Mathematical Inquiries. Let us come to mathematicians. One of them, Maillet, started a first inquiry as to their methods of work. One famous question, in particular, was already raised by him: that of the “mathematical dream,” it having been suggested often that the solution of problems that have defied investigation may appear in dreams.

Though not asserting the absolute non-existence of “mathematical dreams,” Maillet’s inquiry shows that they cannot be considered as having a serious significance. Only one remarkable observation is reported by the prominent American mathematician, Leonard Eugene Dickson, who can positively assert its accuracy. His mother and her sister, who, at school, were rivals in geometry, had spent a long and futile evening over a certain problem. During the night, his mother dreamed of it and began developing the solution in a loud and clear voice; her sister, hearing that, arose and took notes. On the following morning in class, she happened to have the right solution which Dickson’s mother failed to know.

This observation, an important one on account of the personality of the relator and the certitude with which it is reported, is a most extraordinary one. Except for that very curious case, most of the 69 correspondents who answered Maillet on that question never experienced any mathematical dream (I never did) or, in that line, dreamed of wholly absurd things, or were unable to state precisely the question they happened to dream of. Five dreamed of quite naive arguments. There is one more positive answer; but it is difficult to take account of it, as its author remains anonymous.

Besides, in that matter, there is a confusion which raises grave doubts. One phenomenon is certain and I can vouch for its absolute certainty: the sudden and immediate appearance of a solution at the very moment of sudden awakening. On being very abruptly awakened by an external noise, a solution long searched for appeared to me at once without the slightest instant of reflection on my part the fact was remarkable enough to have struck me unforgettably and in a quite different direction from any of those which I had previously tried to follow. Of course, such a phenomenon, which is fully certain in my own case, could be easily confused with a “mathematical dream,” from which it differs.

I shall not dwell any longer on Maillet’s inquiry because a more important one was started, a few years later, by some mathematicians with the help of Claparéde and another prominent Genevese psychologist, Flournoy, and published in the periodical L’Enseignement Mathématique. An extensive questionnaire was sent out, consisting of a few more than 30 questions (See Appendix I). These questions (including “mathematical dream”) belonged to both classes of investigation methods which we have already differentiated, some of them being “objective” (as much as a questionnaire can be). For instance, mathematicians were asked whether they were influenced by noises and to what extent, or by meteorological circumstances, whether literary or artistic courses of thought were considered useful or harmful.

Other questions were of a more introspective character and penetrated more directly and deeply into the nature of the subject. Authors were asked whether they were deeply interested in reading the works of their predecessors or, on the contrary, preferred to study problems directly by themselves; whether they were in the habit of abandoning a problem for a while to resume it again only later on (which I, personally, do in many cases and which I always recommend to beginners who consult me). Above all, they were asked what they could say on the genesis of their chief discoveries.

Some Criticisms. Reading that questionnaire, one may notice the lack of some questions, even when analogous to some which have actually been asked. For instance, when asking mathematicians whether they indulged in music or poetry, the questionnaire did not mention possible interest in sciences other than mathematics. Especially, biology, as Hermite used to observe, may be a most useful study even for mathematicians, as hidden and eventually fruitful analogies may appear between processes in both kinds of study.

Similarly, when inquiring about the influence of meteorological circumstances or the existence of periods of exaltation or depression, no more precise question was asked concerning the influence of the psychical state of the worker and especially the emotions which he may be experiencing. This question is all the more interesting because it has been taken up by Paul Valéry in a lecture at the French Society of Philosophy, in which he suggested that emotions are evidently likely to influence poetical production. Now, however likely it may seem at first glance that some kind of emotions may favor poetry because they more or less directly find their expression in poetry, it is not certain that this reason is the right one or at least the only one. Indeed, I know by personal experience that powerful emotions may favor entirely different kinds of mental creation (e.g., the mathematical one¹); and in this connection, I should agree with this curious statement of Daunou: “In Sciences, even the most rigid ones, no truth is born of the genius of an Archimedes or a Newton without a poetical emotion and some quivering of intelligent nature.”

Moreover, the most essential question I mean the one which concerns the genesis of discovery suggests another one, which is not mentioned in the questionnaire though its interest is obvious. Mathematicians are asked how they have succeeded. Now, there are not only successes but also failures, and the reasons for failures would be at least as important to know.

This is in relation to the most important criticism which can be formulated against such inquiries as Maillet’s or Claparéde and Flournoy’s: indeed, such inquiries are subject to a cause of error which they can hardly avoid. Who can be considered a mathematician, especially a mathematician whose creative processes are worthy of interest? Most of the answers which reached the inquirers come from alleged mathematicians whose names are now completely unknown. This explains why they could not be asked for the reasons of their failures, which only firstrate men would dare to speak of. In the above mentioned inquiries, I could hardly find one or two significant names, such as the physico-mathematician Boltzmann. Such masters as Appell, Darboux, Picard, Painlevé sent no answers, which was perhaps a mistake on their part.

Since most answers to the inquiries of Maillet and of the Enseignement Mathématique were of slight interest for that reason, it occurred to me to submit some of the questions to a man whose mathematical creation is one of the most audacious and penetrating, Jules Drach. Some of his answers were especially suggestive, in the first place, as concerns biology in which, like Hermite, he takes much interest and, chiefly, on the study of previous discoverers. This is a question where it appears that even among men who are born mathematicians, important mental differences may exist. The historians of the amazing life of Evariste Galois have revealed to us that, according to the testimony of one of his schoolfellows, even from his high school time, he hated reading treatises on algebra, because he failed to find in them the characteristic traits of inventors. Now, Mr. Drach, whose work, besides, is closely related to Galois’, has the same way of approach. He always wishes to refer to the very form in which discoveries have appeared to their authors. On the contrary, most mathematicians who have answered Claparéde and Flournoy’s inquiry prefer, when studying any previous work, to think it out and rediscover it by themselves. This is my approach, so that finally I know, in any case, of only one inventor, who is rayself.