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Mathematical anti-realism and explanatory structure

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Notes

  1. I do not however claim that it is the first such step: see, e.g., Rosen (2011).

  2. I am grateful to a referee for raising this.

  3. The axioms of PA2 are listed in the appendix.

  4. See, e.g., Dedekind (1888).

  5. That is, for any sentence \(\alpha \) of the language of PA2, either \(\alpha \) or \(\lnot \alpha \) is a logical consequence of PA2.

  6. More explicitly, if \(\varphi \) is atomic, then \(\varphi _C\) is simply \(\varphi \); if \(\varphi \) is \(\lnot \psi \), then \(\varphi _C\) is \(\lnot \psi _C\); if \(\varphi \) is \(\psi \rightarrow \chi \), then \(\varphi _C\) is \(\psi _C \rightarrow \chi _C\); while if \(\varphi \) is \(\forall x\psi \), then \(\varphi _C\) is \(\forall x(Cx \rightarrow \psi _C)\).

  7. In fact, this circularity worry is more pressing in the case of the subsequent strategies of this section: since it is at least arguable that one can know about the merely logical behaviour of numerical claims (under the superficial use) without knowing what this use amounts to (i.e. which paraphrases capture it). However, for the sake uniformity, I give all of the strategies in terms of the fundamental use.

  8. Thus, officially, our language contains two distinct sentences, \({\varvec{\alpha }}\) and \(\alpha \), where the former corresponds to the fundamental use and the latter to the superficial one.

  9. For attempts to use paraphrases along these lines to elucidate standard mathematical claims, see Putnam (1967) and Hellman (1989, 1996).

  10. For this proposal, see Dorr (2008).

  11. I.e. counterfactuals with metaphysically impossible antecedents.

  12. For a proposal along these lines, see Chihara (1973). (Rosen (1990) gives a similar treatment of claims about possible worlds.) Alternative fictionalist approaches to mathematics are given by Field (1989: pp. 1–52) and Yablo (2001). The latter are not versions of paraphrase anti-realism in the sense of this paper: for one thing, mathematical claims are not true according to these approaches. I believe that versions of the arguments of the paper apply to these, but for reasons of space I do not make that case here.

  13. I should note that I understand the notion of ground in a relatively liberal way: to cover any sort of metaphysical, i.e. non-causal, explanation. For example, I take this notion to cover the explanatory relations that hold between mathematical facts. If one objects to this understanding of the notion, then one can replace my talk of grounding with talk simply of metaphysically explaining—nothing essential would be lost. Note in particular that since it is widely accepted that grounding is at least a species of metaphysical explanation, if a given instance does not metaphysically explain a universal generalization, for example, then this instance does not ground the generalization either. Thus, even understood simply in terms of metaphysical explanation, the counterexamples below to the relevant logico-explanatory principles would amount to counterexamples to the widely held principles about grounding.

  14. For such principles see, e.g., Fine (2010) and Rosen (2010).

  15. Here \({\varvec{N}}\) means: is a number; and \({\varvec{s}}\) is the successor function.

  16. I assume for simplicity that \(\beta \rightarrow \gamma \) is defined as \(\lnot \beta \vee \gamma \). The same point could be made without this assumption, but we would then get a violation not of (\(\vee \)R) but of a similar principle about \(\rightarrow \).

  17. See van Inwagen (1990: pp. 108–14) and Dorr and Rosen (2002).

  18. For this problem with the strategy, see Uzquiano (2004). For doubts about superplurals in general see, e.g., Lewis (1991: pp. 70–71) or Rumfitt (2005: p. 102). (Thanks to Berta Grimau here.)

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Acknowledgements

For comments and discussion, I am grateful to Shamik Dasgupta, Louis deRosset, Gary Kemp, Stephan Leuenberger, Adam Rieger, Raul Saucedo, Martin Smith, Zoltán Szabó, an audience at Glasgow University, and two referees for this journal. This work was supported by the Arts and Humanities Research Council [grant number AH/M009610/1].

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  1. Department of Philosophy, University of Wisconsin-Madison, Madison, WI, 53706, USA

    Bruno Whittle

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Appendix: Axioms of PA2

Appendix: Axioms of PA2

Here N means: is a number; and s denotes the successor function.

  1. (i)

    N0

  2. (ii)

    \(\forall x(Nx \rightarrow Nsx)\)

  3. (iii)

    \(\forall x(Nx \rightarrow sx \ne 0)\)

  4. (iv)

    \(\forall x\forall y(Nx \wedge Ny \wedge sx = sy \rightarrow x = y)\)

  5. (v)

    \(\forall x(Nx \rightarrow x+0 = x)\)

  6. (vi)

    \(\forall x\forall y[Nx \wedge Ny \rightarrow x+sy = s(x+y)]\)

  7. (vii)

    \(\forall x(Nx \rightarrow x\times 0 = 0)\)

  8. (viii)

    \(\forall x\forall y[Nx \wedge Ny \rightarrow x\times sy = (x\times y)+x]\)

  9. (ix)

    \(\forall X[X0 \wedge \forall y(Ny \wedge Xy \rightarrow Xsy) \rightarrow \forall y(Ny \rightarrow Xy)]\)

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Whittle, B. Mathematical anti-realism and explanatory structure. Synthese 199, 6203–6217 (2021). https://doi.org/10.1007/s11229-021-03066-y

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