The practice-based route to anti-nominalism of Burgess and Rosen infers that philosophers ought rationally to believe in mathematical objects from the premises that mathematics is part of science and that core existence theorems are both accepted by mathematicians and acceptable by mathematical standards. Liggins objected that no interpretation of 'acceptable by mathematical standards' keeps the argument sound, and Baker replied with two reconstruction principles: P3, which adds near-universal expert consensus to a permissive reading of mathematical standards, and P4, which reads the standards as requiring professional manifestations of acceptance. Neither principle has been given an independent justification of the normative move it is meant to license. The open problem is to supply, or rigorously rule out, a non-question-begging normative account licensing the move from mathematical acceptability to philosophers' epistemic entitlement; any such account also raises the delimitation of professional manifestations, a scope restriction excluding claims mathematics merely permits by omission, a clarification of the relation between mathematical and scientific standards, and the empirical grounding of the consensus premises.

Background

Nominalism in the philosophy of mathematics is the view that there exist no abstract mathematical objects — no numbers, sets, or functions over and above the concrete physical world; anti-nominalism is its rejection. The best-known argument for anti-nominalism is the indispensability argument: since mathematics is indispensable to our best scientific theories, we ought to believe in the objects those theories quantify over. John Burgess and Gideon Rosen, in their book A Subject with No Objects and in the chapter 'Nominalism Reconsidered' (2005), developed a very different, practice-based route that does not depend on indispensability: roughly, (i) mathematics is part of science; (ii) core existence theorems of mathematics — established theorems asserting that certain mathematical objects exist, such as the infinitude of the primes — are both accepted by working mathematicians and acceptable by mathematical standards; hence (iii) such theorems are acceptable by scientific standards; so (iv) philosophers ought rationally to believe in mathematical objects. David Liggins (2007) objected that no interpretation of the key phrase 'acceptable by mathematical standards' renders the argument sound. On a weak, permissive reading — the standards merely permit acceptance — premise (ii) holds but the inference fails: mere permissibility cannot ground a normative 'ought', the standards may permit mutually incompatible claims, and permissibility would extend far beyond mathematics. On a strong reading — the standards require acceptance — the inference works but premise (ii) becomes implausible, since it would demand that no competent mathematician harbour any private reservations about any established theorem. Alan Baker (2010) replied that two intermediate readings escape this dilemma: (P3) philosophers ought rationally to believe a mathematical claim MM if and only if the standards of mathematics permit the acceptance of MM and all (or nearly all) mathematicians accept MM; and (P4) philosophers ought rationally to believe MM if and only if the standards of mathematics require the professional manifestations of acceptance of MM, where a professional manifestation is an observable act in a recognized professional context — verbal assent in formal settings, use of MM as a premise in proofs, reliance on MM in theoretical and practical mathematical work — so that the standards govern publicly manifested practice rather than private mental states. Two neighbouring positions mark the dialectical space: mathematical naturalism (the view that mathematics is answerable only to its own internal standards, so that external philosophical criticism of established mathematics is misplaced) abandons the normative bridge rather than building it, while the indispensability route grounds belief in mathematical objects in science's use of mathematics rather than in mathematicians' practice. 'Epistemic entitlement' here means warranted belief a philosopher has not earned through their own mathematical inquiry — the standing question of when deference to expert practice yields rational belief for non-practitioners. Baker's published abstract presents the reply as showing how plausible versions of the Burgess–Rosen argument can be constructed on either of the two interpretations of mathematical acceptability; an independent argument for the normative move from mathematical acceptability to philosophers' entitlement has not been given for either principle, and the accompanying questions (delimiting professional manifestations, restricting the permissive reading's scope, clarifying the mathematics–science standards relation, and grounding the consensus premises empirically) remain unaddressed by the cited literature.

Problem Statement

Provide an independently defended answer to the following question, or a rigorous demonstration that no defensible answer exists: under what conditions, if any, does the acceptability of a mathematical claim by mathematical standards ground philosophers' rational entitlement to believe that claim?

The question arises within Baker's reconstruction of the Burgess–Rosen argument. A candidate linking principle (or family of principles) might proceed from the permissive-plus-consensus reading (P3), from the professional-manifestation reading (P4), or from a superior alternative reading, and it must be argued independently, without presupposing anti-nominalism or the truth of any core existence theorem. Depending on the reading adopted, the debate itself raises further questions for any such account: whether a principled restriction can exclude non-mathematical propositions that mathematics permits by omission (propositions about which mathematical standards are simply silent, including contingent empirical claims), or whether a background meta-principle is needed to exclude them; which venues and behaviors count as professional manifestations of acceptance (peer-reviewed publication, lectures, blog postings, use in proofs, reliance in theoretical contexts), and the relevance condition determining in which contexts manifestation of a claim MM is required; how acceptability by mathematical standards relates to the acceptability by scientific standards presupposed by the practice-based argument (the route from mathematicians' acceptance of core existence theorems to philosophers' belief in mathematical objects), in particular whether broad cross-disciplinary standards of science exist and how they relate to discipline-specific methodological norms; and to what extent the premises about mathematicians' actual acceptance or required manifestations — e.g., whether core existence theorems (established theorems asserting that mathematical objects of some kind exist, such as the infinitude of the primes) really are accepted by all or nearly all mathematicians — need empirical grounding, or which parts of the account can stand without it. A principle relying on neither P3 nor P4 still confronts the objections that motivated them: mutually permitted but incompatible claims, and the privately skeptical yet professionally competent mathematician. The objective is to secure, or definitively refute, the practice-based (non-indispensability) route from mathematical practice to anti-nominalism: to determine whether the Burgess–Rosen–Baker argument can be completed soundly along these lines. Any defensible linking principle, or the demonstration that no defensible one exists, must address the four further questions raised above.

References

  1. Alan Baker (2010), 'No Reservations Required? Defending Anti-Nominalism', Studia Logica 96(2): 127–139. DOI: 10.1007/s11225-010-9277-z, https://doi.org/10.1007/s11225-010-9277-z
  2. David Liggins (2007), 'Anti-Nominalism Reconsidered', The Philosophical Quarterly 57(226): 104–111. DOI: 10.1111/j.1467-9213.2007.472.x, https://doi.org/10.1111/j.1467-9213.2007.472.x
  3. Gideon Rosen and John P. Burgess (2005), 'Nominalism Reconsidered', in Stewart Shapiro (ed.), The Oxford Handbook of Philosophy of Mathematics and Logic, Oxford University Press. DOI: 10.1093/oxfordhb/9780195325928.003.0016, https://doi.org/10.1093/oxfordhb/9780195325928.003.0016 (online edition published 2009.)
  4. John P. Burgess and Gideon Rosen (1997), A Subject with No Objects: Strategies for Nominalistic Interpretation of Mathematics, Oxford University Press. DOI: 10.1093/0198250126.001.0001, https://doi.org/10.1093/0198250126.001.0001
  5. John P. Burgess (2022), 'Reconciling Anti-Nominalism and Anti-Platonism in Philosophy of Mathematics', Disputatio. Philosophical Research Bulletin 11(20): 1–19. DOI: 10.63413/disputatio.620, https://doi.org/10.63413/disputatio.620
  6. John P. Burgess (2024), 'Meditations on Second Philosophy: Anti-Nominalist Reflections on Maddy's Semi-Nominalism', in The Philosophy of Penelope Maddy, Outstanding Contributions to Logic, vol. 31, Springer, pp. 201–218. DOI: 10.1007/978-3-031-58425-1_15, https://doi.org/10.1007/978-3-031-58425-1_15
  7. Chris Daly and David Liggins (2011), 'Deferentialism', Philosophical Studies 156(3): 321–337. DOI: 10.1007/s11098-010-9596-y, https://doi.org/10.1007/s11098-010-9596-y (print issue December 2011.)
  8. Jenni Rytilä (2021), 'Social constructivism in mathematics? The promise and shortcomings of Julian Cole's institutional account', Synthese 199(3–4): 11517–11540. DOI: 10.1007/s11229-021-03300-7, https://doi.org/10.1007/s11229-021-03300-7
  9. Athanase Papadopoulos (2024), 'Introduction to Mathematicians' Reflections on Practice and Working Philosophy of Mathematics', in Bharath Sriraman (ed.), Handbook of the History and Philosophy of Mathematical Practice, Springer, Cham, pp. 465–470. DOI: 10.1007/978-3-031-40846-5_128, https://doi.org/10.1007/978-3-031-40846-5_128