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

Most striking is the personality of Evariste Galois whose tragic life, abruptly ended in his early youth, brought to science one of the most capital monuments we know of.  Galois’s passionate nature was captivated by mathematical science from the moment he became acquainted with Legendre’s geometry.  However, he was violently dominated by another overpowering feeling, enthusiastic devotion to republican and liberal ideas, for which he fought in a passionate and sometimes very imprudent way.  Nevertheless, the death he met with at the age of twenty did not occur in that struggle, but in an absurd duel.

Galois spent the night before that duel in revising his notes on his discoveries.  First the manuscript, which had been rejected by the Academy of Sciences as being unintelligible – one may not wonder that such highly intuitive minds are very obscure – then in a letter directed to a friend, scanty and hurried mention of other beautiful views, with the same words hastily and repeatedly inscribed in the margin ‘I have no time’.

Indeed, few hours remained to him before going where death awaited him.

All those profound ideas were at first forgotten and it was only after 15 years  that, with admiration, scientists became aware of the memoir with the Academy had rejected.  It signifies a total transformation of higher algebra, projecting a full light on what had been only glimpsed thus far by the greatest mathematiciens, and, at the same time, connecting that algebraic problem with others in quite different branches of science.

But what especially belongs to our subject is one point in the letter written by Galois to his friend and enunciating a theorem on the ‘periods’ of a certain kind of integrals.  Now this theorem, which is clear for us, could not have been understood by scientists living at the time of Galois; these ‘periods had no meaning in the state of science of that day; they acquired one only by means of some principles in the theory of functions, today classical, but which were not found before something like a quarter of a century after the death of Galois.

It must be admitted therefore that

1.  Galois must have conceived these principles in some way
2. They must have been unconscious in his mind, since he makes no allusion to them, though they by themselves represent a significant discovery