The Mathematician's Mind: The Psychology of Invention in the Mathematical Field

By Jacques Hadamard

Fifty years ago when Jacques Hadamard set out to explore how mathematicians invent new ideas, he considered the creative experiences of some of the greatest thinkers of his generation, such as George Polya, Claude Lévi-Strauss, and Albert Einstein. It appeared that inspiration could strike anytime, particularly after an individual had worked hard on a problem for days and then turned attention to another activity. In exploring this phenomenon, Hadamard produced one of the most famous and cogent cases for the existence of unconscious mental processes in mathematical invention and other forms of creativity. Written before the explosion of research in computers and cognitive science, his book, originally titled The Psychology of Invention in the Mathematical Field, remains an important tool for exploring the increasingly complex problem of mental life.


The roots of creativity for Hadamard lie not in consciousness, but in the long unconscious work of incubation, and in the unconscious aesthetic selection of ideas that thereby pass into consciousness. His discussion of this process comprises a wide range of topics, including the use of mental images or symbols, visualized or auditory words, "meaningless" words, logic, and intuition. Among the important documents collected is a letter from Albert Einstein analyzing his own mechanism of thought.

• Language: English
• Publisher: Princeton University Press
• Release date: May 5, 2020
• ISBN: 9780691212906

Book preview

MIND

I. GENERAL VIEWS AND INQUIRIES

THE SUBJECT we are dealing with is far from unexplored and though, of course, it still holds many mysteries for us, we seem to possess fairly copious data, more copious and more coherent than might have been expected, considering the difficulty of the problem.

That difficulty is not only an intrinsic one, but one which, in an increasing number of instances, hampers the progress of our knowledge : I mean the fact that the subject involves two disciplines, psychology and mathematics, and would require, in order to be treated adequately, that one be both a psychologist and a mathematician. Owing to the lack of this composite equipment, the subject has been investigated by mathematicians on one side, by psychologists on the other and even, as we shall see, by a neurologist.

As always in psychology, two kinds of methods are available: the subjective and the objective methods.¹ Subjective (or introspective) methods are those which could be called observing from the inside, that is, those where information about the ways of thought is directly obtained by the thinker himself who, looking inwards, reports on his own mental process. The obvious disadvantage of such a procedure is that the observer may disturb the very phenomenon which he is investigating. Indeed, as both operations—to think and to observe one’s thought— are to take place at the same time, it may be supposed a priori that they are likely to hamper each other. We shall see, however, that this is less to be feared in the inventive process (at least, in some of its stages) than in other mental phenomena. In the present study, I shall use the results of introspection, the only ones I feel qualified to speak of. In our case, these results are clear enough to deserve, it would seem, a certain degree of confidence. In doing so, I face an objection for which I apologize in advance : that is, the writer is obliged to speak too much about himself.

Objective methods—observing from the outside—are those in which the experimenter is other than the thinker. Observation and thought do not interfere with each other ; but on the other hand, only indirect information is thus obtained, the significance of which is not easily seen. One chief reason why they chance to be difficult to employ in our case is because they require the comparison of numerous instances. In agreement with the general principle of experimental science, this would be an essential condition for arriving at the fact of great yield, as Poincaré says, that is, the fact which penetrates deeply into the nature of the question; but, precisely, these instances cannot be found for such an exceptional phenomenon as invention.

The Mathematics Bump. Objective methods have generally been applied to invention of any kind, no special investigation being devoted to mathematics. One exception, which we shall very briefly mention, is a curious attempt which has been initiated by the celebrated Gall. It depends on his principle of the so-called phrenology, that is, on the connection of every mental aptitude with a greater development not only of some part of the brain, but also of the corresponding part of the skull; a rather unhappy idea, as recent neurologists think, of that man who had other very fruitful ones (he was a forerunner of the notion of cerebral localization). According to that principle, mathematical ability ought to be characterized by a special bump on the head, the localization of which he actually indicates.

Gall’s ideas were taken up (1900)² by the neurologist Möbius, who happened to be the grandson of a mathematician, though he himself had no special knowledge of mathematics.

Möbius’ book is a rather extensive and thorough study of mathematical ability from the naturalist’s standpoint. It contains a series of data which, eventually, are likely to be of interest for that study. They bear, for instance, on heredity (families of mathematicians),³ longevity, abilities of other kinds, etc. Though such an important collection of data may prove useful at a later date, it seems so far not to have given rise to any general rule except as concerns the artistic inclinations of mathematicians. (Möbius confirms the somewhat classic opinion that most mathematicians are fond of music, and asserts that they are also interested in other arts.)

Now, Möbius agrees with Gall’s conclusions in general, considering, however, in the first place, that the mathematical sign, though always present, may assume a greater variety of forms than would be understood from Gall’s description.

However, that bump hypothesis of Gall-Möbius has not met with agreement. Anatomists and neurologists strongly assailed the Gall redivivus, as they called him, because Gall’s phrenological principle, i.e., conformity of skull to brain form, is now considered inaccurate.

Let us not insist any longer on this phase of the question, which is to be left to specialists. But it is not useless to speak of it from the mathematical standpoint. From that point of view also, some objections can be raised, at least at a first glance, against the very principle of such research. It is more than doubtful that there exists one definite mathematical aptitude. Mathematical creation and mathematical intelligence are not without connection with creation in general and with general intelligence. It rarely happens, in high schools, that the pupil who is first in mathematics is the last in other branches of learning; and, to consider a higher level, a great proportion of prominent mathematicians have been creators in other fields. One of the greatest, Gauss, carried out important and classical experiments on magnetism; and Newton’s fundamental discoveries in optics are well known. Was the shape of the head of Descartes or Leibniz influenced by their mathematical abilities or by their philosophical ones ?

Also there is a counterpart. We shall see that there is not just one single category of mathematical minds, but several kinds, the differences being important enough to make it doubtful that all such minds correspond to one and the same characteristic of the brain.

All this would not be contradictory to the principle of Gall interpreted in a general way, i.e., to interdependence of the mathematical functioning of the mind with the physiology and anatomy of the brain ; but the first application of it which Gall and Möbius proposed does not seem to be justified.

Generally speaking, we must admit that mental faculties which seem at first to be simple are composite in an unexpected way. It has been recognized by objective methods (observation of the effects of wounds or other injuries of the head) that such is the case with the best known faculty of all, the language faculty, which consists of several different ones. There are cerebral localizations, as Gall had announced, but without such simple and precise correspondences as he supposed.

There is every reason to think that the mathematical faculty must be at least as composite as has been found for the faculty of language. Though, of course, decisive documents are not and will probably never be available in the former case as they are in the latter, observations on the one phenomenon may help us to understand the other.

Psychologists’ Views on the Subject. Many psychologists have also meditated not especially on mathematical invention, but on invention in general. Among them, I shall mention only two names, Souriau and Paulhan. These two psychologists show a contrast in their opinions. Souriau (1881) was, it seems, the first to have maintained that invention occurs by pure chance, while Paulhan (1901)⁴ remains faithful to the more classic theory of logic and systematic reasoning. There is also a difference in method, which can hardly be accounted for by the small difference in the dates, for while Paulhan has taken much information from scientists and other inventors, there is hardly any to be found in Souriau’s work. It is curious that, operating in such a way, he is led to some very shrewd and accurate remarks; but, on the other hand, he has not avoided one or two serious errors which we shall have to mention.

Later on, a most important study in that line was conducted (1937) at the Centre de Synthèse in Paris, as mentioned in the introduction.

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