202 POPULAR SCIENCE MONTHLY 



For example, I seek to show that some property pertains to some 

 object whose concept seems to me at first indefinable, because it is intui- 

 tive. At first I fail or must content myself with approximate proofs ; 

 finally I decide to give to my object a precise definition, and this enables 

 me to establish this property in an irreproachable manner. 



" And then," say the philosophers, " it still remains to show that 

 the object which corresponds to this definition is indeed the same made 

 known to you by intuition; or else that some real and concrete object 

 whose conformity with your intuitive idea you believe you immediately 

 recognize corresponds to your new definition. Only then could you 

 affirm that it has the property in question. You have only displaced 

 the difficulty." 



That is not exactly so; the difficulty has not been displaced, it has 

 been divided. The proposition to be established was in reality com- 

 posed of two different truths, at first not distinguished. The first was 

 a mathematical truth, and it is now rigorously established. The second 

 was an experimental verity. Experience alone can teach us that some 

 real and concrete object corresponds or does not correspond to some 

 abstract definition. This second verity is not mathematically demon- 

 strated, but neither can it be, no more than can the empirical laws of 

 the physical and natural sciences. It would be unreasonable to ask 

 more. 



Well, is it not a great advance to have distinguished what long was 

 wrongly confused? Does this mean that nothing is left of this objec- 

 tion of the philosophers? That I do not intend to say; in becoming 

 rigorous, mathematical science takes a character so artificial as to strike 

 every one ; it forgets its historical origins ; we see how the questions can 

 be answered, we no longer see how and why they are put. 



This shows us that logic is not enough; that the science of demon- 

 stration is not all science and that intuition must retain its role as 

 complement, I was about to say, as counterpoise or as antidote of logic. 



I have already had occasion to insist on the place intuition should 

 hold in the teaching of the mathematical sciences. Without it young 

 minds could not make a beginning in the understanding of mathe- 

 matics; they could not learn to love it and would see in it only a vain 

 logomachy; above all, without intuition they would never become 

 capable of applying mathematics. But now I wish before all to speak 

 of the role of intuition in science itself. If it is useful to the student, 

 it is still more so to the creative scientist. 



V. 



We seek reality, but what is reality? The physiologists tell us 

 that organisms are formed of cells; the chemists add that cells them- 

 selves are formed of atoms. Does this mean that these atoms or these 



