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Abstraction, equality and univalence.

Gabriel Catren1

  • 1Laboratoire SPHERE (UMR 7219), Université Paris Cité - CNRS 5 rue Thomas Mann, 75205 Paris Cedex 13, France.

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Summary
This summary is machine-generated.

The Univalent Foundations Program, using Martin-Löf type theory and univalence, explores propositional equality and indiscernibility. It analyzes Leibniz

Keywords:
abstractionequalityhomotopy-type theoryindiscerniblesunivalent foundations

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Area of Science:

  • Foundations of mathematics
  • Logic
  • Computer science

Background:

  • The Univalent Foundations Program (UFP) offers a novel approach to the foundations of mathematics.
  • It integrates Martin-Löf type theory with a homotopic interpretation and Voevodsky's univalence axiom.
  • This framework provides a new perspective on fundamental mathematical concepts.

Purpose of the Study:

  • To provide a conceptual discussion of the Univalent Foundations Program.
  • To analyze the relationship between propositional equality and indiscernibility within type theory.
  • To investigate the validity of Leibniz's principle of the identity of indiscernibles in UFP.

Main Methods:

  • Conceptual analysis of Martin-Löf type theory enriched with homotopic interpretation.
  • Examination of the univalence axiom and its implications.
  • Philosophical analysis of Leibniz's principle within the UFP framework.

Main Results:

  • Propositional equality in type theory encodes the notion of indiscernibility.
  • The homotopic interpretation of Martin-Löf type theory is addressed.
  • Univalence is presented as a constructive abstraction resolving Fregean abstraction.

Conclusions:

  • Univalent Foundations provides a framework where propositional equality aligns with indiscernibility.
  • The univalence axiom offers a novel perspective on abstraction and identity.
  • The study contributes to the understanding of identity and individuality in mathematics.