Axiomatic Method and Category Theory

  • Andrei┬áRodin

Part of the Synthese Library book series (SYLI, volume 364)

Table of contents

  1. Front Matter
    Pages i-xi
  2. Andrei Rodin
    Pages 1-12
  3. A Brief History of the Axiomatic Method

    1. Front Matter
      Pages 13-14
    2. Andrei Rodin
      Pages 15-37
    3. Andrei Rodin
      Pages 39-72
    4. Andrei Rodin
      Pages 99-143
    5. Back Matter
      Pages 145-146
  4. Identity and Categorification

    1. Front Matter
      Pages 147-147
    2. Back Matter
      Pages 211-212
  5. Subjective Intuitions and Objective Structures

    1. Front Matter
      Pages 213-213
    2. Andrei Rodin
      Pages 215-234
    3. Andrei Rodin
      Pages 235-263
  6. Back Matter
    Pages 273-285

About this book

Introduction

This volume explores the many different meanings of the notion of the axiomatic method, offering an insightful historical and philosophical discussion about how these notions changed over the millennia.

The author, a well-known philosopher and historian of mathematics, first examines Euclid, who is considered the father of the axiomatic method, before moving onto Hilbert and Lawvere. He then presents a deep textual analysis of each writer and describes how their ideas are different and even how their ideas progressed over time. Next, the book explores category theory and details how it has revolutionized the notion of the axiomatic method. It considers the question of identity/equality in mathematics as well as examines the received theories of mathematical structuralism. In the end, Rodin presents a hypothetical New Axiomatic Method, which establishes closer relationships between mathematics and physics.

Lawvere's axiomatization of topos theory and Voevodsky's axiomatization of higher homotopy theory exemplify a new way of axiomatic theory building, which goes beyond the classical Hilbert-style Axiomatic Method. The new notion of Axiomatic Method that emerges in categorical logic opens new possibilities for using this method in physics and other natural sciences.

This volume offers readers a coherent look at the past, present and anticipated future of the Axiomatic Method.

Keywords

Categorical logic David Hilbert Homotopy Type theory Mathematical Structuralism Topos theory Univalent Foundations of Mathematics Vladimir Voevodsky William Lawvere

Authors and affiliations

  • Andrei┬áRodin
    • 1
  1. 1.Department of Liberal ArtsInstitute of Philosophy, Russian Academy of Sciences, Saint-Petersburg, Russia, and State University of Saint-PetersburgSaint-PetersburgRussia

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-00404-4
  • Copyright Information Springer International Publishing Switzerland 2014
  • Publisher Name Springer, Cham
  • eBook Packages Humanities, Social Sciences and Law
  • Print ISBN 978-3-319-00403-7
  • Online ISBN 978-3-319-00404-4
  • About this book