The Code of Mathematics: Proof and Truth

by Stefan Müller-Stach

Overview

Inspired by recent developments in dependent type theory and infinity categories, this book presents a history of ideas around the topics of truth, proof, equality and equivalence. Besides selected ideas of Platon, Aristoteles, Leibniz, Kant, Frege and others, the results of Gödel and Tarski on incompleteness, undecidability and truth in deductive systems and their semantic models are covered. The main focus of this textbook is on dependent type theory and its recent variant homotopy type theory. Such theories contain identity types, which give a new understanding of equality, symmetry, equivalence and isomorphism in a conceptual way. The interaction of type theory and infinity category theory yields a new paradigm for a structural view on mathematics. This supports the tendencies towards formalising mathematics with the help of proof assistants.

This book was first published in German. The translation was done with the help of artificial intelligence. A subsequent human revision was done primarily in terms of content.

Product Details

ISBN-13:	9783662694824
Publisher:	Springer Berlin Heidelberg
Publication date:	09/04/2024
Series:	Mathematics Study Resources , #11
Edition description:	2024
Pages:	170
Product dimensions:	6.10(w) x 9.25(h) x (d)

About the Author

Stefan Müller-Stach is Vice President for Research and Early Career Academics at Johannes Gutenberg University Mainz and professor for number theory. His main research focus is on algebraic and arithmetic geometry, mathematical physics and history of science.

Table of Contents

Fundamental Questions.- Scientific Languages.- Mathematical Thinking.- Mathematics in our Culture.- Computability and Decidability.- Deductive Systems and Incompleteness.- Category Theory.- Type Theory.- Semantics and Reality.