THE FUTURE OF MATHEMATICS. 
By Henri PoINncars®, 
Member of the Académie des Sciences and the Académie Francaise, Professor 
at the Sorbonne. 
(Translated by permission from Revue générale des Sciences pures et appli- 
quées, Paris, 19th year, No. 28, December, 1908.) 
The true method of forecasting the future of mathematics lies in 
the study of its history and its present state. 
And have we not here, for us mathematicians, a task in some sort 
professional? We are accustomed to extrapolation, that process 
which serves to deduce the future from the past and the present and 
so well know its limitations that we run no risk of being deluded with 
its forecasts. 
In the past there have been prophets incapable of seeing progress, 
those who have so willingly affirmed that all problems capable of 
solution have been solved and that nothing remains for future glean- 
ing. Happily the example of the past reassures us. Often enough, 
already, it has been believed that all problems capable of solution 
have been solved or at least stated. Then the sense of the word 
solution becomes broadened and the insolvable problems become the 
most interesting of all and undreamed-of problems have arisen. ‘To 
the Greeks a good solution must employ only the rule and compass; 
later it became that obtained by the extraction of roots; still later 
that obtained by the use of algebraic or logarithmic functions. 
These prophets of no advance thus always outflanked, always 
forced to retreat, have, I believe, been forced out of existence. 
As they are dead I will not combat them. We know that mathe- 
matics still develops and our task is to find in what sense. Some 
one replies, “in every sense;” and in part that is true. But, if abso- 
lutely true, it would be somewhat startling. Our riches would soon 
¢ Address delivered April 10, 1908, at the general session of the Fourth Inter- 
national Congress of Mathematicians (Rome, April 6-11, 1908); previously 
published in pamphlet form by and at the expense of the Mathematical Society 
of Palermo. M. Poincaré was unable to deliver this lecture and M. Darboux 
graciously undertook the task. To M. Guecia we express our gratitude for the 
authority which he has courteously extended for its reproduction. 
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