Abstract
I submit that in Pincock’s (Mathematics and scientific representation, 2012) structural account [SA] the request of a priori justifiability of mathematical beliefs [AP] follows from the adoption of semantic realism for mathematical statements [SR] combined with a form of internalism about mathematical concepts [INTmc]. The resulting framework seems to clash with Pincock’s proposal of an “extension-based” epistemology for pure mathematics [EBE], in that the endorsement of [EBE] seems to ask for a form of conceptual externalism [EXTmc] that would not provide us with the a priori justifications for mathematical beliefs requested. I claim that Pincock’s overall account of pure and applied mathematics would be made more stable if the assumption of [INTmc] was replaced by [EXTmc]. In eliminating [INTmc], [SA] would not entail any necessary commitment to forms of a priori justification for mathematical beliefs anymore, preventing the tension with [EBE]. Someone could object that the combination of [SR] and [EXTmc] would lead to ontological realism for mathematical objects [OR]. I answer by arguing that the kind of [EXTmc] that can be endorsed within Pincock’s framework takes the content of mathematical concepts to be determined by contingent facts in the historical development of mathematical practice, so that no commitment to the existence of mathematical objects is required.
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Notes
- 1.
Pincock calls his approach “structuralist account” (Pincock 2012, p. 27). I prefer to refer to it as the “structural account” not to confuse it with the positions we call ‘structuralists’ in the classical philosophical debate about mathematics.
- 2.
All the abbreviations in square brackets are mine.
- 3.
This does not mean that a realist ontological position would be incompatible with this framework, but only that it is not implicated by Pincock’s account.
- 4.
For example, the scientific content of the law of universal gravitation is made possible by the constitutive framework provided by Newton’s laws of motion, in which mathematical terms constitutively appear - unless, of course, one agrees with Field (1980).
- 5.
It thus seems fair to say that Pincock could in principle accept a form of indispensability argument concerning the truth of mathematical statements (i.e. an argument of the form of the one stated in Pincock (2012), p. 202). However, as we will see when discussing the a priori justifiability request for mathematical beliefs, he does not think that an a posteriori argument can give us conclusive reasons to believe mathematical claims to be true. For Pincock, in order to give to mathematical statements the central role they have in our scientific representations, we need to have reason to believe them to be true prior to their use in science.
- 6.
Pincock even seems to reject internalism for mathematical terms, when, speaking about the rules for the use of mathematical terms, he claims that “a plausible requirement is that these rules must correspond to the genuine features of the things I wind up referring to” using those terms (Pincock 2012, p. 126).
- 7.
See Goldberg (2007) for a review of the philosophical debate between internalism and externalism in epistemology.
- 8.
In this respect, Pincock refers to [SA]’s constitutive frameworks as both “a priori and relative” (Pincock 2012, p. 138).
- 9.
Obviously we are not dealing with the traditional linguistic nor logical conception of ‘extension’ of a concept.
- 10.
- 11.
- 12.
This line of thought could be taken to be reminiscent of the Quinean claim of the impossibility of a clear-cut distinction between the analytic and the factual component of the truth of a statements (Quine 1951).
- 13.
It could be objected that the uncertainty about the evidence for the justification of mathematical beliefs (a priori or not) would not be a problem for Pincock’s account, in that we did not show that the uncertainty occurs in every case, while we only suggested that it occurs in some case—i.e. in the case of the example. But notice that, from a philosophical point of view, the fact of not having the resources to distinguish between a priori and a posteriori evidence in some case means that the account at issue does not provide the resources to distinguish between the two kinds of evidence in every case—and this is the problem we underscore.
- 14.
Frege famously warned us to distinguish the history of a concept from its definition (Frege 1903, Sect. 56–67). Here I do not deny this distinction; I rather claim that in the context of Pincock’s [EBE] we do not seem to be in the position to make this distinction, given that historical aspects seem to enter into the definition of the content of mathematical concepts.
- 15.
The description of Pincock’s epistemological proposal for pure mathematics is difficult to articulate leaving behind the specific features of the examples proposed; this suggests that in this account some prior experience and understanding of particular cases seems to count in the justification of beliefs involving the mathematical concepts at issue.
- 16.
Remember we think about the history of mathematics as an experiential process within mathematical practice.
- 17.
Pincock himself, in a previous work, claimed that a conception of concepts built on Wilson’s open-ended account would abstain from a priori considerations (Pincock 2010, p. 116).
- 18.
Section 6.7.2 outlines an attempt to characterize this particular kind of externalism.
- 19.
Maddy focused her inquiry on the phenomenon of what she labels “mathematical depth” (Maddy 2011, p. 112).
- 20.
In what I take to be a similar perspective, Shapiro talks about possible “changes in meaning” between the various mathematical theories (Shapiro 2014, pp. 320–325).
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Acknowledgments
I would like to thank Francesca Boccuni, Marco Panza, Andrea Sereni, the audience of the First International Conference of the Italian Network for the Philosophy of Mathematics (Milan 2014), the participants of the seminar Séminaire Inter-Universitaire de Philosophie des Mathématiques de Paris 1 et Paris 4 (Paris 2015), and two anonymous referees for their helpful remarks that led me to significantly improve this paper.
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Imocrante, M. (2016). Epistemology, Ontology and Application in Pincock’s Account. In: Boccuni, F., Sereni, A. (eds) Objectivity, Realism, and Proof . Boston Studies in the Philosophy and History of Science, vol 318. Springer, Cham. https://doi.org/10.1007/978-3-319-31644-4_6
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