Wayne State University
College of Lifelong Learning
Interdisciplinary Studies Program
Winter, 1999
Creativity: Building the New
ISP 5500 Section# 981, Call# 90577, 4 cr and
ISP 5990 Section# 981, Call# 95268, 4 cr
Course web site: http://www.cll.wayne.edu/isp/drbowen/crtvyw99

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Creativity: Building the New

A Description of His Own Creativity
by Henri Poincaré

poincare.gif (7838 bytes) From "The Foundations of Science" by Henri Poincaré, first published in Paris in 1908 and here translated from the French by G.B. Halstead. Poincaré, pictured to the left,  was a well-known mathematician, and time has not diminished his stature. This writing is one of the first and best-known descriptions of creative activity.

"The genesis of mathematical creation is a problem which should intensely interest the psychologist. It is the activity in which the human mind seems to take least from the outside world, in which it acts or seems to act only of itself and on itself, so that in studying the procedure of geometric thought we may hope to reach what is most essential in man's mind.


"To invent, I have said, is to choose [among all of the possible variations in a given area]; but the word is perhaps not wholly exact. It makes one think of a purchaser before whom are displayed a large number of samples, and who examines them, one after the other, to make a choice. Here [in mathematics] the samples would be so numerous that a whole lifetime would not suffice to examine them. This is not the actual state of things. The sterile combinations do not even present themselves to the mind of the inventor. Never in the field of his consciousness do combinations appear that are not really useful, except some that he rejects but which have to some extent the characteristics of useful combinations. All goes on as if the inventor were an examiner for the second [academic] degree who would only have to question the candidates who had passed a previous examination.


"It is time to penetrate further and to see what goes on in the very soul of the mathematician. For this, I believe, I can do best by recalling memories of my own. But I shall limit myself to telling how I wrote my first memoir on Fuchsian functions. I beg the reader's pardon; I am about to use some technical expressions, but they need not frighten him, for he is not obliged to understand them. I shall say, for example, that I have found the demonstration of such a theorem under such circumstances. This theorem will have a barbarous name, unfamiliar to many, but that is unimportant; what is of interest for the psychologist is not the theorem but the circumstances.

"For fifteen days I strove to prove that there could not be any functions like those I have since called Fuchsian functions. I was then very ignorant; every day I seated myself at my work table, stayed an hour or two, tried a great number of combinations and reached no results. One evening, contrary to my custom, I drank black coffee and could not sleep. Ideas rose in crowds; I felt them collide until pairs interlocked, so to speak., making a stable combination. By the next morning I had established the existence of a class of Fuchsian functions, those which came from the hypergeometric series; I had only to write out the results, which took but a few hours.

"Then I wanted to represent these functions by the quotient of two series; this idea was perfectly conscious and deliberate, the analogy with elliptic functions guided me. I asked myself what properties these series must have if they existed, and I succeeded without difficulty in forming the series I have called theta-Fuchsian.

"Just at this time I left Caen, where I was then living, to go on a geologic excursion under the auspices of the school of mines. The changes of travel made me forget my mathematical work. Having reached Coutances, we entered an omnibus to go some place or other. At the moment when I put my foot on the step the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuchsian functions were identical with those of non-Euclidean geometry. I did not verify the idea; I should not have had time, as upon taking my seat in the omnibus, I went on with a conversation already commenced, but I felt a perfect certainty. On my return to Caen, for conscience' sake, I verified the result at my leisure.

"Then I turned my attention to the study of some arithmetical questions apparently without much success and without a suspicion of any connection with me preceding researches. Disgusted with my failure, I went to spend a few days at the seaside, and thought of something else. One morning, walking on the bluff, the idea came to me, with just the same characteristics of brevity, suddenness and immediate certainty, that the arithmetic transformations of indeterminate ternary quadratic forms were identical with those of non-Euclidean geometry.

"Returned to Caen, I meditated on this result and deduced the consequences. The example of quadratic forms showed me that they were Fuchsian functions other than those corresponding to the hypergeometric series; I saw that I could apply to them the theory of theta-Fuchsian functions other than those from the hypergeometric series, the ones I then knew. Naturally I set myself to form all these functions. I made a systematic attack upon them and carried all the outworks, one after another. There was one however that still held out, whose fall would involve that of the whole place. But all my efforts only served at first the better to show me the difficulty, which was indeed something. All this work was perfectly conscious.

"Thereupon I left for Mont-Valerian, where I was to go through my military service; so I was very differently occupied. One day, going along the street, the solution of the difficulty which had stopped me suddenly appeared to me. I did not try to go deep into it immediately, and only after my service did I again take up the question. I had all the elements and had only to arrange them and put them together. So I wrote out my final memoir at a single stroke and without difficulty."

Poincaré then goes on to analyze this raw evidence. He draws the following conclusions:
1. The creations involve a period of conscious work, followed by a period of unconscious work.
2. Conscious work is also necessary after the unconscious work, to put the unconscious results on a firm footing.
3. Earlier in this piece, Poincaré drew the conclusion that mathematical creation cannot be mechanical. Many of the choices are based on grounds of symmetry, mathematical elegance, consistency with other areas of mathematics, and even esthetics. Therefore the unconscious is not simply a mechanical processor; (quoting again from Poincaré) "it is not purely automatic; it is capable of discernment; it has tact, delicacy; it knows how to choose, to divine. What do I say? It knows better how to divine than the conscious self, since it succeeds where that has failed. In a word, is not the subliminal self superior to the conscious self? ... Does it follow that the subliminal self, having divined by a delicate intuition that [certain] combinations would be useful, has formed only these, or has it rather formed many others which were lacking in interest and have remained unconscious?"
4. The unconscious can present the conscious mind with something that is not fruitful, but which is nevertheless elegant or beautiful.
5. What the unconscious presents to the conscious mind is not a full and complete argument or proof, but rather "point of departure" from which the conscious mind can work out the argument in detail. The conscious mind is capable of the strict discipline and logical thinking, of which the unconscious is incapable.

Resuming the quotations, "I shall make a last remark; when above I made certain personal observations, I spoke of a night of excitement when I worked in spite of myself. Such cases are frequent, and it is not necessary that the abnormal cerebral activity be caused by a physical excitant as in that I mentioned. It seems, in such cases, that one is present at his own unconscious work, made partially perceptible to the over-excited consciousness, yet without having changed its nature. Then we vaguely comprehend what distinguishes the two mechanisms or, if you wish, the working methods of the two egos. And the psychologic observations I have been able thus to make seem to me to confirm in their general outlines the views I have given.

"Surely [psychologists] have need of it, for they are and remain in spite of all very hypothetical; the interest of the questions is so great that I do not repent of having submitted them to the reader."

[Commentary: One of the reasons that these passages are quoted so often is surely that they admirably demonstrate the characteristics of science; that is, careful and precise observation, hypothesizing that seeks a uniform account of the whole phenomenon, and a clear presentation of the reasoning. Whether or not Poincaré's analysis is accepted today, here he sets a high standard for anyone wanting to work in this area.]

Picture of Poincaré at http://www-groups.dcs.st-and.ac.uk/~history/Posters/429.html

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