140 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1909. 
If I do not say more upon this subject it is because I have nothing 
to add after the brilliant lecture by M. Darboux.* 
IX. CANTORISM. 
I have already spoken of the need we have of continually going 
back to the first principles of our science and the profit we may thus 
obtain in the study of the human mind. It is this need which has 
inspired two attempts which hold an important place in the more 
recent part of mathematical history. The first is Cantorism, whose 
services to science we all know. One of the characteristic traits of 
Cantorism is that in place of generalizing and building theorems more 
and more complicated on top of each other and defining by means of 
these constructions themselves, it starts out from the genus supremum 
and defines, as the scholastics would have said, per genus proximum 
et differentiam specificam. What horror would have been brought 
to certain minds—that of Hermite, for instance, whose favorite idea 
‘was comparing the mathematical to the natural sciences! With the 
raost of us these prejudices have passed away, but it still happens that 
we come across certain paradoxes, certain apparent contradictions 
which would have overwhelmed Zénon d’Elée and the school of 
Mégore with joy. I think, and I am not the only one who does, that 
it is important never to introduce any conception which may not be 
completely defined by a finite number of words. Whatever may be 
the remedy adopted, we can promise ourselves the joy of the physi- 
cian called in to follow a beautiful pathological case. 
X. THE RESEARCH OF POSTULATES. 
And yet, further, we are trying to enumerate the axioms and postu- 
lates, more or less deceiving, which serve as the foundation stones of 
our various mathematical theories. M. Hilbert has obtained the most 
brillant results. It seems now as if this domain must be very limited 
and that there will not be any more to be done when this inventory is 
finished, and that will be very soon. But when all has been gathered 
together there will be plenty of ways of classifying them, and a good 
librarian will always find something to busy himself with and each 
classification will be instructive to the philosopher. 
I stop this review, which I could not hope to make complete, for 
many reasons, and because I have already drawn too much on your 
patience. I believe that my examples will have been sufficient to show 
you by what means the mathematical sciences have progressed in the 
past and along what paths they must proceed in the future. 
4See G. Darboux: Les origines, les méthodes et les problémes de la Géométrie 
infinitésimale (The origin, methods, and problems of infinitesimal geometry). 
Revue générale des Sciences, 15 Nov., 1908, 
