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An Essay on the Psychology of Invention in the Mathematical Field – Jacques Hadamard

Hermite seemed to shun contact with the world.  He sought ‘within and not without for the vision of truth’.  That Hermite was not used to thinking in the concrete is certain.  He had a positive hatred for geometry and once curiously reproached me with having made a geometrical memoir.

As natural, his memoirs on concrete subjects are very few and not among his most remarkable ones.  So from the point of view of Poncare, Hermite ought to be considered as a logical mathematician.

But to call Hermite a logician!

Nothing can appear to me as more directly contrary to the truth. Methods always seemed to be born in his mind in some mysterious way.

In his lectures at the Sorbonne, which we attended with unfailing enthusiasm, he liked to begin his argument by
“Let us start from the identity”…

And here he was writing a formula the accuracy of which was certain, but whose origin in his brain and way of discovery he did not explain and we could not guess.  The quality of his mind is also evidently illustrated by his celebrated discovery in the theory of quadratic forms.

In that question, two cases are possible in which, as is obvious, things happen quite differently.  In the first one reduction has been known since Gauss.  Nobody, as it seemed, would have thought of the idea of merely carrying out, in the second case, the very calculations which suited the first one and which apparently, had nothing to do with the second case; it seemed quite absurd that they would, that time , lead to the solution; and yet, by a kind of witchcraft, they do……..

I can hardly imagine a more perfect type of an intuitive mind than Hermite.