We thus get a world of conceptions which looks, and is, very different from that which ordinary mathematicians think they see; and perhaps this is the reason why some mathematicians of great eminence, such as Hilbert and Poincaré, have produced such absurd discussions on the fundamental principles of mathematics,[79] showing once more the truth of the not quite original remark of Aunt Jane, who
... observed, the second time
She tumbled off a ’bus:
“The step is short from the sublime
To the ridiculous.”
In their readiness to consider many different things as one thing—to consider, for example, the ratio 2:1 as the same thing as the cardinal number 2—such mathematicians as Peacock, Hankel, and Schubert were forestalled by the Pigeon, who thought that Alice and the Serpent were the same creature, because both had long necks and ate eggs.[80] It is, however, doubtful whether the Pigeon would have followed the example of the mathematicians just mentioned so far as to embrace the creed of nominalism and so to feel no difficulty in subtracting from zero—a difficulty which was pointed out with great acuteness by the Hatter[81] and modern mathematical logicians.
[78] These principles, after many attempts to state them by Peacock, the Red and the White Queen (see [Appendix P]), Hankel, Schröder, and Schubert had been made, were first precisely formulated by Frege in Z. S.; cf. also Chapter VII.
[79] See Couturat, R. M. M., vol. xiv., March, 1906, pp. 208-50, and Russell, ibid., September, 1906, pp. 627-34.
[80] See [Appendix P].
[81] See ibid.