THE VALUE OF SCIENCE 203 



cells constitute reality, or rather the sole reality? The way in which 

 these cells are arranged and from which results the unity of the indi- 

 vidual, is not it also a reality much more interesting than that of the 

 isolated elements, and should a naturalist who had never studied the 

 elephant except by means of the microscope think himself sufficiently 

 acquainted with that animal ? 



Well, there is something analogous to this in mathematics. The 

 logician cuts up, so to speak, each demonstration into a very great 

 number of elementary operations ; when we have examined these opera- 

 tions one after the other and ascertained that each is correct, are we 

 to think we have grasped the real meaning of the demonstration ? Shall 

 we have understood it even when, by an effort of memory, we have be- 

 come able to repeat this proof by reproducing all these elementary 

 operations in just the order in which the inventor had arranged them ? 

 Evidently not; we shall not yet possess the entire reality; that I know 

 not what which makes the unity of the demonstration will completely 

 elude us. 



Pure analysis puts at our disposal a multitude of procedures whose 

 infallibility it guarantees; it opens to us a thousand different ways on 

 which we can embark in all confidence ; we are assured of meeting there 

 no obstacles ; but of all these ways, which will lead us most promptly to 

 our goal? Who shall tell us which to choose? We need a faculty 

 which makes us see the end from afar, and intuition is this faculty. 

 It is necessary to the explorer for choosing his route; it is not less so 

 to the one following his trail who wants to know why he chose it. 



If you are present at a game of chess, it will not suffice, for the 

 understanding of the game, to know the rules for moving the pieces. 

 That will only enable you to recognize that each move has been made 

 conformably to these rules, and this knowledge will truly have very 

 little value. Yet this is what the reader of a book on mathematics 

 would do if he were a logician only. To understand the game is wholly 

 another matter; it is to know why the player moves this piece rather 

 than that other which he could have moved without breaking the rules 

 of the game. It is to perceive the inward reason which makes of this 

 series of successive moves a sort of organized whole. This faculty is 

 still more necessary for the player himself, that is, for the inventor. 



Let us drop this comparison and return to mathematics. For 

 example, see what has happened to the idea of continuous function. 

 At the outset this was only a sensible image, for example, that of a con- 

 tinuous mark traced by the chalk on a blackboard. Then it became 

 little by little more refined; ere long it was used to construct a com- 

 plicated system of inequalities, which reproduced, so to speak, all the 

 lines of the original image; this construction finished, the centering 

 of the arch, so to say, was removed, that crude representation which 



