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

Two Conceptions of Invention

Claparede in his introductory lecture at the Centre de Synthese, observes that there are two kinds of invention:

• One consists of a goal being given, in finding the means to reach it so that the mind goes from the goal to the means, from the question to the solution
• The other consists, on the contrary, in discovering a fact, then imagining what it could be useful for, the answer appears to us before the question.

Now, paradoxical as it seems, that second kind of invention is the more general one and becomes more and more so as science advances.  Practical application is found by not looking for it, and one can say that the whole progress of civilisation rests on that principle.

When the Greeks, some four centuries BC considered an ellipse – the curve generated by the points M in a plane such that the sum MF + MF’ of their distances from two given points F, F’ be a constant – and found many remarkable properties of it, they did not think and could not think of any possible use for such discoveries.  However, without these studies, Kepler could not have discovered, two thousand years later, the laws of motions of planets, and Newton could not have discovered universal attraction.

…..It seldom happens that important mathematical researches are directly undertaken in view of a given practical use; they are inspired by the desire which is the common motive of every scientific work, the desire to know and understand.

And how is the important choice of subjects to be made?  The answer is hardly doubtful; it is the same method which Poincare gave us concerning the means of discovery,-  the same for the drive as for the mechanism.