128 SCIENCE AND METHOD. 



is we who have realized that they did not. In the 

 same way our pupils imagine that they know it when 

 they begin to study mathematics seriously. If, with- 

 out any other preparation, I come and say to them : 

 " No, you do not know it ; you do not understand 

 what you imagine you understand ; I must demon- 

 strate to you what appears to you evident ; " and if, 

 in the demonstration, I rely on premises that seem 

 to them less evident than the conclusion, what will 

 the wretched pupils think ? They will think that the 

 science of mathematics is nothing but an arbitrary 

 aggregation of useless subtleties ; or they will lose 

 their taste for it ; or else they will look upon it as 

 an amusing game, and arrive at a state of mind 

 analogous to that of the Greek sophists. 



Later on, on the contrary, when the pupil's mind 

 has been familiarized with mathematical reasoning 

 and ripened by this long intimacy, doubts will spring 

 up of their own accord, and then your demonstration 

 will be welcome. It will arouse new doubts, and 

 questions will present themselves successively to the 

 child, as they presented themselves successively to 

 our fathers, until they reach a point when only perfect 

 exactness will satisfy them. It is not enough to feel 

 doubts about everything; we must know why we doubt. 



8. The principal aim of mathematical education is 

 to develop certain faculties of the mind, and among 

 these intuition is not the least precious. It is through 

 it that the mathematical world remains in touch with 

 the real world, and even if pure mathematics could 

 do without it, we should still have to have recourse 

 to it to fill up the gulf that separates the symbol 



