We publish, in this special issue of the Annals of Mathematics and Philosophy devoted to the notion of proof, an English translation of Bourbaki’s Rédaction N. 066, a tentative introduction to the Théorie des ensembles written around 1948 by Claude Chevalley. This introduction situates the text: its author, the Bourbaki collective and its long effort to axiomatise abstract structures, and what makes this redaction stand out — its philosophical reflection on mathematical knowledge, organised around the idea of proof, and its metamathematical stance.

English translation, by Jean-Pierre Marquis and Frédéric Patras, of N. Bourbaki’s Rédaction N. 066 — Livre I, Théorie des ensembles, Introduction, written around 1948 by Claude Chevalley (Archives Bourbaki, Donation André Weil, Archives de l’Académie des sciences, côte AWR 002, 34 p.). Organised around the idea of mathematical proof, this tentative introduction moves from the demand for the reader’s complete assent to its consequences for truth, meaning, the role of language, the axiomatic method and formalism.

This paper aims to compare the notion of discours in Henri Poincaré and that of discourse in Anna Sfard’s theory of commognition, with the aim of clarifying the epistemological differences that distinguish them. In particular, we will investigate how, in Poincaré, discours constitutes the condition for the objectivity of science: only what can be transmitted through a shared language can be considered knowledge. Mathematical discours is a rigorous tool that transforms facts in the rough into scientific facts. The theory of commognition reformulates Poincaré’s concept of discours in a communicative key, departing from its cognitive rigour. A synoptic reading of the founding texts of the two authors, separated by a century of history and philosophy of science, allows us to highlight the points of contact and to note the discrepancies.

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, and Hilbert’s hotel, as well as elementary and advanced contemporary case studies, show that proof identity may shift according to context. Rather than proposing universal criteria of individuation, I present a structuralist-semiotic account in which a proof is understood as a fuzzy network of textual and performative proof-presentations related by partial translations. On this view, properties such as rigor and explanatory value are not intrinsic properties of a single presentation, but emerge from the choice of a relevant corpus and from the interpretation of relations among its members. 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 intrinsically plural provides a framework for evaluating mathematical proofs that is more faithful to mathematical practice, to the history of mathematics, and to the interpretive work increasingly foregrounded by formalization and proof assistants.

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.

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.