THE VALUE OF SCIENCE 199 



their legitimate roles. But it is interesting to study more closely in the 

 history of science the part which belongs to each. 



II. 



Strange ! If we read over the works of the ancients we are tempted 

 to class them all among the intuitionalists. And yet nature is always 

 the same; it is hardly probable that it has begun in this century to 

 create minds devoted to logic. If we could put ourselves into the flow 

 of ideas which reigned in their time, we should recognize that many of 

 the old geometers were in tendency analysts. Euclid, for example, 

 erected a scientific structure wherein his contemporaries could find no 

 fault. In this vast construction, of which each piece however is due 

 to intuition, we may still to-day, without much effort, recognize the 

 work of a logician. 



It is not minds that have changed, it is ideas ; the intuitional minds 

 have remained the same; but their readers have required of them 

 greater concessions. 



What is the cause of this evolution ? It is not hard to find. Intui- 

 tion can not give us rigor, nor even certainty; this has been recognized 

 more and more. Let us cite some examples. We know there exist 

 continuous functions lacking derivatives. Nothing is more shocking 

 to intuition than this proposition which is imposed upon us by logic. 

 Our fathers would not have failed to say : " It is evident that every 

 continuous function has a derivative, since every curve has a tangent." 



How can intuition deceive us on this point? It is because when 

 we seek to imagine a curve, we can not represent it to ourselves without 

 width; just so, when we represent to ourselves a straight line, we see it 

 under the form of a rectilinear band of a certain breadth. We well 

 know these lines have no width ; we try to imagine them narrower and 

 narrower and thus to approach the limit ; so we do in a certain measure, 

 but we shall never attain this limit. And then it is clear we can always 

 picture these two narrow bands, one straight, one curved, in a position 

 such that they encroach slightly one upon the other without crossing. 

 We shall thus be led, unless warned by a rigorous analysis, to conclude 

 that a curve always has a tangent. 



I shall take as second example Dirichlet's principle on which rest 

 so many theorems of mathematical physics; to-day we establish it by 

 reasonings very rigorous but very long; heretofore, on the contrary, 

 we were content with a very summary proof. A certain integral de- 

 pending on an arbitrary function can never vanish. Hence it is con- 

 cluded that it must have a minimum. The flaw in this reasoning 

 strikes us immediately, since we use the abstract term function and 

 are familiar with all the singularities functions can present when the 

 word is understood in the most general sense. 



