(1) a × 1 = a.
(2) a × b = [a × (b − 1)] + a.
Like equality (1), equality (2) contains an infinity of definitions; having defined a × 1, it enables us to define successively: a × 2, a × 3, etc.
Properties of Multiplication.—Distributivity.—I say that
(a + b) × c = (a × c) + (b × c).
We verify analytically that the equality is true for c = 1; then that if the theorem is true for c = γ, it will be true for c = γ + 1.
The proposition is, therefore, demonstrated by recurrence.
Commutativity.—1º I say that
a × 1 = 1 × a.
The theorem is evident for a = 1.
We verify analytically that if it is true for a = α, it will be true for a = α + 1.
2º I say that
a × b = b × a.
The theorem has just been proven for b = 1. We could verify analytically that if it is true for b = β, it will be true for b = β + 1.
IV
Here I stop this monotonous series of reasonings. But this very monotony has the better brought out the procedure which is uniform and is met again at each step.
This procedure is the demonstration by recurrence. We first establish a theorem for n = 1; then we show that if it is true of n − 1, it is true of n, and thence conclude that it is true for all the whole numbers.
We have just seen how it may be used to demonstrate the rules of addition and multiplication, that is to say, the rules of the algebraic calculus; this calculus is an instrument of transformation, which lends itself to many more differing combinations than does the simple syllogism; but it is still an instrument purely analytic, and incapable of teaching us anything new. If mathematics had no other instrument, it would therefore be forthwith arrested in its development; but it has recourse anew to the same procedure, that is, to reasoning by recurrence, and it is able to continue its forward march.
If we look closely, at every step we meet again this mode of reasoning, either in the simple form we have just given it, or under a form more or less modified.
Here then we have the mathematical reasoning par excellence, and we must examine it more closely.