Here is, it must be admitted, a striking analogy with the usual procedures of induction. But there is an essential difference. Induction applied to the physical sciences is always uncertain, because it rests on the belief in a general order of the universe, an order outside of us. Mathematical induction, that is, demonstration by recurrence, on the contrary, imposes itself necessarily because it is only the affirmation of a property of the mind itself.
VII
Mathematicians, as I have said before, always endeavor to generalize the propositions they have obtained, and, to seek no other example, we have just proved the equality:
a + 1 = 1 + a
and afterwards used it to establish the equality
a + b = b + a
which is manifestly more general.
Mathematics can, therefore, like the other sciences, proceed from the particular to the general.
This is a fact which would have appeared incomprehensible to us at the outset of this study, but which is no longer mysterious to us, since we have ascertained the analogies between demonstration by recurrence and ordinary induction.
Without doubt recurrent reasoning in mathematics and inductive reasoning in physics rest on different foundations, but their march is parallel, they advance in the same sense, that is to say, from the particular to the general.
Let us examine the case a little more closely.
To demonstrate the equality
a + 2 = 2 + a
it suffices to twice apply the rule
(1) a + 1 = 1 + a
and write
(2) a + 2 = a + 1 + 1 = 1 + a + 1 = 1 + 1 + a = 2 + a.
The equality (2) thus deduced in purely analytic way from the equality (1) is, however, not simply a particular ease of it; it is something quite different.
We can not therefore even say that in the really analytic and deductive part of mathematical reasoning we proceed from the general to the particular in the ordinary sense of the word.