“Teacher: ‘Well, then, once more, Shem, Ham and Japheth were Noah’s sons; who was the father of Shem, Ham and Japheth?’
“A long pause; at last a boy, indignant at what he thought the attempted trick, cried out: ‘It couldn’t have been Mr. Smith.’ These boys had never converted the relation of father and son....”
[77] Trans. Camb. Phil. Soc., vol. x., 1864, part ii., note on page 334.
CHAPTER XXIX
PREVIOUS PHILOSOPHICAL THEORIES OF MATHEMATICS
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.