Multiple Concepts of Equality in the New Foundations of Mathematics by Vladimir Voevodsky

Talk at: FOMUS 2016, for all Talks and more information see: http://fomus.weebly.com/

Multiple Concepts of Equality in the New Foundations of Mathematics by Vladimir Voevodsky (Institute for Advanced Study, Princeton, USA)

Abstract: One of the most confusing aspects of the Univalence Axiom (UF) is that it seems to assert that, which mathematical students learn early in their education to be a mistake - that isomorphic objects are equal. The source of the confusion is in the failure of the earlier attempts to explain UF to emphasize the existence of two classes of equalities in the theories used to formalize it - substitutional equalities and transportational equalities. The concept of transportational equality is the adaptation to the precise requirements of a formal theory of the philosophical equality principle going back to Leibniz. The concept of the substitutional equality is the one that we all learn at school. In the original formal system used for the Univalent Foundations there was one transportational and one substitutional equality. In the more complex formal systems that are being studied now there can be several equalities of each class.

This workshop was organised with the generous support of the Association for Symbolic Logic (ASL), the Association of German Mathematicians (DMV), the Berlin Mathematical School (BMS), the Center of Interdisciplinary Research (ZiF), the Deutsche Vereinigung für Mathematische Logik und für Grundlagenforschung der Exakten Wissenschaften (DVMLG), the German Academic Merit Foundation (Stipendiaten machen Programm), the Fachbereich Grundlagen der Informatik of the German Informatics Society (GI) and the German Society for Analytic Philosophy (GAP).

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