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.
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 Poincare, 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.
FOOTNOTES:
[78] These principles, after many attempts to state them by Peacock, the
Red and the White Queen (see Appendix P), Hankel, Schroeder, 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._