In the same way, I define the operation x + 2 by the relation:
(4) x + 2 = (x + 1) + 1.
That presupposed, we have
| 2 + 1 + 1 = 3 + 1 | (Definition 2), |
| 3 + 1 = 4 | (Definition 3), |
| 2 + 2 = (2 + 1) + 1 | (Definition 4), |
whence
2 + 2 = 4 Q.E.D.
It can not be denied that this reasoning is purely analytic. But ask any mathematician: 'That is not a demonstration properly so called,' he will say to you: 'that is a verification.' We have confined ourselves to comparing two purely conventional definitions and have ascertained their identity; we have learned nothing new. Verification differs from true demonstration precisely because it is purely analytic and because it is sterile. It is sterile because the conclusion is nothing but the premises translated into another language. On the contrary, true demonstration is fruitful because the conclusion here is in a sense more general than the premises.
The equality 2 + 2 = 4 is thus susceptible of a verification only because it is particular. Every particular enunciation in mathematics can always be verified in this same way. But if mathematics could be reduced to a series of such verifications, it would not be a science. So a chess-player, for example, does not create a science in winning a game. There is no science apart from the general.
It may even be said the very object of the exact sciences is to spare us these direct verifications.