2oo POPULAR SCIENCE MONTHLY 



But it would not be the same had we used concrete images, had we, 

 for example, considered this function as an electric potential ; it would 

 have been thought legitimate to affirm that electrostatic equilibrium 

 can be attained. Yet perhaps a physical comparison would have 

 awakened some vague distrust. But if care had been taken to trans- 

 late the reasoning into the language of geometry, intermediate between 

 that of analysis and that of physics, doubtless this distrust would not 

 have been produced, and perhaps one might thus, even to-day, still 

 deceive many readers not forewarned. 



Intuition, therefore, does not give us certainty. This is why the 

 evolution had to happen ; let us now see how it happened. 



It was not slow in being noticed that rigor could not be introduced 

 in the reasoning unless first made to enter into the definitions. For 

 the most part the objects treated of by mathematicians were long ill 

 defined ; they were supposed to be known because represented by means 

 of the senses or the imagination; but one had only a crude image of 

 them and not a precise idea on which reasoning could take hold. It 

 was there first that the logicians had to direct their efforts. 



So, in the case of incommensurable numbers. The vague idea of 

 continuity, which we owe to intuition, resolved itself into a complicated 

 system of inequalities referring to whole numbers. 



By that means the difficulties arising from passing to the limit, or 

 from the consideration of infinitesimals, are finally removed. To-day 

 in analysis only whole numbers are left or systems, finite or infinite, 

 of whole numbers bound together by a net of equality or inequality 

 relations. Mathematics, as they say, is arithmetized. 



III. 



A first question presents itself. Is this evolution ended? Have 

 we finally attained absolute rigor? At each stage of the evolution our 

 fathers also thought they had reached it. If they deceived themselves, 

 do we not likewise cheat ourselves? 



We believe that in our reasonings we no longer appeal to intuition ; 

 the philosophers will tell us this is an illusion. Pure logic could never 

 lead us to anything but tautologies; it could create nothing new; not 

 from it alone can any science issue. In one sense these philosophers 

 are right; to make arithmetic, as to make geometry, or to make any 

 science, something else than pure logic is necessary. To designate 

 this something else we have no word other than intuition. But how 

 many different ideas are hidden under this same word? 



Compare these four axioms: (1) Two quantities equal to a third 

 are equal to one another ; ( 2 ) if a theorem is true of the number 1 and 

 if we prove that it is true of n -f- 1 if true for n, then will it be true 

 of all whole numbers; (3) if on a straight the point C is between A 



