Conclusions

Abstract

In this chapter we shall take stock of the arguments investigated and answer the questions posed in the introduction. Concentrating upon the issue of the intersubjectivity of mathematical knowledge, we have dealt with the three perhaps best known positions that argue for the intuitionistic revision of mathematics: Brouwer’s, Heyting’s, and Dummett’s. While investigating Brouwer’s and Heyting’s conceptions of intuitionistic mathematics, our query was whether the way they conceive of mathematical constructions, the language of mathematical discourse or the status of laws of logic leads to the consequence that mathematical results are incommunicable. This query has a purely philosophical character, as it concerns conceptions of mathematics and their possible philosophical consequences, neglecting altogether questions of how well intuitionists communicate and whether they do so any better than their rivals. Now, there are two tenets of Brouwer’s and Heyting’s philosophy that purportedly rule out the communicability of mathematical results. The first stresses the mental character of mathematical constructions, arguing that these should be created by means of the intuition, rather then introduced in a language-dependent way. The second tenet adds that the exactness of mathematics cannot be secured by linguistic means, seeing that language is an irreparably imperfect medium for both the communication and the description of mathematical constructions.