusion(s) of IA, in any of the versions we have discussed here.
- 34.
Notice that scientific realism, as formulated here, entails that (we are justified to believe that) the entities (both observable and theoretical) which are spoken of in mature scientific theories exist (at least, if we admit that a statement of a scientific theory cannot be true if these entities does not exist). Some remarks are in order. First, one may adopt forms of realism—e.g. structural realism—where the existence of these individual entities is not entailed; this version of realism would still be adequate to motivate premise (i) in all minimal arguments. Second, it would be odd to assume that scientific realism entails either the existence of the mathematical entities mentioned in mathematical statements or that the mathematics used in science is true; assuming scientific realism does not beg the question with regards to the conclusion of neither platonist or realist versions of IA and can be safely assumed in both.
- 35.
To our knowledge, the only other version of IA which is explicitly claimed by its proponent to dispense with naturalism is the one offered by Azzouni (2009). We’ll discuss this below.
- 36.
- 37.
Discussion of the Enhanced Indispensability Argument often pertains to the alleged role that inference to the best explanation (IBE) may have in IA. Still, insofar as the minimal version of this argument, provided by [PE] under the mentioned specifications, is concerned, no appeal to IBE is required, and its validity need then not be presupposed. This does not mean that IBE cannot be involved in any specification of the minimal versions of IA. Indeed, it could be involved in some such specifications in two ways: the equivalence relation ε in the schematic definition of the notion of (in)dispensability could be specified through the notion of explanation in a way that presupposes the validity of IBE or the criterion of virtuosity α in that same definition could itself presuppose the validity of IBE.
- 38.
This is what Resnik himself seems to imply in the quote relative to footnote 11.
- 39.
- 40.
Cf. § Sect. 13.2, above.
- 41.
Azzouni (2009), especially p. 147, footnote 11
- 42.
Cf also Azzouni (2009), p. 140, note 2; p. 147, note 11.
- 43.
Cf. Azzouni (2009, pp. 140–141).
- 44.
Ibidem, p. 141
- 45.
The sense in which we are justified in believing a mathematical statement true is meant, however, to be in some sense ‘stronger’ (cf. Azzouni 2009, p. 147) than that licensed by Resnik’s argument on pragmatic grounds: as Azzouni claims, ‘it isn’t that we’re ‘justified’ in describing an assertorically used sentence as true; Tarski biconditionals make the use of the truth predicate nonnegotiable’. Whatever this distinction comes to in details, it does not seem that from the assertoric use of a statement p, the truth itself of p can follow, over and beyond our commitment to take p as true. Even if this entails that the conclusion of the Assertoric-use QP will be, as a matter of fact, a different, epistemic version of the conclusion of Azzouni’s proposed blueprint (i.e. ‘Those statements are true’), we still see this as the most reasonable outcome of Azzouni’s discussion; we acknowledge, however, that this reading can be subject to controversy depending on how our ‘commitment’ to the truth of a statement is understood.
- 46.
Azzouni (2009, p. 142)
- 47.
Azzouni (2009, p. 141)
- 48.
Azzouni seems to indifferently use in his paper the terms ‘sentence’ and ‘statement’. While maintaining the term ‘sentence’ in all our quotations from Azzouni’s paper where it occurs, we shall, instead, invariably use the terms ‘statement’, as we do throughout our paper.
- 49.
Notice that no particular conception of truth is presupposed in our minimal versions, so that one is at liberty to use whatever notion one prefers in the specification of the schematic arguments (included a disquotational one). Hence, the question here is not whether we, as opposed to Azzouni, make use of some particular conception of truth but whether any notion of truth is involved at all in the relevant versions of IA.
- 50.
Azzouni (2009, p. 144)
- 51.
The reference is, of course, the same as in footnote 50.
- 52.
Cf. footnote 46 above.
- 53.
Notice also that Azzouni explicitly objects to forms of fictionalism that constitute the most obvious strategies for rejecting premise (iii). In the following passage (Azzouni 2009, p. 143), it is easy so read something very close to the suggested specification of premise (iii) of [RE]:
One issue to be explored in this paper is whether the assertoric use of many statements of ordinary science is compatible with one or another construal of the mathematical statements utilized in science as not assertorically used (and therefore, as either not true-apt or as false). I’ll show that a position that takes us as truth-committed to statements in any area where mathematics is applied, while assuming that we aren’t simultaneously truth-committed to that mathematics, is unstable.
This, if needed, seems to be another piece of evidence that premise (iii) can be upheld without appealing to confirmational holism.
- 54.
This is, of course, not intended to suggest that Azzouni’s Assertoric-use QP is already included, in nuce, in [RE]. What we argue is rather that [RE] is schematically general enough in order to provide an argument form that Azzouni’s Assertoric-use QP (which, as a matter of fact, has been offered beforehand and independently of [RE]) can be taken to instantiate via appropriate specifications.
- 55.
[PE] or [PnE] can be seen as instances of a general way in which Quine would draw ontological conclusions. One might object that the minimal formulation would make nothing of the special subject matter of mathematics (thanks to Matti Eklund for raising this). However, we don’t find anything, in Quine’s reluctant acceptance of platonism, like assuming something special about mathematics and building a form of IA on this (contrary to what is suggested by Steiner’s (1978, pp. 19–20) ‘transcendental’ interpretation of IA). The special character of mathematics seems rather to be proved by the very fact that we cannot dispense with it in science. All posits are ontologically on a par until we are faced, as Quine would call it, with an unabridged language of science. Not all posits will come out as indispensable. Propositions and meanings don’t. Mathematics does.
- 56.
Shapiro (2005, pp. 13–14). Shapiro remarks is only cursorily made, and nothing special hinges on it in his discussion; we just take it as an indication of a widespread feeling.
- 57.
- 58.
Radical antinaturalists, like sceptics, would deny that science is any source of knowledge at all.
- 59.
Cf. Putnam (2012, p. 183).
- 60.
Ibid. The ‘in a sense’ qualification concerns Quinean themes (indeterminacy of translation, differences with a standard realist view of language) discussed in Putnam (1988). They do not affect our present point.
- 61.
Quine (1986, p. 400). In later writings, Quine admitted that this would create an unjustifiable asymmetry between different parts of mathematics; hence, he resorted to the idea that we cannot completely deny meaningfulness to unapplied parts of mathematics, but that we can arbitrarily decide whether to call those parts true or false (cf. Quine 1995, pp. 56–57).
- 62.
Cf. Parsons (1983), Maddy (1992), Leng (2002, 2010), and Colyvan (2007). Putnam (1971, pp. 346–347) suggests a view similar to Quine’s on unapplied mathematics. His is however a milder position (unapplied mathematics ‘should today be investigated in an ‘if-then’ spirit’), and he is wary of restricting his claims to ‘the case for ‘realism’ developed in the present section’.
- 63.
According to Putnam (1975b), mathematics could count as quasi-empirical in that we can account for it in terms of quasi-empirical methods of inquiries (other than deductive proof from axioms) based on successful applications. This is for Putnam consistent with a non-platonist interpretation of mathematics.
References
Armstrong, D. 1997. A world of states of affairs. Cambridge: Cambridge University Press.
Azzouni, J. 1998. On ‘On what there is’. Pacific Philosophical Quarterly 79: 1–18.
Azzouni, J. 2004. Deflating existential consequence. A case for nominalism. Oxford/New York: Oxford University Press.
Azzouni, J. 2009. Evading truth commitments: The problem reanalyzed. Logique & Analyse 206: 139–176.
Baker, A. 2005. Are there genuine mathematical explanations of physical phenomena? Mind 114: 223–238.
Baker, A. 2009. Mathematical explanation in science. British Journal for the Philosophy of Science 60: 611–633.
Colyvan, M. 2001. The indispensability of mathematics. Oxford/New York: Oxford University Press.
Colyvan, M. 2007. Mathematical recreation versus mathematical knowledge. In Mathematical knowledge, ed. M. Leng, A. Paseau, and M. Potter, 109–122. Oxford/New York: Oxford University Press.
Craig, W. 1956. Replacement of auxiliary expressions. Philosophical Review 65: 38–55.
Dieveney, P.S. 2007. Dispensability in the indispensability argument. Synthese 157: 105–128.
Dummett, M. 1978. Realism. In Truth and other enigmas, ed. M. Dummett. London: Duckworth.
Field, H. 1980. Science without numbers. Oxford: Blackwell.
Field, H. 1989. Realism, mathematics and modality. Oxford: Blackwell.
Frege, G. 1893–1903. Grundgesetze der Arithmetik, 2 vol. Jena: H. Pohle.
Garavaso, P. 2005. On Frege’s alleged indispensability argument. Philosophia Mathematica (III) 13: 160–173.
Hafner, J., and P. Mancosu. 2005. The varieties of mathematical explanation. In Visualization, explanation and reasoning styles in mathematics, ed. P. Mancosu et al., 215–250. Dordrecht/Norwell: Springer.
Hellman, G. 1989. Mathematics without numbers. Oxford/New York: Oxford University Press.
Hellman, G. 1999. Some ins and outs of indispensability: A modal-structural perspective. In Logic in Florence, ed. A. Cantini, E. Casari, and P. Minari. Dordrecht: Kluwer.
Leng, M. 2002. What’s wrong with indispensability? (Or, the case for recreational mathematics). Synthese 131: 395–417.
Leng, M. 2010. Mathematics and reality. Oxford/New York: Oxford University Press.
Liggins, D. 2008. Quine, Putnam, and the ‘Quine-Putnam’ indispensability argument. Erkenntnis 68: 113–127.
Maddy, P. 1992. Indispensability and practice. Journal of Philosophy 89(1992): 275–289.
Maddy, P. 2005. Three forms of naturalism. In Oxford handbook of philosophy of mathematics and logic, ed. S. Shapiro, 437–460. Oxford/New York: Oxford University Press.
Maddy, P. 2007. Second philosophy. Oxford/New York: Oxford University Press.
Mancosu, P. 2008. Mathematical explanation: Why it matters. In The philosophy of mathematical practice, ed. P. Mancosu. Oxford: Oxford University Press.
Panza, M., and A. Sereni. 2013. Plato’s problem. An historical introduction to the philosophy of mathematics. Houndmills: Palgrave Macmillan.
Panza, M., and A. Sereni. Forthcoming. The varieties of the indispensability argument.
Parsons, C. 1983. Quine on the philosophy of mathematics. In Mathematics in philosophy, ed. C. Parsons. Ithaca: Cornell University Press. Also in Hahn, L., and P. Schilpp (eds.). 1986. The philosophy of W.V. Quine, 369–395. La Salle: Open Court.
Paseau, A. 2007. Scientific realism. In Mathematical knowledge, ed. M. Leng, A. Paseu, and M. Potter, 123–149. Oxford/New York: Oxford University Press.
Peressini, A. 1997. Troubles with indispensability. Applying pure mathematics in physical theory. Philosophia Mathematica (III) 5: 210–227.
Peressini, A. 2003. Critical study of Mark Colyvan’s ‘The indispensability of mathematics’. Philosophia Mathematica 3: 208–223.
Pincock, J. 2004. A revealing flaw in Colyvan’s indispensability argument. Philosophy of Science 71: 61–79.
Psillos, S. 1999. Scientific realism: How science tracks truth. Oxford: Routledge.
Putnam, H. 1967. Mathematics without foundations. The Journal of Philosophy 64: 5–22. Reprinted in Putnam (1975a), pp. 43–59.
Putnam, H. 1971. Philosophy of logic. New York: Harper & Row. Reprinted in Putnam (1975a), Chap. 20.
Putnam, H. 1975a. Mathematics, matter and method, philosophical papers, vol. 1. Cambridge: Cambridge University Press (2nd ed. 1985).
Putnam, H. 1975b. What is mathematical truth? Historia Mathematica 2: 529–543. Reprinted in Putnam (1975a), Chap. 6.
Putnam, H. 1988. The greatest logical positivist. London Review of Books 10.8: 11–13. Reprinted in Putnam, H. 1990. Realism with a human face, ed. J. Conant. Cambridge: Harvard University Press.
Putnam, H. 2012. Indispensability arguments in the philosophy of mathematics. In Philosophy in an age of science, ed. M. De Caro and D. Macarthur, 181–201. Cambridge: Harvard University Press.
Quine, W.V. 1948. On what there is. Review of metaphysics 2: 21–38. Reprinted in Quine (1953), Chap. 1.
Quine, W.V. 1951. Two dogmas of empiricism. Philosophical Review 60: 20–43. Reprinted in Quine (1953), Chap. 2.
Quine, W.V. 1953. From a logical point of view. New York: Harper & Row (2nd ed. 1961).
Quine, W.V. 1975. Five milestones of empiricism. In Quine (1981), pp. 67–72.
Quine, W.V. 1981. Theories and things. Cambridge: Harvard University Press.
Quine, W.V. 1986. Reply to Charles Parsons. In The philosophy of W.V. Quine, ed. L. Hahn and P. Schlipp. La Salle: Open Court.
Quine, W.V. 1995. From stimulus to science. Cambridge: Cambridge University Press.
Resnik, M.D. 1995. Scientific vs mathematical realism: The indispensability argument. Philosophia Mathematica (III) 3: 166–174.
Resnik, M.D. 1997. Mathematics as a science of patterns. Oxford: Clarendon.
Shapiro, S. (ed.). 2005. The Oxford handbook for the philosophy of mathematics and logic. Oxford/New York: Oxford University Press.
Sereni, A. 2013. Frege, indispensability, and the compatibilist Heresy. Philosophia Mathematica. doi:10.1093/philmat/nkt046.
Sober, E. 1993. Mathematics and indispensability. The Philosophical Review 102: 35–57.
Steiner, M. 1978. Mathematics, explanation, and scientific knowledge. Noûs 12: 17–28.
van Fraassen, B. 1980. The scientific image. Oxford/New York: Oxford University Press.
Weir, A. 2005. Naturalism reconsidered. In The Oxford handbook of the philosophy of mathematics and logic, ed. S. Shapiro, 460–482. Oxford/New York: Oxford University Press.
Acknowledgements
Earlier versions of this paper have been presented on several occasions in seminars and conferences. The authors wish to thank all the audiences for their helpful comments. Special thanks go to: Andrea Bianchi, Francesca Boccuni, Jacob Busch, Annalisa Coliva, Matti Eklund, Mario De Caro, Jan Lacki, David Liggins, Paolo Mancosu, Sebastiano Moruzzi and Eva Picardi. Many thanks also to two anonymous referees for this volume, who provided precious suggestions.
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Panza, M., Sereni, A. (2015). On the Indispensable Premises of the Indispensability Argument. In: Lolli, G., Panza, M., Venturi, G. (eds) From Logic to Practice. Boston Studies in the Philosophy and History of Science, vol 308. Springer, Cham. https://doi.org/10.1007/978-3-319-10434-8_13
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