the same way I define the operation x + 2 by the relation; (4) x + 2 = (x + 1) + 1. Given this, we have:
| 2 + 2 | =(2 + 1) + 1; | (def. 4). |
| (2 + 1) + 1 | = 3 + 1 | (def. 2). |
| 3 + 1 | =4 | (def. 3). |
| whence 2 + 2 | = 4 | Q.E.D. |
It cannot be denied that this reasoning is purely analytical. But if we ask a mathematician, he will reply: "This is not a demonstration properly so called; it is a verification." We have confined ourselves to bringing together one or other of two purely conventional definitions, and we have verified their identity; nothing new has been learned. Verification differs from proof precisely because it is analytical, and because it leads to nothing. It leads to nothing because the conclusion is nothing but the premisses translated into another language. A real proof, on the other hand, is fruitful, because the conclusion is in a sense more general than the premisses. The equality 2 + 2 = 4 can be verified because it is particular. Each individual enunciation in mathematics may be always verified in the same way. But if mathematics could be reduced to a series of such verifications it would not be a science. A chess-player, for instance, does not create a science by winning a piece. There is no science but the science of the general. It may even be said that the object of the exact sciences is to dispense with these direct verifications.
