attention; it is of a particular nature which distinguishes it even at this stage from the purely logical definition; the equality (1), in fact, contains an infinite number of distinct definitions, each having only one meaning when we know the meaning of its predecessor.
PROPERTIES OF ADDITION.
Associative.—I say that a + (b + c) = (a + b) + c; in fact, the theorem is true for c = 1. It may then be written a + (b + 1) = (a + b) + 1; which, remembering the difference of notation, is nothing but the equality (1) by which I have just defined addition. Assume the theorem true for c = γ, I say that it will be true for c = γ + 1. Let (a + b) + γ = a + (b + γ), it follows that [(a + b) + γ] + 1 = [a + (b + γ)] + 1; or by def. (1)—(a + b) + (γ + 1) = a + (b + γ + 1) = a + [b + (γ + 1)], which shows by a series of purely analytical deductions that the theorem is true for γ + 1. Being true for c = 1, we see that it is successively true for c = 2, c = 3, etc.
Commutative. (1) I say that a + 1 = 1 + a. The theorem is evidently true for a = 1; we can verify by purely analytical reasoning that if it is true for a = γ it will be true for a = γ + 1.[1] Now, it is true for a = 1, and therefore is true for a = 2, a = 3, and so on. This is what is meant by saying that the proof is demonstrated "by recurrence."
(2) I say that a + b = b + a. The theorem has just
- ↑ For (γ + 1) + 1 = (1 + γ) + 1 = 1 + (γ + 1).—[Tr.]
