Reflections on the Foundations of Mathematics

1. Front Matter

   Pages i-xxviii

   PDF

2. Current Challenges for the Set-Theoretic Foundations

   1. Front Matter

      Pages 1-1

      PDF

   2. Interview With a Set Theorist

      • Mirna Džamonja, Deborah Kant

      Pages 3-26

   3. How to Choose New Axioms for Set Theory?

      • Laura Fontanella

      Pages 27-42

   4. Maddy On The Multiverse

      • Claudio Ternullo

      Pages 43-78

   5. Proving Theorems from Reflection

      • Philip D. Welch

      Pages 79-97

3. What Are Homotopy Type Theory and the Univalent Foundations?

   1. Front Matter

      Pages 99-99

      PDF

   2. Naïve Type Theory

      • Thorsten Altenkirch

      Pages 101-136

   3. Univalent Foundations and the Equivalence Principle

      • Benedikt Ahrens, Paige Randall North

      Pages 137-150

   4. Higher Structures in Homotopy Type Theory

      • Ulrik Buchholtz

      Pages 151-172

   5. Univalent Foundations and the UniMath Library

      • Anthony Bordg

      Pages 173-189

   6. Models of HoTT and the Constructive View of Theories

      • Andrei Rodin

      Pages 191-219

4. Comparing Set Theory, Category Theory, and Type Theory

   1. Front Matter

      Pages 221-221

      PDF

   2. Set Theory and Structures

      • Neil Barton, Sy-David Friedman

      Pages 223-253

   3. A New Foundational Crisis in Mathematics, Is It Really Happening?

      • Mirna Džamonja

      Pages 255-269

   4. A Comparison of Type Theory with Set Theory

      • Ansten Klev

      Pages 271-292

   5. What Do We Want a Foundation to Do?

      • Penelope Maddy

      Pages 293-311

5. Philosophical Thoughts on the Foundations of Mathematics

   1. Front Matter

      Pages 313-313

      PDF

   2. Formal and Natural Proof: A Phenomenological Approach

      • Merlin Carl

      Pages 315-343

   3. Varieties of Pluralism and Objectivity in Mathematics

      • Michèle Friend

      Pages 345-362

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.

• 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

“The volume succeeds in presenting a pleasantly non-biased, balanced, and multi-faceted selection of papers which does not convey the impression that any one particular approach or school would be too dominant. … The volume is well-edited and curated. The many cross-references amongst the contributions help one appreciate the volume as something more than the mere sum of its parts, after all, conveying the feel of an actual ongoing discussion.” (Hans-Christoph Kotzsch, Philosophia Mathematica, Vol. 30 (1), February, 2022)

• Technical University of Berlin, Berlin, Germany

  Stefania Centrone

• University of Konstanz, Konstanz, Germany

  Deborah Kant

• University of Hamburg, Hamburg, Germany

  Deniz Sarikaya

Stefania Centrone is currently Privatdozentin at the University of Hamburg and was deputy Professor of Theoretical Philosophy at the Georg-August-Universität Göttingen. In 2012 she was awarded a DFG-Eigene Stelle for the project Bolzanos und Husserls Weiterentwicklung von Leibnizens Ideen zur Mathesis Universalis at the Carl-von-Ossietzky University of Oldenburg. She is author/editor, among others, of the volume Logic and philosophy of Mathematics in the Early Husserl (Springer 2010) and Studien zu Bolzano (Academia Verlag 2015).

Deborah Kant studied mathematics at Free University and Humboldt University in Berlin, and specialized in set theory and logic. At the DMV Students' Conference 2015 in Hamburg, her talk about her master's thesis “Cardinal Sequences in ZFC” was being awarded. Since 2015, she is a PhD student at the Humboldt University Berlin under the supervision of Karl-Georg Niebergall with a project on naturalness inset theory.

Deniz Sarikaya is PhD-Student of Philosophy (BA: 2012, MA: 2016) and studies Mathematics (BA: 2015) at the University of Hamburg with experience abroad at the Universiteit van Amsterdam and Universidad de Barcelona. He stayed a term as a Visiting Student Researcher at the University of California, Berkeley developing a project on the Philosophy of Mathematical Practice concerning the Philosophical impact of the usage of automatic theorem prover and as a RISE research intern at the University of British Columbia. He is mainly focusing on philosophy of mathematics and logic.

Policies and ethics