Read more
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.
List of contents
Introduction : Giovanni SommarugaPart I: Senses of foundations of mathematics Bob Hale, The Problem of Mathematical ObjectsGoeffrey Hellman, Foundational FrameworksPenelope Maddy, Set Theory as a FoundationStewart Shapiro, Foundations, Foundationalism, and Category Theory
Part II: Foundations of classical mathematicsSteve Awodey, From Sets to Types, to Categories, to SetsSolomon 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 mathematicsJohn Bell, The Axiom of Choice in the Foundations of MathematicsJim Lambek and Phil Scott, Reflections on a Categorical Foundations of Mathematics
Part IV: Foundations of constructive mathematicsPeter Aczel, Local Constructive Set Theory and Inductive DefinitionsDavid McCarty, Proofs and ConstructionsJohn Mayberry, Euclidean Arithmetic: The Finitary Theory of Finite SetsPaul Taylor, Foundations for Computable TopologyRichard Tieszen, Intentionality, Intuition, and Proof in Mathematics
Summary
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.
