Introduction: The Three Foundational Programmes -- Introduction: The Three Foundational Programmes -- Logicism and Neo-Logicism -- Protocol Sentences for Lite Logicism -- Frege’s Context Principle and Reference to Natural Numbers -- The Measure of Scottish Neo-Logicism -- Natural Logicism via the Logic of Orderly Pairing -- Intuitionism and Constructive Mathematics -- A Constructive Version of the Lusin Separation Theorem -- Dini’s Theorem in the Light of Reverse Mathematics -- Journey into Apartness Space -- Relativization of Real Numbers to a Universe -- 100 Years of Zermelo’s Axiom of Choice: What was the Problem with It? -- Intuitionism and the Anti-Justification of Bivalence -- From Intuitionistic to Point-Free Topology: On the Foundation of Homotopy Theory -- Program Extraction in Constructive Analysis -- Brouwer’s Approximate Fixed-Point Theorem is Equivalent to Brouwer’s Fan Theorem -- Formalism -- “Gödel’s Modernism: On Set-Theoretic Incompleteness,” Revisited -- Tarski’s Practice and Philosophy: Between Formalism and Pragmatism -- The Constructive Hilbert Program and the Limits of Martin-Löf Type Theory -- Categories, Structures, and the Frege-Hilbert Controversy: The Status of Meta-mathematics -- Beyond Hilbert’s Reach? -- Hilbert and the Problem of Clarifying the Infinite.

The period in the foundations of mathematics that started in 1879 with the publication of Frege's Begriffsschrift and ended in 1931 with Gödel's Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I can reasonably be called the classical period. It saw the development of three major foundational programmes: the logicism of Frege, Russell and Whitehead, the intuitionism of Brouwer, and Hilbert's formalist and proof-theoretic programme. In this period, there were also lively exchanges between the various schools culminating in the famous Hilbert-Brouwer controversy in the 1920s. The purpose of this anthology is to review the programmes in the foundations of mathematics from the classical period and to assess their possible relevance for contemporary philosophy of mathematics. What can we say, in retrospect, about the various foundational programmes of the classical period and the disputes that took place between them? To what extent do the classical programmes of logicism, intuitionism and formalism represent options that are still alive today? These questions are addressed in this volume by leading mathematical logicians and philosophers of mathematics. The volume will be of interest primarily to researchers and graduate students of philosophy, logic, mathematics and theoretical computer science. The material will be accessible to specialists in these areas and to advanced graduate students in the respective fields.