Mathematicians, as I have said before, always endeavour to generalise the propositions they have obtained. To seek no further example, we have just shown the equality, a + 1 = 1 + a, and we then used it to establish the equality, a + b = b + a, which is obviously more general. Mathematics may, therefore, like the other sciences, proceed from the particular to the general. This is a fact which might otherwise have appeared incomprehensible to us at the beginning of this study, but which has no longer anything mysterious about it, since we have ascertained the analogies between proof by recurrence and ordinary induction.
No doubt mathematical recurrent reasoning and physical inductive reasoning are based on different foundations, but they move in parallel lines and in the same direction—namely, from the particular to the general.
Let us examine the case a little more closely. To prove the equality a + 2 = 2 + a......(1), we need only apply the rule a + 1 = 1 + a, twice, and write a + 2 = a + 1 + 1 = 1 + a + 1 = 1 + 1 + a = 2 + a......(2).
The equality thus deduced by purely analytical means is not, however, a simple particular case. It is something quite different. We may not therefore even say in the really analytical and deductive part of mathematical reasoning that we proceed from the general to the particular in the ordinary sense of the words. The two sides of
