L. E. J. Brouwer and Karl Popper: Two Perspectives on Mathematics

Abstract

This article provides an appraisal of Karl Popper's criticism of L. E. J. Brouwer's intuitionist mathematics. Although the discussion presented below concerns primarily the problem of intuition (or rather immediate perception) in relation to mathematics, the problem of such pre-linguistic perceptions discussed here is not limited to this. Both Popper and Brouwer agreed that difficulty of sequential or rational reasoning concerning immediate perceptions is a crucial problem for human knowledge. For Popper, language or sequential rationality distorts our very immediate perceptions. Brouwer on the other hand, viewed a greater private accessibility of such intuitions or immediate perceptions, however language, including mathematical language is incapable of providing the means of communicating them to others. Where Brouwer had the objects of mathematical thought in mind, Popper had in mind the theories of scientists as seen from an evolutionary psycho-linguistic perspective. For Popper, such theories are organically related to all acts of rational problem-solving, and indeed, "in-born” or dispositional pre-cognitive problem solving behaviour. As such the problem of intuition or as Popper preferred "unconscious expectations” and its relationship to knowledge is not restricted to mathematics nor the hard sciences as its implications have reverberations in other fields such as aesthetics and morality.