THE PSYCHOLOGY OF INVENTION IN THE MATHEMATICAL FIELD ESSAY OJY The Psychology of Invention in the Mathematical Field BY JACQUES DOVER PUBLICATIONS, INC., Copyright., 1945, by Princeton University Press This new Dover edition, first published in 1954, is an unaltered and unabridged reprint of tifcxe first edition by special arrangement with Princeton University I^ress Library of Congress Catalog Card. Number; 54-4731 Manufactured in the United States of America J>over Publications, Inc. 180 Varick Street New York 14, N. Y. ~ia ccrmpftcpne tie met trie et cle mm* FOREWORD **Je dirai que j*ai trouvS la demonstration de tel theorems dans teffles cirConstances ; ce t'heore'me aura, tin notn barbare, que beaiicotip (Fentre von* ne connattront pa*; mai* cela n*a pas d*importance: ce qzi4 est interes&ant pour le psychologue* ce n*est pa* le tTieoreme* ce sont les cirConstances," Henri Poincare THIS study, like everything which could be writ- ten on mathematical invention, was first inspired by Henri l?oincare's famous lecture before the Societe de Psychologie in Paris. I first came back to the subject in a meeting at the Centre de Syn- th&se in Paris (1937). But a more thorough treatment of it has been given in an extensive course of lectures delivered (1943) at the Ecole .Libre des Hautes Etudes, New York City, I wish to express my gratitude to Princeton University Press, for the interest taken in this work and the careful help brought to its pub- lication. 1944 CONTENTS INTRODUCTION Xi I. GENERAL VIEWS AND INQ.UIRIES 1 U. DISCUSSIONS ON UNCONSCIOUSNESS 21 HI. THE UNCONSCIOUS AND DISCOVERY 29 IV. THE PREPARATION STAGE. LOGIC AND CHANCE 43 V. THE LATER CONSCIOUS WORK 56 VI. DISCOVERY AS A SYNTHESIS. THE HELP OF SIGNS 64 VH. DIFFERENT KINDS OF MATHEMATICAL MINDS 1OO VUJL. PARADOXICAL CASES OF INTUITION 116 IX. THE GENERAL DIRECTION OF RESEARCH 124 FINAL REMARKS 138 APPENDIX I 137 APPENDIX H 142 APPENDIX III 144 INTRODUCTION CONCERNING the title of this study, two remarks are use- ful. We speak of invention: it would be more correct to speak of discovery. The distinction between these two words is well known : discovery concerns a phenomenon, a law, a being which already existed, but had not been per- ceived. Columbus discovered America: it existed before him ; on the contrary, Franklin invented the lightning rod : before him there had never been any lightning rod. Such a distinction has proved less evident than appears at first glance. Toricelli has observed that when one inverts a closed tube on the mercury trough, the mercury ascends to a certain determinate height: this is a discovery; but, there are in doing this, he has invented the barometer ; and plenty of examples of scientific results which are just as much discoveries as inventions. Franklin's invention of the lightning rod is hardly different from his discovery of the electric nature of thunder. This is a reason why the afore- said distinction does not truly concern us; and, as a mat- ter of fact, psychological conditions are quite the same for both cases. On the other hand, our title is "Psychology of Invention in the Mathematical Field," and not "Psychology of Mathematical Invention." It may be useful to keep in mind that mathematical invention is but a case of invention in take in several general, a process which can place domains, whether it be in science, literature, in art or also tech- nology. have M6dern philosophers even say more. They per- xii INTRODUCTION ceived that intelligence is perpetual and constant inven- that life is 1 tion, perpetual invention. As Bibot says, "Invention in Fine Arts or Sciences is but a special case. In practical life, in mechanical, military, industrial, com- mercial inventions, in religious, social, political institu- tions, the human mind has spent and used as much imagina- 2 tion as else" with a still anywhere ; and Bergson, higher and more general intuition, states : "The inventive effort which is found in all domains of life by the creation of new species has found in mankind alone the means of continuing itself by individuals on whom has been bestowed, along with intelligence, the fac- 5 ulty of initiative, independence and liberty/ Such an audacious comparison has its analogue in Met- schnikoff, who observes, at the end of his book on phagocy- tosis, that, in the human species, the fight against microbes is the work not only of phagocytes, but also of the brain, by creating bacteriology. One cannot say that various kinds of invention proceed exactly in the same way. As the psychologist Souriau has noticed, there is, between the artistic domain and the scien- tific one, the difference that art enjoys a greater freedom, since the artist is governed only by his own fantasy, so that works of art are truly inventions. Beethoven's sym- phonies and even Racine's tragedies are inventions. The scientist behaves quite otherwise and his work properly concerns discoveries. As my master, Hermite, told me: **We are rather servants than masters in Mathematics." Although the truth is not yet known to us, it preexists and i Sec Delacroix, L'lnvention et le G6nle (in G. Dumas* Nova>ea& Trait* d* Ptychologie^ VoL VI), p. 449. p. 447. INTRODUCTION xiii inescapably imposes on us the path we must follow under penalty of going astray. This does not preclude many analogies between these two activities, as we shall have occasion to observe. These analogies appeared when, in 1937, at the Centre de Syn- these in Paris, a series of lectures was delivered on inven- tion of various kinds, with the help of the great Genevese psychologist, Claparede. A whole week was devoted to the various kinds of invention, with one session for mathemat- ics. Especially, invention in experimental sciences was treated by Louis de Broglie and Bauer, poetical invention by Paul Valery. The comparison between the circum- stances of invention in these various fields may prove very fruitful. It is all the more useful, perhaps, to deal with a special case such as the mathematical one, which I shall discuss, since it is the one I know best. Results in one sphere (and we shall see that important achievements have been reached in that field, thanks to a masterly lecture of Henri Poin- care*) may always be helpful in order to understand what happens in other ones. 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, consider- ing 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 sub- ject 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 psycholo- gists on the other and even, as we shall see, by a neurolo- gist. As always in psychology, two kinds of methods are avail- 1 able: the "subjective** and the "objective" methods. Sub- 1 1 speak of objective or introspective methods. I see that the modern behaviorist distinguishes between objective psychology and introspective psychology (the latter being said to belong to the past since the death of William James and Titchener), as though these were two different dences, differing as to their object, while it seems to me that botk kinds of methods of observation could be applied and even help each other for the study of the same psychological processes. I understand, however, that for the behaviorist, the object of introspection, Le-, thought and consciousness, is to be ignored. Already, in older times, the prominent biologist Le Dantec eliminated consciousness by qualifying it as an "epipbenomenon." I have always considered that an unscientific attitude, because if consciousness were an epipbenomenon, it would be the only epiphenomenon in nature, where everything reacts on everything else. But, epiphenomenon or not, it exists and can be observed. We are not unjustified in presenting such observm- 2 GENERAL VIEWS jective (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, re- ports 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 men- tal 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. interfere each other Observation and thought do not with ; tkras, made by ourselves or by others, as I shall do in the course of this study.