The Mathematician's Mind

The Psychology of Invention in the Mathematical Field

Jacques Hadamard

• Publisher: Princeton University Press
• Year: 2020
• Print ISBN: 9780691029313
• eBook ISBN: 9780691212906
• Topic: Mathematics
• Subtopic: History & Philosophy of Mathematics
• Index: Mathematics

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.

VI. DISCOVERY AS A SYNTHESIS. THE HELP OF SIGNS

Synthesis in Discovery. Souriau, in his Théorie de l’Invention, writes, “Does the algebraist know what becomes of his ideas when he introduces them, in the form of signs, into his formulae? Does he follow them throughout every stage of the operations which he performs? Undoubtedly not: he immediately loses sight of them. His only concern is to put in order and to combine, according to known rules, the signs which he has before him ; and he accepts with a full confidence the result thus obtained.”

We have said that this author hardly seems to have gathered information from professional men. Probably, if he had, he would not have expressed himself in that way. One cannot say, however, that his statement is completely false. It can be admitted to be true, roughly speaking, as far as concerns the final phase of verifying and “precising” already mentioned in the preceding section ; but, even then, things do not occur as he states. The mathematician does not so blindly confide in the results of the rules which he uses. He knows that faults of calculation are possible and even not infrequent; if the purpose of the calculation is to verify a result which unconscious or subconscious inspiration has foreseen and if this verification fails it is by no means impossible that the calculation be at first false and the inspiration be right.

If applied not to that final phase but to the total research work, the behavior which Souriau describes is that of the pupil, and even of the rather bad pupil ; our efforts aim to have him change it. The true process of thought in building up a mathematical argument is certainly rather to be compared with the process we mentioned in Section II, I mean the act of recognizing a person. An intermediate case which illustrates the analogy between the two processes is afforded by psychological studies on chess players, some of whom, as is well known, can play ten or twelve games simultaneously without seeing the chess boards. Inquiries were started, especially by Alfred Binet, in order to understand how this was possible : their results¹ may be summed up by saying that for many of these players, each game has, so to say, a kind of physiognomy, which allows him to think of it as a unique thing, however complicated it may be, just as we see the face of a man.

Now, such a phenomenon necessarily occurs in invention of any kind. We saw it mentioned in Mozart’s letter (see Section I) ; similar statements are issued by artists like Ingres or Rodin (quoted by Delacroix, L’Invention et le Génie, p. 459). Only, while the happily gifted Mozart does not seem to have needed any effort in order to see the unity of his work, Rodin writes, “Till the end of his task, it is necessary for him [the sculptor] to maintain energetically, in the full light of his consciousness, his global idea, so as to reconduct unceasingly to it and closely connect with it the smallest details of his work. And this cannot be done without a very severe strain of thought.”

Similarly, any mathematical argument, however complicated, must appear to me as a unique thing. I do not feel that I have understood it as long as I do not succeed in grasping it in one global idea and, unhappily, as with Rodin, this often requires a more or less painful exertion of thought.

The Use of Signs. Let us now examine a question which, as I intend to show below, is connected with the preceding one : the help which is afforded to thought by concrete representations. Such an investigation, belonging to the field of direct introspection, is possible thanks only to that fringe-consciousness which we mentioned at the end of Section II. However, we shall see that its chief results most probably also subsist in the deeper unconscious, though the latter is not directly known to us.

Words and Wordless Thought. The most classic kind of signs spoken of as cooperating with thought consists of words. Here we face a curious question on which quite divergent opinions are held.

I had a first hint of this when I read in Le Temps (1911) : “The idea cannot be conceived otherwise than through the word and only exists by the word.”² My feeling was that the ideas of the man who wrote that were of a poor quality.

But it was even more surprising for me to see such a man as Max Müller, the celebrated philologist and orientalist, maintain³ that no thought is possible without words⁴ and even write this sentence, fully unintelligible to me: “How do we know that there is a sky and that it is blue? Should we know of a sky if we had no name for it?”, admitting not only, with Herder, that “without language, man could never have come to his reason,” but also adding that, without language, man could never have come even to his senses. Are animals, which do not speak, devoid of senses?

That statement of Max Müller is the more curious because he claims to find in the fact that thought is impossible without words an argument against every evolutionary theory, a proof that man cannot be descended from any animal species. The deduction, even admitting the premise, is contestable. But it could more legitimately be reversed against Max Müller’s thesis if we take into account, for instance, Köhler’s Mentality of Apes⁵ and the actions of his chimpanzees, which do imply reasoning.

Max Müller gives a historical review, which we shall reproduce in its essential parts, of opinions expressed on the question of words in thought: a review which is not devoid of interest, first in itself, next, because of the standpoint of Max Müller toward it. We hear, in the first place, that the Greeks originally used one and the same word, “logos,” to denote language and thought and only later on were led to distinguish both meanings by epithets—on which, of course, the author declares them better inspired in the former case than in the latter.

Medieval scholastics, by a similitude which perhaps lies in the nature of things, agree with the beginning of Greek philosophy. Abelard, in the twelfth century, said that “Language is generated by the intellect and generates intellect.” An analogous statement is to be found in a more modern philosopher, Hobbes (who, generally, keeps in sympathy with the scholastics).

But, as a rule, ideas took a different course, on that subject as on many others, with the stream of thought initiated by Descartes. There is only one period in Germany, around 1900 (Humboldt, Schelling, Hegel, Herder) when philosophical minds were near to “truth,” that is, to Max Müller’s opinion. Hegel summarily says, “We think in nouns,” as if nobody had ever doubted it.

But the other great philosophers of modern times are not so sure of the identity of language and reason. Precisely, the greatest of them—be it Locke, Leibniz or even Kant or Schopenhauer, or, more recently, John Stuart Mill—agree in a methodic doubt. Not that Leibniz does not think in words, but he does not recognize that without openly regretting it.⁶ One philosopher, Berkeley, is absolutely categorical—but in the opposite direction. He is convinced that words are the great impediment to thought.

Max Müller’s passionate view of the subject leads him to qualify as “lack of courage” that general attitude of modern thinkers, which everybody else would call scientific prudence, as though no sincere opinion other than his own might exist.

Whether he admits it or not, it does exist. Immediately after the Lectures on Science of Thought were delivered, contradictions arose ; indeed they came from most various parts.⁷ Above all, there came the authorized voice of another first-rank scholar, Francis Galton, the great geneticist, who, moreover, after having begun as an explorer, has done important work in psychological matters. The latter’s great habit of introspection allows him to assert that his mind does not behave at all in the way supposed by Max Müller to be the only possible one. Whether he is playing billiards and calculating the course of his ball or investigating higher and more abstract questions, his thought is never accompanied by words.

Galton adds that he sometimes happens, while engaged in thinking, to catch an accompaniment of nonsense words, just “as the notes of a song might accompany thought.” Of course, nonsense words are something quite different from real words ; we shall see later to what kind of images they may be reasonably compared.

That disposition of mind in Galton is not devoid of inconvenience for him. “It is,” he says, “a serious drawback to me in writing, and still more in explaining myself, that I do not so easily think in words as otherwise. It often happens that after being hard at work, and having arrived at results that are perfectly clear and satisfactory to myself, when I try to express them in language I feel that I must begin by putting myself upon quite another intellectual plane. I have to translate my thoughts into a language that does not run very evenly with them. I therefore waste a vast deal of time in seeking for appropriate words and phrases, and am conscious, when required to speak on a sudden, of being often very obscure through mere verbal maladroitness, and not through want of clearness of perception. That is one of the small annoyances of my life.”

I have wanted to reproduce at length that statement of Galton, because in his case I exactly recognize mine, including the rather regrettable consequence which I experience just as he does.

The fact that it is impossible for Max Müller to recall lightning without thinking of its name does not mean that “we” are unable to do so. As for myself, if I remember lightning, I see in my mind the flash of light which I have admired several times, and I should need an instant of reflection—a short one, of course, but certainly an instant— if I should wish the corresponding word to recur to me. Just as for Galton, such a translation from thought to language always requires on my part a more or less difficult effort. Whether the verses of Boileau

“Ce qui se conçoit bien s’énonce clairement

Et les mots pour le dire arrivent aisément,”

are justified or not concerning other people, it is certain that they are not true for me. I have a tangible proof of that—-an “objective” one, I could say—in the fact that it is difficult for me to deliver a lecture on anything but mathematical subjects without having written down practically every part of it, the only means of avoiding constant and painful hesitation in the expression of thought which is very clear in my mind.

Galton legitimately points out how strange it is that Max Müller has utterly failed to understand that other people’s minds may be different from his own : a most common error, but one which it is surprising to find among men accustomed to psychological studies. Differences between minds being, on the contrary, undeniable according to what we have just found, the question ought to be settled not by polemics but by inquiries relating to every human race and every class of men and, if possible (we shall see that there may be some difficulty in this), not only among intellectual people. Galton, inquiring, as he says, as much as occasion has allowed him, finds a certain percentage, though a minor one, of persons whose thought is habitually carried on without the use of mental or spoken words. One may wonder that a man as well acquainted with statistical operations as Galton does not give a precise percentage ; a possible reason for that will appear below.⁸

Mental Pictures in Usual Thought. Thought can be accompanied by concrete representations other than words. Aristotle admitted that we cannot think without images. Taine’s well-known work On Intelligence is chiefly devoted to the importance, in the constitution of ideas, of images, which he defines, at the beginning of his Volume II, as recurring, surviving and spontaneously resurging sensations. However, he is now believed to have exaggerated that importance and described it as a too exclusive one.

At about the same time, Alfred Binet was making an important improvement in the study of that question by attacking it in the experimental way.⁹ He investigated some twenty persons, but chiefly two young girls (aged thirteen and fourteen) in his own family, whose valuable help in somewhat delicate psychological researches, at such a youthful age, is a very remarkable thing. He submits them sometimes to pure experiments, but more often to experiment combined with introspection. For instance, asking a question or pronouncing a word, he inquires what ideas, images, etc., this has suggested to the subj...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Dedication
• Contents
• Preface to the Paperback Edition
• Foreword
• Introduction
• I. General Views and Inquiries
• II. Discussions on Unconsciousness
• III. The Unconscious and Discovery
• IV. The Preparation Stage. Logic and Chance
• V. The Later Conscious Work
• VI. Discovery as a Synthesis the Help of Signs
• VII. Different Kinds of Mathematical Minds
• VIII. Paradoxical Cases of Intuition
• IX. The General Direction of Research
• Appendix I
• Appendix II
• The Princeton Science Library