THE FUTURE OF MATHEMATICS—POINCARE. 129 
Among the words which have had this happy result I will mention 
the group and the invariant. They make us perceive the gist of many 
mathematical demonstrations; they make us realize how often mathe- 
maticians of the past must have run across groups without recogniz- 
ing them and how, believing these groups such isolated things, they 
have found them in close relationship without knowing why. 
To-day we would say that they were looking right in the face 
of isomorphic groups. We feel now that in a group the substance 
interests us but very little; it is the form alone which matters, and 
so, when we once know well a single group, then we know through it 
all the isomorphic groups; thanks to the words groups and isomor- 
phism, which sum in a few syllables this subtle law and make it at once 
familiar to us all, we take our step at once and in so doing economize 
all effort of thought. The idea of group, moreover, is bound up with 
that of transformation. Why then do we attach so much value to 
the invention of a new transformation? Because from a single 
theorem we may deduce ten or twenty; it has a value similar to the 
addition of a zero at the right of an integral number. 
We now realize what has determined the direction of the advance 
of mathematics in the past and the present and it is as certain what 
will determine it in the future. But the nature of the problems 
which come up will contribute equally. We must not forget what 
should be our goal; according to me that end is double. Our science 
confines itself at the same time to philosophy and to physies, and it is 
for these two neighbors that we work. And so we have always seen 
and always will see mathematics progressing in two opposite 
directions. 
In one sense mathematics must return upon itself and that is use- 
ful, for in returning upon itself it goes back to the study of the human 
mind which has created it rather than to those creations which bor- 
row the least bit from the external world. That is why certain 
mathematical speculations are useful, such as those whose aim is the 
study of postulates, of unusual geometries, of functions having 
peculiar values. The more these speculations depart from our com- 
mon conceptions and consequently from nature or practical applica- 
tions, the better they show us the working of the human mind which 
constructs them when it becomes freed from the tyranny of the exter- 
nal world, and the better, in consequence, it comes to know itself. 
But it is to the opposite side—the side of nature—against which we 
must direct the main corps of our army. 
There we meet the physicist or the engineer who says to us: “ Can 
you integrate for me such a differential equation? I must have it 
within eight days because of a certain construction which must be 
finished by that time.” “That equation,” we reply, “is not of an 
integrable type; you know there are many like it.” “ Yes, I know 
