equal numbers will give identical results.
All these modes of reasoning, whether or not they be reducible to the syllogism properly so called, retain the analytic character, and just because of that are powerless.
II
The discussion is old; Leibnitz tried to prove 2 and 2 make 4; let us look a moment at his demonstration.
I will suppose the number 1 defined and also the operation x + 1 which consists in adding unity to a given number x.
These definitions, whatever they be, do not enter into the course of the reasoning.
I define then the numbers 2, 3 and 4 by the equalities
(1) 1 + 1 = 2; (2) 2 + 1 = 3; (3) 3 + 1 = 4.
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