Thirty Years of Foundational Studies Lectures on the Development of Mathematical Logic and the Study of the Foundations of Mathematics in 1930–1964

Abstract

Publisher Summary This chapter presents sixteen lectures on the development of mathematical logic and the study of the foundations of mathematics in the years 1930–1964, delivered by the author in the Summer School in Vaasa, Finland in the summer of 1964. The chapter distinguishes three major movements in the philosophy of mathematics: the intuitionism of Brouwer, the logicism of Frege and Russell, and the formalism of Hilbert. The logicism of Frege and Russell tries to reduce mathematics to logic. This seemed to be an excellent program, but when it was put into effect, it turned out that there is simply no logic strong enough to encompass the whole of mathematics. Thus what remained from this program is a reduction of mathematics to set theory. The formalism of Hilbert sets up a program which requires that the whole of mathematics be axiomatized and, that these axiomatic theories be then proved consistent by using very simple combinatorial arguments. The chapter discusses the changes that the three schools underwent during the years 1930–960.