The most puzzling and intriguing aspect of intuitionism as a philosophy of mathematics is its claim that classical deductive reasoning in mathematics is illegitimate. The two most well-known proponents of this position are L. E. J. Brouwer and Michael Dummett. Both of their criticisms of the use of classical logic in mathematics have, by and large, been taken to depend on the thesis that the principle of bivalence does not apply to mathematical statements; and the difference between these critic…
Read moreThe most puzzling and intriguing aspect of intuitionism as a philosophy of mathematics is its claim that classical deductive reasoning in mathematics is illegitimate. The two most well-known proponents of this position are L. E. J. Brouwer and Michael Dummett. Both of their criticisms of the use of classical logic in mathematics have, by and large, been taken to depend on the thesis that the principle of bivalence does not apply to mathematical statements; and the difference between these criticisms is taken to consist in their grounds for this thesis. Brouwer bases this thesis on an account of the nature of mathematics: it is an activity of mental construction essentially independent of language. Dummett, in contrast, bases the thesis on his anti-realist view of the nature of meaning. ;My first principal argument in this essay shows that the conception of anti-realism just outlined is problematic as a reading of Dummett. To begin with, Dummett explicitly rejects, not only the evidentiary restrictions of this conception, but also, more seriously, the use of specific epistemological theses in a philosophical account of meaning. Moreover, I give an alternative account of anti-realism, consisting of alternative interpretations of the manifestation requirement and of the rejection of realist truth conditions, that do not rely on an epistemology of meaning, and hence, a fortiori, not on evidential restrictions. ;My second principal argument in this essay consists of drawing a consequence of this conception of anti-realism, namely, that the failure of bivalence cannot, by itself, support a cogent criticism of classical logic. The basic reason is that, given my view of anti-realism, the claim that the legitimacy of applying classical reasoning to a class of statements depends on whether bivalence applies to that class already implies that the system of deductive reasoning applying to these statements is non-classical. Hence an argument against classical logic based on this claim is question-begging. ;Finally, I conclude by showing that Dummett's criticism of classical reasoning, as reconstructed by my third argument, fails
