NUMBEK AND MAGNITUDE 635 



syllogisms as we wished. It is only when it is a question of a single 

 formula to embrace an infinite number of syllogisms that this prin- 

 ciple breaks down, and there, too, experiment is powerless to aid. 

 This rule, inaccessible to analytical proof and to experiment, is the 

 exact type of the a priori synthetic intuition. On the other hand, we 

 cannot see in it a convention as in the case of the postulates of geom- 

 etry. 



Why then is this view imposed upon us with such an irresistible 

 weight of evidence? It is because it is only the affirmation of the 

 power of the mind which knows it can conceive of the indefinite repe- 

 tition of the same act, when the act is once possible. The mind has a 

 direct intuition of this power, and experiment can only be for it an 

 opportunity of using it, and thereby of becoming conscious of it. 



But it will be said, if the legitimacy of reasoning by recurrence 

 cannot be established by experiment alone, is it so with experiment 

 aided by induction ? We see successively that a theorem is true of the 

 number 1, of the number 2, of the number 3, and so on the law is 

 manifest, we say, and it is so on the same ground that every physical 

 law is true which is based on a very large but limited number of ob- 

 servations. 



It cannot escape our notice that here is a striking analogy with the 

 usual processes of induction. But an essential difference exists. In- 

 duction applied to the physical sciences is always uncertain, because 

 it is based on the belief in a general order of the universe, an order 

 which is external to us. Mathematical induction i.e., proof by re- 

 currence is, on the contrary, necessarily imposed on us, because it 

 is only the affirmation of a property of the mind itself. 



Mathematicians, as I have said before, always endeavor to generalize 

 the propositions they have obtained. To seek no further example, we 

 have just shown the equality, a+l=l+a, and we then used it to estab- 

 lish the equality, a+&=&+a, which is obviously more general. Math- 

 ematics may, therefore, like the other sciences, proceed from the 

 particular to the general. This is a fact which might otherwise have 

 appeared incomprehensible to us at the beginning of this study, but 

 which has no longer anything mysterious about it, since we have 

 ascertained the analogies between proof by recurrence and ordinary 

 induction. 



No doubt mathematical recurrent reasoning and physical inductive 

 reasoning are based on different foundations, but they move in par- 

 allel lines and in the same direction namely, from the particular to 

 the general. 



Let us examine the case a little more closely. To prove the equality 



a+2=2-f-a (1), we need only apply the rale a-f-l^l+a, twice, 



and write a+2=a+l+l=l+a+l=l+l+a=2+a (2). 



The equality thus deduced by purely analytical means is not, how- 



