This proof which is not one: the problem of individuating mathematical proofs and its impact on their evaluation
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.
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