Mathematicians usually try to found mathematics on two principles:[78] one is the principle of confusion between the sign and the thing signified (they call this principle the foundation-stone of the formal theory), and the other is the Principle of the Identity of Discernibles (which they call the principle of the permanence of equivalent forms).

But the truth is that if we set sail on a voyage of discovery with Logic alone at the helm, we must either throw such principles as “the identity of those conceptions which have in common the properties that interest us” and “the principle of permanence” overboard, or, if we do not like to act in such a way to old companions with whom we are so familiar that we can hardly feel contempt for them, at least recognize them clearly as having no logical validity and merely as psychological principles, and reduce them to the humble rank of stewards, to minister to our human weaknesses on the voyage. And then, if we adopt the wise policy of keeping our axioms down to the minimum number, we must refrain from creating or thinking that we are creating new numbers to fill up gaps among the older ones, and thence recognize that our rational numbers are not particular cases of “real” numbers, and so on.

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.