TEE VALUE OF SCIENCE 197 



Intuition and Logic in Mathematics 



It is impossible to stud}' the works of the great mathematicians, or 

 even those of the lesser, without noticing and distinguishing two op- 

 posite tendencies, or rather two entirely different kinds of minds. The 

 one sort are above all preoccupied with logic; to read their works, one 

 is tempted to believe they have advanced only step by step, after the 

 manner of a Vauban who pushes on his trenches against the place 

 besieged, leaving nothing to chance. The other sort are guided by 

 intuition and at the first stroke make quick but sometimes precarious 

 conquests, like bold cavalrymen of the advance guard. 



The method is not imposed by the matter treated. Though one 

 often says of the first that they are analysts and calls the others 

 geometers, that does not prevent the one sort from remaining analysts 

 even when they work at geometry, while the others are still geometers 

 even when they occupy themselves with pure analysis. It is the very 

 nature of their mind which makes them logicians or intuitionalists, 

 and they can not lay it aside when they approach a new subject. 



Xor is it education which has developed in them one of the two 

 tendencies and stifled the other. The mathematician is born, not made, 

 and it seems he is born a geometer or an analyst. I should like to cite 

 examples and there are surely plenty; but to accentuate the contrast I 

 shall begin with an extreme example, taking the liberty of seeking it 

 in two living mathematicians. 



M. Meray wants to prove that a binomial equation always has a 

 root, or, in ordinary words, that an angle may always be subdivided. 

 If there is any truth that we think we know by direct intuition, it is 

 this. Who could doubt that an angle may always be divided into any 

 number of equal parts? M. Meray does not look at it that way; in 

 his eyes this proposition is not at all evident and to prove it he needs 

 several pages. 



On the other hand, look at Professor Klein: he is studying one of 

 the most abstract questions of the theory of functions to determine 

 whether on a given Eiemann surface there always exists a function 

 admitting of given singularities. What does the celebrated German 

 geometer do? He replaces his Eiemann surface by a metallic surface 

 whose electric conductivity varies according to certain laws. He con- 

 nects two of its points with the two poles of a battery. The current, 

 says he, must pass, and the distribution of this current on the surface 

 will define a function whose singularities will be precisely those called 

 for by the enunciation. 



Doubtless Professor Klein well knows he has given here only a 



