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. (shrink)

1. Pohlers and The Problem. I first met Wolfram Pohlers at a workshop on proof theory organized by Walter Felscher that was held in Tübingen in early April, 1973. Among others at that workshop relevant to the work surveyed here were Kurt Schütte, Wolfram’s teacher in Munich, and Wolfram’s fellow student Wilfried Buchholz. This is not meant to slight in the least the many other fine logicians who participated there.2 In Tübingen I gave a couple of survey lectures on results (...) and problems in proof theory that had been occupying much of my attention during the previous decade. The following was the central problem that I emphasized there: The need for an ordinally informative, conceptually clear, proof-theoretic reduction of classical theories of iterated arithmetical inductive definitions to corresponding constructive systems. As will be explained below, meeting that need would be significant for the then ongoing efforts at establishing the constructive foundation for and proof-theoretic ordinal analysis of certain impredicative subsystems of classical analysis. I also spoke in Tübingen about. (shrink)