cerned only with reckoning; or of the notion, almost equally vulgar, that "the mathematician's mind is but a syllogistic mill and that his life resolves itself into a weary repetition of A is B, B is C, therefore A is C; and Q.E.D." (pp. 298-9).
Now, I wish to urge gently that, in so far as the mathematician's mind is not a 'syllogistic mill' he is not 'rigorously' thinking. Mr. Keyser explains, quite truly, I think, (pp. 294-5) that the mathematician is, in his own domain, "matching, point for point, the processes, methods, and experience familiar to the devotee of natural science"; "making guesses regarding the order and internal harmony of the data observed and collocated; testing the hypotheses, not merely by the complete induction peculiar to mathematics, but, like his colleagues of the outer world, resorting also to experimental tests and incomplete induction; frequently finding it necessary, in view of unforeseen disclosures, to abandon a once hopeful hypothesis or to transform it by retrenchment or enlargement." Furthermore (p. 301), "the master mathematician is constantly required to exercise judgment—judgment, that is, in matters not admitting of certainty"—"a function analogous to that of the comparative anatomist like Cuvier." I shall pass over the larger question, whether 'rigorous thinking' is not a contradiction in terms whether, namely, a rigorous nexus of things and a reflective consideration of things do not stand for tendencies in contrary direction; and I will simply ask how one who is engaged in "making guesses," in resorting to "incomplete induction," in correcting "once hopeful" hypotheses, and, finally, in exercising "judgment," "in matters not admitting of certainty," can claim any especial distinction for his thinking on the score of rigor. Mr. Keyser, I fear, has given the whole case away. It seems that the mathematician's mind, so far as it is a genuine human mind and not a 'syllogistic mill,' is just like any other human mind.
Princeton University.
