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Conference Papers Year : 2022

From the Universality of Mathematical Truth to the Interoperability of Proof Systems

Abstract

The development of computerized proof systems, such as Coq, Matita, Agda, Lean, HOL 4, HOL Light, Isabelle/HOL, Mizar, etc. is a major step forward in the never ending quest of mathematical rigor. But it jeopardizes the universality of mathematical truth [5]: we used to have proofs of Fermat's little theorem, we now have Coq proofs of Fermat's little theorem, Isabelle/HOL proofs of Fermat's little theorem, PVS proofs of Fermat's little theorem, etc. Each proof system: Coq, Isabelle/HOL, PVS, etc. defining its own language for mathematical statements and its own truth conditions for these statements. This crisis can be compared to previous ones, when mathematicians have disagreed on the truth of some mathematical statements: the discovery of the incommensurability of the diagonal and side of a square, the introduction of infinite series, the non-Euclidean geometries, the discovery of the independence of the axiom of choice, and the emergence of constructivity. All these past crises have been resolved.
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Dates and versions

hal-03959359 , version 1 (27-01-2023)

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Gilles Dowek. From the Universality of Mathematical Truth to the Interoperability of Proof Systems. IJCAR 2022 - International Joint Conference on Automated Reasoning, Aug 2022, Haifa, Israel. ⟨10.1007/978-3-031-10769-6_2⟩. ⟨hal-03959359⟩
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