DocWIP: Gareth Pearce - “How do you pick your axioms?”.

January 18th

The talk will take place on Wednesday, January 18th, 16:45 at 3A, NIG and online. (information about online participation below)

Abstract: In theory, all mathematical proofs are derivations from agreed upon axioms. So the classical definition goes, axioms are self evident truth, the unquestionable foundations of mathematics. For at least a century or so, this classical picture has been untenable. Gödel's first incompleteness theorem has shown us that no (finite and consistent) axiomatic theory can ever resolve all mathematical questions. Moreover, modern Set Theory has given us tools for exploring and understanding numerous rival axiomatic theories and Homotopy Type Theory offers an entirely different approach to axiomatisation all together. Modern mathematicians engaged with the foundations of mathematics face a choice between dozens of rival axiomatic theories. This, then, raises a number of philosophical questions. What are the criteria one should use when picking one's axioms? Are certain axioms objectively true or false, and does this matter for mathematical practice? Do axioms merely need to entail the mathematical truths, or also explain them? Are there normative considerations relevant for axiom selection? Do we even need a single foundation, or would plural foundations be better? In this talk, I'll try and answer as many of these questions as we have time for.


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