117. Geoffrey Hellman | Math Without Numbers

Can we keep everything that makes mathematics rigorous by treating it as the study of possible structures, without committing to a mysterious realm of abstract objects?My links: https://linktr.ee/frictionphilosophy.1. GuestGeoffrey Hellman is Professor of Philosophy at the University of Minnesota. His work focuses on the philosophy of mathematics, logic, science and metaphysics.2. Interview SummaryGeoffrey Hellman frames the philosophy of (pure) mathematics as a contest between a default “objectivist” or Platonist picture—where mathematics is about abstract objects—and a structuralist impulse that treats mathematics as primarily about patterns of relations rather than special entities. They note how, after Richard Dedekind and David Hilbert, this structuralist thought was often sidelined by approaches that re-centered “definite objects,” most notably in Principia Mathematica and related work by Gottlob Frege and Bertrand Russell. Hellman’s guiding complaint is the familiar epistemic worry: if mathematical objects are outside space, time, and causation, what could ground our knowledge of them—so the philosophical pressure is to keep what’s powerful in mathematics without buying a mysterious ontology.Against that background, the Hellman reconstructs the route to “modal structuralism” by weaving together historical foundations work and a modal turn. They highlight the need for a rigorous account of the structuralist idea that axioms characterize any system of objects interrelated in the right way (rather than a single privileged domain), an idea they connect to Howard Stein and especially to Hilary Putnam’s suggestion that foundations should explicitly use necessity/possibility talk—mathematics as the study of possible structures. They then use set theory as the main case study, sketching the standard ZFC story associated with Ernst Zermelo (plus choice) and Abraham Fraenkel (replacement), and motivating an “indefinite extensibility” picture: any given model can be extended to a bigger one, so there’s no coherent “standing outside” all of them at once.Finally, Hellman contrasts this with “face-value” Platonist structuralisms—especially Stuart Shapiro’s ante rem approach—and argues that they inherit a version of Paul Benacerraf’s permutation/uniqueness worries: even if you grant abstract positions, you can permute them (e.g., swap 2 and 3) and get an equally good “number structure,” with no non-arbitrary fact to pick the intended one. On the Hellman’s modal view, the point of arithmetic, set theory, etc. is preserved by talking only about what could exist (structures satisfying the axioms), not about a realm of abstract objects—making the view attractive to nominalist-leaning philosophers and close in spirit to functionalist attitudes about theoretical commitments.3. Interview Chapters00:00 - Introduction00:42 - Overview08:41 - History and development59:21 - Motivating the view1:25:38 - Modal structuralism1:48:53 - Proposed semantics1:54:30 - Other abstracta2:03:20 - Logical possibilities2:11:08 - Logical truths and nominalism2:20:36 - Compatibility with other views2:23:50 - Indispensability arguments2:37:47 - Non-classical mathematics2:58:30 - Classical mathematics with non-classical logic3:01:56 - Value of philosophy3:13:30 - 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