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Reflections on the Foundations of Mathematics

Part of the Synthese Library Book Series (volume 407)

Editors: Centrone, Stefania; Kant, Deborah; Sarikaya, Deniz

DOI: 10.1007/978-3-030-15655-8

ISBN: 978-3-030-15654-1

Publisher: Springer International Publishing - Springer Journals

Nov 12, 2019

1 - 10 of 22 Chapters

Available

Interview With a Set Theorist

Džamonja, Mirna; Kant, Deborah

Pages: 3 –26

[The status of independent statements is the main problem in the philosophy of set theory. We address this problem by presenting the perspective of a practising set theorist. We thus give an authentic insight in the current state of thinking in set-theoretic practice, which is to a large extent...
Available

How to Choose New Axioms for Set Theory?

Fontanella, Laura

Pages: 27 –42

[We address the problem of the choice of new axioms for set theory. After discussing some classical views about the notion of axiom in mathematics, we present the most currently debated candidates for a new axiomatisation of set theory, including Large Cardinal axioms, Forcing Axioms and...
Available

Maddy On The Multiverse

Ternullo, Claudio

Pages: 43 –78

[Penelope MaddyMaddy has recently addressed the set-theoretic multiverseset-theoretic multiverse, and expressed reservations on its status and merits (Maddy, Set-theoretic foundations. In: Caicedo et al (eds) Foundations of mathematics. Essays in honor of W. Hugh Woodin’s 60th birthday....
Available

Proving Theorems from Reflection

Welch, Philip D.

Pages: 79 –97

[We review some fundamental questions concerning the real line of mathematical analysis, which, like the Continuum Hypothesis, are also independent of the axioms of set theory, but are of a less ‘problematic’ nature, as they can be solved by adopting the right axiomatic framework. We contend that...
Available

Naïve Type Theory

Altenkirch, Thorsten

Pages: 101 –136

[We introduce Type Theory, in its latest incarnation of Homotopy Type Theory, as an alternative to set theory as a foundation of Mathematics. We emphasize the naïve, intuitive understanding of Type Theory.]
Available

Univalent Foundations and the Equivalence Principle

Ahrens, Benedikt; North, Paige Randall

Pages: 137 –150

[In this paper, we explore the ‘equivalence principle’ (EP): roughly, statements about mathematical objects should be invariant under an appropriate notion of equivalence for the kinds of objects under consideration. In set theoretic foundations, EP may not always hold: for instance,...
Available

Higher Structures in Homotopy Type Theory

Buchholtz, Ulrik

Pages: 151 –172

[The intended model of the homotopy type theories used in Univalent Foundations is the ∞-category of homotopy types, also known as ∞-groupoids. The problem of higher structures is that of constructing the homotopy types needed for mathematics, especially those that aren’t sets. The current...
Available

Univalent Foundations and the UniMath Library

Bordg, Anthony

Pages: 173 –189

[We give a concise presentation of the Univalent Foundations of mathematics outlining the main ideas, followed by a discussion of the UniMath library of formalized mathematics implementing the ideas of the Univalent Foundations (Sect. 8.1), and the challenges one faces in attempting to design a...
Available

Models of HoTT and the Constructive View of Theories

Rodin, Andrei

Pages: 191 –219

[Homotopy Type theory and its Model theory provide a novel formal semantic framework for representing scientific theories. This framework supports a constructive view of theories according to which a theory is essentially characterised by its methods. The constructive view of theories was earlier...
Available

Set Theory and Structures

Barton, Neil; Friedman, Sy-David

Pages: 223 –253

[Set-theoretic and category-theoretic foundations represent different perspectives on mathematical subject matter. In particular, category-theoretic language focusses on properties that can be determined up to isomorphism within a category, whereas set theory admits of properties determined by...

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