The Multiple-Foundations Challenge to the Indispensability Argument: Can the Indispensable Role of Mathematics in Science Determine Which Mathematical Objects Exist?

The Quine–Putnam indispensability argument infers the existence of mathematical objects from the indispensable role of mathematics in our best empirical scientific theories. Alan Baker's 'The Indispensability Argument and Multiple Foundations for Mathematics' (The Philosophical Quarterly, 2003) shows conditionally that if there exist multiple distinct, genuinely adequate, ontologically non-overlapping foundations for the mathematics science uses — for example, ZFC-style set theory (Zermelo–Fraenkel set theory with the axiom of choice, the standard set-theoretic foundation) and a category-theoretic foundation — then the argument cannot establish the existence of any specific collection of mathematical objects, only a disjunctive commitment. Deciding this conditional turns on factors the paper raises but does not settle: the truth of the multiple-foundations (MF) hypothesis itself, including whether category theory can serve as a foundation autonomous of set-theoretic presuppositions; the minimal structural and logical repertoire that scientific theories actually require; the justificatory status of strong mathematical naturalism (whether mathematicians' internal judgments can carry ontological weight sufficient to select a unique foundational ontology); whether such naturalism can be derived from indispensability; and whether internal choices among set-theoretic axioms — large-cardinal axioms, which assert the existence of infinite sets so large that ZFC cannot prove they exist, and the continuum hypothesis, the claim that no set is strictly intermediate in size between the integers and the real numbers — could ever have consequences outside pure mathematics. Later literature has refined but not resolved them: Linnebo and Pettigrew (2011) concede logical and conceptual autonomy for categorical foundations while leaving justificatory autonomy open; Maddy (2019) and Džamonja (2017) treat the choice of foundation as purpose-relative or pluralist; Hamkins (2012), Hunt (2016), Jonas (2024), and Berry (2025) keep the pluralism–indeterminacy interaction actively contested; and reverse mathematics — the program that determines exactly which axioms are needed to prove particular theorems of ordinary mathematics — has begun calibrating the logical strength of the analysis used in physics (Normann and Sanders, 2019) without determining science's minimal structural repertoire. Whether the argument can yield a determinate ontology rather than only a disjunctive commitment remains open.

Background

The indispensability argument is the most influential argument for mathematical Platonism, the view that abstract mathematical objects exist. In its canonical Quine–Putnam form it combines Quinean naturalism — the doctrine that philosophy is continuous with empirical science, so that we ought to accept the ontological commitments of our best scientific theories — with confirmational holism, the thesis that mathematical statements are confirmed along with the scientific theories that indispensably use them. Since our best scientific theories cannot be stated without mathematics (real analysis, infinite-dimensional Hilbert spaces, tensor calculus, group representations), the argument concludes that mathematical objects exist. Hartry Field's 'Science Without Numbers' (1980) launched the nominalist counter-programme of reformulating science without abstract objects; the shape of the whole debate is surveyed in the Stanford Encyclopedia entries on indispensability arguments and on category theory. A second, older worry concerns which mathematical objects we should believe in: Paul Benacerraf (1965) observed that arithmetic can be reduced to set theory in many equally adequate ways that disagree about which sets the numbers are, so multiple reducibility threatens the determinacy of mathematical ontology at the level of identification. Alan Baker (2003) elevated this worry from individual identifications to whole foundations and tied it directly to the indispensability argument. Let $T$ range over adequate formulations of our best scientific theories and $M$ over collections of mathematical objects. The argument's quantificational premise yields only

$$2^*:\ \forall T\, \exists M\,(T\ \text{quantifies over}\ M),$$

while the Platonist conclusion requires the stronger

$$P^*:\ \exists M\, \forall T\,(T\ \text{quantifies over}\ M).$$

Suppose the multiple-foundations (MF) hypothesis holds: there are at least two distinct foundational systems — for example ZFC-style set theory $S$ (Zermelo–Fraenkel set theory with the axiom of choice, the standard membership-based foundation) and a category-theoretic foundation $C$ (a foundation organizing mathematics through categories — algebraic structures of objects and structure-preserving maps — together with functors between categories and universal properties of constructions, rather than through set membership; the best-developed candidates are the Elementary Theory of the Category of Sets, ETCS, and topos- or type-theoretic frameworks, a topos being a category-theoretic generalization of the universe of sets) — each adequate to underpin all the mathematics that science indispensably uses, whose basic ontologies are not mere notational variants of one another. Then for any specific candidate $M$ there is an equally adequate formulation of science avoiding it, so $2^*$ holds while $P^*$ fails: indispensability cannot single out any particular mathematical ontology, and the standard route to Platonism is blocked. Baker argues further that Quine's three strategies for singling out set theory — necessity (nothing else can play the role), sufficiency (sets suffice for everything), and pragmatic superiority — each fail under MF, and that what science actually requires are mathematical structures (the ordered field of the reals, Hilbert-space structure, tensor structure, group structure) rather than particular objects, so that one escape route is mathematical structuralism, in which ontology is relocated from objects to structures. The negative result is explicitly conditional: if MF is false, the indispensability argument may still yield a specific ontology. Whether indispensability, alone or together with demonstrable foundational facts or with mathematical naturalism — the view, associated with Penelope Maddy's 'Naturalism in Mathematics' (1997), that mathematics is answerable to its own internal methodological standards — can yield determinate ontological conclusions therefore depends on a cluster of unresolved questions raised by the paper and sharpened, but not settled, by the subsequent literature.

Problem Statement

Determine whether the indispensable role of mathematics in empirical science can, by itself or in conjunction with demonstrable foundational facts or with mathematical naturalism, yield determinate ontological conclusions about mathematical objects — that is, whether the stronger quantificational claim $P^*$ (a single collection of mathematical objects quantified over by every adequate formulation of our best scientific theories) can be established, and if so, on what grounds. The following interconnected questions, drawn from the topics of Baker (2003) and developed in the subsequent literature, bear on the umbrella question and frame the main lines along which it is contested. (1) The multiple-foundations hypothesis: do there exist at least two distinct, genuinely adequate, ontologically non-overlapping foundational systems — for example, ZFC-style set theory (Zermelo–Fraenkel set theory with the axiom of choice) and a category-theoretic foundation such as ETCS (the Elementary Theory of the Category of Sets, which axiomatizes the category of sets and functions rather than the membership relation) or a univalent foundation (which bases mathematics on a typed formal language rather than on sets) — each capable of underpinning all the mathematics that science indispensably uses, without set-theoretic presupposition? This includes whether categorical foundations are autonomous from set theory in the logical, conceptual, and justificatory senses; whether adequacy for all of applied mathematics can be demonstrated; and whether mutual interpretability of two foundations (each being translatable into the other in a theorem-preserving way) makes them notational variants rather than genuinely distinct ontologies. (2) The structural repertoire of science: which mathematical structures, and how much logical strength, do current scientific theories actually require (at least the structures of real analysis are assumed here, for argument's sake, to be part of that repertoire), and what is the minimal and necessary structural repertoire of current — and insofar as foreseeable, future — scientific theories? (3) Strong mathematical naturalism: can it be shown that mathematicians' internal methodological and pragmatic judgments legitimately carry ontological weight sufficient to select a unique foundational ontology, rather than merely regulating mathematical practice? (4) The derivation question: can some form of indispensability itself justify deferring to mathematicians' internal reasons for ontological purposes, or are indispensability and naturalism independent routes to specificity? (5) The external import of set-theoretic choices: do internal choices among set-theoretic axioms — large-cardinal axioms (axioms asserting the existence of infinite sets so large that their existence cannot be proved in ZFC) and the continuum hypothesis (the claim that there is no set strictly intermediate in cardinality between the integers and the real numbers) — have consequences outside pure mathematics, and could they ever produce empirically relevant differences? A resolution of the umbrella question would settle the determinacy question: whether indispensability yields a determinate ontology, yields only a disjunctive or structural commitment, or yields no ontological conclusion at all.

References

1. Alan Baker, 'The Indispensability Argument and Multiple Foundations for Mathematics', The Philosophical Quarterly 53(210):49–67, 2003. DOI: 10.1111/1467-9213.00295. https://doi.org/10.1111/1467-9213.00295
2. Paul Benacerraf, 'What Numbers Could Not Be', The Philosophical Review 74(1):47–73, 1965. DOI: 10.2307/2183530. https://doi.org/10.2307/2183530
3. Hartry Field, 'Science Without Numbers: A Defence of Nominalism', Princeton University Press, 1980 (2nd edition, Oxford University Press, 2016). DOI: 10.1093/acprof:oso/9780198777915.001.0001. https://doi.org/10.1093/acprof:oso/9780198777915.001.0001
4. Penelope Maddy, 'Naturalism in Mathematics', Clarendon Press, Oxford, 1997. DOI: 10.1093/0198250754.001.0001. https://doi.org/10.1093/0198250754.001.0001
5. Øystein Linnebo and Richard Pettigrew, 'Category Theory as an Autonomous Foundation', Philosophia Mathematica 19(3):227–254, 2011. DOI: 10.1093/philmat/nkr024. https://doi.org/10.1093/philmat/nkr024
6. Joel David Hamkins, 'The Set-Theoretic Multiverse', The Review of Symbolic Logic 5(3):416–449, 2012. DOI: 10.1017/S1755020311000359. https://doi.org/10.1017/S1755020311000359
7. Mirna Džamonja, 'Set Theory and its Place in the Foundations of Mathematics: A New Look at an Old Question', Journal of the Indian Council for Philosophical Research 34(2):415–424, 2017. DOI: 10.1007/s40961-016-0082-6. https://doi.org/10.1007/s40961-016-0082-6
8. Marco Panza and Andrea Sereni, 'The varieties of indispensability arguments', Synthese 193(2):469–516, 2016. DOI: 10.1007/s11229-015-0977-9. https://doi.org/10.1007/s11229-015-0977-9
9. Josh Hunt, 'Indispensability and the problem of compatible explanations', Synthese 193(2):451–467, 2016. DOI: 10.1007/s11229-015-0667-7. https://doi.org/10.1007/s11229-015-0667-7
10. Dag Normann and Sam Sanders, 'On the mathematical and foundational significance of the uncountable', Journal of Mathematical Logic 19(1):1950001, 2019. DOI: 10.1142/S0219061319500016. https://doi.org/10.1142/S0219061319500016
11. Penelope Maddy, 'What Do We Want a Foundation to Do? Comparing Set-Theoretic, Category-Theoretic, and Univalent Approaches', in 'Reflections on the Foundations of Mathematics' (Synthese Library, vol. 407), Springer, 2019, pp. 293–311. DOI: 10.1007/978-3-030-15655-8_13. https://doi.org/10.1007/978-3-030-15655-8_13
12. Silvia Jonas, 'Mathematical Pluralism and Indispensability', Erkenntnis 89(7):2899–2923, 2024. DOI: 10.1007/s10670-023-00659-6. https://doi.org/10.1007/s10670-023-00659-6
13. Sharon Berry, 'Explanatory indispensability and the set theoretic multiverse', Synthese 205(6), 2025. DOI: 10.1007/s11229-025-05059-7. https://doi.org/10.1007/s11229-025-05059-7
14. Chanwoo Lee, 'Category theory as an explanatory foundation', Synthese 207(1), 2026. DOI: 10.1007/s11229-025-05394-9. https://doi.org/10.1007/s11229-025-05394-9
15. Jean-Pierre Marquis, 'Category Theory', Stanford Encyclopedia of Philosophy (first published 1996; substantive revision 29 August 2019). https://plato.stanford.edu/entries/category-theory/
16. Mark Colyvan, 'Indispensability Arguments in the Philosophy of Mathematics', Stanford Encyclopedia of Philosophy (first published 1998; substantive revision 6 March 2023). https://plato.stanford.edu/entries/mathphil-indis/
17. Penelope Maddy, 'Indispensability and Practice', The Journal of Philosophy 89(6):275–289, 1992. DOI: 10.2307/2026712. https://doi.org/10.2307/2026712
18. Elaine Landry, 'Category Theory: The Language of Mathematics', Philosophy of Science 66(S3):S14–S27, 1999. DOI: 10.1086/392712. https://doi.org/10.1086/392712
19. Colin McLarty, 'The Uses and Abuses of the History of Topos Theory', The British Journal for the Philosophy of Science 41(3):351–375, 1990. DOI: 10.1093/bjps/41.3.351. https://doi.org/10.1093/bjps/41.3.351