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prove, rigidly prove, almost any proposition about
whole numbers. For example, in order to prove
X + Y = Y + X whatever whole numbers
X and Y may be, we first prove that it is so
when Y = 1. That is we first prove X + 1 =
1 + X. To prove this we begin by remarking that it necessarily
is so when X = 1. For then it is merely 1 + 1 = I + 1. Now suppose it to be true of any
whole number N that N + 1 = 1 + N. Then
it is true also of 1 + N. That is (1 + N) + 1 = 1 + (1 + N).
For (1 + N) + 1 = 1 + (N + 1). I suppose that to
be proved already. And since N + 1 = 1 + N,
1 + (N + 1) = 1 + (1 + N). So then X + 1 = 1 + X
if true for any one value of X is true of the next greater value.
But it is true when X = 1. Hence it is true for all values.