Abstract
The indispensability argument (IA) comes in many different versions that all reduce to a general valid schema. Providing a sound IA amounts to providing a full interpretation of the schema according to which all its premises are true. Hence, arguing whether IA is sound results in wondering whether the schema admits such an interpretation. We discuss in full details all the parameters on which the specification of the general schema may depend. In doing this, we consider how different versions of IA can be obtained, also through different specifications of the notion of indispensability. We then distinguish between schematic and genuine IA, and argue that no genuine (non-vacuously and non-circularly) sound IA is available or easily forthcoming. We then submit that this holds also in the particularly relevant case in which indispensability is conceived as explanatory indispensability.
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Notes
In what follows, we shall distinguish between genuine (or fully determinate) arguments and argument schemas. We shall generally speak of arguments tout court in order to refer either to the former or to the latter or to both at once.
As Bueno and Shalkowski make clear, “the indispensability argument per se does not establish the nature of [...][the relevant] objects (whether such objects are contingent or not) nor does it establish the form of knowledge one may have [...] (whether such knowledge is a priori or not). The argument [...] only establishes that we ought to be ontologically committed to [...][these] objects. This leaves open the possibility of using indispensability considerations in support of an ontology of non-contingent objects whose knowledge is justified on a priori grounds.” As we also have stressed somewhere else (cf. Panza and Sereni 2015), although the indispensability argument, being based on premises that may be established on a posteriori grounds—such as the form that proper formulations of scientific theories must take in order to be suitable for description, explanation or prediction—is usually a powerful tool in the hands of empiricists willing to defend the existence of mathematical abstract objects on empirical grounds, the argument per se does not rule out the availability of other a priori reasons for believing in those objects—in which case, it will at most offer some auxiliary, defeasible, and less than ideal evidence for the wanted conclusion.
Something akin to a form of indispensability argument for abstract entities in semantics can be traced back to Church (1951). Thanks to an anonymous reviewer for pointing us to Church’s classic paper in this connection. As noticed by Psillos (1999), pp.10–11, the indispensability of the use of theoretical terms in the formulation of “efficacious” systems of laws is claimed in Carnap (1939), p. 64.
Thanks to Maria Paola Sforza Fogliani for bringing this to our attention.
This thesis (or similar ones) is usually identified with mathematical platonism tout court. We take it to stand for ontological mathematical platonism for the reason that we take mathematical platonism tout court to stand for the more general thesis that mathematical statements are about some objects. It seems possible to argue for this latter claim without arguing for the existence of these objects (at least if the notion of existence is considered, as it usually is, as an universal and primitive notion already clear in itself). This issue goes beyond the scope of the present paper, however.
This thesis is often referred to by labels such as ‘mathematical realism’, ‘semantic realism in mathematics’, or similar ones. We adopted this convention in Panza and Sereni (2013), but we opt here for ‘mathematical veridicalism’ in order to avoid confusion with other theses often referred to as ‘realism’, such as the thesis that mathematical statements have a determinate truth value independently of any sort of justification one may have for them. On some occasions, the bare term ‘realism’ is used to refer to ontological realism, i.e. platonism, e.g. in Field (1982). We avoid this use. We will later identify another possible conclusion for IA’s beyond ontological mathematical platonism and veridicalism, which will be labeled “externality”; cf. Sect. 3.6.
Cf. footnote 6, above.
We shall prefer the latter passive formulation (‘p is justified’) to the active one (‘We are justified in believing in p’), but we do not mean to attach any particular relevance here to the distinction between doxastic and propositional justification. Notice that the conclusions of many ia’s are stated in terms of ought’s: namely, that we ought to do something, e.g. to believe in the relevant thesis, or to be commited to the relevant objects. For our present purpose, we just assume that if we are justified in believing a given thesis, we ought to believe it, that we ought to believe it just in case we are justified in believing it, and that being committed to some objects is the same as believing that they exist.
But cf. footnote (14), below.
We call an argument ‘vacuously sound’ if all its premises are true, but at least one of them is vacuously so. We call it ‘circular’ if one of its premises cannot be argued for if its conclusion is not previously granted. A circular argument can of course be sound: this happens if all its premises (and then also the conclusion) are true. In this case, it is said to be ‘circularly sound’. We shall give examples in Sect. 4.
This use of ‘some’ in premise (iii) of \({{\mathbf {\mathsf{{Sc.IA}}}}}_{0}\) might appear odd. We shall discuss the reason for adopting it in Sect. 3.5.
This becomes evident when \(^{*}{{\mathbf {\mathsf{{Sc.SIA}}}}}_{0}\).iii is written in prenex normal form.
Cf. Putnam (1971), p. 347:
So far I have been developing an argument for realism roughly along the following lines: quantification over mathematical entities is indispensable for science, both formal and physical, therefore we should accept such quantification; but this commits us to accepting the existence of the mathematical entities in question. This type of argument stems, of course, from Quine [...]
We discuss the role of naturalism with respect to ia in Panza and Sereni (2015).
In what follows we shall generically speak of ia’s either to refer both to proper and strengthened ia’s at once, or in contexts where it is clear that the reference is to proper ones. We shall never specifically refer to strengthened ia’s as ia’s tout court.
Cf. footnote (13), above.
The recent discussion on ia’s usually places these arguments in a scientific realist framework, where scientific theories are roughly conceived as literally, or at least approximately (if literally understood), true descriptions of some external reality. This can be rendered in our schematic arguments by choosing an appropriate substitution for ‘P’, as well as by giving some appropriate further characterization of the features of scientific theories. It must certainly be acknowledged that some of the views of earlier proponents of ia, like Quine and Putnam, can be seen as conflicting with a scientific realist position (e.g. Quine’s views on ontological relativity or Putnam’s views on internal realism). And, indeed, appropriate determinations of the parameters in our schemas, apt to reflect appropriate conceptions of what scientific theories are intended to be, could also deliver genuine ia’s compatible with those non-realist views. We thank an anonymous reviewer for prompting a clarification of this point.
Some may suppose this import to be stronger when one considers scientific theories at some ideal final stage of development. Cf. footnote 52 for some remarks on this point.
It goes without saying that arguments in which ‘P’ is replaced by an ontological property are likely to presuppose some robust realist understanding of scientific theories as delivering faithful representations of some external reality in agreement with some experimental practice, while arguments in which ‘P’ is replaced by an ontological property may be compatible with weaker conceptions, where scientific theories are conceived, for instance, as giving reliable descriptions limited to the observable world and/or as allowing for sufficiently reliable predictions of empirical phenomena.
Assessment of the virtues of scientific theories is commonly made through an array of properties, such as simplicity, ontological parsimony, unificatory power, familiarity of principles, fruitfulness, and so on. It is then plausible to consider a scientific theory to be one of our best insofar as it possesses some or all of these virtues to some specified degree. Still, the problem in the present context is not just on which of these grounds a scientific theory is taken to be among our best. Rather, the problem is to consider whether its being among the best entails that it has, or depends on its having, a certain property. A crude example will clarify the matter here. Suppose we suggest to enlist a scientific theory among our best merely because of its extreme simplicity. However we evaluate this choice, it should be clear that this will not be enough for a mathematical theory indispensable to it to be true, or for the objects this latter theory is putatively about to exist. Something more than mere simplicity should be required for this to obtain.
Resnik’s argument is hard to accommodate not only with \({{\mathbf {\mathsf{{Sc.IA}}}}}_{0}\) but also with any valid argument schema. By eliminating a number of logically unessential ingredients, we take it to be as follows:
We discussed Resnik’s argument in some more details in Panza and Sereni (2015).
This seems to be the case, for instance, in Colyvan (2001), p. 16.
Before Baker’s argument, Colyvan’s discussion of the role of confirmation theory (cf. Colyvan 2001, pp. 78–81) displayed awareness that the notion of indispensability may be somehow relational in character. We take our following discussion to improve on that suggestion. It still remains, however, that whatever relation character indispensability may be thought to have by Colyvan, it was not displayed in the very formulation of arguments he discusses, contrary to what happens in Baker’s argument and, more explicitly, in our versions of ia.
Notice that we leave open how recourse of a scientific theory to a mathematical theory is to be understood.
In order to avoid confusion, let us make clear from the beginning that we do not take a family of theories as the mere conjunction of the elements that these theories are formed by, but rather as a system of separate (though possibly related) theories, each of which is considered as self-standing. If we suppose that theories are nothing but bodies of statements, the former option would consist in taking families of theories to be in turn theories. The latter option consists in taking them to be systems of bodies of statements, each of which remains distinct from each other, even when the family is considered as a whole. Clearly, the latter is the only plausible option if we want to consider families of theories composed by rival, or even incompatible theories. Moreover, it is only under this option that it makes sense to distinguish between the case in which ia concerns theories from that in which it rather concerns families of theories.
This is the only option we have taken into account in Panza and Sereni (2013).
Cf. footnote 52, below, for the sense in which we are talking of task here.
Baker’s argument is meant to be enanched with respect to the argument put forward by Colyvan (2001), p. 11:
Baker’s argument is meant to enhance Colyvan’s argument exactly in so far as it makes it explicit that there is a specific task that scientific theories, and mathematical theories (or, allegedly, objects) indispensable to them, are meant to accomplish, and that this task is an explanatory one.
Concerning the circularity of this attitude, cf. Sect. 3.4.
It is assumed here that we have an ability of recognising canonical instances of the relevant theories. This is indeed required if we are to be able to identify these theories in the scientific and mathematical practice in which they should be immersed, in accordance with what have been said in Sects. 3.3 and 3.4.
The former option mentioned in footnote (26) would probably support a different intuition. But under this option, T would be h -k-indispensable to a family of scientific theories including both S and \(\textsf {S}^{\star }\) just for its being h -k-indispensable to S, so that there would be no point in considering \(\textsf {S}^{\star }\). Moreover, in the case considered here it should be possible that S and \(\textsf {S}^{\star }\) be rival theories; this is compatible with considering a family including both theories only under the first option of footnote (26). Taking T as h -k-dispensable to a family of scientific theories including both S and \(\textsf {S}^{\star }\) expresses thus the idea that having recourse to T is not indispensable for accomplishing the relevant task in the appropriate way; and this is, of course, all that is relevant in the case considered.
The example is well suited for emphasizing the importance of relativizing indispensability as we are suggesting. For it would be natural to argue that the pervasive use of differential and integral calculus (especially differential and partial differential equations) in so many scientific theories (not only physical ones) makes real analysis (or whatever appropriate theory of mathematical continuum) indispensable for these very theories. The point would be certainly well taken, since no scientific theory akin to classical mechanics, in its usual formulation coming from Newton, Lagrange, and Hamilton, could, for example, avoid appealing to the differential and integral calculus without losing its very identity. Still, this form of indispensability is far from supporting appropriate versions of ia, since appealing to it in order to draw conclusions about mathematics would reduce to arguing that mathematics is as these conclusions claim it to be because it happened that some scientific theories are just as they happen to be, which is certainly not what ia is intended to conclude (though one could imagine other essentially different arguments along this line). Notice, incidentally, that, as convincingly maintained in Maddy (1997), Chap. II.6, the use of differential and integral calculus in science goes typically together with the appeal to different sorts of “idealizations” (cf. also Maddy 1992 on idealizations, and Colyvan (2001, Chap. 5) and Leng (2010, Chap. 5) for discussions). And this is enough to undermine the possibility of taking the “indispensable appearance of an entity in our best scientific theory to warrant the ontological conclusion that it is real”, since “for this conclusion, the appearance must be in a hypothesis that is not legitimately judged a ‘useful fiction’, in other words, in one that has been ‘experimentally verified’ [...], and it must be in the context that is not an explicit idealization” (ibid., p. 152).
And thus also on the exact meaning to be ascribed to the predicate ‘ScTh’ and ‘Q’ in \(^{*}{{\mathbf {\mathsf{{Sc.IA}}}}}_{0}\) and \(^{*}{{\mathbf {\mathsf{{Sc.SIA}}}}}_{0}\).
And thus whether ‘ScTh’ and ‘Q’ in \(^{*}{{\mathbf {\mathsf{{Sc.IA}}}}}_{0}\) and \(^{*}{{\mathbf {\mathsf{{Sc.SIA}}}}}_{0}\) are to stand respectively for the properties of being a single scientific theory and a single (mathematical) theory, or for the properties of being a family of scientific theory and a family of (mathematical) theories.
For premise (i), things seem to be simpler, since it is reasonable to admit that a family of scientific theories is P if and only if any theory of this family is individually P (the underlying thought being that those properties for which this would not be the case would not be allowed for as possible replacements for ‘P’).
Also in the cases just considered the first option mentioned in footnote (26) would probably suggest a different intuition. But, then, a family of theories would be \(\mathfrak {L}\)-indispensable to a family of scientific theories, or a theory to a family of scientific theories, or a family of theories to a scientific theory, just in case this were the case for one theory of the former family and one theory of the latter. Hence, for example, mathematics as a whole would be \(\mathfrak {L}\)-indispensable to a science as a whole, just in case that a mathematical theory were so for a scientific theory; but then it would somehow be misleading to speak of science and mathematics as wholes, rather than speaking of specific theories
See, however, remarks at pp. 38–38, below.
Cf. footnote 22, above.
If ‘a’ is replaced with ‘entities’ or ‘(putative) objects’, then: the a’s of the relevant quantifications will be the entities or putative objects that the corresponding quantifiers range on; the a’s of the relevant entities or putative objects will be these very entities or putative objects; the a’s of the relevant terms will be the entities or putative objects that these terms putatively refer to; the a’s of the relevant constants will be the entities or putative objects that these constants are used to speak of (either by putatively referring to them, or by designating some properties of, or relation among them, or some functions defined on them); finally, the a’s of the relevant putative truths will be the entities or putative objects of which these putative truths are supposed to be true. If ‘a’ is replaced with ‘statements’ or ‘consequences’, then: the a’s of the relevant quantifications will be the corresponding quantified statements; the a’s of the relevant entities or putative putative objects will be the relevant statements concerning these entities or putative objects; the a’s of the relevant terms will be the relevant statements involving these terms; the a’s of the relevant constants will be the relevant statements involving these constants; finally, the a’s of the relevant putative truths will be these very putative truths.
The adjectives ‘internal’ and ‘external’ are reminiscent of other well-known debates. They are reminiscent of the distinction between internalism and externalism about knowledge and justification in epistemology; and they are reminiscent of the distinction between internal and external ontological questions as advanced in Carnap (1950). Although similiarities between our use of these expressions and their use in these other debates may be suggested, and may have driven us in choosing them, our terminology should merely be understood here as technical jargon with no other meaning than the one we have explained.
This is in no way an argument against the possibility of endorsing ia in a different setting, where mathematics is rather conceived as imprecise and/or contingent as empirical sciences. We have no intention to argue against such an approach. We rather confine ourselves to considering more traditional and usual forms of ia, where ‘\(\mathcal {A}\)’ may stand either for \(\vartheta \)-truth or for \(\vartheta \)-justification. We shall just take it that what we shall say in the rest of this paper about these forms of ia also applies, mutatis mutandis, to other forms, agreeing with this alternative setting.
In what follows we shall take the verb ‘to exist’ and its cognates in their external sense, as appears to be customary in the debate on ia.
Again, this is no argument against the possibility of endorsing ia in a different setting. Far for excluding this possibility, we merely confine ourselves to consider more traditional and usual forms of ia, where ‘\(\mathcal {A}\)’ may stand either for existence, or for justifiably ascribed existence, and we take it that what we shall say about these forms of ia also applies, mutatis mutandis, to other forms, agreeing with this alternative setting.
This will become clearer in Sect. 4.1
In first-order language:
In first-order language:
In first-order language:
In first-order language:
If Q’s are not taken to be theories, the forms of these premises have to be slightly changed, but it should not be difficult to see how this could be done in other cases.
Clearly, most of the mathematical theories that we currently take to be indispensable to our scientific theories would not have been such before the seventeenth century, and it is absurd to think that the putative obejcts or statements corresponding to these mathematical theories would not have met the relevant condition then, but now do. It is easy to generalize this line of reasoning: as the well-know argument of the pessimistic meta-induction on scientific theories suggests (cf. Laudan 1981), what we now take to be our best scientific theories are inductively much more likely to be false than true. One could, perhaps, envisage something as absolute indispensability, or indispensability in principle, depending on some final stage of scientific development, where all scientific and mathematical theories will be available at their ultimate level of precision and perfection. Taking this sort of indispensability to be necessary for THE existence of mathematical objects or FOR \(\vartheta \)-truth of mathematical statement may not be so implausible AS SUCH. But this very notion of indispensability in principle will be highly implausible AS AN INGREDIENT OF IA, and contrary to how we have suggested scientific and mathematical theories should be understood TO BE FAITHFUL TO the spirit of THE ARGUMENT.
It is not at all our intention to suggest that the relevant tasks are practical in nature, or limited to some pragmatic purposes. Descriptive tasks may, for example, well be identifed with the task of delivering a literally true description of some external reality. Also in this regard, one could evoke the notion of indispensability in principle mentioned in footnote 51 above. Even if some plausibility were given to this notion, however, appealing to it would be problematic for our construal of the indispensability relation only if it were admitted that such indispensability in principle is independent of the specification of any particular task. But, why should one admit this? Could one not, rather, take it that providing a final description or explanation of what the relevant scientific theories are about is a particular task, namely either a descriptive or an explanatory one? If this is conceded, as we think it should be, our charaterization of indispensability can be made perfectly compatible with this rather implausible (at least in our opinion) conception of science. It would be enough to specify the parameter ‘h’ in agreement with this particular task, and to consider the appropriate class of scientific theories (i.e. those at their ultimate stage) in the argument’s premises.
Putnam’s ia given above (cf. p. 10) can be seen as a paradigmatic example for ia’s where \(\mathfrak {D}_{\beta }\)-indispensability is involved. Resnik’s pragmatic argument can be seen as a way of focusing on \(\mathfrak {P} _{\beta }\)-indispensability. Baker’s version of ia (cf. p. 19, above) is clearly based on \(\mathfrak {S}_{\beta }\)-indispensability (as is the version of ia suggested by Field 1989, p. 15) and will be discussed in more details below. Colyvan’s argument (cf. p. 19, above) is not explicitly relying on a particular notion of indispensability: this seems to us partly due to the fact that the notion of indispensability occurs in it in a schematic way, that should be then made precise according to different determinations. In Panza and Sereni (2015) we suggest how the versions of ia commonly discussed can be retrieved by determination of schematic premises from schematic ia’s cognate to the ones discussed here.
Compare what Baker (2003), p. 58, claims:
Category theory is not an extension of set theory, nor vice versa, and the ontologies of the two theories are entirely non-overlapping. Thus neither set theory nor category theory is indispensable for science, because neither provides a unique foundation for mathematics. Hence we are not rationally compelled to believe in the existence of sets, nor are we rationally compelled to believe in the existence of categories. Our ontological commitment to mathematical objects cannot be made more specific than a disjunctive commitment to sets-or-categories. [...] Commitment to a specific ontology of abstract objects cannot be derived from the indispensability argument alone.
This is not to say, of course, that some known versions of ia could not be read as such. For one, Quine’s long exposition in Chapt. 5 of Quine (1960) can easily be read, and has indeed been read, as outlining a version of indispensability argument for classes. The main reason offered by Quine for being committed just to classes, rather than, say, to classes and natural numbers, is that classes neatly allow for “the explication of the various other sorts of abstract objects” (p. 267). This would make Quine’s argument irrelevant in the present context, where we are looking for arguments based on the descriptive or predictive power of set theory beyond the fact that it allows for the reformulation of other mathematical theories. Quine also adds that “the power of the notion [of class] on other counts, [...] keeps it in continuing demand in mathematics and elsewhere as a working notion in its own”, so that it “confers a power that it is not known to be available through less objectionable channels”. These additional reasons would be relevant here only if it turned out that they did not depend on the power of set theory (the notion of class) to systematize and unify different (mathematical and non mathematical) theories by allowing for their reformulation in terms of classes. Here we just want to stress that it is not so clear that this is the case. Therefore, it is all but clear that Quine’s remarks provide an argument that a genuine ia based on descriptive or predictive indispensability and concerned with set theory as such is forthcoming.
In this scenario, it would still be possible for set theory to be part of a family of mathematical theories which, as such, are ‘\(\mathfrak {D}_{\beta }\)’ or ‘\(\mathfrak {P}_{\beta }\)’-indispensable to a given scientific theory, or to a family of such theories; one could, then, argue that some genuine, non-vacuously and non-circularly sound ia concerned with this family of mathematical theories, whose premise (iii) is intended disjunctively, as explained above at pp. 38–38, may be available. Here, however, the same considerations raised there apply too.
This means, of course, that the relevant specification of the parameters ‘h’ and ‘k’ capture what it is for a theory to provide the best explanation of the relevant accepted claims E’s.
Pincock objects to Baker (cf. Pincock 2012, pp. 212–213) that his alleged explanation fails what Pincock calls the ‘weaker alternatives problem’: for an agent that does not already believe that there are infinitely many primes, there seems to be no reason for preferring an explanation based on the claim that prime periods minimize intersections with predators’ life-cycles, rather than an explanation based on the claim that prime periods less than 100 years minimize intersection. Hence, there is no reason for accepting, on the basis of an application of IBE to the cicada case, that there exist infinitely many prime numbers, rather that merely those less than 100. The point seems well taken in the framework of Pincock’s discussion, and is reminiscent of a similar point raised by Peressini (1997, pp. 220–223) as the problem of “the unit of indispensability”. However, we believe it is possible to object to Baker’s argument even if it is admitted that one can decide whether the theory to be taken into account in the case at issue is arithmetic as such, or a fragment of it only dealing with small natural numbers, for example with those less than 100. Hence, for the sake of our present argument again, we take for granted that a decision between these two alternative theories is possible.
A way of providing such a number-free description has been suggested by Rizza (2011). Baker (2009), pp. 618–622, had previsously suggested that such a description is not possible, since mathematical terms such as ‘prime’ cannot be eliminated from the statement expressing the explanandum in the periodical cicadas case. Notice, in passing, that Baker there also answers to the challenge raised by Bangu (2008) according to which he would be begging the question against the nominalist, since the explanandum must be formulated in mathematical terms (and obviously believed true).
Pincock makes his point in connection with the adoption of a contrastive account of IBE. Our present remarks are indipendent of the adoption of such an account.
References
Baker, A. (2003). The indispensability argument and multiple foundations for mathematics. Philosophical Quarterly, 53(210), 49–8211.
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.
Bangu, S. I. (2008). Inference to the best explanation and mathematical realism. Synthese, 160(1), 13–20.
Benacerraf, P. (1965). What numbers could not be. Philosophical Review, 74(1), 47–73.
Bueno, O., & Shalkowski, S. A. (2015). Empirically grounded philosophical theorizing. In C. Daly (Ed.), The Plagrave handbook of philosophical methods. New York: Palgrave.
Carnap, R. (1939). Foundation of logic and mathematics. International Encyclopedia of Unified Science, The University of Chicago Press, Chicago, I/3.
Carnap, R. (1950). Empiricism, semantics, and ontology. Revue Internationale De Philosophie, 4(2), 20–40.
Church, A. (1951). The need for abstract entities. American Academy of Arts and Sciences Proceedings, 80, 100–113.
Colyvan, M. (2001). The indispensability of mathematics. New York: Oxford U. P.
Enoch, D., & Schechter, J. (2008). How are basic belief-forming methods justified? Philosophy, Philosophy and Phenomenological Research, 76(3), 547–579.
Field, H. (1982). Realism and anti-realism about mathematics. Philosophical Topics, 13(1), 45–69.
Field, H. (1989). Realism, mathematics & modality. Oxford: Basil Blackwell.
Laudan, L. (1981). A confutation of convergent realism. Philosophy of Science, 48(1), 19–49.
Leng, M. (2010). Mathematics and reality. Oxford: Oup Oxford.
Maddy, P. (1992). Indispensability and practice. Journal of Philosophy, 89(6), 275–289.
Maddy, P. (1997). Naturalism in mathematics. Oxford: Oxford University Press.
Maddy, P. (2011). Defending the axioms: On the philosophical foundations of set theory. Oxford: Oxford University Press.
Molinini, D. (2015). Evidence, explanation and enhanced indispensability. In F. Pataut, D. Molinini, A. Sereni (eds.) Synthese, S.I “Indispensability and explanation”, this volume.
Panza, M., & Sereni, A. (2013). Plato’s problem. An introduction to mathematical platonism. Basingstoke (UK): Palgrave MacMillan.
Panza, M., & Sereni, A. (2015). On the indispensable premises of the indispensability argument. In G. Lolli, M. Panza, & G. V (Eds.), From logic to practice (pp. 241–276). Dordrecht: Springer.
Peressini, A. (1997). Troubles with indispensability: Applying pure mathematics in physical theory. Philosophia Mathematica, 5(3), 210–227.
Pincock, C. (2012). Principled limitations on inference to the best explanation. Manuscipt version presented at the Paris workshop on Indispesability and Expnation, November 19th–20th, 2012.
Psillos, S. (1999). Scientific realism. How science tracks truths. New York: Routledge.
Putnam, H. (1971). Philosophy of logic. New York: Harper and Row.
Quine, W. V. O. (1960). Word & object. Cambridge: The Mit Press.
Resnik, M. (1995). Scientific vs. mathematical realism: The indispensability argument. Philosophia Mathematica, 3(3), 166–174.
Resnik, M. (1997). Mathematics as a science of patterns. Oxford: Clarendon Press.
Rizza, D. (2011). Magicicada, mathematical explanation and mathematical realism. Erkenntnis, 74(1), 101–114.
Sereni, A. (2015). Equivalent explanations and mathematical realism. Reply to “Evidence, explanation, and enhanced indispensability”. In D. Molinini, F. Pataut, & A. Sereni (Eds.), Synthese, S.I. “Indispensability and Explanation”, this volume.
Wagner, S. J. (1996). Prospect for platonism. In A. Morton & S. P. Stich (Eds.), Benacerraf and his critics (pp. 73–99). Oxford: Blackwell.
Acknowledgments
The authors would like to thank the projects which financially supported their research for this article. Marco Panza wishes to thank the ANR-DFG Project “Mathematical objectivity by representation”. Andrea Sereni wishes to thank the Italian National Project PRIN 2010 “Realism and Objectivity”.
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Many thanks to the audience of the Indispensability and Explanation workshop held at IHPST in Paris in November 19–20, 2012, where this paper was first presented. We are particularly grateful to Henri Galinon for his useful rejoinder at the conference and for subsequent discussion. This paper has greatly benefitted from the insightful comments and suggestions of the anonymous reviewers of this journal.
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Panza, M., Sereni, A. The varieties of indispensability arguments. Synthese 193, 469–516 (2016). https://doi.org/10.1007/s11229-015-0977-9
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DOI: https://doi.org/10.1007/s11229-015-0977-9