107. Mary Leng | Mathematics

Can we keep the predictive power of mathematics in science while refusing to believe in mathematical objects at all?My links: https://linktr.ee/frictionphilosophy.1. GuestMary Leng is a professor at the University of York, specializing in the philosophy of mathematics and philosophy of science.2. Interview SummaryMary Leng begins by laying out her ‘mathematical functionalist’ (or fictionalist-leaning) view: ordinary mathematical talk looks like it’s about objects—numbers, functions, real-valued magnitudes, infinitely many primes—but she thinks we shouldn’t automatically read that surface grammar as a commitment to a realm of abstract entities. In the interview’s opening framing (including her background in philosophy of mathematics and science and her book Mathematics and Reality), she contrasts this stance with more realist options like Platonism and with structuralist approaches that try to treat mathematics as “about” abstract structures rather than particular objects.A central motivation, she says, is epistemic: the standard “negative characterization” of mathematical objects (not spatiotemporal, not causal, not mental, etc.) makes it hard to give any satisfying story about how we could know truths about them—yet mathematical knowledge is supposed to be among our most secure. So her proposal is to rethink what we’re doing when we do mathematics: instead of aiming at literal truths about abstract objects, we speak as if there are such objects and investigate what would be the case if there were. Along the way she presses familiar trouble for robust Platonism—like “embarrassment of riches” (many distinct set-theoretic reductions can equally play the natural-number role), and the way working mathematicians tend to be relaxed about identity conditions (they don’t worry whether “2” is this set or that set so long as the axioms are satisfied). She also locates herself on the “revolutionary” side of the revolutionary/hermeneutic divide: even if mathematicians often proceed as if they’re talking about objects, that doesn’t settle what the best philosophical interpretation or reform should be.In the later part of the conversation, the focus shifts to science: why mathematical language is so effective, and whether that effectiveness supports realism about mathematical objects. Leng argues that much of mathematics’ role in empirical theory is representational—letting us index and describe patterns in concrete reality—so it’s not surprising that scientific practice could keep working even if there were no mathematical objects “behind” the discourse. That thought underwrites her resistance to “no-miracles”-style arguments for mathematical entities (including a recurring thought experiment about mathematical objects “popping in and out of existence” without affecting successful science). She does grant that mathematics sometimes seems explanatory, and discusses examples meant to push that point (like prime-number life cycles in cicadas), but she maintains that the best lesson is structural: the explanation can run via the way concrete systems instantiate patterns to which theorems apply, without requiring numbers themselves to be causally or ontologically doing explanatory work.3. Interview Chapters00:00 - Introduction00:32 - Mathematical fictionalism09:03 - Characterizing platonism13:42 - Revolutionary vs. hermeneutic17:13 - Empirical semantics22:06 - Ontological commitment27:58 - Thick and thin discourse34:46 - Thin objects42:36 - Progress49:46 - Chess example52:42 - Benacerraf55:03 - Structuralism59:44 - Are structures objects?1:03:10 - Irrelevance of abstracta?1:09:43 - Acausal but not independent?1:16:03 - No miracles argument1:22:06 - Making a difference1:31:33 - Explanation without truth1:34:38 - Value of philosophy1:38:48 - 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