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.

Introduction into fundamental concepts of mathematics Explains the main three foundations of mathematics: set theory, category theory and homotopy type theory Depicts the motivation behind homotopy type theory, also with links to philosophical questions of identity

Autorentext

Richard Dedekind (1831-1916) was one of the most significant mathematicians of the 19th century. His work had far-reaching effects on the foundations of mathematics, particularly in algebraic number theory and algebra. His contributions have influenced many other scientists up to the present day and are indispensable in mathematics. Stefan Müller-Stach, works at the Institute of Mathematics at Johannes Gutenberg University Mainz. He specializes in arithmetic and algebraic geometry.

Inhalt

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