From Brouwerian Counter Examples to the Creating Subject

The earliest doubts of the validity of the Principle of the Excluded Middle (PEM) can be found in Brouwer’s notes for his dissertation. The actual formulation had to wait for the “Unreliability of the logical principles” paper one year later. It is a curious fact that in the dissertation Brouwer shrugged off the PEM as obvious and ‘uninformative’. He denied it mathematical significance, because “a function is either differentiable or not differentiable says nothing; it expresses the same as the following: ‘If a function is not differentiable, then it is not differentiable.’ ”1 This formulation is all the more surprising as Brouwer rejected at the same time Hilbert’s dogma of the solvability of mathematical problems. Moreover, in the notebooks which he compiled for the dissertation, he explicitly noted that “tertii exclusi” does not always hold”.2 Whatever his reasons may have been for balking at this final step, within a year he clearly and decisively formulated his objections to PEM. He observed that “the question of validity of the principium tertii exclusi is equivalent to the question whether unsolvable mathematical problems can exist. There is not a shred of proof for the conviction, which has sometimes been put forward that there exist no unsolvable mathematical problems.”3 The paper ends with the first Brouwerian counterexamples, i.e. examples of mathematical statements which hold on the basis of PEM, but which cannot be considered as proved from a constructive point of view: