Foundational Theories of Classical and Constructive Mathematics

ISBN-13: 9789400735613 / Angielski / Miękka / 2013 / 316 str.

Giovanni Sommaruga

Opis

The book "Foundational Theories of Classical and Constructive Mathematics" is a book on the classical topic of foundations of mathematics. Its originality resides mainly in its treating at the same time foundations of classical and foundations of constructive mathematics. This confrontation of two kinds of foundations contributes to answering questions such as: Are foundations/foundational theories of classical mathematics of a different nature compared to those of constructive mathematics? Do they play the same role for the resp. mathematics? Are there connections between the two kinds of foundational theories? etc. The confrontation and comparison is often implicit and sometimes explicit. Its great advantage is to extend the traditional discussion of the foundations of mathematics and to render it at the same time more subtle and more differentiated. Another important aspect of the book is that some of its contributions are of a more philosophical, others of a more technical nature. This double face is emphasized, since foundations of mathematics is an eminent topic in the philosophy of mathematics: hence both sides of this discipline ought to be and are being paid due to.

Spis treści

Introduction : Giovanni Sommaruga
Part I: Senses of ‚foundations of mathematics’
Bob Hale, The Problem of Mathematical Objects
Goeffrey Hellman, Foundational Frameworks
Penelope Maddy, Set Theory as a Foundation
Stewart Shapiro, Foundations, Foundationalism, and Category Theory

Part II: Foundations of classical mathematics
Steve Awodey, From Sets to Types, to Categories, to Sets
Solomon Feferman, Enriched Stratified Systems for the Foundations of Category TheoryColin McLarty, Recent Debate over Categorical Foundations

Part III: Between foundations of classical and foundations of constructive mathematics
John Bell, The Axiom of Choice in the Foundations of Mathematics
Jim Lambek and Phil Scott, Reflections on a Categorical Foundations of Mathematics

Part IV: Foundations of constructive mathematics
Peter Aczel, Local Constructive Set Theory and Inductive Definitions
David McCarty, Proofs and Constructions
John Mayberry, Euclidean Arithmetic: The Finitary Theory of Finite Sets
Paul Taylor, Foundations for Computable Topology
Richard Tieszen, Intentionality, Intuition, and Proof in Mathematics

Blurb

The book “Foundational Theories of Classical and Constructive Mathematics” is a book on the classical topic of foundations of mathematics. Its originality resides mainly in its treating at the same time foundations of classical and foundations of constructive mathematics. This confrontation of two kinds of foundations contributes to answering questions such as: Are foundations/foundational theories of classical mathematics of a different nature compared to those of constructive mathematics? Do they play the same role for the resp. mathematics? Are there connections between the two kinds of foundations? Etc. The confrontation and comparison is often implicit and sometimes explicit. Its great advantage is to extend the traditional discussion of foundations of mathematics and to render it at the same time more subtle and more differentiated.
Another important aspect of the book is that some of its contributions are of a more philosophical, others of a more technical nature. This double face is emphasized, since foundations of mathematics is an eminent topic in the philosophy of mathematics: hence both sides of this discipline ought to be and are being paid due to.