Exploring Human-Computer Interaction in Mathematics: From Voevodsky’s Univalent Foundations of Mathematics to Mochizuki’s IUT-Theoretic Proof of the ABC Conjecture

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AbstractHuman-Computer Interaction research features the human-in-the-loop view of computer science and engineering, or system science in general. Here we argue that the same view applies to mathematics as well as computer science, especially in the context of Voevodsky’s univalent foundations programme, which involves the human-in-the-loop view of mathematics. At the same time we explicate and articulate the difference between Hilbert’s programme and Voevodsky’s programme on foundations of mathematics. We elucidate, inter alia, the foundational significance of Voevodsky’s programme, which arguably aims at structural realist foundations of mathematics rather than reductive idealist foundations as problematised in Hilbert’s programme. Voevodsky’s univalent foundations programme pursues the conceptually transparent and logically certified treatment of mathematics qua Big Data, the whole web of knowledge of which goes far beyond a single mathematician’s comprehension bound by the finitary nature of the human mind, just as ordinary Big Data does; the fundamental reason we need statistical machine learning for Ordinary Big Data analytics is essentially the same as the reason we need univalent foundations for Mathematical Big Data analytics. Voevodsky’s programme is concerned with the computational methodology of ensuring and maintaining the certainty and objectivity of mathematical knowledge, which would arguably help deal with puzzling issues in the mathematical community, such as the recent case of Mochizuki’s Inter-Universal Teichimüller Theoretic proof of the abc conjecture, the formal computer verification of which could resolve the perplexing complications and firmly support its mathematical truth.KeywordsVoevodsky’s Univalent FoundationsHomotopy Type TheoryVoevodsky’s Programme as opposed to Hilbert’s ProgrammeMathematical big dataThe Human-in-the-Loop view of mathematicsInteractive theorem proving/proof assistantFormal verification of mathematicsMochizuki’s Inter-Universal Teichimüller Theoretic Proof of the ABC ConjectureThe nature of mathematical knowledgeThe objectivity and certainty of mathematical truth