104. Michael Resnik | Mathematics

Can mathematics be indispensable to science without forcing us to believe in a realm of abstract objects?My links: https://linktr.ee/frictionphilosophy.1. GuestMichael Resnik is Professor Emeritus at UNC Chapel Hill, and his work has focused on the philosophy of mathematics, logic, decision theory, and more.2. Interview SummaryMichael Resnik begins by discussing the indispensability strategy associated with W. V. Quine and developed in an influential way by Hilary Putnam: if our best scientific theorizing essentially uses mathematics, then (given standard views about truth and ontological commitment) we end up committed to mathematical objects. Resnik emphasizes a practical, discourse-focused version of the point: to use the mathematics in scientific (and scientific-adjacent) reasoning, we typically have to assert mathematical claims, and that assertion is what drives commitment—even in idealized modeling or in theories we suspect are literally false. He illustrates this with a simple geometrical case about comparing travel routes (New York → Pittsburgh → DC vs. New York → DC): once you model cities as points and appeal to a theorem about triangles, you’ve already imported mathematical structure and quantification over geometrical entities.From there, the interview turns to structuralism and why Resnik has moved toward a non-ontological form of it. Structuralism, on his view, is best treated as a guiding slogan—mathematics studies patterns/structures, and mathematical “objects” (like numbers) don’t have any identity beyond their place in a structure. He sketches several ways philosophers have tried to make that slogan precise, ranging from set-theoretic reductions to views that locate structural instantiation in the physical world, and then to more explicit “positions-in-structures” accounts (e.g., Stewart Shapiro) and modal structuralism (e.g., Geoffrey Hellman). What pushes him away from settling on a single ontology is his appeal to ontological relativity, underwritten by what he calls the “same size theorem”: roughly, if a theory has a model at all, it can be reinterpreted with a model in any domain of the same cardinality—so the most we can robustly preserve across reinterpretations is structure, not a unique answer to “what the objects really are.”In the final portion, Resnik pivots to logic and defends a comparably deflationary stance: logic has a descriptive side (studying validity relations in a mathematically precise way) but its normative side—talk about what we ought to infer or what counts as a good argument—shouldn’t be conflated with psychology, and may be closer in spirit to something like “applied ethics.” He then distinguishes ordinary acceptance of particular logical claims from logical realism about “logical truth” as an objective, practice-independent status; his “logical anti-realism” denies that further realist step, and he’s open to a kind of logical non-cognitivism on which some familiar logical utterances function more as tools for regulating inferential practice than as straightforward fact-stating claims. He closes with a meta-philosophical note: philosophy matters (and philosophy of math in particular) because it trains clarity about abstract, often-vague questions—even when the “final” metaphysical picture remains elusive.3. Interview Chapters00:00 - Introduction00:54 - Indispensability arguments10:22 - Against Quine12:10 - Indispensability and scientific anti-realism14:49 - Balaguer and denying the truth of theories19:34 - A world without mathematical entities22:50 - Folk discourse30:41 - Structuralism43:01 - Structuralism and ontological relativity44:21 - Same size theorem50:05 - Changes to view51:51 - What are structures?54:14 - Foundations56:30 - Realism and structuralism1:01:17 - What structures are there?1:05:21 - Epistemic concern1:09:35 - Logic as normative/descriptive1:20:05 - Logical anti-realism1:26:42 - Logical non-cognitivism1:30:21 - Motivation for view1:31:49 - Necessity of mathematics1:34:04 - Value of philosophy1:36:05 - Conclusion This is a public episode. If you'd like to discuss this with other subscribers or get access to bonus episodes, visit fric.substack.com/subscribe