Thursday, July 6, 2017

The Mathematical Genius of Henri Poincare 1912

Henri Poincare, Mathematical Genius, article in the Journal of the American Medical Association 1912

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The recent death of Henri Poincare has been the occasion of a number of biographic articles, the most important of which is undoubtedly that which appeared in the Revue des Deux Mondes, Sept. 15, 1912. Not only was Poincaré probably the greatest mathematical genius of our time, but he was besides a great physicist, astronomer and philosopher. Between the careful study of Poincaré made some years ago by Dr. Toulouse, and Poincaré’s revelation with regard to the working of his own mind as be analyzed its processes and reflected on the methods by which he secured results, we have probably more intimate data with regard to Poincaré's mental characteristics than with regard to those of any other man of genius that ever lived.

Of his genius there could be no possible doubt, hailed as it was not only by the French, but also by the Germans, the English and the Americans. He added an important chapter to mathematics by his discovery and development of the Fuchsian functions, which simplified marvelously the solution of all curves and all linear differential equations that could be expressed in algebraic coefficients. Humbert of the French Academy of Sciences declared that Poincaré had in this provided “the keys of the algebraic world.” In astronomy be deduced by mathematics that the rings of Saturn must be composed of small separate bodies—an agglomeration as it were, in circular form of a series of planets. This mathematical deduction was afterward demonstrated physically by means of the spectroscope. This mathematical discovery on Saturn has been compared to the discovery of Neptune by pure mathematics. It was as a philosopher, however. that Poincare was best known by the general public, and in this field some of his suggestions are considered as among the most valuable that had been made in centuries. Men have compared him in mathematics to Newton, in philosophy to Descartes.

The question how so great a mind works is extremely interesting. Poincare has told the story of how he reached his great discovery of the Fuchsian functions. It was not reached all at once but by several steps. The first and most important development came to him one evening when, contrary to his custom, having taken a cup of black coffee at dinner, he could not sleep and the idea of this new mathematical mode took form little by little under these unusual circumstances. The problems which were involved came clearly before his mind and seemed too difficult for solution; so gradually he put them away and succeeded in falling asleep. The successive steps of the solution came to him subsequently not as the result of deliberate study of the problems, but long afterward and under most diverse circumstances, at moments when he was not thinking about them. They came to him as flashes of light, almost inspirations, as it were—once when he was just about to put his foot on the step of an omnibus, again when he was crossing a boulevard, a third time in the midst of a geologic excursion with some friends when the conversation was about ordinary subjects and had no relation at all to mathematics.

Ordinarily mathematics at least is supposed to be eminently intellectual and its developments are connected by the most rigid logic. It might be expected, then, that it would be only in the midst of deep thinking, even absorption of mind in mathematical subjects, that great new ideas would come; but Poincaré believed that it was a subconscious mind that solved the problems. His explanation of this, which resembles that so often heard with regard to the inspiration of the poet or the musician, is that certain thoughts are passing through the unconscious mind all the time and that, as in daydreaming, we are never without groups of thoughts. Whenever one of these thoughts proves to be a particularly beautiful or strikingly novel conception of some kind it attracts the attention of the conscious mind and then is retained. According to Poincaré’s experience, then, like poetry and music, the sciences, including mathematics, owe their development not to the rational conscious mind so much as to the unconscious and involuntary faculties. There would seem to be a tirelcss force in man, a part of him and yet not a part of him, working, thinking. developing, which brings to the conscious entity, man, his best thoughts and discoveries.

Poincare, for all his genius, was a sane and simple father of a family; he himself taught his four children to read, often took part in their sports and found his favorite relaxation in the family life. Moments of abstraction in deep thinking he had, but they did not interfere with pleasant family intercourse or disturb his interests in the practical every-day life around him. It will add to the value of the study of Poincare's mind made by himself to know that Dr. Toulouse after most careful investigations on a number of occasions by himself and a series of collaborators, including tests, measurements, investigation of reflexes, physical and mental, found him to be perfectly normal from the psychophysiologic point of view, with faculties perfectly harmonious and in complete equilibrium. The failure to find in the great scientist the slightest trace of a neurotic condition of any kind, flatly contradicts, in this instance at least, the theory of an intimate relation between genius and insanity.

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