Overview of the foundational debate

Abstract

In this chapter, I give an overview of the foundational crisis in mathematics in the beginning of the 20th century. In order to determine the temporal extension and other characteristics of the foundational crisis, we have to look at four factors. There has to be a group of influential mathematicians who (1) reflect on the mathematical process, (2) express doubts concerning the validity of certain methods or results, and (3) request changes in the mathematical process. Furthermore, there has to be (4) a sense of crisis among the participants to the debate.³ The first three criteria have been met from the end of the 19th century on, when Kronecker and others criticised existing mathematical practices and results.⁴ The fourth criterion, however, was only met once intuitionism was presented as a full-blown alternative to classical mathematics, threatening its very existence. From that moment on, the debate on the foundations of mathematics became more emotional and polemical, as seen among other things in the use of metaphors.⁵ Therefore, I have taken the reactions to Brouwer’s intuitionism as a main indicator of the course of the foundational debate. The overview of the foundational crisis presented below is thus limited to those characteristics which can be derived from the reactions to intuitionism.