Overview
- Presents some of the most important scholars in the fields of set theory, univalent foundations and philosophy of mathematics
- Considers criteria for a suitable foundation in mathematics, fostering interdisciplinary discussion
- Brings readers up to date with contemporary work in mathematics, philosophy and computer science
Part of the book series: Synthese Library (SYLI, volume 407)
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About this book
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.
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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
Table of contents (22 chapters)
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Current Challenges for the Set-Theoretic Foundations
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What Are Homotopy Type Theory and the Univalent Foundations?
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Comparing Set Theory, Category Theory, and Type Theory
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Philosophical Thoughts on the Foundations of Mathematics
Reviews
Editors and Affiliations
About the editors
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.
Bibliographic Information
Book Title: Reflections on the Foundations of Mathematics
Book Subtitle: Univalent Foundations, Set Theory and General Thoughts
Editors: Stefania Centrone, Deborah Kant, Deniz Sarikaya
Series Title: Synthese Library
DOI: https://doi.org/10.1007/978-3-030-15655-8
Publisher: Springer Cham
eBook Packages: Religion and Philosophy, Philosophy and Religion (R0)
Copyright Information: Springer Nature Switzerland AG 2019
Hardcover ISBN: 978-3-030-15654-1Published: 20 November 2019
Softcover ISBN: 978-3-030-15657-2Published: 20 November 2020
eBook ISBN: 978-3-030-15655-8Published: 11 November 2019
Series ISSN: 0166-6991
Series E-ISSN: 2542-8292
Edition Number: 1
Number of Pages: XXVIII, 494
Number of Illustrations: 24 b/w illustrations
Topics: Philosophy of Mathematics, Mathematics of Computing, Mathematical Logic and Foundations, Theoretical, Mathematical and Computational Physics, Mathematical Logic and Formal Languages