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Reflections on the Foundations of Mathematics

Univalent Foundations, Set Theory and General Thoughts

  • Stefania Centrone
  • Deborah Kant
  • Deniz Sarikaya
Book

Part of the Synthese Library book series (SYLI, volume 407)

Table of contents

  1. Front Matter
    Pages i-xxviii
  2. Current Challenges for the Set-Theoretic Foundations

    1. Front Matter
      Pages 1-1
    2. Mirna Džamonja, Deborah Kant
      Pages 3-26
    3. Laura Fontanella
      Pages 27-42
    4. Claudio Ternullo
      Pages 43-78
    5. Philip D. Welch
      Pages 79-97
  3. What Are Homotopy Type Theory and the Univalent Foundations?

    1. Front Matter
      Pages 99-99
    2. Thorsten Altenkirch
      Pages 101-136
    3. Benedikt Ahrens, Paige Randall North
      Pages 137-150
    4. Ulrik Buchholtz
      Pages 151-172
    5. Anthony Bordg
      Pages 173-189
  4. Comparing Set Theory, Category Theory, and Type Theory

    1. Front Matter
      Pages 221-221
    2. Neil Barton, Sy-David Friedman
      Pages 223-253
    3. Penelope Maddy
      Pages 293-311
  5. Philosophical Thoughts on the Foundations of Mathematics

    1. Front Matter
      Pages 313-313
    2. Roy Wagner
      Pages 381-396
  6. Foundations in Mathematical Practice

    1. Front Matter
      Pages 397-397
    2. Bernhard Fisseni, Deniz Sarikaya, Martin Schmitt, Bernhard Schröder
      Pages 417-436
    3. Lawrence C. Paulson
      Pages 437-453

About this book

Introduction

This edited work presents contemporary mathematical practice in the foundational mathematical theories, in particular set theory and the univalent foundations. It shares the work of significant scholars across the disciplines of mathematics, philosophy and computer science. Readers will discover systematic thought on criteria for a suitable foundation in mathematics and philosophical reflections around the mathematical perspectives.

The volume is divided into three sections, the first two of which focus on the two most prominent candidate theories for a foundation of mathematics. Readers may trace current research in set theory, which has widely been assumed to serve as a framework for foundational issues, as well as new material elaborating on the univalent foundations, considering an approach based on homotopy type theory (HoTT). The third section then builds on this and is centred on philosophical questions connected to the foundations of mathematics. Here, the authors contribute to discussions on foundational criteria with more general thoughts on the foundations of mathematics which are not connected to particular theories.

This book shares the work of some of the most important scholars in the fields of set theory (S. Friedman), non-classical logic (G. Priest) and the philosophy of mathematics (P. Maddy). The reader will become aware of the advantages of each theory and objections to it as a foundation, following the latest and best work across the disciplines and it is therefore a valuable read for anyone working on the foundations of mathematics or in the philosophy of mathematics.

Keywords

Foundations of Mathematics Philosophy of Mathematics Philosophy of Set Theory Univalent foundations Philosophy of Mathematical Practice Advantages univalent foundations New Axioms in Set Theory Comparing foundations of mathematics Advantages set theory foundation Constructivism Martin Löf's intuitionistic type theory Homotopy Type Theory Voevodsky's Revolution Revolution in Mathematics Foundations of automated theorem proving Univalent foundations homotopy type theory Objections to homotopy type theory as a foundation Category theory Maddy on category theory Objections to set theory as a foundation

Editors and affiliations

  • Stefania Centrone
    • 1
  • Deborah Kant
    • 2
  • Deniz Sarikaya
    • 3
  1. 1.Technical University of BerlinBerlinGermany
  2. 2.University of KonstanzKonstanzGermany
  3. 3.University of HamburgHamburgGermany

Bibliographic information