Catarina Dutilh-Novaes - Mistakes in mathematical proofs - Perspectives on Scientific Error 2024
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Imre Lakatos famously claimed that mathematical knowledge is produced by a dialectic of ‘proofs and refutations’, whereby a proof-concept is proposed which is then scrutinized to ascertain whether there are counterexamples to individual steps in the proof (local counterexamples) or else to the whole proof (global counterexamples). In my book The Dialogical Roots of Deduction, I further develop this insight in terms of Prover-Skeptic dialogue where Prover tries to prove a conclusion from given premises, while Skeptic critically examines the proof not only looking for counterexamples but also for steps in the proof that are not sufficiently clear.
I submit that the Prover-Skeptic model provides a compelling account of practices of mathematical proof in real-life mathematics. In this talk, I present the Prover-Skeptic model and show how it illuminates practices of proof in mathematics, in particular peer review and how proofs are certified within the relevant mathematical community. I then discuss three examples of Prover-Skeptic interaction in mathematical research: Wiles’ proof of Fermat’s last theorem, a failed proof of the inconsistency of Peano Arithmetic, and a purported proof of the ABC conjecture whose status as a valid or invalid proof has been under debate for over 10 years now.
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