Abstract
The indispensability argument is a method for showing that abstract mathematical objects exist (call this mathematical Platonism). Various versions of this argument have been proposed (§1). Lately, commentators seem to have agreed that a holistic indispensability argument (§2) will not work, and that an explanatory indispensability argument is the best candidate. In this paper I argue that the dominant reasons for rejecting the holistic indispensability argument are mistaken. This is largely due to an overestimation of the consequences that follow from evidential holism. Nevertheless, the holistic indispensability argument should be rejected, but for a different reason (§3)—in order that an indispensability argument relying on holism can work, it must invoke an unmotivated version of evidential holism. Such an argument will be unsound. Correcting the argument with a proper construal of evidential holism means that it can no longer deliver mathematical Platonism as a conclusion: such an argument for Platonism will be invalid. I then show how the reasons for rejecting the holistic indispensability argument importantly constrain what kind of account of explanation will be permissible in explanatory versions (§4).
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Notes
If most mathematical realists are Platonists, this is because most people agree that it “appears to make no sense to ask where numbers or sets are located, or when they came into existence” (Hale 1994, p. 299). One exception is Penelope Maddy, who attributes physical/causal properties to some mathematical objects. Maddy maintains that sets can be perceivable (Maddy 1990, pp. 58–63). Michael Resnik (Resnik 1997, pp. 93–95) critically discusses her position.
Clearly, it’s not enough just to stipulate that some claims are ontologically committing while others are only instrumental—appealing to scientific practice might allow us to invoke such a distinction (see Maddy 1990, p. 280), but it doesn’t licence drawing this line arbitrarily. Such a distinction is a consequence of prior philosophical theory; various arguments can be given for putting particular cases of apparent ontological commitment into either the ‘tool’ or ‘result’ categories. For example, perhaps, of all the propositions indispensable to science, the ontologically committing propositions are just those that can act as explanations, or that identify causes, or that are falsifiable, etc. Regardless of how such a distinction is drawn, the present point is that various types of indispensability argument attempt to either remove it or redraw it. (Thanks to an anonymous referee for encouraging this point).
Strictly speaking there are only two distinct theses here, since I show that PT and FT are contapositives.
I do not mean to imply that Hilary Putnam should be understood as having argued for the existence of abstract mathematical objects. Commentators often refer to indispensability arguments as ‘Quine-Putnam indispensability arguments’; David Liggins (2008) makes it perfectly clear that the elision is incorrect, and that any overlap is illusory.
Quine clearly does not place too much emphasis in what scientists think that they are up to when espousing his holism: “the scientist thinks of his experiment as a test specifically of his new hypothesis, but only because this was the sentence he was wondering about and is prepared to reject.” (1990a, p. 14). So while he was concerned to give an accurate explanation of scientific practice, it was not necessary that it be consonant with the descriptions that scientists might use to describe their own activities.
It might be that various Quinean doctrines are supposed to yield CT. Jerry Fodor and Ernie Lepore suggested that something like CT is a consequence of Quine’s dissolution of the analytic/synthetic distinction or his semantic holism, but this seems unlikely (Fodor and Lepore 1992). Samir Okasha (2000) explains why this gets the direction of fit the wrong way round. In general, the Quinean doctrines which could plausibly be employed to produce CT tend to be more controversial than CT itself, so such arguments won’t be straightforwardly motivating to most.
Maddy’s mathematical practice objection has little force if mathematicians either do look to developments in science to see which of their theories is confirmed, or if they have good reasons for not doing so. I have not argued that mathematicians do not take considerations of the applied parts of their theories into account, or that mathematicians should look for empirical support for their theories. Rather, the argument here proceeds by parallel steps with the case against holistic indispensability arguments: if we do take Maddy’s concerns about mathematical practice seriously enough to repudiate holistic indispensability arguments, then the fact that explanatory indispensability arguments offend against those same concerns is equally problematic.
Mathematicians might still employ IBE within mathematics for inferring the confirmation of mathematical theories or propositions from their status as best explanations of mathematical phenomena. Such applications of IBE wouldn’t be sufficient to compel scientific realists to accept the existence of mathematical abstracta since in these inferences the explananda aren’t sufficiently connected to the general motivations of scientific realists (see also Leng 2005; Bangu 2008; Baker 2009). (Thanks to an anonymous referee for encouraging this point).
Penelope Maddy represents the debate this way in her 2005a (see p. 358)
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Acknowledgments
Thanks to Jacob Busch for his feedback on earlier drafts of this work and ongoing encouragement. I’m indebted to the anonymous reviewers for this journal for having provided such useful, clear and incisive comments. I’m also grateful to the editors of this journal for having been so responsive with keeping communications clear, timely and relevant. Many thanks to my colleagues and the audience at the University of Birmingham, where this work was presented to a research seminar, May 24th 2010. Further thanks for discussion and comments go to Darragh Byrne, Sean Cordell, Paul Faulkner, Chris Hookway, Gerry Hough, Mary Leng, David Liggins, Arash Pessian, Joe Melia, Bob Plant, Duncan Pritchard, Kirk Surgener, David Walker, Nick Wiltsher and Rich Woodward.
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Morrison, J. Evidential Holism and Indispensability Arguments. Erkenn 76, 263–278 (2012). https://doi.org/10.1007/s10670-011-9300-4
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DOI: https://doi.org/10.1007/s10670-011-9300-4