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.

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.