THE VALUE OF SCIENCE 201 



and B and the point D between A and C, then the point D will be 

 between .-1 and B; (4) through a given point there is not more than 

 one parallel to a given straight. 



All four are attributed to intuition, and yet the first is the enuncia- 

 tion of one of the rules of formal logic; the second is a real synthetic 

 a priori judgment, it is the foundation of rigorous mathematical induc- 

 tion; the third is an appeal to the imagination; the fourth is a dis- 

 guised definition. 



Intuition is not necessarily founded on the evidence of the senses; 

 the senses would soon become powerless ; for example, we can not repre- 

 sent to ourselves a chiliagon, and yet we reason by intuition on polygons 

 in general, which include the chiliagon as a particular case. 



You know what Poncelet understood by the principle of continuity. 

 What is true of a real quantity, said Poncelet, should be true of an 

 imaginary quantity; what is true of the hyperbola whose asymptotes 

 are real, should then be true of the ellipse whose asymptotes are imag- 

 inary. Poncelet was one of the most intuitive minds of this century; 

 he was passionately, almost ostentatiously, so ; he regarded the principle 

 of continuity as one of his boldest conceptions, and yet this principle 

 did not rest on the evidence of the senses. To assimilate the hyperbola 

 to the ellipse was rather to contradict this evidence. It was only a sort 

 of precocious and instinctive generalization which, moreover, I have no 

 desire to defend. 



We have then many kinds of intuition ; first, the appeal to the senses 

 and the imagination; next, generalization by induction, copied, so to 

 speak, from the procedures of the experimental sciences; finally, we 

 have the intuition of pure number, whence arose the second of the 

 axioms just enunciated, which is able to create the real mathematical 

 reasoning. I have shown above by examples that the first two can not 

 give us certainty ; but who will seriously doubt the third, who will doubt 

 arithmetic ? 



Now in the analysis of to-day, when one cares to take the trouble 

 to be rigorous, there can be nothing but syllogisms or appeals to this 

 intuition of pure number, the only intuition which can not deceive us. 

 It may be said that to-day absolute rigor is attained. 



IV. 



The philosophers make still another objection: " What you gain in 

 rigor," they say, " you lose in objectivity. You can rise toward your 

 logical ideal only by cutting the bonds which attach you to reality. 

 Your science is infallible, but it can only remain so by imprisoning 

 itself in an ivory tower and renouncing all relation with the external 

 world. From this seclusion it must go out when it would attempt the 

 slightest application." 



