Dummettʼs Anti-Realism about Mathematical Statements

Abstract

Just as the accuracy of scientific theories is best tested in extreme physical conditions, it is advisable to verify the accuracy of a recognized conception of language on its extreme parts. Mathematical statements meet this role, thanks to the notion of truth and proof. Michael Dummett’s anti-realism is an enterprise that has attempted on this basis to question the notion of the functioning of language-based primarily on the principle of bivalence, the truth-condition theory of meaning, and the notion that the speaker must be able to demonstrate his knowledge of meaning publicly. In common language practice, one can observe assertions that we can neither verify nor refute in principle. On these so-called undecidable statements, Dummett tried to show that if we apply the traditional description to them, we inevitably reach paradoxical conclusions. Mathematical statements referring to an infinite number may be examples of these assertions. In the submitted paper, I will present Dummett’s position resulting primarily in a manifestation and acquisition argument, according to which it should not be possible to understand undecidable statements at all. In conclusion, however, I will show that his intention – despite many valuable comments – fails, i.e. that there is a way to avoid both arguments while preserving the realistic description of the language in general.

Key words: anti-realism, mathematical statements, meaning, Michael Dummett, truth, truth-condition theory of meaning