Reflections on the Foundations of Mathematics: Univalent Foundations, Set Theory and General Thoughts

Description

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. Front Matter ....Pages i-xxviii Front Matter ....Pages 1-1 Interview With a Set Theorist (Mirna Džamonja, Deborah Kant)....Pages 3-26 How to Choose New Axioms for Set Theory? (Laura Fontanella)....Pages 27-42 Maddy On The Multiverse (Claudio Ternullo)....Pages 43-78 Proving Theorems from Reflection (Philip D. Welch)....Pages 79-97 Front Matter ....Pages 99-99 Naïve Type Theory (Thorsten Altenkirch)....Pages 101-136 Univalent Foundations and the Equivalence Principle (Benedikt Ahrens, Paige Randall North)....Pages 137-150 Higher Structures in Homotopy Type Theory (Ulrik Buchholtz)....Pages 151-172 Univalent Foundations and the UniMath Library (Anthony Bordg)....Pages 173-189 Models of HoTT and the Constructive View of Theories (Andrei Rodin)....Pages 191-219 Front Matter ....Pages 221-221 Set Theory and Structures (Neil Barton, Sy-David Friedman)....Pages 223-253 A New Foundational Crisis in Mathematics, Is It Really Happening? (Mirna Džamonja)....Pages 255-269 A Comparison of Type Theory with Set Theory (Ansten Klev)....Pages 271-292 What Do We Want a Foundation to Do? (Penelope Maddy)....Pages 293-311 Front Matter ....Pages 313-313 Formal and Natural Proof: A Phenomenological Approach (Merlin Carl)....Pages 315-343 Varieties of Pluralism and Objectivity in Mathematics (Michèle Friend)....Pages 345-362 From the Foundations of Mathematics to Mathematical Pluralism (Graham Priest)....Pages 363-380 Does Mathematics Need Foundations? (Roy Wagner)....Pages 381-396 Front Matter ....Pages 397-397 Foundations for the Working Mathematician, and for Their Computer (Nathan Bowler)....Pages 399-416 How to Frame a Mathematician (Bernhard Fisseni, Deniz Sarikaya, Martin Schmitt, Bernhard Schröder)....Pages 417-436 Formalising Mathematics in Simple Type Theory (Lawrence C. Paulson)....Pages 437-453 Dynamics in Foundations: What Does It Mean in the Practice of Mathematics? (Giovanni Sambin)....Pages 455-494