THE FUTURE OF MATHEMATICS. 45 



the natural sciences. For the greater number of us 

 these prejudices had been dissipated, but it has come 

 about that we have run against certain paradoxes and 

 apparent contradictions, which would have rejoiced 

 the heart of Zeno of Elea and the school of Megara. 

 Then began the business of searching for a remedy, 

 each man his own way. For my part I think, and I 

 am not alone in so thinking, that the important thing 

 is never to introduce any entities but such as can be 

 completely defined in a finite number of words. What- 

 ever be the remedy adopted, we can promise ourselves 

 the joy of the doctor called in to follow a fine patho- 

 logical case. 



The Search for Postulates. 



Attempts have been made, from another point of 

 view, to enumerate the axioms and postulates more 

 or less concealed which form the foundation of the 

 different mathematical theories, and in this direction 

 Mr. Hilbert has obtained the most brilliant results. 

 It seems at first that this domain must be strictly 

 limited, and that there will be nothing more to do 

 when the inventory has been completed, which cannot 

 be long. But when everything has been enumerated, 

 there will be many ways of classifying it all. A good 

 librarian always finds work to do, and each new classi- 

 fication will be instructive for the philosopher. 



I here close this review, which I cannot dream of 

 making complete. I think that these examples will 

 have been sufficient to show the mechanism by which 

 the mathematical sciences have progressed in the past, 

 and the direction in which they must advance in the 

 future. 



