Abstract
Mary Leng has published many spirited, insightful defences of mathematical fictionalism, the view that the claims of mathematics are not literally true. I offer as an alternative an anti-realist approach to mathematics that preserves many of Leng’s valuable insights while ridding fictionalism of its most unpalatable feature, the claim that substantive mathematical claims are “in error”. In making my argument, I first present the virtues of Leng’s fictionalism by considering how she defends it against influential objections due to John Burgess. Leng’s view is roughly that indispensability in science is necessary but not sufficient for believing in the reality of something, and that philosophical analysis can make clear why some things, including mathematics, are necessary for science but not real. I suggest we can accept this without adopting error theory. Marrying features of Leng’s view with constructivism, a quite different sort of anti-realism about mathematics, allows us to: maintain that mathematical assertions are (at least often) literally true, but that it is a mistake to understand them as referring to abstract entities; to be anti-realists about mathematics; and to make use of the fictionalist toolkit Leng supplies for explaining why mathematics is indispensable, even if not real.
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- 1.
The idea of “corrections” to mathematics seems a bit unclear, but I imagine the notion might be captured in this passage from Yablo (2005, 88): “Trembling at the implications, you return to civilization to spread the concrete gospel. Your first stop is [your university here], where researchers are confidently reckoning validity in terms of models and insisting on I-I functions as a condition of equinumerosity. Flipping over some worktables to get their attention, you demand that these practices be stopped at once. These entities do not exist, hence all theoretical reliance on them should cease. They, of course, tell you to bug off and am-scray.”
- 2.
Burgess’s own emphasis.
- 3.
I have chosen to focus on Dummett as I feel mathematical constructivism is a nice complement to fictionalism, which I briefly address later in this section. This is not to say that this is the only way for fictionalism to avoid a commitment to error theory. It has, for instance, been suggested to me that a coherentist approach to truth may also be able to furnish fictionalism with the ability to avoid a commitment to error theory. Although I don’t believe that coherentism can actually furnish fictionalism with this ability, I don’t have the space to address that issue here.
References
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Bériault, P. (2018). A Non-error Theory Approach to Mathematical Fictionalism. In: Zack, M., Schlimm, D. (eds) Research in History and Philosophy of Mathematics. Proceedings of the Canadian Society for History and Philosophy of Mathematics/ Société canadienne d’histoire et de philosophie des mathématiques. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-90983-7_12
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