M×Φ

Annals of Mathematics and Philosophy

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This proof which is not one: the problem of individuating mathematical proofs and its impact on their evaluation online version, March 30, 2026

Roy Wagner

Mathematical proofs are often evaluated as individuated entities, represented by a printed text, a lecture, a diagram, or some other single presentation. I question this assumption by showing that the identity of a proof is context-dependent and frequently underdetermined by mathematical content alone. Historical cases such as Cauchy’s proof of Euler’s polyhedron formula, Lakatos’ topological reconstruction of it, Hilbert’s hotel, and Bradwardine’s argument against actual infinity, as well as contemporary cases involving diagrammatic, inductive, and set-theoretic proofs, the Arnold conjecture, and the Poincaré conjecture, show that proof identity may shift with notation, material imagination, audience, background theory, and standards of practice. Rather than proposing universal criteria of individuation, I develop a structuralist-semiotic account in which a proof is understood as a fuzzy network of textual and performative proof-presentations related by partial translations, substitutions, similarities, and differences. On this view, properties such as rigor and explanatory value are not intrinsic properties of a single presentation. They emerge from the choice of a relevant corpus and from the interpretation of relations among its members: formal translations, diagrammatic variants, algebraic or set-theoretic renderings, pedagogical reformulations, complementary arguments, and broader theoretical embeddings. Historical practices in Arabic geometry, Chinese mathematical commentaries, and Sanskrit mathematics further show that proofs need not be conceived as single arguments from premises to conclusion. Treating proofs as relational and sometimes intrinsically plural provides a framework for evaluating mathematical proof that is more faithful to mathematical practice, to the history of mathematics, and to the interpretive work increasingly foregrounded by formalization and proof assistants.

The Philosophical Prospects of Large Language Models in the Future of Mathematics online version, Dec. 31, 2025

Fenner Stanley Tanswell, Ásgeir Berg

In this article, we examine the philosophical implications Large Language Models might have on mathematical practice in the near future. Some prominent researchers argue that Large Language Models will soon have the ability to generate or check proofs, lifting a great burden of human mathematicians. We claim, however, that the implementation of LLM technologies in mathematics is not merely a neutral tool that assists mathematicians to continue on as before, but instead entails a radical change to the practices of mathematics with important philosophical implications. We will argue that we cannot be confident such tools will continue to work as expected, even if they become arbitrarily more reliable than they currently are, and that the kind of justification we get from LLM-generated proofs can never be properly mathematical. We will evaluate solutions to this problem involving either computer verification or human checking and argue that these cannot fix the philosophical gap to give us proper mathematical justification.

Explanations of Mathematical Statements

Joseph Y. Halpern

A definition of what counts as an explanation of a mathematical statement, and when one explanation is better than another, is given. Since all mathematical facts must be true in all causal models, and hence known by an agent, mathematical facts cannot be part of an explanation (under the standard notion of explanation). This problem is solved using impossible possible worlds.

The philosophical significance of algebraic geometry

Alberto Peruzzi

This paper explores the philosophical significance of algebraic geometry by addressing Federigo Enriques' question on the relation between logic and intuition. Through a historical and conceptual analysis, it traces the transition from the Italian school of algebraic geometry to the abstract frameworks developed by Grothendieck and Lawvere. The article highlights how key categorical notions — such as schemes, sheaves, and toposes — transform the interplay between geometry and logic, allowing logical principles to be internalized within geometric structures. It argues that the philosophy of mathematics cannot be reduced to meta-mathematical reflection alone, since algebraic geometry itself generates conceptual innovations with direct philosophical import. Ultimately, the paper shows that algebraic geometry reshapes the foundations of mathematics by dissolving the separation between formal rigor and spatial intuition and providing support to the reasonable effectiveness of "conceptual mathematics". This approach not only provides an answer to Enriques' question but also defines a new sense for the foundations of mathematics, where logical principles are intrinsically linked to the geometric structure of a mathematical universe.

Human mathematics in the age of reasoning machines

Akshay Venkatesh

This essay intertwines reflection on current mathematical practice with discussion of how it may be reshaped by automation. I speculate that the conceptual language of mathematics might undergo drastic rewritings, and look to historical examples for guidance.

Introduction to the English translation of « Il significato della critica dei principi »

Frédéric Patras

In this article, Enriques focuses on the critique of principles and their role in the development of mathematics, a role whose analysis and understanding are inseparable from the adoption of a historical perspective. The questioning and reworking of fundamental concepts emerge as one of the essential components of mathematical progress, providing it with ever more refined and profound tools.