The Psychology of Invention in the Mathematical Field

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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 cir Constances ; 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 cir Constances,"

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,
in doing this, he has invented the barometer ; and there are
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
general, a process which can take place in several domains,
whether it be in science, literature, in art or also tech-
nology.

M6dern philosophers even say more. They have per-



xii INTRODUCTION

ceived that intelligence is perpetual and constant inven-
tion, that life is perpetual invention. As Bibot says, 1
"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-
tion as anywhere else" ; and Bergson, 2 with a still 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-
ulty of initiative, independence and liberty/ 5

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-
able: the "subjective** and the "objective" methods. 1 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.
Observation and thought do not interfere with each other ;

tkras, made by ourselves or by others, as I shall do in the course of this
study.

It must be noticed that most instances considered by behaviorists (I
found them in J. B. Watson's Behaviorism) are very different from those
which may concern us, being generally taken from thoughts having a
direct relation with our bodily sensations and which are more easily
interpreted hi terms of the doctrine than others. In such cases, cor-
respondences between bodily phenomena and states of consciousness
are easily seen and are more or less known things. They are more hid-
den for cases of abstract thought, such as those we are going to study;
but there is no reason why they should not be discovered at some future
date. This may happen, for instance, with the help of the electric
waves which accompany cerebral processes (a suggestion which I take
from an article of Henri Laugier in the Revue Modern*, reproduced
hi his book Service de Franc* au Canada).



GENERAL VIEWS 3

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 numer-
ous 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 Poincare 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 excep-
tion, which we shall very briefly mention, is a curious at-
tempt 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
Mobius, who happened to be the grandson of a mathema-
tician, though he himself had no special knowledge of
mathematics.

Mobius* book is a rather extensive and thorough study
of mathematical ability from the naturalist's standpoint

*Die Anlage zur Maikematik (l>ipzig).



4 GENERAL VIEWS

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), 5 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 con-
cerns the artistic inclinations of mathematicians. (Mobius
confirms the somewhat classic opinion that most mathema-
ticians are fond of music, and asserts that they are also
interested in other arts.)

Now, Mobius agrees with Gall's conclusions in general,
considering, however, in the first place, that the mathe-
matical 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-Mobius 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 ques-

* There bad beta, some years earlier (1869), an Important work of
Francis Gallon on Hereditary Oennu (London and New York). An
extensive chapter is devoted to men of science.

In connection with the methods to which Mobius' book was generally
devoted, interesting data are contained in Leonard George Guthrie's
Contribution* to the Study of Precocity t Children, concerning early
inclinations of prominent men. To speak only of mathematicians,
Galilei's first calling was toward painting, after which, when seventeen,
be began to study medicine, and only later mathematics. William
Herscbell's first education was as a musician- Besides, it is known that
Gauss hesitated between mathematics and philology.

Similar Instances exist as concerns contemporary men. I heard from
Ptal Painkv himself that he hesitated greatly between devoting himself
to matbematks or to political life. He at first adopted the former
activity, bat^ as is well known, finally engaged in both of them.



GENERAL VIEWS 5

tion, 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 appli-
cation of it which Gall and Mobius 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 un-
expected way. It has been recognized by objective methods



6 GENERAL VIEWS

(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 cor-
respondences 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 9 Views on the Subject. Many psycholo-
gists have also meditated not especially on mathematical in-
vention, 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
thus, Souriau 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

* Souriau, TMorie de I'lnvention (Paris, 1881). Paulhan, Psychologi*
d* F Invention,



GENERAL VIEWS 7

conducted (1937) at the Centre de Synthese 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 prob-
lems 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 prom-
inent 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 Dick-
son'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 mathemat-
ical dream (I never did) or, in that line, dreamed of wholly
absurd things, or were unable to state precisely the ques-
tion 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.



8 GENERAL VIEWS

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 uwakened by an ex-
ternal noise, a solution 5 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 Claparede and an-
other prominent Genevese psychologist, Flournoy, and
published in the periodical UEnseignement Mathema-
tique. 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, mathe-
maticians were asked whether they were influenced by
noises and to what extent, or by meteorological circum-
stances, whether literary or artistic courses of thought
were considered useful or harmful.

Other questions were of a more introspective character

* For technicians, the beginning of No. 27 (pp. 199-200) in Journal de
Mathtmatiqve* pure* et appliqvte*, Series 4, Vol. IX, 1898 (valuation
of a determinant).



GENERAL VIEWS

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 prede-
cessors 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 mete-
orological 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 Val6ry in a lecture
at the French Society of Philosophy, in which he sug-
gested 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



10 GENERAL VIEWS

poetry, it is not certain that this reason is the right one or
at least the only one. Indeed, I know by personal ex-
perience that powerful emotions may favor entirely dif-
ferent kinds of mental creation (e.g., the mathematical
one 1 ) ; 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 Claparede 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 in-
terest? 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 first-
rate 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

i The above mentioned finding of a solution on a sudden awakening
occurred during such & period of emotion.



GENERAL VIEWS 11

masters as Appell, Darboux, Picard, Painleve sent no
answers, which was perhaps a mistake on their part.

Since most answers to the inquiries of Maillet and of
the Enseignement MathematiqiM 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 differ-
ences 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 in-
ventors. Now, Mr. Drach, whose work, besides, is closely
related to Galois', has the same way of approach. He al-
ways wishes to refer to the very form in which discoveries
have appeared to their authors. on the contrary, most
mathematicians who have answered Claparede and Flour-
noy*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 myself.

Poincarfs Statements. Again, we shall put aside the in-
quiry of the EnseigTiement Mathematique. While it failed,
as we have said, to distinguish adequately between those
who replied, it did, on the other hand, provoke, somewhat
later, a testimony which was the most authoritative one
could wish to obtain. Conditions of invention have been



12 GENERAL VIEWS

investigated by the greatest genius which our science has
known during the last half century, by the man whose
impulse is felt throughout contemporary mathematical
science. I allude to the celebrated lecture of Henri Poin-
care at the Societe de Psychologic in Paris. 6 Poincare's
observations throw a resplendent light on relations be-
tween the conscious and the unconscious, between the logi-
cal and the fortuitous which lie at the base of the problem.
Notwithstanding possible objections which will be dis-
cussed in due time, the conclusions which he reaches in
that lecture seem to me fully justified and, at least in the
first five sections, I shall follow him 7 throughout.

Poincare's example is taken from one of his greatest
discoveries, the first which has consecrated his glory, the
theory of fuchsian groups and fuchsian functions. In the
first place, I must take Poincare's own precaution and
state that we shall have to use technical terms without the
reader's needing to understand them. "I shall say, for
example," he says, "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." 8

So, we are going to speak of fuchsian functions. At first,
Poincare attacked the subject vainly for a fortnight, at-

6 "Mathematical Creation," in The Foundations of Science. Translated
by G. Bruce Halsted (New York: The Science Press, 1913), p. 387.

i Quotations without an author's name which will be found in the
following pages are taken from Poincare''s lecture.

* Poincare' deals with the case of mathematics. As Dr. de Saussure,
to whom I am indebted for various interesting remarks, suggested
to me, independence between the process of invention and tie invented
thing may be less in more concrete subjects (see below, Section IX,
p, 131).



GENERAL VIEWS 13

tempting to prove that there could not be any such func-
tions : an idea which was going to prove to be a false one.

Indeed, during a night of sleeplessness and under con-
ditions to which we shall come back, he builds up one first
class of those functions. Then he wishes to find an expres-
sion for them.

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

"Just at this time, I left Caen, where I was living, to go
on a geologic excursion under the auspices of the School
of Mines. The incidents of the 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-Euclidian 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 com-
menced, but I felt a perfect certainty. on my return to
Caen, for conscience 5 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 my pre-
ceding researches. Disgusted with my failure, I went to
spend a few days at the seaside and thought of something



14 GENERAL VIEWS

else. one morning, walking on the bluff, the idea came to
me, with just the same characteristics of brevity, sudden-
ness and immediate certainty, that the arithmetic trans-
formations of indefinite ternary quadratic forms were
identical with those of non-Euclidian geometry/'

These two results showed to Poincare that there existed
other fuchsian groups and, consequently, other fuchsian
functions than those which he had found during his sleep-
lessness. The latter constituted only a special case: the
question was to investigate the most general ones. In this
he was stopped by most serious difficulties, which a persist-
ent conscious effort allowed him to define more adequately,
but not, at first, to overcome. Then, again, the solution
appeared to him as unexpectedly, as unpreparedly as in
the other two instances, while he was serving his time in
the army.

And he adds : "Most striking at first is this appearance
of sudden illumination, a manifest sign of long, uncon-
scious prior work. The role of this unconscious work in
mathematical invention appears to me incontestable."

Looking at one's Own Unconsciousness. Before examin-
ing the latter conclusion, let us resume the history of that
sleepless night which initiated all that memorable work,
and which we set aside in the beginning because it offered
very special characteristics.

one evening," Poincare says, "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."

That strange phenomenon is perhaps the more interest-
ing for the psychologist because it is more exceptional.
Poincare lets us know that it is rather frequent as concerns



GENERAL VIEWS 15

himself: "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 distin-
guishes the two mechanisms or, if you wish, the working
methods of the two egcs."

But that extraordinary fact of watching passively, as
if from the outside, the evolution of subconscious ideas
seems to be quite special to him. I have never experienced
that marvelous sensation, nor have I ever heard of its hap-
pening to others.

Instances in Other Fields. What he reports in the re-
mainder of his lecture is, on the contrary, absolutely gen-
eral and common to every student of research. Thus Gauss,
referring to an arithmetical theorem which he had unsuc-
cessfully tried to prove for years, writes: "Finally, two
days ago, I succeeded, not on account of my painful efforts,
but by the grace of God. Like a sudden flash of lightning,
the riddle happened to be solved. I myself cannot say what
was the conducting thread which connected what I previ-
ously knew with what made my success possible."**

It is unnecessary to observe that what happened to me
on my awakening is perfectly similar and typical, as the
solution which appeared to me: (1) was without any rela-
tion to my attempts of former days, so that it could not
have been elaborated by my previous conscious work; (2)
appeared without any time for thought, however brief.

The same character of suddenness and spontaneousness
had been pointed out, some years earlier, by another great
scholar of contemporary science. Helmholtz reported it in
an important speech delivered in 1896. Since Helmholtz
and Poincare, it has been recognized by psychologists as

sa Letter mentioned in Revue des questions tcientifiques, October 1886,
p. 575.



16 GENERAL VIEWS

being very general in every kind of invention. Graham
Wallas, in his A rt of Thought, suggested calling it "il-
lumination," this illumination being generally preceded by
an "incubation" stage wherein the study seems to be com-
pletely interrupted and the subject dropped. Such an
illumination is even mentioned in several replies on the in-
quiry of UEnseignement Mathematique. Other physicists
like Langevin, chemists like Ostwald, tell us of having ex-
perienced it. In quite different fields, let us cite a couple
of instances. one has attracted the attention of the psy-
chologist Paulhan. It is a celebrated letter of Mozart : 8b

"When I feel well and in a good humor, or when I am
taking a drive or walking after a good meal, or in the
night when I cannot sleep, thoughts crowd into my mind
as easily as you could wish. Whence and how do they come?
I do not know and I have nothing to do with it. Those which
please me, I keep in my head and hum them ; at least others
have told me that I do so. once I have my theme, another
melody comes, linking itself to the first one, in accordance
with the needs of the composition as a whole : the counter-
point, the part of each instrument, and all these melodic
fragments at last produce the entire work. Then my soul
is on fire with inspiration, if however nothing occurs to
distract my attention. The work grows ; I keep expanding
it, conceiving it more and more clearly until I have the
entire composition finished in my head though it may be
long. Then my mind seizes it as a glance of my eye a beauti-
ful picture or a handsome youth. It does not come to me
successively, with its various parts worked out in detail, as
they will be later on, but it is in its entirety that my imag-
ination lets me hear it.

"Now, how does it happen, that, while I am at work, my

*b See Paulhan, Psychologic de I'invention^ p. 97, who refers to Seailles,
Le Qinie dans Far*, p. 177.



GENERAL VIEWS 17

compositions assume the form or the style which charac-
terize Mozart and are not like anybody else's? Just as it
happens that my nose is big and hooked, Mozart's nose and
not another man's. I do not aim at originality and I should
be much at a loss to describe my style. It is quite natural
that people who really have something particular about
them should be different from each other on the outside as
well as on the inside."

Poetical inspiration is reported to have been as spon-
taneous with Lamartine who happened to compose verses
instantly, without one moment of reflection ; and we have
the following most suggestive statement made at the
French Philosophical Society by our great poet Paul
Valery c 80 "In this process, there are two stages.

iC There is that one where the man whose business is writ-
ing experiences a kind of flash for this intellectual life,
anything but passive, is really made of fragments; it is
in a way composed of elements very brief, yet felt to
be very rich in possibilities, which do not illuminate the
whole mind, which indicate to the mind, rather, that there
are forms completely new which it is sure to be able
to possess after a certain amount of work. Sometimes I
have observed this moment when a sensation arrives at
the mind ; it is as a gleam of light, not so much illuminat-
ing as dazzling. This arrival calls attention, points, rather
than 31ii.TniTifl.tes, and in fine, is itself an enigma which car-
ries with it the assurance that it can be postponed. You say,
*I see, and then tomorrow I shall see more.' There is an
activity, a special sensitization ; soon you will go into the
dark-room and the picture will be seen to emerge.

"I do not affirm that this is well described, for it is ex-
tremely hard to describe. . . ."

sc B+Ueti* 8oc. Philosophic, VoL 28, 1928.



18 GENERAL VIEWS

Similarly, as Catherine Patrick has noticed in the article
cited below, footnote 10 to Section III, the English poet
A. E. Housman in a lecture delivered at Cambridge, Eng-
land (see his valuable booklet The Name and Nature of
Poetry) also describes that spontaneous and almost in-
voluntary creation, eventually alternating with conscious
effort.

Similar observations occur even in ordinary life. Does
it not frequently happen that the name of a person or of a
place which you have vainly tried to remember, recurs to
you when you are no longer thinking of it?

That this fact is more analogous to the process of inven-
tion than would be believed at first is shown by a remark
of Remy de Gourmont : he notices that the right word to
express an idea is also very often, after long and fruitless
search, found in the same way, viz., when one is thinking
of something else. This case is interesting as it presents an
intermediate character, being obviously analogous to the
preceding one and nevertheless already belonging to the
field of invention.

No less similar to the above observation is the well-known
proverb: "Sleep on it." This again may be considered as
belonging to the realm of invention if we follow modern
philosophers and take the word in a broad sense, as we
said in the Introduction.

The Chance Hypothesis. The biologist Nicolle 9 also men-
tions creative inspirations and even strongly insists on
them. But it is necessary to discuss the way in which he
interprets them.

For Poincare, as we saw, they are evident manifestations

Biologit dt Flnvention, pp. 5-7.



GENERAL VIEWS 19

of a previous unconscious work, and here I must say that
I do not see how this view can be seriously disputed.

However, Nicolle does not seem to agree with it ; or, more
exactly, he does not speak of the unconscious. <4 The in-
ventor," he writes, "does not know prudence nor its junior
sister, slowness. He does not sound the ground nor quibble.
He at once jumps into the unexplored domain and by this
sole act, he conquers it. By a streak of lightning, the
hitherto obscure problem, which no ordinary feeble lamp
would have revealed, is at once flooded with light. It is like
a creation. Contrary to progressive acquirements, such an
act owes nothing to logic or to reason. The act of discovery
is an accident."

This is, in its most extreme form, the theory of chance
which psychologists like Souriau also set forth.

Not only can I not accept it, but I can hardly under-
stand how a scientist like Nicolle could have conceived
of such an idea. Whatever respect we must have for the
great personality of Charles Nicolle, explanation by pure
chance is equivalent to no explanation at all and to assert-
ing that there are effects without causes. Would Nicolle
have been contented to say that diphtheria or typhus,
which he so admirably investigated are the result of
pure chance? Even if we do not, at this juncture, enter into
the analysis which we shall try to give in the next section,
chance is chance: that is, it is the same for Nicolle or
Poincare or for the man in the street. Chance cannot ex-
plain that the discovery of the cause of typhus was made
by Nicolle (that is, by a man having pondered scientific
subjects and the conditions of experiments for years and
also having shown his marvellous ability) rather than by
any of his nurses. And as to Poincare, if chance could ex-



20 GENERAL VIEWS

plain one of the brilliant intuitions which he describes in his
lecture which I cannot even believe how would that
explanation account for all those which he successively
mentions, not to speak of all those which have occurred
throughout the various theories which constitute his im-
mense work and have transformed almost every branch
of mathematical science? You could as well imagine, ac-
cording to well-known comparison, a monkey striking a
typewriter and fortuitously printing the American Con-
stitution.

This does not mean that chance has no role in the in-
venting process. 10 Chance does act. We shall see in Section
III how it acts inside unconsciousness.

10 Dealing with the mathematical domain, we speak only of psycho-
logical chance, Le., fortuitous mental processes. As Claparede points
out (Meetings of 1937 at the Centre de Synthese), it must be distinguished
from external hazards, such as occur in the well-known case of Galvani's
frogs and which, of course, are likely to play the initial role in experi-
mental discovery.



//. DISCUSSIONS on UNCONSCIOUSNESS



THOUGH unconsciousness is, strictly speaking, a business
of professional psychologists, it is so closely connected with
my main subject that I cannot help dealing scantily with it.

That those sudden enlightenments which can be called
inspirations cannot be produced by chance alone is already
evident by what we have said : there can be no doubt of the
necessary intervention of some previous mental process
unknown to the inventor, in other terms, of an unconscious
one. Indeed, after having seen, as we shall at many places
in the following, the unconscious at work, any doubt as to
its existence can hardly arise.

Although observations in everyday life show us this
existence, and although it has been recognized since the
time of St. Augustine and by masters such as Leibniz, the
unconscious has by no means remained unquestioned. The
very fact that it is unknown to the usual self gives to it such
an appearance of mystery that it has experienced, at the
hands of various authors, equally excessive disgraces and
favors. Several authors have been stubbornly opposed to
admitting any unconscious phenomenon. To speak of a
case directly connected with invention, it is difficult to un-
derstand how, as late as 1852, after centuries of psycho-
logical studies, one could read in a work on invention, 1 such
a statement as the following: "These seeming divinations,
these almost immediate conclusions are to be explained
most naturally by known laws[?] : mind thinks either by
analogy or by habit; thus, mind jumps over intennedi-
i Desdotuts, Theorie de ^Invention.



22 UNCONSCIOUSNESS

aries," as though the fact of jumping over intermediaries
which one cannot but know of were not, by definition, an
unconscious mental process ! one cannot help remembering
Pierre Janet's patients, who, obeying his suggestion, "did
not see" the cards which were marked with a cross . . . which,
however, they must necessarily have seen in order to elim-
inate them. But, of course, Pierre Janet's conclusion was
not to deny unconsciousness, whose intervention is, in such
cases, grossly evident.

In order to ignore unconscious ideas by any means, the
philosopher Alfred Fouillee uses two contrary attitudes:
either he will contend that, under any conditions, there
must be consciousness, only very feeble and indistinct ; or,
if this hypothesis does not give him a means to avoid the
one he seems to be afraid of, he withdraws in the opposite
direction by invoking reflex actions, i.e., such actions the
existence of which has been undoubtedly recognized by
physiologists operating, for instance, on beheaded frogs,
and which do not imply the intervention of mental centers
but only of more or less peripheral and inferior nervous
elements.

There are many well-known acts of mind which do not
admit of either one or the other of these opposite explana-
tions. Let us only mention the so-called "automatic writ-
ing," which has been thoroughly studied in the case of some
psychical patients, but which is by no means an exclusive
feature of such abnormal people. Many of us, if not all of
us, have experienced automatic writing; at least I have
very often in my life. once, when I was in the high school
and had before me a task which did not interest me very
much, I suddenly perceived that I had written at the top
of my sheet of paper "Mathematiques." Could that be con-



UNCONSCIOUSNESS 23

sidered as a reflex motion? Can such reflex motions imply
the rather complicated gestures of handwriting; and are
the corresponding inferior centers aware that "Mathema-
tiques" wants an "h" after the "t"? on the other hand, if
I had given one instant's thought, however short, to what
I was writing, I should never have written that word, as
the paper was devoted to a quite different subject.

The Manifold Character of Unconsciousness. Today,
the existence of the unconsciousness seems to be rather gen-
erally admitted, although some philosophical schools still
wish to exclude it.

Indeed, very ordinary facts illustrate with full evidence
not only the intervention of unconscious phenomena, but
one of their important properties : I allude to the familiar
which does not mean simple fact of recognizing a hu-
man face. Identifying a person you know requires the help
of hundreds of features, not a single one of which you
could explicitly mention (if not especially gifted or trained
for drawing) . Nevertheless, all these characters of the face
of your friend must be present in your mind in your un-
conscious mind, of course and all of them must be present
at the same instant. Therefore, we see that the unconscious
has the important property of being manifold ; several and
probably many things can and do occur in it simultane-
ously. This contrasts with the conscious ego which is
unique.

We also see that this multiplicity of the unconscious
enables it to carry out a work of synthesis. In the above
case, the numerous details of a physiognomy result, for
our consciousness, in only one sensation, viz., recognition. 1

2 1 understand that, In the contemporary Gestalt psychology, there is a
unique sensation of physiognomy, independent of the ideas of the various



24 UNCONSCIOUSNESS

Fringe-Consciousness. Not only is it impossible to doubt
the reality of the unconscious, but we must emphasize that
there is hardly any operation of our mind which does not
imply it. At a first glance, ideas are never in a more posi-
tively conscious state than when we express them in speak-
ing. However, when I pronounce one sentence, where is
the following one? Certainly not in the field of my con-
sciousness, which is occupied by sentence number one ; and
nevertheless, I do think of it, and it is ready to appear the
next instant, which cannot occur if I do not think of it un-
consciously. But, in that case, we have to deal with an un-
conscious which is very superficial, quite near to conscious-
ness and at its immediate disposal.

It seems that we can identify this with what Francis
Galton 3 calls the "ante-chamber" of consciousness, beauti-
fully describing it as follows :

details. I am not qualified to discuss that important conception; how-
ever, as the question is closely connected with what we are going to
say later on (especially in Section VI), I must state precisely that
whether we admit it or not, there is certainly something corresponding
to the individual effect of light on each point of our retina (at least at
the very first moment when this effect is transmitted to the brain) and
that these individual sensations are unconscious. Is that unconscious-
ness generally a very remote one, probably because the corresponding
mechanism has been acquired hi earliest childhood analogous to those
which will interest us in the following sections? This would be another
question; but I must add that this identity of nature is hardly doubtful
for me: there is a chain of intermediaries, some of which are even
described by the Gestaltists themselves (as I see by Paul Guillaume's
Paychologie de la Forme) and above all, such cases as learning to ride a
bicycle (as has been noticed by several authors). Personally, having
learned that when an adult, I did not master it until everything, from
having been conscious in the beginning, became fully unconscious so
fuHy unconscious that I hardly knew why my motions had a better
success finally than originally.

* Inquiries into Human Faculty, p. 203 of the first edition, 1883 (Lon-
don, New York: Macmillan) ; p. 146 of the 1908 edition (London: J. M.
Dent; New York: E. P. Button).



UNCONSCIOUSNESS 25

"When I am engaged in trying to think anything out,
the process of doing so appears to me to be this : The ideas
that lie at any moment within my full consciousness seem to
attract of their own accord the most appropriate out of a
number of other ideas that are lying close at hand, but im-
perfectly within the range of my consciousness. There
seems to be a presence-chamber in my mind where full con-
sciousness holds court, and where two or three ideas are at
the same time in audience, and an ante-chamber full of
more or less allied ideas, which is situated just beyond the
full ken of consciousness. Out of this ante-chamber the
ideas most nearly allied to those in the presence-chamber
appear to be summoned in a mechanically logical way, and
to have their turn of audience, 5 *

The word "subconsciousness" might be distinguished
from "unconsciousness" in order to denote such superficial
unconscious processes and there is, moreover, the word
"fringe-consciousness," created by William James and
then used by Wallas with that same meaning, as much as I
understand, and which is even very expressive in that
sense.* These subconscious states are valuable for psy-
chology by being accessible to introspection, which, at least
in general, 5 is not possible for more remote processes. In-
deed, it is thanks to them that introspection is possible. To
describe them, psychologists srach as Wallas use a com-
parison drawn from the facts of eyesight. "The field of
vision of our eyes consists of a small circle of full or *f ocaP

* "Foreconsckms," used by Varendonck and other writers of Freud's
school for special psychic states, is doubtfully available for us.

*An exception is Poincar6*s sleepless night: see p. 14. Another one
possibly (though less certainly) occurred in the case of a technical
inventor mentioned by Qaparede in the meeting of the Centre de
Synthese.



26 UNCONSCIOUSNESS

vision, surrounded by an Irregular area of peripheral
vision, which is increasingly vague and imperfect as the
limit of vision is neared. We are usually unaware of the
existence of our peripheral vision, because as soon as any-
thing interesting presents itself there we have a strong
natural tendency to turn the focus of vision in its direction.
Using these terms, we can say that one reason why we tend
to ignore the mental events in our peripheral consciousness
is that we have a strong tendency to bring them into focal
consciousness as soon as they are interesting to us, but that
we can sometimes, by a severe effort, keep them in the
periphery of consciousness, and there observe them."

The observation of the distinction between consciousness
and fringe-consciousness is generally difficult; but the
difficulty happens to be much less in the case of invention,
which interests us. The reason for that is that invention
work by itself implies that thought be inflexibly directed
toward the solution of the problem : when obtaining the lat-
ter, and only then, the mind can perceive what takes place
in the "fringe-consciousness," a fact which will be of a
great interest to us in this study.

Successive Layers in the Unconscious. We see that there
are at least two kinds more precisely, two degrees of
unconsciousness.

It can hardly be doubted, and we shall be able to con-
firm this later on, that there must even be, in the uncon-
scious, several successive layers, the most superficial one
being the one we just considered. More remote is the uncon-
scious layer which acts in automatic writing; still more
those which allow inspirations such as we reported in the
preceding section. Even deeper ones will appear to us at
the end of this study. There seems to be a kind of con-



UNCONSCIOUSNESS 27

tinuity between full consciousness and more and more hid-
den levels of the unconscious : a succession which seems to be
especially well described in Taine's book on Intelligence,
when he writes : 6

<4 You may compare the mind of a man to the stage of a
theatre, very narrow at the footlights but constantly
broadening as it goes back. At the footlights, there is
hardly room for more than one actor. ... As one goes
further and further away from the footlights, there are
other figures less and less distinct as they are more distant
from the lights. And beyond these groups, in the wings and
altogether in the background, are innumerable obscure
shapes that a sudden call may bring forward and even
within direct range of the footlights. Undefined evolutions
constantly take place throughout this seething mass of
actors of all kinds, to furnish the chorus leaders who in
turn, as in a magic lantern picture, pass before our eyes." 7

That striking description is quite similar to the Book X
of St. Augustine's Confessions. only St. Augustine speaks
of memory ; but, as it seems to me, he fully realizes the for
me undoubted fact that memory belongs to the domain
of the unconscious.

Fringe-subconscious evidently offers some analogy with
the very vaguely conscious ideas which Fouillee supposes,
while, at the other end of the chain, the succession of un-

Added by Taine for the first time in the edition of 1897 (Vol. I),
p. 278.

7 Some recent psychological schools, such as Freud's, would seem, at
first, to disagree with the above point of view and to speak only of one
kind of (proper) unconscious. As I am informed by a competent col-
league and friend, this would be a misinterpretation of Freud's thought.

We already have seen (Note 2, p. 23) that ideas have a tendency
to become more and more unconscious by influence of time: a circum-
stance we shall again meet with in Section VII (see p. 101).



28 UNCONSCIOUSNESS

conscious layers is more probably, as Spencer states (The
Principles of Psychology, Vol. I, Chap. IV) , in continuity
with reflex phenomena. Thus, the two states which Fouillee
wants to oppose to unconsciousness seem to be nothing else
than the extreme cases of it: a double conclusion, which,
however, Fouillee rejects (L*Evolutionisme des I dees-
Forces, Introduction, p. xiv and end of p. xix) by argu-
ments the discussion of which it is useless to inflict upon the
reader.



///. THE UNCONSCIOUS AND
DISCOVERY



Combination of Ideas. What we just observed concerning
the unconscious in general will be seen again from another
angle, when speaking of its relations with discovery.

We shall see a little later that the possibility of imputing
discovery to pure chance is already excluded by Poincare's
observations, when more attentively considered.

On the contrary, that there is an intervention of chance
but also a necessary work of unconsciousness, the latter
implying and not contradicting the former, appears, as
Poincare shows, when we take account not merely of the
results of introspection, but of the very nature of the
question.

Indeed, it is obvious that invention or discovery, be it in
mathematics or anywhere else, takes place by combining
ideas. 1 Now, there is an extremely great number of such
combinations, most of which are devoid of interest, while,
on the contrary, very few of them can be fruitful. Which
ones does our mind I mean our conscious mind per-
ceive? only the fruitful ones, or exceptionally, some which
could be fruitful.

However, to find these, it has been necessary to construct
the very numerous possible combinations, among which the
useful ones are to be found.

It cannot be avoided that this first operation take place,



Miiller observes that the Latin verb "cogito," for "to think,"
etymologically means "to shake together." St. Augustine had already
noticed that and also observed that "intelligo" means to "select among,"
in a curious connection with what we say in the text.



80 THE UNCONSCIOUS AND DISCOVERY

to a certain extent, at random, so that the role of chance
is hardly doubtful in this first step of the mental process.
But we see that that intervention of chance occurs inside
the unconscious: for most of these combinations more
exactly, all those which are useless remain unknown to us.
Moreover, this shows us again the manifold character of
the unconscious, which is necessary to construct those
numerous combinations and to compare them with each
other.

The Following Step. It is obvious that this first process,
this building up of numerous combinations, is only the
beginning of creation, even, as we should say, preliminary
to it. As we just saw, and as Poincare observes, to create
consists precisely in not mating useless combinations and
in examining only those which are useful and which are
only a small minority. Invention is discernment, choice.

To Invent Is to Choose. This very remarkable conclusion
appears the more striking if we compare it with what Paul
Valery writes in the NouveUe Revue Franfaise: "It takes
two to invent anything. The one makes up combinations ;
the other one chooses, recognizes what he wishes and what
is important to him in the mass of the things which the
former has imparted to him.

*<What we call genius is much less the work of the first
one than the readiness of the second one to grasp the value
of what has been laid before him and to choose it."

We see how beautifully the mathematician and the poet
agree in that fundamental view of invention consisting of a
choice.

Esthetics in Invention. How can such a choice be made?
The rules which must guide it "are extremely fine and
delicate. It is almost impossible to state them precisely;



THE UNCONSCIOUS AND DISCOVERY 31

they are felt rather than formulated. Under these condi-
tions, how can we imagine a sieve capable of applying them
mechanically ?"

Though we do not directly see this sieve at work, we can
answer the question, because we are aware of the results it
affords, i.e., the combinations of ideas which are perceived
by our conscious mind. This result is not doubtfuL fit The
privileged unconscious phenomena, those susceptible of
becoming conscious, are those which, directly or indirectly,
affect most profoundly our emotional sensibility.

"It may be surprising to see emotional sensibility in-
voked a propos of mathematical demonstrations which, it
would seem, can interest only the intellect. This would be
to forget the feeling of mathematical beauty, of the har-
mony of numbers and forms, of geometric elegance. This
is a true esthetic feeling that all real mathematicians know,
and surely it belongs to emotional sensibility."

That an affective element is an essential part in every
discovery or invention is only too evident, and has been in-
sisted upon by several thinkers ; indeed, it is clear that no
significant discovery or invention can take place without
the mil of finding. But with Poincare, we see something
else, the intervention of the sense of beauty playing its part
as an indispensable means of finding. We have reached the
double conclusion :

that invention is choice

that this choice is imperatively governed by the
sense of scientific beauty.

Coming Back to the Unconscious. In what region a
word not to be taken in a too literal, but, so to say, in a
symbolic meaning in what region of the mind does that



32 THE UNCONSCIOUS AND DISCOVERY

sorting take place? Surely not in consciousness, 2 which,
among all possible combinations, only knows of the right
ones.

Poincare, at first, sets forth the idea that the uncon-
scious itself should exclusively perceive the interesting
combinations. He does not insist on that first hypothesis
and, indeed, I cannot consider it as deserving further ex-
amination. It is, as it seems, nothing but recording before
making a second jump and would be only a verbal question,
a question of definition: we should have to find in what
"region" uninteresting combinations could be eliminated,
and there would be no reason not to call that other region,
if existing, another part of the unconscious.

So there remains only Poincare's final conclusion, viz.,
that to the unconscious belongs not only the complicated
task of constructing the bulk of various combinations of
ideas, but also the most delicate and essential one of select-
ing those which satisfy our sense of beauty and, conse-
quently, are likely to be useful.

Other Views on Incubation. These a priori reasons would
be by themselves sufficient to justify Poincare's conclusion.
However, that conclusion has been assailed by various au-
thors, some of whom 3 again seem to be moved by that same

2 Mozart's letter (see p. 16) suggests that, in his mind, choice is
partly conscious, though probably, as we should think, preceded by a
preliminary unconscious one: otherwise we should be re-conducted to the
hypothesis of pure chance.

sRossman (Psychology of the Inventor, VI, p. 86) writes: "The
assumption that the subconscious is responsible for the final condition
is no answer to the problem. It merely amounts to giving a name to a
thing which puzzles and mystifies us." Now, unconscious phenomena are
not mere names, but realities. Though they are less easy to observe
directly, they exist, as we have seen, whether they are the cause of
illumination or not To invoke "physiological and chemical conditions
in the body," as the author does without trying, of course, to specify



THE UNCONSCIOUS AND DISCOVERY 83

fear of the unconscious which we met with in the preceding
section. Let us see how they endeavor to account for the
striking fact of illumination.

It is not necessary to speak again of the doctrine of pure
chance, which we have discussed in Section I and at the
beginning of this one. Therefore, we consider it as granted
that incubation generally precedes illumination. 4 In this
period of incubation, no work of the mind is consciously
perceived ; but, instead of admitting that there is an un-
conscious work, could we not admit that nothing at all
occurs ? Two chief hypotheses have been set forth. 5

A. It has been supposed that an explanation for the new
state of mind which makes illumination possible could lie
in a freshness or lack of brain fatigue. This is what we can
call the rest-hypothesis. Poincare, though not adopting it,
has thought of it but (see below) in a special case.

B. It can be admitted that an essential cause of illumina-
tion may be the absence of interferences which block prog-
ress during the preparation stage. **When, as must often
happen, the thinker makes a false start, he slides insensibly
into a groove and may not be able to escape at the moment.
. . . Incubation would consist in getting rid of false leads
and hampering assumptions so as to approach the prob-
lem with an 'open mind. 5 " We can call this the forgetting-
hypothesis.

what they arc incurs, at least as modi, the same objection of merely
giving a name to our ignorance. That statement of Rossman, if It means
anything, merely gives another name for the rest-hypothesis.

4 Exceptions seem to occur in cases such as Mozart's and, as Catherine
Patrick observes Jn the work mentioned below hi footnote 10, in A. B.
Housman's (Cf. p. 87). For the reason given in the text, it seems to me
certain that these exceptions must be apparent ones, doe to a special
rapidity of incubation.

* See Woodworth's Exp*r*m.0ntal Ptyckology, p. 828.



34 THE UNCONSCIOUS AND DISCOVERY

Discussion of Tliese Ideas. Helmholtz's testimony has
been invoked in favor of the rest-hypothesis. Helmholtz
says that "happy ideas" (his word for illumination) never
come to him when his mind is fatigued or when he is seated
at his work table; 6 and, mentioning his preparatory study
of the question investigated, adds that "Then, after the
fatigue resulting from this labor has passed away, there
must come an hour of complete physical freshness before
the good ideas arrive" a statement which perhaps doubt-
fully supports the hypothesis in question, as illumination is
not said to occur the moment mind is rested, but about an
hour later.

Besides, the case of Helmholtz is not a universal one and
there are observations to the contrary. Professor K. Fried-
richs writes to me that "creative ideas" (if any) "come
mostly of a sudden, frequently after great mental exertion,
in a state of mental fatigue combined with physical relaxa-
tion"; and a similar statement has been given to me by a
critic of art, Dr. Sterling, who has to solve problems con-
cerning the authenticity of pictures. Dr. Sterling tells me
he has noticed that, after long conscious effort, inspiration
usually comes when he is fatigued, as though it were neces-
sary for the conscious self to be weakened in order that un-
conscious ideas may break through.

The question whether mathematicians usually stand or sit when
working is one of those asked in the questionnaire of L'Enseignement
MafkSmatique. Habits of that kind are certainly among those that vary
most among individuals. Helmholtz's and Poincar6*s statements seem
to suggest that they often used to sit at a work table. I never do so,
except when I am obliged to effectuate written calculations (for which
I have a certain reluctance). Except in the night when I cannot sleep,
I never find anything otherwise than by pacing up and down the room.
I feel exactly like the character of Emile Augier who says: "Legs are the
wheels of thought"



THE UNCONSCIOUS AND DISCOVERY SS

This instance shows, and we have to bear it in mind, that
rules about such matters are not necessarily invariable
ones. Processes may differ not only with individuals, but
even in one and the same man. Indeed, the very observa-
tions of Poincare show us three kinds of inventive work
essentially different if considered from our standpoint,
viz.,

a. fully conscious work.

b. illumination preceded by incubation.

c. the quite peculiar process of his first sleepless night. 7
Moreover, we have said that Poincare has thought of the

rest-hypothesis. He has thought of it (without being in-
clined to adopt it even then) in connection with a special
case, even different, as it seems, from the three mentioned
above and which is quite similar to Helmholtz's description,
viz., a first and unsuccessful work-period, a rest a new
half -hour of work (Helmholtz speaks of one hour) after
which discovery comes. As he himself says, this could be
explained by the rest-hypothesis, but with the same ob-
jection as in the case of Helmholtz.

Still other processes are possible. A very curious one is
reported by the chemist J. Teeple 8 who, after half an hour,
realized that he had been working on a question without
being aware that he was doing so, and in such an abstracted
state of mind that during that time he forgot that he had
already taken a bath and was taking a second one : a special
case of unconscious process, as the thinker was not con-

T Unhappily, Poincare, when telling -as that that night was not the
only one when he has been able to watch the action of his subconscious
self, does not enter into further particulars. More details would have
been interesting about the first night in question.

s See Platt and Baker, Jonm. Chemical Educ^ Vol. VIII (1931),
pp. 1969-2002,



86 THE UNCONSCIOUS AND DISCOVERY

scious of his mental work while it was going on, but per-
ceived it when it ended.

Thus there are several possibilities, and some of them
might be explained by the rest or forgetting hypotheses. I
willingly admit that a "fresh" or "open" mind that is,
forgetting of some unsuccessful attempts can account for
discovery when the latter is separated from the first con-
scious period by a long interval, say some months. I speak,
in that case, of discovery, not of illumination, because this
is not, properly speaking, illumination : the solution does
not appear suddenly and unexpectedly, but is given by a
new work.

Also, forgetting could, perhaps, as Professor Wood-
worth notices, 8 explain illuminations which would reveal
solutions of an especially great simplicity, it happening
that such may be at first overlooked. Any other ones, in the
hypotheses we are dealing with, would require a new effort
of the mind in a new condition (whether on account of rest
or on account of forgetting previous ideas) to resume the
subject. Such explanations are, therefore, to be rejected
unhesitatingly in those cases of suddenness which are re-
ported by Poincare and confirmed by other authors. Poin-
care was not working when he boarded the omnibus of
Coutances: he was chatting with a companion; the idea
passed through his mind for less than one second, just the
time to put his foot on the step and enter the omnibus. It
cannot even be said that the idea was simple enough not to
require any work. Poincar informs us that he had to work
it out for verification on his return to Caen. Though not as
strictly limited in time, other illuminations reported by

See the above cited p. 823 of his Experimental Psychology.



THE UNCONSCIOUS AND DISCOVERY $7

Poincare occur with the same suddenness and unprepared-
ness which exclude the proposed explanations.

If we admitted any one of them, illuminations ought to
come as a result of a new work of the same kind as the pre-
liminary one, from which it could differ only by elimination
of the result paradoxically, an exclusively harmful one
(be it fatigue or a misleading idea) of that preliminary
work. Now, observation shows that the illumination process
is not of the same nature as the previous conscious work.
The latter is really work, implying a more or less severe
tenseness of mind and often multiplied attempts ; the for-
mer occurs at once, without any perceptible effort.

Besides, if we should accept the forgetting hypothesis,
that is, that several previous and inappropriate ideas pre-
cede and for a while block the right one, we ought to be
aware of them (as these theories precisely claim to exclude
any role of the unconscious). on the contrary, most often,
instead of seeing several ways supposedly likely to lead to
the solution, we conceive no such ideas at all.

To sum it up, we see that such explanations as the rest or
the forgetting hypotheses can be admitted in some cases,
but that in other instances, especially typical illuminations
as noted by Poincare and others, they are contradicted by
facts. 10

10 Catherine Patrick (Archives of Psychology, Vol. 26, 1935, No. 178)
has carried out an experimental study of poetical invention: she has
asked several persons some being professional poets, others not to
write a poem suggested by a picture shown them. Contrary to her
opinion, I cannot consider the process she observed as being comparable
to those we investigate. The time of the experiment hardly more than
twenty minutes shows that she deals with a quite different question j
and her so-called incubation, where subjects are required to relate aloud
the course of their thoughts during their attempt to do the required work,
has nothing to do with incubation as Helmholtz or Poincard consider it,



38 THE UNCONSCIOUS AND DISCOVERY

Other Views on Illumination. An Intimation Stage. An
objection of another kind, one less essential, has been
raised 11 against Helmholtz's and Poincare's descriptions of
the phenomenon of illumination. Does not a part of it take
j)lace in what we have previously called the fringe-con-
sciousness ?

Now, fringe-consciousness and proper consciousness are
so close to each other, exchanges between them are so con-
tinuous and so rapid, that it seems hardly possible to see
how they divide their roles in that sudden lightninglike
phenomenon of illumination. However, in consequence of
what we shall find in Section VI, we shall be led to some
probable views on that subject.

Another most curious circumstance has been noted. For
some thinkers, while engaged in creative work, illumination
may be preceded by a kind of warning by which they are
made aware that something of that nature is imminent
without knowing exactly what it will be. Wallas has given
the name of "intimation" to that peculiar phenomenon. I
never experienced any such sensation and Poincare does
not speak of it, which he certainly would have done if it had
happened to him. An inquiry on that point among scien-
tists would be useful. lla

A last objection against Poincare's ideas has been raised
by Wallas. This author admits that Poincare may be right
in saying "that without a rather high degree of this aes-
thetic instinct no man will ever be a great mathematical dis-
coverer," but adds that "it is extremely unlikely that the

in which that course of thought, inasmuch as it refers to the research,
may remain totally unknown to the subject.

11 Graham Wallas, The Art of Thought, p. 96.

11* Later inquiry brought a few affirmative answers.



THE UNCONSCIOUS AND DISCOVERY 39

aesthetic instinct alone was the 'power* driving the *ma-
chine 5 of his thought."

I can but reproduce textually that passage of Wallas,
because I hardly understand it. It seems to me to rest on
an evident confusion between two things which must be
distinguished 12 from each other: the "drive" (i.e., "what to
do") and the "mechanism" 15 ("how to do it"). We shall
speak of "drive" in Section IX (and again find that the
sense of beauty is the moving power) : for the time being,
we deal with mechanism ; and what that mechanism is, I can
hardly doubt. Wallas suspects Poincare of thinking thus
because he was a friend of Boutroux, who was a friend of
William James, or because he was influenced by the ideas of
the "Mechanist" school. Not only is Poincare's statement
the result of his own observations and not anybody else's
suggestions ; but personally, I feel exactly as he does, not
because of having been a special friend of Boutroux or
William James, or having studied the Mechanist school
(which I have not) or any other one, but on account of my
own auto-observation because the ideas chosen by my
unconscious are those which reach my consciousness, and I
see that they are those which agree with my aesthetic sense.

As a matter of fact, I consider that every mathematician,
if not every scientist, would agree to that opinion. I may
add that actually some of them, writing to me on the gen-
eral subject of this work, spontaneously (i.e., without a
question from me on that special point) expressed them-
selves in the same sense, in the most positive way.

12 See Woodworth's Dynamic Psychology.

is I son obliged to use almost simultaneously both words "mechanism*
and (see below) "mechanist,** the meanings of which are quite different
and Independent. I hope this will not make any confusion in the reader's
mind.



40 THE UNCONSCIOUS AND DISCOVERY

Further Theories on the Unconscious. Certainly, our
above conclusions are, to a certain extent, surprising. They
raised for Poincare a disconcerting question. The uncon-
scious is usually said to be automatic and, in a certain sense,
it undoubtedly is so, as it is not subjected to our will, at
least not to the direct action of our will, and even is sub-
tracted from our knowledge. But now we find an opposite
conclusion. The unconscious self "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?" 14

Now, such an idea has been in favor among metaphysi-
cians in recent times and even in more ancient ones. The
very fact that unconsciousness, though manifesting itself
from time to time, is not really known to us, has given it a
mysterious character and, on account of that mysterious
character, it has often been endowed with superior powers. 15
That unconsciousness may be something not exclusively
originating in ourselves and even participating in Divinity
seems already to have been admitted by Aristotle. In Leib-
niz's opinion, it sets the man in communication with the
whole universe, in which nothing could occur without its
repercussion in each of us ; and something analogous is to

"Here Poincar alludes to Emile Boutroux, whose influence on him
certainly existed in this case.

15 Similar cases obviously occur as concerns dreams, which, since the
world exists, have been supposed to enjoy a kind of prophetic value. But
ft would be too disrespectful to compare metaphysical theories of the
unconscious with the popular "Clue to Dreams" books.



THE UNCONSCIOUS AND DISCOVERY 41

be found in Schelling; again, Divinity is invoked by
Fichte; etc.

Even more recently, a whole philosophical doctrine has
been built on that principle in the first place by Myers,
then by William James himself (although the great phi-
losopher, in an earlier work, his Principles of Psychology^
expresses himself, at times, as though doubting the very
existence of unconsciousness). According to that doctrine,
the unconscious would set man in connection with a world
other than the one which is accessible to our senses and with
some kinds of spiritual beings.

Meanwhile, unconsciousness appears to other authors as
being the trace of prior existence; and others again sug-
gest the possibility that it is due to the action of disem-
bodied spirits.

Since celestial causes are thus set forth, we must not
wonder too much at hearing that infernal ones can be in-
voked; and that is what has actually happened. An auda-
cious philosopher, the German von Hartmann, considers
the unconscious as a universal force, specifically an evil one,
which influences things and beings in a constantly harmful
way; and such is the pessimism generated in him by the
fear of that terrible unconscious that he advises not indi-
vidual suicide, which he considers as insufficient, but "cos-
mic suicide," hoping that powerful forces of destruction
will be devised by mankind, enabling it to destroy at once
the whole planet : a kind of foresight of what is taking place
as these lines are written.

We have seen how some authors had a kind of fear of the
unconscious and were even unwilling to admit its very exist-
ence. Perhaps this unwillingness is a kind of reaction
against the flights of imagination which some others have



42 THE UNCONSCIOUS AND DISCOVERY

indulged in about it. The same mysterious character im-
puted to it seems to have repelled some and overexcited
others.

The question is whether there is any mystery, or more
exactly, any special mystery. The true mystery lies in the
existence of any thoughts, of any mental processes what-
ever, those mental processes being connected in a way
about which we hardly know anything more than mankind
did thousands of years ago with the functioning of some
of our brain cells. The existence of several kinds of such
processes is hardly more mysterious than the existence of
one kind of them.

As to the unconscious mind being "superior" or "in-
ferior" to the conscious one, I should deny any meaning to
such a question and consider that no question of "superior-
ity" or "inferiority" is a scientific one. When you ride a
horse, is he inferior or superior to you? He is stronger than
you are and can run more quickly than you can ; however,
you make him do what you want him to do. I do not know
what would be meant by saying that oxygen is superior or
inferior to hydrogen; nor is the right leg superior or in-
ferior to the left one : they cooperate for walking. So do the
conscious and the unconscious, a cooperation which we shall
consider presently.



IV. THE PREPARATION STAGE.
LOGIC AND CHANCE



Throughout Conscious Work. Reviewing Poincare's lec-
ture, the literary critic Emile Faguet wrote : "A problem
. . . reveals itself suddenly when it is no longer investigated,
probably because it is no longer investigated and when one
only expects, for a short time, to rest and relax: a fact
which would prove lazy people, it is to be feared, might
make ill use of it that rest is the condition of work."

It would, of course, be very easy if, being told of the
terms of a problem, we could simply think that it would be
very nice to find it, then go to bed and find the solution
ready on awakening the next morning. Indeed, we could
think it to be perhaps too easy from the moral point of
view.

As a matter of fact, things by no means behave that way.
In the first place, it often happens that, in some of its parts,
the work is perfectly conscious. 1 This has been the case for
some parts of Poincare's work itself, as has been shown in
the beginning: for instance, in the step just after the ini-
tial one.

Very typical, from that point of view, is Newton's dis-
covery of universal attraction. He is reported to have been
asked how he had obtained it and to have answered, "By
constantly thinking it over" ; but we do not need that anec-
dote, which may not be authentic, to understand that his

i However, the word "conscious" ought not, perhaps, to be understood
In too strict a sense. A more attentive analysis will show us (see Section
VI) a cooperation between perfect consciousness and that superficial
subconscious or "f ringe-consciousness" we spoke of in Section II.



44 THE PREPARATION STAGE

discovery was a work of high and inflexible logic, the main
and essential idea, i.e., that the moon must really fall to-
ward the earth, being a necessary and unavoidable conse-
quence of the fact that any material body (be it an apple
or not) does so. A tenacious continuity of attention, "a
consented, a voluntary faithfulness to an idea" 2 was neces-
sary for that.

Must we, then, agree with Buffon's thesis that genius
may often be nothing else than a long patience? This idea
is obviously contrary to all that we have noticed so far. I
confess that I cannot share the admiration for it, nor even
approve of it. In Newton's case, one can certainly see,
from the beginning, a continuous course of thought con-
stantly directed toward its goal. But this process was
started by the initial recognition that the subject was
worthy of this continuity of attention, of the consented and
voluntary faithfulness we have just spoken of. This is
again an inspiration, a choice ; only, this takes place in the
conscious will.

Conscious Work as Preparatory. Let us now consider the
opposite case, the unexpected inspirations which repeatedly
illuminated Poincare's mind. We have acquired the notion
that they are the consequence of a more or less intense and
lengthy unconscious work. But is that unconscious work
itself an effect without a cause? We should be utterly mis-
taken in thinking so ; we have only to come back to Poin-
care's report to be led to the contrary conclusion. His first
inspiration on getting into the car at Coutances follows a
preliminary period of deliberate labor ; and after that, we
see him studying arithmetical questions "apparently with-
out much success" and finally, "disgusted with his failure" ;

* Delacroix, L' Invention et U Gtnie.



THE PREPARATION STAGE 45

upon which new fruitful steps reveal themselves to him.
Then he makes a systematic attack upon the chief remain-
ing question, "carrying all the outworks, one after the
other. 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 indeed was something." And he again notices that all
this work was perfectly conscious.

Only then, and after having been compelled even to set it
aside for a while, the solution of the difficulty suddenly
appeared.

In all these successive steps, as we see, "sudden inspira-
tions (and the examples already cited sufficiently prove
this), never happen except after some days of voluntary
effort which has appeared absolutely fruitless and whence
nothing good seems to have come, where the way taken
seems totally astray. These efforts then have not been as
sterile as one thinks. They have set going the unconscious
machine and without them it would not have moved and
would have produced nothing."

Helmholtz had similarly observed that what we have
called incubation and illumination must be preceded by this
stage of preparation. Its existence has been, after Helm-
holtz and Poincare, recognized by psychologists as a gen-
eral fact, and probably it exists even when it is not ap-
parent, as in the case of Mozart (who does not mention
incubation either) .

It is not useless to notice that independently of the rea-
sons we have already given, this, by itself, is sufficient to
settle the question whether discovery is a matter of pure
chance. Discovery cannot be produced only by chance, al-
though chance is to some extent involved therein, any more



46 THE PREPARATION STAGE

than does the inevitable role of chance in artillery dispense
with the necessity for the gunner to take aim, and to aim
very precisely. Discovery necessarily depends on prelimi-
nary and more or less intense action of the conscious.

Not only does this answer the question of the chance-
hypothesis, but at the same time it prevents us from admit-
ting the other hypotheses which we have examined in the
preceding section. It ought, indeed, to be noticed that the
rest and forgetting hypotheses have one feature in com-
mon ; whether it be in one or in the other of them, the pre-
paratory work, if not bringing directly the solution by
itself, is assumed to be completely useless and even harmful.
Then, discovery would happen just as if there had been no
preparation work at all; that is, we should be again com-
pelled to go back to the inadmissible hypothesis of pure
chance.

Poincare's View on the Mode of Action of Preparatory
Work. Having recognized this, we cannot any longer think
of the conscious as being subordinated to the unconscious.
On the contrary, it starts its action and defines, to a greater
or lesser extent, the general direction in which that uncon-
scious has to work.

To illustrate that directing action, Poincare uses a strik-
ing and remarkably fruitful comparison. He imagines that
the ideas which are the future elements of our combinations
are "something like the hooked atoms of Epicurus. During
the complete repose of the mind, these atoms are motionless ;
they are, so to speak, hooked to the wall ; so this complete
rest may be indefinitely prolonged without the atoms meet-
ing, and consequently without any combination between
them." The act of studying a question consists of mobiliz-
ing ideas, not just any ones, but those from which we might



THE PREPARATION STAGE 47

reasonably expect the desired solution. It may happen
that that work has no immediate result. **We think we have
done no good, because we have moved these elements a thou-
sand different ways in seeking to assemble them and have
found no satisfactory aggregate." But, as a matter of fact,
it seems as though these atoms are thus launched, so to
speak, like so many projectiles and flash in various direc-
tions through space. "After this shaking-up imposed upon
them by our will, these atoms do not return to their primi-
tive rest. They freely continue their dance."

Consequences can now be foreseen. ''The mobilized atoms
undergo impacts which make them enter into combinations
among themselves or with other atoms at rest, which they
struck against in their course." In those new combinations,
in those indirect results of the original conscious work, lie
the possibilities of apparently spontaneous inspiration.

Logic and Chance. Though Poincare presents that com-
parison as a very rough one and it could hardly avoid being
such, it proves, as a matter of fact, to be highly instructive.

We shall now see that, by pursuing it, other points can
be elucidated. Let us consider, from that point of view, the
question of logic and chance in discovery, on which au-
thors are most divided. Several of them, though not as ex-
treme as we have seen to be the case with Nicolle, insist on
the importance of chance, while others emphasize the pre-
eminence of logic. Among the two psychologists whom we
mentioned in the beginning, Paulhan belongs to the latter
school, while Souriau represents the former. It seems to me,
in accordance with my personal introspection, that we can
get a good understanding of that question by using Poin-
care's comparison of projected atoms : a comparison which
I shall complete by assimilating that projection with that



48 THE PREPARATION STAGE

which is produced by a hunting cartridge. It is well known
that good hunting cartridges are those which have a proper
scattering. If this scattering is too wide, it is useless to
aim ; but if it is too narrow, you have too many chances to
miss your game by a line. I see quite similar circumstances
in our subject. Again comparing ideas to Poincare's atoms,
it may happen that the mind projects them, exactly or al-
most exactly, in certain determinate directions. Doing so
has this advantage that the proportion of useful meetings
between them happens to be relatively great compared to
the sterile ones; but we may fear lest these meetings be
insufficiently different from each other. on the contrary, it
may happen that the atoms are launched in a rather dis-
orderly manner. If so, most of the meetings will be uninter-
esting ones ; but on the other hand, as in a kind of lottery,
that disorder can be highly valuable, because the few meet-
ings which are useful, being of an exceptional nature and
between seemingly very remote ideas, will probably be the
most important ones.

This is what Souiiau expresses by the quite striking
phrase: "In order to invent, one must think aside 55 ; 3 and,
even in mathematics though, in that realm, its meaning is
rather different from what it is in experimental sciences
we can remember Claude Bernard 5 s statement, "Those who
have an excessive faith in their ideas are not well fitted to
make discoveries."

Errors and Failures. The reason for the difference be-
tween the meanings of Claude Bernard 5 s sentence in math-
ematics and in experimental sciences is that, in the latter
case, too stubbornly following an idea once conceived may
lead to errors: that is, to inaccurate conclusions.
* "Pour inventer, il f ant penser



THE PBEPABATION STAGE 49

On the contrary, in our domain, we do not need to ponder
on errors. Good mathematicians, when they make them,
which is not infrequent, soon perceive and correct them. As
for me (and mine is the case of many mathematicians), I
make many more of them than my students do ; only I al-
ways correct them so that no trace of them remains in the
firml result. The reason for that is that whenever an error
has been made, insight that same scientific sensibility we
have spoken of warns me that my calculations do not look
as they ought to.

There are, however, celebrated exceptions concerning
some delicate points of reasoning; those may sometimes
prove more fruitful than accurate results, as has been an
insufficient proof of Riemann for "Dirichlet's principle."

But, in both domains, the mathematical and the experi-
mental, the fact of not sufficiently "thinking aside" is a
most ordinary cause of failure i.e., the lack of success in
finding a solution which may appear to better inspired
thinkers a circumstance which is at least as interesting
as discovery for psychology.

This, especially, often explains the failures which may
be called '^paradoxical," viz., the failure of a research
scholar to perceive an important immediate consequence
of his own conclusions.

Of course, we must insist on speaking of immediate con-
sequences. When the discoverer of a certain fact hears that
another scholar has found a notable consequence of it, if
this improvement has required some effort, the former wfll
consider it not a failure but a success : he has the right to
daim his part in the new discovery.

Such paradoxical failures are reported by Clapar&de in
the lecture referred to (p. viii) and they are, in my opinion,



50 THE PREPABATION STAGE

explained as we have just said. The most striking in-
stance which he gives, concerns the invention of the oph-
thalmoscope. The physiologist Briicke had investigated the
means of illuminating the back part of the eye and suc-
ceeded in doing so ; but it was Helmholtz who, while pre-
paring a lecture on that result of Briicke, conceived the
idea that optical images could be generated by the rays
thus reflected by the retina : an almost obvious idea, which
as it seems, Briicke could hardly have overlooked. In that
case, most evidently at least to me Briicke's mind was
too narrowly directed toward his problem.

Similarly, as Claparede also reports, de la Rive failed
to invent the galvanoplastic method ; Freud missed finding
the application of cocaine to the surgery of the eye.

Personal Instances. Every scientist can probably record
similar failures. In my own case, I have several times hap-
pened to overlook results which ought to have struck me
blind, as being immediate consequences of other ones which
I had obtained. Most of these failures proceed from the
cause which we have just mentioned, viz., from attention
too narrowly directed.

The first instance I remember in my life had to do with
a formula which I obtained at the very beginning of my
research work. I decided not to publish it and to wait till
I could deduce some significant consequences from it. At
that time, all my thoughts, like many other analysts', were
concentrated on one question, the proof of the celebrated
"Picard's theorem." Now, my formula most obviously
gave one of the chief results which I found four years later
by a much more complicated way: a thing which I was
never aware of until years after, when Jensen published
that formula and noted, as an evident consequence, the re-



THE PREPARATION STAGE 51

suits which, happily for my self-esteem, I had obtained
in the meanwhile. It is clear that, in 1888, I had thought
too exclusively of Picard's theorem.

My next work was my thesis. Two theorems, important
to the subject,* were such obvious and immediate con-
sequences of the ideas contained therein that, years later,
other authors imputed them to me, and I was obliged to
confess that, evident as they were, I had not perceived
them.

Some years later, I was interested in generalizing to
hyperspaces the classic notion of curvature of surfaces. I
had to deal with Riemann's notion of curvature in hyper-
spaces, which is the generalization of the more elementary
notion of the curvature of a surface in ordinary space.
What interested me was to obtain that Riemann curvature
as the curvature of a certain surface S, drawn in the con-
sidered hyperspace, the shape of S being chosen in order
to reduce the curvature to a minimum. I succeeded in show-
ing that the minimum thus obtained was precisely Rie-
mann's ｅxpression; only, thinking of that question, I
neglected to take into consideration the circumstances un-
der which the minimum is reached, i.e., the proper way of
constructing S in order to reach the minimum. Now, inves-
tigating that would have led me to the principle of the
so-called "Absolute Differential Calculus," the discovery
of which belongs to Ricci and Levi Civita.

Absolute differential calculus is closely connected with
the theory of relativity ; and in this connection, I must con-

* For technicians : "If the coefficients of a Maclaurin series are real
positive numbers, the radius of convergence being R, x = R must be a
singular point"; "A Maclaurin series with a finite radius of convergence
generally admits its whole circle of convergence as an essentially singular
line."



52 THE PREPARATION STAGE

fess that, having observed that the equation of propaga-
tion of light is invariant under a set of transformations
(what is now known as Lorentz's group) by which space
and time are combined together, I added that "such trans-
formations are obviously devoid of physical meaning."
Now, these transformations, supposedly without any phys-
ical meaning, are the base of Einstein's theory.

To continue about my failures, I shall mention one which
I particularly regret. It concerns the celebrated Dirichlet
problem which I, for years, tried to solve in the same initial
direction as Fredholm did, i.e., by reconducting it to a sys-
tem of an infinite number of equations of the first degree
in an infinite number of unknowns. But physical interpre-
tation, which is in general a very sure guide and had been
most often such for me, misled me in that case. It suggested
an attempt to solve the problem by a "potential of simple
layer" in that question, a blind alley while the solution
was to be looked for in the introduction of a "potential
of a double layer." This shows how justified Claude Ber-
nard is in the above-mentioned sentence, and that one
ought not to follow too stubbornly a determinate principle,
however justifiable and fruitful in general.

In all these examples, as we see, the reason for the fail-
ures was" basically the same. But the opposite case occurred
when I overlooked the fact that a problem in "inversive
geometry" could be indeterminate a fact which leads to
the beautiful properties discovered by Andre Bloch. It is
not, this time, a consequence of having too strictly followed
my original direction, which would precisely have led me to
discuss more thoroughly the problem which I had solved,
and therefore, to notice the possibility of indetermination.



THE PREPARATION STAGE 53

That Case is exactly contrary to the preceding ones. I was
unsufficiently faithful to my main idea.

I must close the enumeration of these failures with one
which I can hardly explain : having found, for construct-
ing conditions of possibility for a problem in partial dif-
ferential equations, 5 a method which gives the result in a
very complicated and intricate form, how did I fail to
notice, in my own calculations, a feature which enlightens
the whole problem, and leave that discovery to happier
and better inspired successors ? That is what is difficult for
me to conceive.

The Case of Pascal. It is probable that many scholars,
if not all of them, can remember similar experiences. It is a
comforting thing to think that the same may happen to
some of the greatest ones.

In his Art de Persuader, Pascal has stated a principle
which is fundamental for method not only in mathematics,
but in any deductive subject or any matter of reasoning,
viz.:

one must substitute definitions instead of the defined."

On the other hand, in another place, he points out the ob-
vious fact that, for the same reason that it is not possible
to prove everything, it is also impossible to define every-
thing. There are primitive ideas which it is not possible
to define.

If he had only thought of juxtaposing these two state-
ments, he would have found himself before a great problem
of logic which not only enables us to understand the true

s For technicians: see pp. 257-260 of my Yale Lectures on Cauchy's
Problem-^ pp. 351-355 of the French edition. The improved answer is
given in Hilbert-Courant's Metkoden der Mathematischen Physik (pp.
425-430), following works of John and Asgueiersson.



54 THE PREPARATION STAGE

meaning of the celebrated Euclid postulate, but, more gen-
erally, has produced a profound revolution which, as we see
it, might have taken place three centuries earlier.

However, he did not connect both ideas. Whether the
reason for this was that his thoughts were too intensely
directed toward theological consequences, as a friend of
mine suggested to me, is a question which it would be diffi-
cult to elucidate. 6

Attempts to Govern our Unconscious. Such instances
show us that, in research, it may be detrimental to scatter
our attention too much, while overstraining it too strongly
in one particular direction may also be harmful to dis-
covery.

What should we do in order to avoid these opposite
objections?

Of course, there is the obvious influence of the way in
which we direct our preparation work which gives the im-
pulse to the unconscious work : and in fact, especially with
reference to Poincar6's conception, this can be considered
as a way to educate our unconscious. The formula of
Souriau, "To invent, one must think aside," is to be under-
stood in that sense.

But this is not yet completely satisfactory: in this way
we shall think of expected directions for "aside" thoughts,
but not of those unexpected and all the more interesting
for this very reason. We must notice, in that direction, that
it is important for him who wants to discover not to confine
himself to one chapter of science, but to keep in touch with
various others.

Could we find other means of influencing our uncon-
scious ? That would be of great importance, in fact not only
for invention but also for the whole conduct of life and es-

e See Appendix III.



THE PREPARATION STAGE 55

pecially for education. The study of that question, which
deserves to be pursued, has been undertaken at least In one
periodical. La Psychologic et la Vie ; a whole fascicle was
devoted to it by that review in 1932, with contributions
of several authors. Particularly, Dwelshauvers suggests an
analysis of the conditions of the phenomenon, such as the
time of the day at which it takes place, how much time
elapses between voluntary preparation and solution;
whether such incubation lasts for hours or for days,
whether its duration is in proportion to the difficulty of the
question, etc.

Pending the results of such studies, one rule proves evi-
dently useful: that is, after working on a subject and see-
ing no further advance seems possible, to drop it and try
something else, but to do so provisionally, intending to re-
sume it after an interval of some months. This is useful
advice for every student who is beginning research work.

There is another direction in which that education of the
unconscious could be pursued, though I cannot undertake
to speak of it. Indeed, as Dr. de Saussure suggested to me,
very powerful means for that purpose may be supplied by
the methods of psychoanalysis.



V. THE LATER CONSCIOUS WORK



The Fourth Stage. We have now become acquainted with
the three stages in invention which Helmholtz and Poincare
have taught us to distinguish: preparation, incubation and
illumination. But Poincare shows the necessity and impor-
tance of a fourth and final one, which again occurs in con-
sciousness. This new intervention of consciousness, after
the unconscious work, is necessary not only for the obvious
purpose of expressing the results by language or writing,
but at least for three other reasons which are, however,
closely dependent on each other :

(1) To Verify Them. The feeling of absolute certitude
which accompanies the inspiration generally corresponds
to reality; but it may happen that it has deceived us. 1
Whether such has been the case or not must be ascertained
by our properly so-called reason, a task which belongs to
our conscious self.

(2) To "Precise" Them. That is, to state them precisely.
It never happens, as Poincare observes, that the uncon-
scious work gives us the results of a somewhat long calcu-
lation already solved in its entirety. If we should, as con-
cerns the unconscious, retain the original idea which is
suggested by the quality "automatic" imputed to it, we
should suppose that, thinking of an algebraic calculation
before falling asleep, we might hope to find its result ready
made upon our awakening; but nothing of the sort ever
happens, and indeed we begin to understand that automa-

i Polncare" notices that this happens to him especially in regard to
ideas coming to him in the morning or evening, or in bed while in a
semihypnagogic state.



LATER CONSCIOUS WORK 57

tism of the unconscious must not be understood in that
way. on the contrary, effective calculations which require
discipline, attention and volition, and, therefore, conscious-
ness, depend on the second period of conscious work which
follows the inspiration.

Thus, we come to the paradoxical-looking conclusion
to which, besides, we shall have to bring a correction as we
already have done in Newton's case that this intervention
of our will, i.e., of one of the highest faculties of our soul,
happens in a rather mechanical part of the work, where it
is in some way subordinated to the unconscious, though
supervising it. The second operation is inseparable from
the first, from verification. The conscious mind performs
them both at the same time.

A Statement by Paid Valery, What we just met with in
the domain of mathematical research, especially that co-
ordination of the "precising" work to the original inspira-
tion, is once more in agreement with what Paul Valery says
on quite a different kind of invention, except that the very
description of Paul Valery suggests that facts may be even
more complicated or delicate than he himself or Poincare
saw them, and would deserve a more thorough study. Paul
Valery says, in the passage 2 of which we have already
quoted the beginning (see Section I, p. 17) :

'There is the period of dark-room. There must be no ex-
cessive zeal at this moment, or you would spoil your plate.
You must have your reagents, you must work as your own
employee, your own foreman. The master has provided the
spark, it is your job to make something of it. A very curi-
ous thing is the disappointment that may follow. There are
misleading gleams of light ; when the foreman comes to the

2 Bullet** 8oc. PhUosopkie, VoL 28 (1928), p. 16.



58 LATER CONSCIOUS WORK

result, he perceives that there is no authentic product, that
it would have been good if it had been true. Sometimes a
series of judgments intervene which cancel each other out.
A kind of irritation follows; you say yourself you will
never succeed in recording what appears before you."

This "precising" state in invention is again quite a gen-
eral one, and even the most spontaneous creators experi-
ence it. The same Lamartine whom we have seen to answer
so rapidly, unhesitatingly and as though almost unwill-
ingly when asked for a couple of verses, is reported by his
biographers to have repeatedly and indefatigably cor-
rected his work, as appears from his manuscripts.

Numerical Calculators. The process seems to be slightly
different, on one point, in a case which one is often tempted
to mix with that of mathematicians : I mean those prodi-
gious calculators frequently quite uneducated men who
can very rapidly make very complicated numerical calcu-
lations, such as multiplications of numbers in ten or more
figures, who will want only one instant of reflection to tell
you how many minutes or seconds have elapsed since the
beginning of our era.

Such a talent is, in reality, distinct from mathematical
ability. Very few known mathematicians are said to have
possessed it : one knows the case of Gauss and Ampere and
also in the seventeenth century, Wallis. Poincare confesses
that he is a rather poor numerical calculator, and so am I.

Exceptional calculators often present remarkable psy-
chological peculiarities. 8 The one I want to mention as be-
longing to our subject is that, contrary to what we just
heard from Poincare, it happens that calculation results,

We mention the very curious one that, in several of them, that ability
has been temporary and disappeared after some years.



LATER CONSCIOUS WORK 59

or at least partial ones, appear to them without willful
effort and by inspiration elaborated in their unconscious.

Perhaps the most outspoken testimony is afforded by a
letter written to Mobius 4 by the calculator Ferrol: "If I
was asked any question, rather a difficult one by itself, the
result immediately proceeded from my sensibility without
my knowing at the first moment how I had obtained it;
starting from the result, I then sought the way to be fol-
lowed for this purpose. That intuitive conception which,
curiously enough, has never been shaken by an error, has
developed more and more as needs increased. Even now, I
have often the sensation of somebody beside me whispering
the right way to find the desired result ; it concerns some
ways where few people have entered before me and which
I should certainly not have found if I had sought for them
by myself.

"It often seems to me, especially when I am alone, that
I find myself in another world. Ideas of numbers seem to
live. Suddenly, questions of any kind rise before my eyes
with their answers."

It must be added that Ferrol was attracted not only by
numerical calculations, but also and even more strongly by
algebraic ones. It is the more striking that, also in that
case, he brings calculations to an effective end in an uncon-
scious way. 5

* See Die Anlage fur Mathematik, pp. 74-76.

5 Unconscious interventions in numerical calculations are also reported
by Scripture, American Journal of Ptych^ Vol. IV (1891). See also
Binet, Psychologic det Grands Calculateurs et Joueurs d'Echecs. How-
ever, these statements are not as positive and precise as Ferrol's, and
confusion would be possible between partial results unconsciously ob-
tained and results known in advance by heart.



60 LATER CONSCIOUS WORK

Appreciation of one's OISTI Work. once we have ob-
tained our result, what do we think of it?

Very often research which has deeply interested me while
I was investigating it loses its interest for me just after I
have the solution, unhappily at a time which coincides with
the period when I have to record it. After a while, say a
couple of months, I come to a more just appreciation of it.

Paul Valery was asked the same question about his feel-
ing toward his own work after its completion at a meeting
of the Societe de Philosophic in Paris ; he answered : "It
always turns out badly ; Je divorce" ; and he already gave
an indication in the same sense when describing the inven-
tion process, as we have seen.

(3) The Continuation of the Work. Relay-Results. The
double operation of verifying and "precising 55 the result
assumes another meaning when, as happens most fre-
quently, we regard it not as the end of the research, but
as one stage of it we have met with such successive stages
in Poincare's report so that we think of utilizing it.

Such a utilization not only requires that the result be
verified, but that it be "precised." Indeed, since we know
that our unconscious work, showing us the way to obtain
the result, does not offer it in its precise form, it may hap-
pen, and it actually happens in many cases, that some
features in that precise form, which we could not fully
foresee, wield a capital and even total influence on the
continuation of the thought.

Such has already been the case with Poincare's initial
stage (though not for the following ones) . We hear from
him that he originally supposed that the functions which
he has called fuchsian could not exist, and it was only the
fact of having discovered, in his sleepless night, the op-



LATER CONSCIOUS WORK 61

posite conclusion which gave his following thoughts the
course they took.

That each planet moves around the sun as being at-
tracted by it with a force proportional to the inverse square
of a distance, was found by Newton to be the interpretation
of the first two laws of Kepler. But there is a coefficient of
proportionality the ratio between the force of attraction
and the inverse square of the distance, which ratio does not
vary during the motion and the meaning of that coeffi-
cient is to be deduced from the third law of Kepler, which
concerns the comparison between the motions of different
planets. The conclusion is that this coefficient is the same
for all of them. All planets obey the same law of attraction ;
that conclusion does not arise from the general and syn-
thetic view of the question, but from a precise and careful
calculation. one may doubt whether Newton can have
reached that last conclusion otherwise than by calculating
pen in hand. Now, if the result of those calculations had
been a different one, the last step of the discovery, that of
identifying the force which keeps the moon revolving
around the earth with the one which makes a heavy body
(an apple, if we believe the legend) fall down, would not
have existed.

Perhaps it is imprudent to imagine how Newton's mind
functioned; but it may be noticed that the identification
which he viewed required not only an algebraic but even
a numerical verification, using the observed values of the
magnitudes involved in the formulae (a verification which
even, as is well known, was temporarily believed by Newton
to be wrong) , and if, strictly speaking, there could remain
a doubt as to Newton's example, others are completely be-
yond doubt. For instance, it is certain that Georg Cantor



62 LATER CONSCIOUS WORK

could not have foreseen a result of which he himself says
"I see it, but I do not believe it."

In any case, moreover, the continuation of the work, just
as was the case for its beginning, requires the preparation
work we have spoken of. After a first stage of research has
been brought to an end, the following one requires a new
impulse, which can be originated and directed only when
our consciousness takes account of the first precise result.

To take a rather familiar example, everybody under-
stands that, intersecting two parallel straight lines by two
other parallel ones, the segments thus determined are equal
two by two ; everybody knows that, consciously 01 not. But
as long as it is not consciously enunciated, none of its con-
sequences, such as similitude, can be deduced.

One of the possible cases is that the new part of the re-
search be one which is to be carried out by exclusively con-
scious work, as Poincare reports (more exactly, as I should
say, by conscious work with the cooperation of fringe-
consciousness) ; or even, as in Newton's example, one which
deserves and requires a systematic and exhaustive work of
that kind. To recognize such cases is again a task of our
volition and the precise result is essential for that.

To sum up, every stage of the research has to be, so to
speak, articulated to the following one by a result in a
precise form, which I should propose to call a relay-resvlt
(or a relay-formula if it is a formula, as in Newton's inter-
pretation of Kepler's third law). When reaching such a
joining, somewhat analogous to railroad bifurcations, the
new direction in which further research will follow must be
decided, so that they clearly illustrate the directing action
of that conscious ego which we were tempted to consider as
"inferior" to unconsciousness.



LATER CONSCIOUS WORK 63

The above remarks may seem to a certain extent obvious,
if not childish ; but it is not useless to notice that, besides
the processes in the mind of any individual researchmen,
they help us to understand the structure of mathematical
science in general. Its improvement would have been im-
possible not only without verification of the results, but
especially without the systematic use of what we have just
called relay-results, which are very often intensely and ex-
haustively utilized as much as possible to the extreme end
of their consequences. Such is, for instance, the role of the
simple and classic fact that cutting a triangle by a parallel
to one of its sides, we obtain another triangle similar to the
former: a self-evident fact, but one which needed to be pre-
cisely enunciated in order to yield the long series of prop-
erties which proceed from it.

Incubation and Relay-results, The above has a connec-
tion with what we said in Section III. There is undoubtedly
(at least in my own case) a process different from those
mentioned in Section III which takes place during incu-
bation, when a relay-result occurs. It most often happens
that such a result needs to be digested, or to say it differ-
ently, to be classed in our fringe-consciousness, so as to be
"ready for use." Then it can easily and rapidly find its
place in the synthetic scheme of the deduction (see Section
VI) . That such a process is unconscious and that it corre-
sponds to an incubation stage cannot be denied. Having
reached an intermediate result which seems to be useful
for further investigation, in many cases I deliberately
leave the whole work sleeping till the following day, when
I find it "ready for use."



VI. DISCOVERY S Jl SYNTHESIS.
THE HELP OF SIGNS



Synthesis in Discovery. Souriau, in his Theorie de Vlnven-
tion, 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 in-
spiration 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 re-
search work, the behavior which Souriau describes is that
of the pupil, and even of the rather bad pupil ; our efforts



DISCOVERY AS A SYNTHESIS 65

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 proc-
esses 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. In-
quiries were started, especially by Alfred Binet, in order
to understand how this was possible : their results 1 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
Genie, 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 with-
out a very severe strain of thought. 35

Similarly, any mathematical argument, however com-
plicated, 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

i See, e.g^ Binefs article in the Revue des Deux Mondes, Series 3, VoL
117 (May-June, 1893), pp. 826-859, especially Section IV.



66 DISCOVERY AS A SYNTHESIS

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 rep-
resentations. 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 di-
vergent 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 feel-
ing 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 Miiller, the celebrated philologist and orien-
talist, maintain 3 that no thought is possible without words*
and even write this sentence, fully unintelligible to me:
"How do we know that there is a sky and that it is blue?

2 1 have also seen the following topic (a deplorable subject, as far as I
can judge) proposed for an examination an elementary one, the "bac-
calaupdaf in philosophy in Paris: 4< To show that language is as neces-
sary for us to think as it is to communicate our thougnts."

a Three Introductory Lecture* on the Science of Thought, delivered in
London in 1887: Chicago, 1888; and also his more extensive work, The
Science of Thought, published the same year.

*It is quite possible that M. Miiller's unlimited confidence in words
may be doe to the linguistic work of his whole life.



DISCOVERY AS A SYNTHESIS 67

Should we know of a sky if we had no name for it?**, ad-
mitting 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 Miiller is the more curious be-
cause he claims to find in the fact that thought is impos-
sible without words an argument against every evolution-
ary 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 Miiller's thesis if we take into ac-
count, for instance, Kohler's Mentality of Apes* and the
actions of his chimpanzees, which do imply reasoning.

Max Miiller 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 stand-
point of Max Miiller toward it. We hear, in the first place,
that the Greeks originally used one and the same word,
6 *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 in-
tellect." An analogous statement is to be found in a more

e N.Y. 1923. See, e.g., the experiment of the "jointed stick," p. 132.



68 DISCOVEEY AS A SYNTHESIS

modern philosopher, Hobbes (who, generally, keeps in
sympathy with the scholastics).

But, as a rule, ideas took a different course, on that sub-
ject as on many others, with the stream of thought initi-
ated by Descartes. There is only one period in Germany,
around 1800 (Humboldt, Schelling, Hegel, Herder) when
philosophical minds were near to *truth," that is, to Max
MuUer'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. Pre-
cisely, 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 abso-
lutely categorical but in the opposite direction. He is
convinced that words are the great impediment to thought.

Max Miiller's passionate view of the subject leads him to
qualify as "lack of courage" that general attitude of mod-
ern 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/ Above all, there came the authorized voice of an-
other first-rank scholar, Francis Galton, the great ge-



an the Connection Between Tkfags and Words-. "It troubles
me greatly [Hoc unum me male habet] to find that I can never acknowl-
edge, discover or prove any truth except by using in my mind words or
crtber things."

* See tbe exchange of letters at the end of the Introductory Lecture*.



DISCOVERY AS A SYNTHESIS 69

neticist, who, moreover, after having begun as an explorer,
has done important work in psychological matters. The
latter^ great habit of introspection allows him to assert
that his mind does not behave at all in the way supposed
by Max Miiller 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 in-
convenience 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 hap-
pens 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 maladroit-
ness, 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, in-



70 DISCOVERY AS A SYNTHESIS

eluding the rather regrettable consequence which I ex-
perience just as he does.

The fact that it is impossible for Max Miiller 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 re-
flection 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 confoit bien s'enonce clairement
Et les mots pour le dire arrivent aisement,"

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 math-
ematical 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 MuHer has utterly failed to understand that other
people's minds may be different from his own : a most com-
mon error, but one which it is surprising to find among
men accustomed to psychological studies. Differences be-
tween minds being, on the contrary, undeniable according
to what we have just found, the question ought to be set-
tled not by polemics but by inquiries relating to every
human race and every class of men and, if possible (we



DISCOVERY AS A SYNTHESIS 71

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 per-
centage ; a possible reason for that will appear below. 8

Mental Pictures in Usual Thought. Thought can be ac-
companied 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 re-
curring, surviving and spontaneously resurging sensa-
tions. 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. Por instance,
asking a question or pronouncing a word, he inquires what
ideas, images, etc., this has suggested to the subject. The

s Galton, in his Inquiries into Human Faculty, has conducted inquiries,
interpreted according to the rules of statistics, on mental imagery con-
sidered in itself. The question for us would be to have a similar inquest
on imagery as a help to thought,

Etude ExptrimentaU de V Intelligence, Paris (1903).



72 DISCOVERY AS A SYNTHESIS

method has been criticized and indeed there is the objection
which can be advanced against almost every kind of psy-
chological experiment, viz., an involuntary suggestion
from the experimenter himself. This, however, is not to be
feared when the results are of an unexpected nature as
some have been for Binet. As a matter of fact, Binet's
method is considered by psychologists as not being invali-
dated by that or analogous objections, to which convincing
answers have been given by Biihler ; 10 and a somewhat sim-
ilar method was used, a little later, by the so-called Wurtz-
bourg school. Its creation, however, belongs to Binet.

In Binet's experiments, the question of words is dealt
with incidentally. The answer on that point is favorable
to Galton against Max Muller. To one of the girls, 11 an
answer in words appears as "an image which cuts thought."
Thought is something which appears to her suddenly like
any kind of feeling.

What is more unexpected is that even the intervention of
images is minimized, contrary to Taine's theory. The pre-
cision of the answer is striking. 12 "In order to get images,
I must no longer have anything to think of. They [ideas
and images] are separated from each other and never come
together. I never have any images when a word suggests to
me a very great number of thoughts. I must wait for a
while. When, with respect to this word, I have exhausted
every thought, then images come and if thoughts begin
again, the images fade, and alternate."

On that point, Binet himself concludes: "Later on, I

10 Ureter /. die Get. P*yck^ Vol. IX (1907); Vol. XII (1908) es-
pecially pp. 93-123. See G. Dumas' Traiti de Psyckologie, Vol. I, Chap
IV and VoL II, pp. 113 ff. P '

11 Etude Experimental de VlnteUigence^ p. 107.

12 ibid^ p. 124.



DISCOVERY AS A SYNTHESIS 73

was able to convince myself that Armande was quite right ;
I admit that there exists a kind of antagonism between
image and reflection, the more so when the image is very
intense. It is in revery and dream that the finest images
arise." There is also the fact, observed by Galton and
others, that women and children have finer images than
adult men who are superior in reflection.

Later experiments by Dwelshauvers (Les Mecanismes
Subconscients) , carried on with students, led to the same
main conclusions as Binet's about the conditions of the ap-
parition of images. He finds that images appear only if we
give our ideas uncontrolled freedom, i.e., when we are
dreaming while awake. As soon as full consciousness, volun-
tary consciousness, returns, images weaken, darken; they
seem to withdraw into some unknown region.

Mental Pictures in Tense Thought. More recent authors
(Delacroix, James Angell, Titchener, Varendonck, etc.)
have also treated that same subject of words and images in
thought. Most of their works, however, will not directly
occupy us, on account of a distinction which is especially
necessary in our subject*

Psychologists have already distinguished between two
kinds of thought. There is "free" thought, which takes
place when you let your thoughts wander, without directing
them toward any special goal; and there is "controlled**
thought, when such a direction is given. 18 The second term
is not precise enough for our purpose. There is already a
direction in your thought when you are asked what is the
date ; but the case of inventive thought is obviously differ-

i R, S. Woodworth, Psychology (4th edition, 1940), p. 83. However,
Woodworth speaks of your being asked a difficult question, which would
mean our third case of the tense thought, rather than the simply con-
trolled one.



74 DISCOVERY AS A SYNTHESIS

ent. It requires a certain effort of concentration ; it is not
only controlled, but tense.

There is no reason why the processes of those three kinds
of thought should be the same; and actually they are not.
The last case is the only one which directly concerns us.

Binefs View. As a conclusion to his series of experi-
ments, Binet is 1 * inclined to think that words or sensorial
images may be useful for giving a precise form to feelings
or thoughts which, without both these helps, would remain
too vague; even, to give us a full consciousness of that
thought which would, otherwise, remain an unconscious act
of mind ; in order to allow the passage of ideas from uncon-
scious to conscious : more precisely, from the unconscious
where they are somewhat vague to consciousness where they
acquire precision.

I was myself inclined, for a while, to admit that concep-
tion of Binet. Indeed, it satisfied to a certain esient the
double and seemingly contradictory condition :

(a) That the help of images is absolutely necessary for
conducting my thought.

(b) That I am never deceived and even never fear to be
deceived by them.

However, further reflection led me to a different concep-
tion. Indeed, the case of Binet's or Dwelshauvers 9 experi-
ments is not ours ; it deals with a controlled thought, but
not with a tense one. The two girls are asked such questions
as, "What appears in your mind when you think of what
you did yesterday?" The most difficult question, as far as
I could see on examining Binet's book, was, "Think of what
you would like to do if you could remain three hours by
yourself, being completely free in your actions?"

i* Etnde ExptrimentdU de F Intelligence, p. 108.



DISCOVERY AS A SYNTHESIS 75

Personal Observations. The case of research work is, of
course, very different, for which reason I have wished to
understand what takes place in my own mind when I under-
take to build up or to comprehend I have said in the be-
ginning that there is no essential difference a mathemati-
cal argument.

I insist that words are totally absent from my mind when
I really think and I shall completely align my case with
Galton's in the sense that even after reading or hearing a
question, every word disappears at the very moment I am
beginning to think it over ; words do not reappear in my
consciousness 15 before I have accomplished or given up the
research, just as happened to Galton; and I fully agree
with Schopenhauer when he writes, 'Thoughts die the mo-
ment they are embodied by words."

I think it also essential to emphasize that I behave in this
way not only about words, but even about algebraic signs.
I use them when dealing with easy calculations ; but when-
ever the matter looks more difficult, they become too heavy

is It is quite possible, and rather probable, that words are present
In fringe-consciousness. Such is the case, I imagine, for me as concerns
words used in mathematics. I doubt, however, that it is so for some other
kind of thought because, if it were, I should have less difficulty in finding
them. An evident misunderstanding as to the meaning of the question oc-
curs when, after citing William Hamilton who observes that "a cognition
must have been already there before it could receive a sign," so that the
Idea must necessarily precede the word, Max Mutter claims to be in
agreement (sic) with him, because William Hamilton's statement means
an "almost contemporaneous" progress of thinking and naming. Though
being only an occasional psychologist, I know enough to understand that
mental processes are often rapid ones and that it would be absurd to
study them without distinguishing between "almost contemporaneous**
and simultaneous states. Besides, William Hamilton expresses his feeling
on the question in a striking way, saying: "Speech is thus not the mother,
but the godmother of knowledge."



76 DISCOVERY AS A SYNTHESIS

a baggage for me. I use concrete representations, but of a
quite different nature.

One example of this kind is already known in the history
of science. It was given by Euler, in order to explain to a
Swedish princess the properties of syllogism. He represents
general ideas by circles ; then if we are to think of two cate-
gories of things, A and B, such that every A is a B, we
shall imagine a circle A lying inside a circle B. If, on the
contrary, we are told that no A is a B, we shall imagine the
circle A as lying completely outside B ; while if some A's
are B's and some not, the two circles ought to be intersect-
ing ones. Now, personally, if I had to think of any syllo-
gism, I should not think of it in terms of words words
would hardly allow me to see whether the syllogism would
be right or wrong but with a representation analogous
to Euler's, only not using circles, but spots of an undefined
form, no precise shape being necessary for me to think of
spots lying inside or outside of each other.

To consider a slightly less simple case, let us take an
elementary and well-known proof in arithmetic, the theo-
rem to be proved being: 'The sequel of prime numbers is
unlimited." I shall repeat the successive steps of the classic
proof of that theorem, writing, opposite each of them, the
corresponding mental picture in my mind. We have, for
instance, to prove that there is a prime greater than 11,

STEPS IN THR PROOF MY MENTAL PICTURES

I consider all primes from 2 to I see a confused mass.

11, say 2, 3, 5, 7, II.
I form their product N being a rather large num-

2X3X5X7XH = N ber?I ima S ine a P oint ratner

remote from the confused
mass.



DISCOVERY AS A SYNTHESIS 77

I increase that product by 1, I see a second point a little

say N plus 1. beyond the first.

That number, if not a prime, I see a place somewhere be-

must admit of a prime divi- tween the confused mass and

sor, which is the required the first point,
number.

What may be the use of such a strange and cloudy image-
ry? Certainly, it is not meant to remind me of any prop-
erty of divisibility, prime numbers and so on. This is most
important because any such information which it could
give me would be likely to be more or less inaccurate and
to deceive me. Thus, that mechanism satisfies condition (b)
previously required. on the contrary, this condition is but
partly satisfied by Binet's hypothesis : giving precision to
unconscious ideas would always risk altering them.

But at the same time, one can easily realize how such a
mechanism or an analogous one may be necessary to me for
the understanding of the above proof. I need it in order to
have a simultaneous view of all elements of the argument,
to hold them together, to make a whole of them in short,
to achieve that synthesis which we spoke of in the beginning
of this section and give the problem its physiognomy. It
does not inform me on any Hnlc of the argument (i.e., on
any property of divisibility or primes) ; but it reminds me
how these links are to be brought together. If we still fol-
low Poincare's comparison, that imagery is necessary in
order that the useful bookings, once obtained, may not get
lost.

Indeed, every mathematical research compels me to
build such a schema, which is always and must be of a vague
character, so as not to be deceptive. I shall give one less
elementary example from my first researches (my thesis) . I



78 DISCOVERY AS A SYNTHESIS

had to consider a stun of an infinite number of terms, in-
tending to valuate its order of magnitude. In that case,
there is a group of terms which chances to be predominant,
all others having a negligible influence. Now, when I think
of that question, I see not the formula itself, but the place
it would take if written: a kind of ribbon, which is thicker
or darker at the place corresponding to the possibly im-
portant terms ; or (at other moments) , I see something like
a formula, but by no means a legible one, as I should see it
(being strongly long-sighted) if I had no eye-glasses on,
with letters seeming rather more apparent (though still
not legible) at the place which is supposed to be the impor-
tant one.

I have been told by some friends that I have a special
way of looking when indulging in mathematical research. I
hardly doubt that this especially accompanies the construc-
tion of the schema in question.

This is in connection with the question of intellectual
fatigue. I have asked some prominent physiologists, espe-
cially Louis Lapicque, how intellectual work can produce
fatigue, as no **work," in the physicist's meaning of the
word, seems to be produced. Lapicque's opinion is that in-
tellectual work ought to be comparable to nothing more
than the act of turning the pages of a book. However, intel-
lectual fatigue exists: from the objective and physiological
point of view, it has been studied in an important book of
Binet and Victor Henri. That part of the question is be-
yond my sphere. From the psychological point of view, it
can be considered as certain that, similarly to what we have
said of Rodin's case, fatigue corresponds to the effort for
synthesis, to the fact of giving the research its unity and,



DISCOVERY AS A SYNTHESIS 79

therefore, at least in my case, to the constitution of a
proper schema.

One or two observations can be added:

If I should use a blackboard and write the ｅxpression
2X3X5X7X11, the above described schema would dis-
appear from my mind as having obviously become useless,
and would be automatically replaced by the formula which
I should have before my eyes.

Then I must observe that I distinctly belong to the audi-
tory type ; 16 and precisely on that account my mental pic-
tures are exclusively visual. The reason for that is quite
clear to me : such visual pictures are more naturally vague,
as we have seen it to be necessary in order to lead me with-
out misleading me.

I also add that the case we have just examined especially
concerns arithmetical, algebraic or analytic studies. When
I undertake some geometrical research, I have generally a
mental view of the diagram itself, though generally an
inadequate or incomplete one, in spite of which it affords
the necessary synthesis a tendency which, it would ap-

le I have a rather bad memory of physiognomies and am much exposed
to failures of recognition or false recognitions; on the contrary, I am
very sensitive to the sound of names, being much more desirous to see
such rivers as the Mohawk or the Mattawamkeag than even their beauty
would warrant, because their mere names call to my mind the idea, of
forests and Indian life. Also, I am rather less sensible than others to like-
nesses in faces and rather more sensible to likenesses in voices.

Many mistakes by automatic writing (see Section II, p. 22) are due
to auditory mental images. Examples of that kind are classic. Personally,
they are frequent in my case: my conscious ego dictates to my uncon-
scious one, which writes a word instead of another one if their sounds
are somewhat alike. While writing the present work, I remember to have
written "simple" instead of "same place" and "will she" instead of "we
shall." I should think that such auditive mistakes are more frequent on
my part when I am writing in English than in French quite naturally.



SO DISCOVEEY AS A SYNTHESIS

pear, results from a training which goes back to my very
earliest childhood.

Paradoxical as it seems, it very often happens that, in
those geometrical problems, I use successfully a process
quite opposite to the synthesis I have explained in what
precedes. I happen to abstract some special part of the dia-
gram and consider it apart from the rest, this consideration
leading to a "relay-result." The whole argument is, never-
theless, even in that case, grasped as a unique entity, as a
synthesis into which such a relay-result, if existent, is in-
cluded. That is a process which, according to Pierre Bou-
troux 17 (see below), Descartes says is frequent in Greek
geometry.

Respective Roles of Full Consciousness and Fringe-Con-
sciou&ness. The above observations concern the functioning
of thought when it is intensely concentrated, be it in an en-
tirely conscious work or in a conscious preliminary work.

Now, as we have explained at the end of Section II, that
very concentration gives us a possibility of distinguishing
between full consciousness and fringe-consciousness, a dis-
tinction which is rather difficult in other circumstances, but
which, in that case, is rather easily accessible to observation.

What does observation give as to the phenomena just
described?

It could be supposed a priori that the links of the argu-
ment exist in full consciousness, the corresponding images
being thought of by the subconscious. My personal intro-
spection undoubtedly leads me to the contrary conclusion:
my consciousness is focused on the successive images, or
more exactly, on the global image; the arguments them-
selves wait, so to speak, in the antechamber (see p. 25) to

IT P. Bontronx, however, gives no precise reference.



DISCOVERY AS A SYNTHESIS 81

be introduced at the beginning of the "precising" phase.

This instance most clearly illustrates the nature and role
of the fringe-consciousness, which is, so to speak, at the
service of full consciousness, being ready to appear in it
whenever wanted.

Other Stages of Research. What happens when there is
a period of incubation: in other words, an action of the
deeper unconscious? Of course, no direct answer is avail-
able ; but a strong presumption that there is some mecha-
nism of an analogous kind at work results from the fact
that it seems to be the fittest one to satisfy the double con-
dition (a), (b) which is to be fulfilled.

I should even interpret in a similar way the case of illu-
mination. When I think of the example mentioned in Sec-
tion I (see page 8) I see a schematic diagram: a square
of whose sides only the verticals are drawn and, inside of it,
four points being the vertices of a rectangle and joined by
(hardly apparent) diagonals a diagram the symbolic
meaning of which will be clear for technicians. It even seems
to me that such was my visualization of the question in
1892, as far as I can recollect. Of course, remembrances
going back half a century are not quite reliable ; however,
we recognized symbolic diagrams as being essential to a
synthetic view of questions, and it seems to me that such a
synthetic view is at least as necessary in cases of illumina-
tion as in conscious work. If we admit this line of reason-
ing, illumination would be transmitted from a lesser or
greater depth of unconsciousness to fringe-consciousness,
which would have it represented by a symbolic diagram in
the conscious ego.

That image and its meaning are in some way connected
and, at the same time, independent, is observed by Watt,



82 DISCOVERY AS A SYNTHESIS

Archiv. f.d. Ges. Psych., 1904, Vol. IV (see G. Dumas'
Traite de Psychologie, Vol. I, Chap. IV). It seems to me
that such a kind of simultaneous connection and independ-
ence is enlightened by the intervention of fringe-conscious-
ness.

Then comes the verifying and "precising" stage. In that
final phase of the work, I may use algebraic symbols ; but,
rather often, I do not use them in the usual and regular
way. I do not take time to write the equations completely,
only caring to see, so to speak, how they look. These equa-
tions, or some terms of them, are often disposed in a pecul-
iar and funny order like actors in a scenario, by means of
which they "speak" to me, as long as I continue to consider
them. But if, after having been interrupted in my calcula-
tions, I resume them on the following day, what I have writ-
ten in that way is as if "dead" for me. Generally, I can do
nothing else than throw the sheet away and begin every-
thing anew, except if, in the first day, I have obtained one
or two formulae which I have fully verified and can use as
relay-formulae.

As to words, they remain absolutely absent from my
mind until I come to the moment of communicating the
results in written or oral form, or (very exceptionally) for
relay-results; in the latter case, they may, as William
Hamilton observes, be the intermediary Necessary to give
stability to our intellectual process, to establish each step
in our advance as a new starting point for our advance to
another beyond" in which William Hamilton is right but
for the fact that any relay-result can play such a role. 18

ia William Hamilton uses an interesting comparison to the process of
tending through a sand bank. "In this operation, it is impossible to
swxced unless erery foot, nay, almost every inch in our progress be



DISCOVERY AS A SYNTHESIS 83

AnotJier Conception. After having acquired some infor-
mation about the behaviorist school, I wondered how behav-
iorist doctrine dealt with our present question and whether
it agreed with my observations. I understand that, for be-
haviorism, we do not necessarily think in words, but that,
otherwise, our thought may consist of muscular motions,
such as a shrug of the shoulders, motions of the eyelids or
the eyes, etc.

I have no recollection of such kind of motions connected
with my research work. Of course, I cannot watch my mo-
tions while deeply engaged in research, but witnesses of my
daily life and work can assert that they never saw anything
of that kind. They have only observed the special "inside"
look which I often happen to have when plunged in deeply
concentrated reflection. What I can say is that I do not see
what kind of motions could help me to get a clear view of
more or less difficult reasonings, while, on the contrary,
we have seen that proper mental pictures can evidently be
helpful for that.

An Inquiry among Mathematicians. It is natural to
investigate, with respect to our present subject, the behav-
ior of mathematicians in general. Unhappily, I was unable
to learn about French mathematicians, having thought of
the question only after my departure from Europe.

About the mathematicians born or resident in America,
whom I asked, phenomena are mostly analogous to those

secured by an arch of masonry before we attempt the excavation of
another. Now, language is to the mind precisely what the arch is to the
tunnel. The power of thinking and the power of excavation are not de-
pendent on the words in the one case, on the mason-work in the other;
but without these subsidiaries, neither process could be carried on beyond
its rudimentary commencement."

More generally, the function thus described belongs to what we have
called relay-results. In inventive effort, those do not always imply words.



8* DISCOVERY AS A SYNTHESIS

which I have noticed in my own case. 19 Practically all of
them contrary to what occasional inquiries had suggested
to Galton as to the man in the street avoid not only the
use of mental words but also, just as I do, the mental use
of algebraic or any other precise signs ; also as in my case,
they use vague images. There are two or three exceptional
cases, the most important of which is the mathematician
George D. Birkhoff , one of the greatest in the world, who is
accustomed to visualize algebraic symbols and to work with
them mentally. Norbert Wiener's answer is that he happens
to think either with or without words. Jessie Douglas gen-
erally thinks without words or algebraic signs. Eventually,
his research thought is in connection with words, but only
with their rhythm, a kind of Morse language where only
the numbers of syllables of some words appear. This, of
course, has notliing in common with Max Mailer's thesis
and is rather analogous to Galton's use of meaningless
words.

G. Polya's case I intend to speak only of men who
have made quite significant discoveries is different. He
does make an eventual use of words. "I believe," he writes
to me, 'that the decisive idea which brings the solution of
a problem is rather often connected with a well-turned
word or sentence. The word or the sentence enlightens the
situation, gives things, as you say, a physiognomy. It
can precede by little the decisive idea or follow on it im-
mediately ; perhaps, it arises at the same time as the deci-
sive idea The right word, the subtly appropriate word,

helps us to recall the mathematical idea, perhaps less com-
pletely and less objectively than a diagram or a mathe-

i At the moment of printing, a letter from Professor Einstein reaches
me, containing information of capital interest See Appendix II.



DISCOVERY AS A SYNTHESIS 85

matical notation, but in an analogous way. ... It may con-
tribute to fix it in the mind." Moreover, lie finds that a
proper notation that is, a properly chosen letter to de-
note a mathematical quantity can give him similar help ;
and some kind of puns, whether of good or poor quality,
may be useful for that purpose. For instance, Polya, teach-
ing in German at a Swiss university, usually made his
junior students observe that z and w are the initials of the
German words "Zahl" and "Wert," which precisely denote
the respective roles which z and w had to play in the theory
which he was explaining.

That case of Polya seems to be quite exceptional (I did
not meet with any similar one among the other men who
answered me) . 20 Even he, however, does not use words as
equivalents of ideas, since he uses one word or one or two
letters to symbolize a whole line of thought ; his psycholog-
ical process would be in agreement with Stanley's state-
ment 21 that language, as an indicator, can only indicate
by suggesting to our consciousness what is indicated, as
object, thought or feeling, even in most summary and un-
self-conscious form to wliich it is brought by practice."

The mental pictures of the mathematicians whose answers
I have received are most frequently visual, but they may
also be of another kind for instance, kinetic. There can
also be auditive ones, but even these, as the example of
J. Douglas shows, quite generally keep their vague char-
acter. 22 For B. O. Koopman, "images have a symbolic rather

20 I have just heard of the rather analogous case of Professor Chevalley.

21 Psychological Review, VoL IV (1891), p. 71. In that place, Stanley
chiefly deals with poetical invention, where the role of words is evidently
more important than elsewhere.

22 one of my colleagues of Columbia writes to me that his mathematical
thought is usually accompanied by visual images and hardly ever by
words other than vague exclamations of surprise, irritation, elation, etc.



86 DISCOVERY AS A SYNTHESIS

than a diagrammatic relation to the mathematical ideas"
which are considered, a description whose analogy with the
above is evident. Professor Koopman's observations also
agree with mine on the fact that such images appear in full
consciousness while the corresponding arguments provi-
sionally remain in the "antechamber. 5 *

We can say as much of the observations which Ribot 28
has gathered by questioning mathematicians. Some of them
have told him that they think in a purely algebraic way,
with the help of signs ; others always need a "figurated rep-
resentation," a "construction," even if this is "considered
as pure fiction."

Some Ideas of Descartes. In the ReguLae ad Directionem
Ingenii, which, in their second half (from the 14th rule
on), deal with the role of imagination in science, Descartes
seems to have conceived the idea of processes similar to
those we speak of. At least, this can be induced from some
places of the analysis of the Regulae made by Pierre Bou-
troux.* 4 For instance, he is reported by Pierre Boutroux to
have said that "Imagination, by itself, is unable to create
Science, but we must, in certain cases have recourse to it.
First, by focusing it on the object we want to consider, we
prevent it from going astray and, moreover, it can be use-
ful in awakening within us certain ideas." Again, "Imagi-
nation will chiefly be of great use in solving a problem by
several deductions, the results of which need to be coordi-
nated after a complete enumeration. Memory is necessary
to retain the data of the problem if we do not use them all
from the beginning. We should risk forgetting them if the



a Evolution &* 7<W G6*4rakt, p. 143.

L'Ima$mation ei let Mathtmatiqut* *elo* Deicartet, Bibl. de la
Facaltt dcs Lettrcs de Paris, VoL 10 (1900).



DISCOVERY AS A SYNTHESIS 87

image of the objects under consideration were not con-
stantly present to our mind and did not offer all of them to
us at each instant."

This is the role of images as we described it above. How-
ever, Descartes distrusts that intervention of imagination
and wishes to eliminate it completely from science. He even
reproaches ancient geometry for having used it. He wants
to eliminate imagination from every branch of science by
reducing all of them to mathematics (which he tried to do,
but did not succeed in doing) , mathematics consisting, more
than any other science, of pure understanding.

To see what we must think of such an idea, we need only
recall how Descartes* program has been carried out by
modern mathematicians. First, as is well known, geometry
can be completely reduced to numerical combinations by
the help of analytical geometry which Descartes himself
created. But we have just seen that deductions in the realm
of numbers may be, at least in several mathematical minds,
most generally accompanied by images.

More recently, another rigorous treatment of the princi-
ples of geometry, which, logically speaking, has been fully
freed from any appeal to intuition, has been developed on
quite different bases by the celebrated mathematician Hil-
bert. His beginning, which is now classic among mathema-
ticians, is "Let us consider three systems of things. The
things composing the first system, we will call points; those
of the second, we will call straight lines, and those of the
third system, we will call planes" clearly meaning that we
ought by no means to inquire what those "things" may
represent.

Logically, of course and this is all that is essential
the result announced is fully attained and every interven-



88 DISCOVERY AS A SYNTHESIS

tion of geometrical sense eliminated : that is, theoretically
unnecessary to follow the reasoning from the beginning to
the end. Is it the same from the psychological point of view?
Certainly not. There is no doubt that Hilbert, in working
out his Principles of Geometry, has been constantly
guided by his geometrical sense. If anybody could doubt
that (which no mathematician will), he ought simply to
cast one glance at Hilbert's book. Diagrams appear at
practically every page. They do not hamper mathematical
readers in ascertaining that, logically speaking, no con-
crete picture is needed. 25

This is again a case where one is guided by images with-
out being enslaved by them, and it is again possible (at
least in my own case) thanks to the same division of work
between proper consciousness and fringe-consciousness. 26

Similarly, Descartes censures the habit noticed by him in
Greek geometers (see above) of considering separately one
part of a diagram. There is no reason for that objection.
We meet with the same confusion between logical and psy-
chological processes. The method in question no more com-
promises the rigor of the argument than the image men-
tioned above compromises the proof of the fact that prime
numbers form an unlimited sequel.

There is already a paradox, as Klein notices, in the fact that we can
reason on an angle equal to the millionth part of a second, though we are
completely unable to distinguish between the sides of such an angle; and
its discussion (Rerve dc Mttaphysique et de Morale [1908], p. 923) by
Winter, one of the philosophers who had the best understanding of
scientific subjects, shows the analogy of that circumstance with our ob-
serrations in the text.

** Another instance, which we shall meet in Section VII (see p. 103
note) wiH illustrate this even more dearly and convincingly Indeed,
in the latter instance of Section VII, no doubt can subsist (at least as
concerns my own mind) on the way the division of work mentioned in
the text takes place.



DISCOVERY AS A SYNTHESIS 89

Other Thinkers. We do not have many data on that ques-
tion in fields other than the mathematical one. It is curious
that, according to the above-mentioned work of Binet
(Etude Experimentale de V Intelligence, pp. 127-129),
even in free thought, vague images may occur as repre-
sentatives of more precise ideas.

An instance quite analogous to our above description is
that of the economist Sidgwick, which he himself reported
at the International Congress of Experimental Psychol-
ogy, in 1892. His reasonings on economic questions were
almost always accompanied by images, and "the images
were often curiously arbitrary and sometimes almost unde-
cipherably symbolic. For example, it took him a long time
to discover that an odd, symbolic image which accompanied
the word Value' was a faint, partial image of a man put-
ting something on a scale." Also a most curious process
occurs among musical composers, according to Julius
Bahle (Der Musikcdische S chaff ensprozess y Leipzig, Hir-
zel, 1936 ; quoted by Delacroix, ^Invention et le Genie, p.
520) . Several of them see their creations in their initial con-
ception, in a visual form (inspiration is what he calls a
"Tonvision"). one of them perceives in that way, without
any precise musical presentation, "the main line and main
characteristics of his music. Besides, it is perhaps rather
difficult to say to what extent music is absent from that
formal schema." 27

I have asked only a few men belonging to other branches
of mental activity. The answers have been various and I

ZT A painter tells me that, in the first phase of composition, his visual
images are voluntarily vague.



90 DISCOVERY AS A SYNTHESIS

cannot assert that results could not differ from our previ-
ous ones. 28

Some scientists have told me of mental pictures quite
analogous to those which we have described. For instance, 28
Professor Claude Levi-Strauss, when thinking about a diffi-
cult question in his ethnographic studies, sees, as I do, un-
precise and schematic pictures which, moreover, have the
remarkable character of being three-dimensional. Also,
asking a few chemists, all of them reported absolutely
wordless thought, with the help of mental pictures.

The physiologist Andre Mayer's mind behaves quite
differently. He tells me that his thought immediately ap-
pears to him in a fully formulated form, so that no effort
whatever is necessary to him in order to write it down,

It would be interesting to know how medical men behave,
in that respect, in the difficult act of arriving at a diag-
nosis. I had the opportunity of asking- a prominent one,
and he told me that he thinks without words in that case,
though his thought uses words in theoretical and scientific
studies.

A type of thinking which seems surprising at first has
been discovered by the psychologist Ribot, 30 who finds it

Our statesman Aristkle Briand, according to what was observed by
one of his closest collaborators who was able to see him often at work,
did not think in words when he planned his speeches. Words appeared
only at the very moment of pronouncing them.

It would certainly be worth while to know about the opinion of some
important military leaders. There could be no case where a simultaneous
view of the synthesis and of every detail would be more essentiaL

Such ?s also the case of Professor Roman Jakobson (see p. 96).

** L* Evolution de* Idte* Gtntraleg, p. 143. Jean Pen-in, according to
information given to me by his son, had intermittently pictures of the
typographic-visual type; Francis Perrin generally thinks without words;
bat, from time to time, a word appears to him. Sidgwick's ideas appeared
in typographic-visual representations when, instead of economic subjects,
he was thinking of mathematics or logic.



DISCOVERY AS A SYNTHESIS 91

to be more frequent than would be expected. It is what he
calls the "typographic visual type" and consists in seeing
mentally ideas in the form of corresponding printed words.
The first discovery of this by Ribot was the case of a man
whom he mentions as a well-known physiologist. For that
man, even the words "dog, animal" (while he was living
among dogs and experimenting on them daily) were not
accompanied by any image, but were seen by him as being
printed. Similarly, when he heard the name of an intimate
friend, he saw it printed and had to make an effort to see
the image of this friend. It was the same with the word
"water," and carbonic acid or hydrogen appeared to his
mind either by their printed full names or by their printed
chemical symbol. Being strongly surprised by that state-
ment, the sincerity and accuracy of which were not to be
doubted, Ribot later on observed that that case was by no
means a unique one and similar ones were to be found in
several people.

Moreover, according to Ribot, men belonging to that
typographic-visual type cannot conceive how other peo-
ple's thought can proceed differently.

This is the state of mind which we have already noticed
inMaxMuller himself as concerns, more generally, thought
in words and which is really stupefying when we find it
among men accustomed to dealing with philosophical mat-
ters. How can we wonder that people have been burned
alive on account of differences in theological opinions, when
we see that a first-rate man like Max Miiller, apropos of a
harmless question of psychology, uses scornful words to-
ward his old master Lotze, for having written that the



92 DISCOVEEY AS A SYNTHESIS

logical meaning of a given proposition is in itself inde-
pendent of the form in which language expresses it?* 1

Thus, we have let ourselves be induced to deal with a
chapter of psychology rather different from the one which
is our main object. Some parts of this section could be
called "A case of psychological incomprehension."

This is not at all the only instance of the double fact:
(1) that the psychology of different individuals may dif-
fer in some essential points; (2) that, if so, it may be al-
most impossible for the one to conceive the state of mind of
the other. 82

Is Thought in words without Inconvenience? Of course,
I must myself be on my guard against the same lack of

*i Max Milller concedes that he can "with some effort" enter into the
mind of a decided adversary like Berkeley, "a kind of philosophical hal-
lucination," to use his own words. But he cannot understand the opinion
that most of our thoughts are carried on hi language but not quite all, or
that mott people think in words but not all. That some of the greatest
writers may have said that not from "lack of courage" but because facts
are like that, visibly lies beyond his imagination.

** Paradoxical as it seems, there are two such instances in the domain
of mathematics. Some years before World War I, a question which,
though a mathematical one, was contiguous to metaphysics raised a lively
discussion among some of us, especially between myself and one of my
best and most respected friends, the great scientist Lebesgue. We could
not avoid the conclusion that evidence that starting point of certitude
in every order of thinking did not have the same meaning for him and
for me. only, of course, we were never tempted to despise each other
merely because we recognized the impossibility of understanding each
other,

IT* subject in question belonged to the theory of "sets." Now, when,
in 1879-1884, Georg Cantor communicated his fundamental results on
that theory (now one of the bases of contemporary science), one of them
looked so paradoxical and upset so radically all our fundamental notions
tfeat it unleashed the decided hostility of Kroneker, one of the leading
mathematicians in that time, who prevented Cantor from getting any
new appointment in German universities and even from having any
Eaemotr published in German periodicals. Of course, the proof of that
resatt k as dear and rigorous as any other proof in mathematics, leav-
ing no possibility of not admitting it



DISCOVERY AS A SYNTHESIS 93

comprehension. Certainly, I must confess my incomprehen-
sion of the fact that the typographic-visual type or other
verbal types are possible, and I can hardly refrain from
thinking of Goethe's verses :

"Denn wo Begriffe fehlen,

Da stellt ein Wort zur rechten Zeit sich ein."

But I cannot forget that such men as Max Miiller and
others, not mediocre, think in such a way, though I do not
succeed in understanding it. In this connection, I regret
that Ribot has not published the name of the physiologist
he speaks of, and that we are therefore unable to form any
opinion of the value of his work.

For those of us who do not think in words, the chief dif-
ficulty in understanding those who do lies in our inability
to understand how they can be sure they are not misled by
the words they use see our condition (b), page 74*. As
Ribot says, 33 "The word much resembles paper money
(banknotes, checks, etc.), having the same usefulness and
the same dangers."

Such a danger has not remained unnoticed. Locke men-
tions many men as using words instead of ideas, and we
have seen that Leibniz cannot help experiencing a certain
anxiety as to the influence of thinking in words on the
course of his thought.

Curiously enough, Max Miiller himself says that indi-
rectly. To Kant, he opposes his friend Hamann, whom he

S3 The Psychology of Attention, p, 52 (translation of 1911; p. 85 of the
French edition of 1889). Ribot, in that same place, also describes the
evolution of that function of the word. He writes, "Learning how to
count in the case of children, and, better still, in the case of savages,
clearly shows how the word, at first firmly clinging to objects, then to
images, progressively detaches itself from them, to live an independent
life of its own."



94 DISCOVERY AS A SYNTHESIS

highly praises, and he cites the latter as having written
"Language is not only the foundation for the whole fac-
ulty of thinking, but the central point also from which
proceed the misunderstandings of reason by herself. , . .
The question with me is not what is the reason, but what is
the language? And here, I suspect, is the ground of the
paralogisms and antinomies with which the reason is
charged."

This would be all right if, consequently, Max Miiller
warned us to beware of such misunderstandings caused by
language; but on the contrary, he maintains that words,
by themselves, could never produce any error. "The word
itself is clear and simple and right ; we ourselves only de-
range and huddle and muddle it."

I should not have spoken once more of Max Miiller if
the statement expressed in that passage of thelntrodtictory
Lectures did not go even beyond the hitherto examined
question of words in thought. Immediately after that cita-
tion of Hamann, apparently yielding to a professional de-
formation, he speaks to us of "the science of thought,
founded as it is on the science of language." Will he have
us believe that language not only must accompany thought,
but must govern it?

Unhappily for his thesis, such a tendency seems not to
have been always harmless to him. It requires a man who
identifies thought with words to attack Darwin's theory 84
by only taking account of the word "selection" and neg-
lecting, as allegedly a "metaphorical disguise," the mean-
ing of it in Darwin's ideas.

The thinker using mental words may, on the contrary,
understand that not only words, but every kind of auxiliary

&**<* of T*o*$H VoL I, p. 97.



DISCOVERY AS A SYNTHESIS 95

signs only play the role of kinds of labels attached to ideas.
He will, more or less consciously, apply proper methods
(about which it would be interesting to inquire) in order to
give them that role and no other one. We have seen that
Polya himself, the only one among the mathematicians I
have consulted who makes so much use of words, introduces
a single one in a whole course of thought so as to remind
himself of a central idea, while Jessie Douglas represents
some of them by their mere syllabic rhythm. Similarly, one
of my colleagues in literary matters thinks in words, but
from time to time introduces a nonexistent word. This proc-
ess, to be compared with Jessie Douglas's or Galton's,
seems to me to be evidently useful for the same purpose.

In the thought of Leibniz, we can be sure that such mis-
understandings as were dreaded by Hamann could not
occur: first, because he is Leibniz, then, because he is aware
of the danger. But, though being little acquainted with the
theories of metaphysicians, I am rather disquieted on read-
ing in Ribot's Evolution des Idees Generates that among
them the typographic-visual type seems to be overwhelm-
ingly the most frequent one.

Among philosophers, there seems, indeed, to be a certain
tendency to confuse logical thought with the use of words.
For instance, it is difficult not to recognize it in William
James when he complains 85 that, **we are so subject to the
philosophical tradition which treats logos or d,iscursive
thought generally as the sole avenue to truth, that to fall
back on raw unverbalized life has more of a revealer, etc."
the word Unverbalized" hardly leaving any doubt that
he has used the word logos in the ancient Greek sense.

Is not that tendency likely eventually to mislead those

as A Pluralistic Universe, p. 272.



96 DISCOVERY AS A SYNTHESIS

who let themselves be governed by it? Reading FouilleVs
objections concerning the unconscious in his Evolutionisme
des Idees-Forces (see Section II, p. 28), I wonder whether
he does not mistake words for reasons.

I even feel some uneasiness when I see that Locke and
similarly John Stuart Mill consider the use of words neces-
sary whenever complex ideas are implied. I think, on the
contrary, and so will a majority of scientific men, that the
more complicated and difficult a question is, the more we
distrust words, the more we feel we must control that dan-
gerous ally and its sometimes treacherous precision.

A Valuable Description. Though, in the question of
words in thought, divergent opinions still occasionally
arise, it is now rather generally admitted that words do not
need to be present. on the other hand, several recent psy-
chologists, even while insisting on words, 36 have noticed, as
we do, the intervention of vague images which do not truly
represent but symbolize ideas. 87

I shall not undertake to review these works ; but I cannot
resist the temptation to reproduce the highly interesting
communication which has been kindly addressed to me by
Professor Roman Jakobson, who, besides his well-known
linguistic work, takes a fruitful interest in psychological
subjects. It reads thus :

"Signs are a necessary support of thought. For social-
ized thought (stage of communication) and for the thought
which is being socialized (stage of formulation), the most
usual system of signs is language properly called ; but in-
ternal thought, especially when creative, willingly uses

s Sec Delacroix, Lt Langage et la Penste, pp. 384 ff. and compare the
footnote of p, 406.

*T See also Titchener's Experimental Psychology of the Thought Proc-
***, especially Lecture I and corresponding Notes.



DISCOVERY AS A SYNTHESIS 97

other systems of signs which are more flexible, less stand-
ardized than language and leave more liberty, more dy-
namism to creative thought. . . . Amongst all these signs or
symbols, one must distinguish between conventional signs,
borrowed from social convention and, on the other hand,
personal signs which, in their turn, can be subdivided into
constant signs, belonging to general habits, to the individ-
ual pattern of the person considered and into episodical
signs, which are established ad hoc and only participate in
a single creative act."

This remarkably precise and profound analysis beau-
tifully enlightens our observations such as we have re-
ported above. That there should be such an agreement be-
tween minds working in quite different branches is a
remarkable fact.

Comparison with Another QuestionConcerning Imagery.
Images constitute, as we could say, the chief subject of
Taine's celebrated work on Intelligence. He treats them
from points of view rather different from ours (as tense
thought is rarely, if ever, considered) . However, there is,
concerning them, a question which particularly interests
hi' and on which the above observations might possibly
throw some light. As he persistently notices, it ought to be
explained how images appear to us, often very vividly, and
nevertheless remain distinct from real sensations ; how our
mind can generally differentiate between images and hallu-
cinations. 88

But, in our case, we have also a sequel of images develop-
ing parallel to thought properly called. Both mental

88 The same question also assumes great importance in some psycho-
logical studies of Varendonck. See his book, Psychology of Day
especially Chap. II, pp. 75-86.



98 DISCOVERY AS A SYNTHESIS

streams, images and reasonings, constantly guide each
other though keeping perfectly distinct and even, to a
certain extent, independent ; and we have found this to be
due to a cooperation between proper consciousness and
fringe-consciousness. It may be supposed that there is some
analogy between the two phenomena and that one could
help us to understand the other.

Can Imagery be Educated? The above considerations
suggest a question analogous to one which has been raised
at the end of Section IV. Is it possible if desirable for
our volition to influence the nature of the auxiliary signs
used by your thought? Now, this has been done. Titchener
has carried out a most remarkable attempt in that sense. As
he explains to us, 38 his natural tendency would have been to
employ internal speech ; but he has always tried and always
succeeded in having a wide range and a great variety of
imagery, "fearing that, as one gets older, one tends also to
become more and more verbal in type."

That too great importance of verbal intervention in his
thought is thus prevented by a constant renewing of image-
ry. What is more curious, he uses for such a purpose not
only visual images but, above all, auditory, viz., musical
ones.

But he uses also the help of visual imagery, "which is
always at my disposal," he says, "and which I can mould
and direct at will." "Reading any work, I instinctively ar-
range the facts or arguments in some visual pattern and I
am as likely to think in terms of this pattern as I am to
think in words," and the better the work suits such a pat-
tern, the better it is understood.

Such an auto-education of mental processes seems to me

Evperian#*tal Ptychology of the Thought Procettes, pp. 7 ff.



DISCOVERY AS A SYNTHESIS 99

to be one of the most remarkable achievements in psy-
chology.

Using Relay-results. It is obvious that the necessary
synthesis is greatly simplified when some parts of the de-
duction are replaced by the corresponding relay-results,
which are in this case especially important. 40

General Remarks. All this concerns men engaged in in-
tellectual work. Investigation among other groups seems
to meet the difficulty that, as we have seen, the laws of tense
thought may be and seem to be very different from those of
usual and common ideation, which is the only frequent one
among ordinary people. This is probably the reason why
Galton, though seeing the necessity of a more~ general
inquiry, was unable to make it.

At any rate, we see that, while what we said in the pre-
ceding sections seems to be common to various creative
minds, the nature of auxiliary concrete representations
may vary considerably from one mind to the other.

Addendum. The distinction between full consciousness
and fringe-consciousness has not been easily perceived by
several scientists, and I feel it necessary to elaborate this
point. Just after having constructed a deduction, I can
easily remember what has taken place during its develop-
ment ; and, so doing, I generally realize that at no instant
had the properly-called reasoning been present to my con-
scious ego, which had considered nothing else than the
above-described schema, only "fishing" each piece of the
logical construction when the proper time had come.

This is a peculiar feature of mathematical thought, by which it dif-
fers from other kinds of meditation. It is clear that only mathematical
deduction allows us to leave a proof completely aside and replace it by
its conclusion. In other cases it does not happen that the latter may be
"mathematically" equivalent to its premises.



VII. DIFFERENT KINDS OF
MATHEMATICAL MINDS



THE PHENOMENA considered in Sections I-V seem to hap-
pen similarly in many mathematical research scholars. on
the contrary, concrete representations studied in the pre-
ceding section were far from the same for everybody. This
section will also be devoted to differences among various
ways of mathematical thinking. It will have to our former
considerations the same relation which the distinction be-
tween various zoological genera and species has to general
physiology.

The Case of Common Sense, Let us start from the begin-
ning, that is from the case of people simply reasoning with
their common sense. In that case, we can say that much is
afforded by the unconscious and little is asked from further
conscious elaboration.

It often happens, besides, that that unconscious is a very
superficial one and its data are not essentially different
from regular reasoning. Thus, Spencer, alluding to the
classic syllogism "every man is mortal; now, Peter is a
man ; therefore, Peter is mortal," supposes that you hear
of a ninety-year-old man who undertakes to build a new
house for himself. Spencer has no difficulty in proving that
the syllogism is really present in your fringe-consciousness
and that, between it and the stream of thought an instan-
taneous one, as is general in unconsciousness which leads
you to speak of the man as being unreasonable, there is but
a difference of form. Things can happen similarly in the
case of many simple mathematical deductions.

In other instances, however, ways followed by common



KINDS OF MATHEMATICAL MINDS 101

sense may be very different from those which we can f ormu-
late by explicit reasoning. It happens especially in ques-
tions of a concrete nature say, geometrical or mechanical
ones. Our ideas on such subjects, acquired in early child-
hood, seem to be relegated to a remote unconsciousness ; we
cannot know them exactly and it is probable that they
often imply empirical reasons, taken not from true reason-
ing but from the experience of our senses. Let us take one
or two examples.

Let us imagine that I throw what is called a "material
point" that is, a very small body, such as a very small
marble which will go on moving on account of its initial
velocity and its weight. Common sense tells us that the
motion must take place in a vertical plane, the vertical
plane P drawn through the initial line of projection. In
that case, it is hardly doubtful that the subconscious rea-
soning uses the "principle of sufficient reason," there being
no reason why the movable point should go to the right
rather than to the left side of the aforesaid plane P.

The mathematical proof, such as classically given in
courses of rational mechanics, proceeds in an utterly dif-
ferent way, with the use of several theorems of the differen-
tial and integral calculus. It is to be noticed, however, that
the proof which occurs to common sense could be trans-
formed into a perfectly rigorous one, by applying a gen-
eral theorem (also belonging to the integral calculus)
which says that under the aforesaid conditions (the direc-
tion and magnitude of the initial velocity being given) the
motion is uniquely determined. That theorem, in its turn,
can be rigorously proved; but the latter proof takes place
only in higher courses of calculus, so that, in regular teach-
ing, the way suggested by common sense to reach our con-



102 KINDS OF MATHEMATICAL MINDS

elusions appears actually less elementary than the other
one.

Let us now consider two examples in geometry. If I
think of drawing a curve in a plane, by the continuous
motion of a point, it is a fact of common sense that at all of
its points (some exceptional ones being perhaps excepted)
that curve will admit of a tangent (in other words, that, at
every instant, the motion must take place in some deter-
mined direction). We do not know how our common sense,
i.e., our unconscious, reaches such a conclusion : perhaps by
empiricism, i.e., by memory of the lines we are accustomed
to see or, as F. Klein supposes, by a confusion of geometri-
cal curves, which have no thickness whatever, with the lines
which we can actually draw and which always have some
thickness. As a matter of fact, the conclusion is false;
mathematicians can construct continuous curves which
have no tangent at any point.

In the second place, let us consider a plane closed curve
which has no "double point," that is, which nowhere inter-
sects itself. It is evident to common sense that such a curve,
whatever its shape may be, divides the plane (a) into two
different regions ; (b) into not more than two.

How common sense elaborates that conclusion, is not
positively known, intervention of empiricism being again
probable. This time, the conclusion (Jordan's theorem) is
correct ; but, evident as it is for our common sense, its proof
is of great difficulty.

By such examples as these two, it has been realized that,
at least in a certain class of questions relating to prin-
ciples, 1 we cannot surely rely on our ordinary space-intui-



questions we are allnding to depend on arithmetization rather
than on Hflberf s ideas such as mentioned in Section VI.



KINDS OF MATHEMATICAL MINDS 103

tion : as geometrical properties can always be reconducted
to numerical ones, thanks to the invention of analytical
geometry, arguments should always be fully arithmetized,
or, at least, it must be ascertained that this arithmetization,
if not given at length for brevity's sake, is possible. Pascal's
word "Tout ce qui passe la Geometric nous passe" is re-
placed, for the modern mathematician, by 'Tout ce qui
passe PArithmetique nous passe."

For instance, a proof of Jordan's theorem, which we
have just enunciated, is not satisfactory if not fully arith-
metizable. 2

Second Step: the Student in Mathematics. After that
common-sense state of human thought, there has come the
scientific state. We have seen that it is characterized by the
intervention of the threefold operation of verifying the
result ; "precising" it ; and, above all, making it utilizable,
which, as we have seen, requires the enunciation of relay-
results. We have seen that that too is essential, first for the
certitude of the knowledge thus acquired; then, for its
fruitfulness and the possibility of extending it.

These characters can help us to understand what takes
place, psychologically speaking, in the passage from the

2 Same remark on this subject as on Hubert's Principle t. I have given
a simplified proof of part (a) of Jordan's theorem. Of course, my proof
is completely arithmetizable (otherwise it would be considered non-
existent) ; but, investigating it, I never ceased thinking of the diagram
(only thinking of a very twisted curve), and so do I still when remember-
ing it. I cannot even say that I explicitly verified or verify every link of
the argument as to its being arithmetizable (hi other words, the arith-
metized argument does not generally appear in my full consciousness).
However, that each link can be arithmetized is unquestionable as well for
me as for any mathematician who will read the proof: I can give it in-
stantly in its arithmetized form, which proves that that arithmethed form
is present in my fringe-consciousness.



104 KINDS OF MATHEMATICAL MINDS

former state to the latter: in other words, what concerns
the case of the student of mathematics.

How commonly total misunderstandings and failures oc-
cur in that case, is well known. I shall, besides, be very
brief on that subject, because it has been profoundly
treated by Poincare (Les Definitions dans VEnseignement
in Science et Meihode) . Even before quoting him, it is not
useless to observe that that case of the mathematical stu-
dent already belongs to our subject of invention. Between
the work of the student who tries to solve a problem in
geometry or algebra and a work of invention, one can say
that there is only a difference of degree, a difference of
level, both works being of a similar nature.

Now, how does it happen that so many are incapable of
that work, incapable of understanding mathematics ? That
is what Poincare examines and of which he in a striking
way shows the true reason which lies in the meaning that
ought to be given to the word "to understand."

**To understand the demonstration of a theorem, is that
to examine successively each syllogism composing it and
ascertain its correctness, its conformity to the rules of the
game? . . . For some, yes ; when they have done this, they
will say, I understand.

"For the majority, no. Almost all are much more exact-
ing; they wish to know not merely whether all the syllo-
gisms of a demonstration are correct, but why they link
together in this order rather than another. In so far as to
them they seem engendered by caprice and not by an intel-
ligence always conscious of the end to be attained, they do
n0t believe they understand.

"Doubtless they are not themselves just conscious of
what they crave, and they could not formulate their desire,



KINDS OF MATHEMATICAL MINDS 105

but if they do not get satisfaction, they vaguely feel that
something is lacking."

The connection of this with our former considerations is
easy to understand. For the purpose of teaching be it
oral or written every part of the argument is brought
into its entirely conscious form, corresponding to the simul-
taneous verifying and "precising" stages which we have
described above. Even, in view of further consequences,
there is a tendency to increase the number of relay-results.
In this way of working, which seems to be the best one of
getting a rigorous and clear presentation for the beginner,
nothing remains, however, of the synthesis, the importance
of which we have underlined in the preceding section. But
that synthesis gives the leading thread, without which one
would be like the blind man who can walk but would never
know in what direction to go.

Those to whom such a synthesis appears "understand
mathematics." In the contrary case, there are the two atti-
tudes mentioned by Poincare. The rather general one is the
second one: the student feels that something is lacking, but
cannot realize what is wrong ; if he does not overcome that
difficulty, he will get lost.

In the first case mentioned by Poincare, the student, not
finding any synthetic process, will do without it. Although
this allows him to pursue his studies, often for long years,
his case is, from a certain point of view, worse than the
other one in which at least the existence of some difficulty
was understood. on account of the mathematical knowledge
more and more required for entrance to several careers, one
frequently meets with such a student. I have seen a case in
which a candidate, guided by his common sense, knew the
right answer to my question, but did not think he was al-



106 KINDS OF MATHEMATICAL MINDS

lowed to give it and did not realize that the suggestion of
his subconscious could be very easily translated into a cor-
rect and rigorous proof.

Curious instances of this sort are not uncommon among
students in differential and integral calculus. Most often
the question is whether such and such a theorem or formula
is appropriately invoked, whether the conditions for its
application are satisfied or not. Students sometimes indus-
triously investigate that question when common sense in-
dicates the answer to be a practically evident one and on
the other hand neglect to study it in the case where it is
a delicate one and does deserve a careful examination. Such
remark or analogous ones could be eventually useful in



Logic and Intuitive Minds. A Political Aspect of the
Question. After having spoken of students, let us now deal
with mathematicians themselves, able not only to under-
stand mathematical theories, but also to investigate new
ones. Not only do these differ from ordinary students, but
they also profoundly differ from each other. A capital dis-
tinction has been emphasized: some mathematicians are
"intuitive" and others "logical." Poincare has dealt with
that distinction and so has the German mathematician
Klein. Poincare's lecture on the subject begins as follows:

"The one sort are above all preoccupied by logic ; to read
their works, one is tempted to believe they have advanced
only step by step, after the manner of a Vauban who pushes
on his trenches against the place besieged, leaving nothing
to chance. The other sort are guided by intuition and at
the first stroke, make quick but sometimes precarious con-
quests, like bold cavalrymen of the advance guard."

With Klein, even politics has been introduced into the



KINDS OF MATHEMATICAL MINDS 107

question: he asserts 3 that "It would seem as if a strong
naive space intuition were an attribute of the Teutonic
race, while the critical, purely logical sense is more de-
veloped in the Latin and Hebrew races." That such an as-
sertion is not in agreement with facts will appear clearly
when we come to examples. It is hardly doubtful that, in
stating it, Klein implicitly considers intuition, with its
mysterious character, as being superior to the prosaic way
of logic (we have already met with such a tendency in
Section III) and is evidently happy to claim that superi-
ority for his countrymen. We have heard recently of that
special kind of ethnography with Nazism : we see that there
was already something of this kind in 1893.

One will find such tendentious interpretations of facts
whenever nationalistic passions enter into play. At the be-
ginning of the First World War, one of our greatest
scientists and historians of sciences, the physicist Duhem,
was misled by them just as Klein had been, only in the op-
posite sense. In a rather detailed article,* he depicts Ger-
man scientists, especially mathematicians, as lacking intui-
tion or even deliberately setting it aside. It is especially
hard to understand how he can characterize in that way
Bernhard Riemann who is undoubtedly one of the most
typical examples of an intuitive mind. Duhem's assertion
of 1915 seems to me as unreasonable as Klein's in 1893. If
one or the other were right, the reader will realize by all
that precedes that either Frenchmen or Germans would
never have made any significant discovery. The only thing
for which I should think of reproaching the German mathe-
matical school in that line is a systematic, though hardly

s The Evanston Colloquium, p. 46.

* Revue des Deux Monde* (January-February, 1915), p. 657.



108 KINDS OF MATHEMATICAL MINDS

defendable and somewhat pedantic claim, chiefly under
Klein's influence, that, for certain proofs in analysis and
its arithmetical applications, "series" must be used in
preference to "integrals." Precisely in those questions, the
use of series looks more logical and the use of integrals
more intuitive. Perhaps there is again some nationalism in
such a tendency, because series are used by the celebrated
Weierstrass a most evident logician whose reputation
and influence have been enormous among German scholars,
while, into similar subjects, Cauchy or Hermite introduced
integrals 5 (though this was also the case of Riemann).

Poincar&s View of the Distinction. Poincare, more wisely
I think, does not connect the matter with politics. on the
contrary, he implicitly shows how doubtful such a connec-
tion is : for, in order to illustrate the opposition between
the two kinds of mind, he opposes to each other in the first
place two Frenchmen and, then, two Germans.

However, after having fully accepted and faithfully fol-
lowed Poincare's ideas in Sections I to V, I shall, this time,
disagree with him. We have cited the first paragraph of his
lecture ; let us reproduce the second. It reads as follows :

"The method is not imposed by the matter treated.
Though one often says of the first that they are analysts
and calls the others geometers, that does not prevent the
one sort from remaining analysts even when they work at
geometry, while the others are still geometers even when

a According to that doctrine, Klein thinks it necessary to modify the
proof of a celebrated theorem of Hermite; and even, on reaching a
certain point, says "the proof is not yet perfectly simple: something
stm remains of the ideas of Hermite," this inducing him to a further
modification. As a matter of fact, these "simplifications" are superficial
ooesy and after them just as before them, everything absolutely every-
thing essential rests on Hermite's fundamental idea.



KINDS OF MATHEMATICAL MINDS 109

they occupy themselves with pure Analysis. It is the very
nature of their mind which makes them logicians or in-
tuitionalists, and they cannot lay it aside when they ap-
proach a new subject."

What must we think of the comparison between those
two paragraphs? Both times, a distinction is made between
intuition and logic, but on bases quite different, though
somewhat related to each other. 6

This appears even more clearly in the examples set forth
by Poincare. To Joseph Bertrand, who visibly had a con-
crete, spatial view of every question, he opposes Hermite
whose eyes "seem to shun contact with the world" and who
seeks "within, not without, the vision of truth."

That Hermite was not used to thinking in the concrete
is certain. He had a kind of positive hatred for geometry
and once curiously reproached me with having made a
geometrical memoir. As natural, his own memoirs on con-
crete subjects are very few and not among his most remark-
able ones. So, from the second point of view of Poincare,
Hermite ought to be considered as a logical mathematician.

But to call Hermite a logician ! Nothing can appear to
me as more directly contrary to the truth. Methods always
seemed to be born in his mind in some mysterious way. In
his lectures at the Sorbonne, which we attended with* un-
failing enthusiasm, he liked to begin his argument by : "Let
us start from the identity . . ." and here he was writing a
formula the accuracy of which was certain, but whose
origin in his brain and way of discovery he did not explain
and we could not guess. This quality of his mind is also
most evidently illustrated by his celebrated discovery in
the theory of quadratic forms. In that question, two cases

6 See the remarks at the beginning of this section (p. 101).



110 KINDS OF MATHEMATICAL MINDS

are possible in which, as is obvious, things happen quite
differently. In the first one, "reduction" has been known
since Gauss. Nobody, as it seemed, would have thought of
the idea of merely carrying out, in the second case, the very
calculations which suited the first one and which appar-
ently, had nothing to do with that second case; it seemed
quite absurd that they would, that time, lead to the solu-
tion ; and yet, by a kind of witchcraft, they do. The mecha-
nism of that extraordinary phenomenon was, some years
later, partly explained by a geometrical interpretation (of
course, given not by Hermite, but by Klein) ; but it did
not become entirely clear to me before reading Poincare's
conception of it, in one of his early notes. 7 1 can hardly im-
agine a more perfect type of an intuitive mind than Her-
mite's, if not taking account of the extreme cases which

T Poincare* himself, in spite of the inspiration phenomena which we
hare mentioned, does not make that same impression on me. Reading one
of his great discoveries, I should fancy (evidently a delusion) that, how-
ever magnificent, one ought to have found it long before, while such
memoirs of Hermite as the one referred to in the text arouse in me the
idea: "What magnificent results! How could he dream of such a thing?"

There is obviously something subjective in such a judgment. A deduc-
tion which will seem to be a logical one for me that is, congenial to my
mind, one which I should be naturally inclined to think of may appear
as intuitive to some other man. Perhaps, almost every mathematician
would be a logician according to his own judgment For instance, I have
been asked by what kind of guessing I thought of the device of the
"finite part of infinite integral," which I have used for the integration of
partial differential equations. Certainly, considering it in itself, it looks
typically like "thinking aside." But, in fact, for a long while my mind
refused to conceive that idea until positively compelled to. I was led to it
step by step as the mathematical reader will easily verify if he takes
the trouble to consult my researches on the subject, especially my
Recherche* tur let Solutions Fondamentales et VInUgration des tiqwt-
tiont Lineavret a*x Derwte* Partielle*, 2nd Memoir, especially pp. 121 ff.
(Annafes scientifiques de FEcole Normale Supe"rieure, Vol. XXII,,
1906). I could not avoid it any more than the prisoner in Poe's tale
The Pit and tke Pendulum could avoid the hole at the center of his
celL



KINDS OF MATHEMATICAL MINDS 111

will be mentioned in the next section. Hermite's example
undoubtedly shows that the two definitions of intuition and
logic given by Poincare do not agree or do not necessarily
agree, which Poincare finally admits to a certain extent on
account of that case.

The two German mathematicians whom Poincare com-
pares are Weierstrass and Riemann, That, as he concludes,
Riemann is typically intuitive and Weierstrass typically
logical is beyond contestation. But as to the latter, Poin-
care says <c You may leaf through all his books without
finding a figure." It strikes me that there happens to be
there an error of fact. 8 It is true that almost no memoir of
Weierstrass implied any figure: there is only one excep-
tion ; but there is one, and this exception occurs in one of
his most masterly and clear-cut works, one giving the most
complete impression of perfection : I mean Ids fundamental
method in the calculus of variations. Weierstrass draws a
simple diagram 8 * and, after that initial step is taken, every-
thing goes on in the profoundly logical way which is un-
doubtedly his characteristic, so that, by merely looking at
that diagram, anyone sufficiently acquainted with mathe-
matical methods could have rebuilt the whole argument.
But of course there was an initial intuition, that of con-
structing the diagram. This was the more difficult and the
more evidently an act of genius because it meant breaking
from the general methods which had continued to become
more and more successful after the invention of infinites!-

s An error for which, however, Poincare* is not to be reproached (see
next footnote).

s* Whether Weierstrass himself actually drew the diagram (or simply
described it in words) cannot be said because he did not develop his
method elsewhere than in his oral lectures. That method remained un-
known for years, except to his former students.



112 KINDS OF MATHEMATICAL MINDS

mal calculus, which had been beautifully successful in the
hands of Lagrange for obtaining the first stage of the
solution, though not enabling anybody to complete it cor-
rectly. Weierstrass showed that abandoning these methods
and operating directly was the right way for that.

In reality, as we see, this is an undeniable case of the
general fact of logic following an initial intuition.

Application of Our Previous Data. We are thus com-
pelled to admit that there is not a single definition of in-
tuition vs. logic, but there are at least two different ones.
Now, for elucidating this, why should we not make use of
what we have found in our former analysis of the phe-
nomena?

Summing up the results of that analysis, let us remem-
ber that every mental work and especially the work of
discovery implies the cooperation of the unconscious, be
it the superficial or (fairly often) the more or less remote
one ; that, inside of that unconscious (resulting from a pre-
liminary conscious work), there is that starting of ideas
which Poincare has compared to a projection of atoms and
which can be more or less scattered ; that concrete represen-
tations are generally used by the mind for the maintenance
and synthesis of combinations.

This carries, in the first place, the consequence that,
strictly speaking, there is hardly any completely logical
discovery. Some intervention of intuition issuing from the
unconscious is necessary at least to initiate the logical
work.

With this reservation, we immediately see that the proc-
esses such as those described above can behave differently
in different minds.

( A) More or Less Depth in the Unconscious. As we know



KINDS OF MATHEMATICAL MINDS 113

that there must be several layers in the unconscious, some
quite near consciousness while some may lie more and more
remote, it is clear that the levels at which ideas meet and
combine may be deeper or, on the contrary, more super-
ficial; and it is not unreasonable to admit that there is a
usual behavior of every single mind from that point of
view.

It is quite natural to speak of a more intuitive mind if
the zone where ideas are combined is deeper, and of a
logical one if that zone is rather superficial. This manner
of facing the distinction is the one I should believe to be
the most important.

If that zone is deeper, there will be more difficulty in
bringing the result to the knowledge of consciousness and
it is likely to happen that the mind will have a tendency
to do so only for what is strictly necessary. I should think
this to be the case with Hermite, who certainly did not omit
anything strictly essential in the results of his reflections,
so that his methods were quite correct and rigorous, but
without letting any trace remain of the way in which he
had been led to them.

The contrary may happen : some minds may be such that
ideas elaborated in the depth of the unconscious are never-
theless integrally brought to the light of consciousness. I
should fancy that as happening with Poincare, whose ideas,
inspired as they may have been by farsighted intuitions,
generally seemed to follow a quite natural way. one sees
that there can be apparent logicians, who are logical in the
enunciation of their ideas, after having been intuitive in
their discovery. 9

& Generally speaking, as some authors observe (see Meyerson D*
Chemnwment de la Penste, Vol. 1, cited in Delacroix LVwo et U



114 KINDS OF MATHEMATICAL MINDS

(B) More or Less Narrowly Directed Thought. In the
second place, we have seen that the projection of Poincare's
atoms the starting of ideas, to use a less metaphorical
language can be more or less scattered. This is another
reason why we can have the sensation of an intuitive mind
(which will happen if there is much scattering) or (in the
contrary case) of a logical mind; and the second reason
can, at least a priori, be without any connection with the
first one: the direction of thought may be narrower or
wider, be it at one level of unconsciousness or at another.
A priori, we do not know whether there is not a connection
between these two kinds of "intuitive tendencies" ; but in
fact, an example (see below the case of Galois) will show
us independence.

(C) Different Auxiliary Representations. We have seen
how differently scientists behave as to the way in which
their thought is helped by mental pictures or other concrete
representations: differences can bear either on the nature
of representations or on the way they influence the work of
the mind. It is evident that some of these kinds of repre-
sentations may give the thought a rather logical course,
some others a rather intuitive one. But this side of the
question is much less accessible to study, precisely because
phenomena are not always comparable in different minds.

We can understand the use of geometrical images in in-
tuition with the help of reference to Section II (especially
p. 24 and the footnote of p. 23) and Section VI (p. 65).
A most evident fact of observation is that a quite perfect
organization and synthesis of usual sensations occurs in

<?****, p. 480), there is often a great difference between the discovery of
n idea and its enunciation.



KINDS OF MATHEMATICAL MINDS 115

our earliest childhood, taking place in the very deep layers
of our unconscious, and consequently with very great
rapidity (the time required lying far below the reach of
our measuring methods). That such a process should be
especially adapted to intuition, considered from our point
of view (A), is quite natural.

It would have been most interesting to get the auto-
observations of Hermite, who, though a highly intuitive
mind, seems to be so completely distant from geometrical
and, generally speaking, concrete considerations.

Other Differences in Mathematical Minds. The above
question is the only one which has been examined so far,
concerning different kinds of mathematical minds ; but, of
course, there is no doubt that mathematicians can differ
from each other from various other points of view.

For instance, there exists a theory, the theory of groups,
the importance of which, in our science, grew increasingly
for more than one century, especially since the work of
Sophus Lie at the end of the nineteenth century. Some
mathematicians, especially contemporary ones, have im-
proved it by most beautiful discoveries. Some others I
confess that I belong to the latter category though being
eventually able to use it for simple appli cations, feel in-
superable difficulty in mastering more than a rather ele-
mentary and superficial knowledge of it. Psychological
reasons for that difference, which seems to me incontesta-
ble, would be interesting to find.



VIII. PARADOXICAL CASES OF
INTUITION



IF, IN SOME exceptionally intuitive minds, ideas may evolve
and combine in still deeper unconscious layers than in the
above-mentioned cases, then even important links of the
deduction may remain unknown to the thinker himself who
has found them. The history of science offers some remark-
able examples.

Fermat (1601-1661), Pierre de Fermat was a magis-
trate, a counselor at the Parliament of Toulouse. It was a
time when life was less complicated than nowadays, and the
requirements of his duties apparently did not hamper him
in his mathematical researches, which were considerable.
Besides having participated in the origins of infinitesimal
calculus and even in the creation of calculus of probabili-
ties, he dealt actively with arithmetical questions. Among
the ancient mathematicians whose works were in his posses-
sion, he owned a translation of the work of Diophantes, a
Greek author who had dealt with such arithmetical sub-
jects. Now, at Fermat's death, his copy of Diophantes*
work was found to bear in the margin the following ob-
servation (in Latin) :

"I have proved that the relation x m -j- y m = z m is impos-
sible in integral numbers (x, y, z different from 0; m
greater than 2) ; but the margin does not leave me room
enough to inscribe the proof."

Three centuries have elapsed since then, and that proof
which Fermat could have written in the margin had the
latter been a little broader, is still sought for. However,
Fermat does not seem to have been mistaken, for partial



PARADOXICAL CASES OF INTUITION 117

proofs have been found, viz., proofs for some extended
classes of values of the exponent m : for instance, the proof
has been obtained for every m not greater than 100. But
the work an immense one which made it possible to
get these partial results could not be accomplished by
direct arithmetical considerations i 1 it required the help of
some important algebraic theories of which no knowledge
existed in the time of Fermat and no conception appears
in his writings. After several fundamental principles of
algebra had been laid down during the eighteenth century
and at the beginning of the nineteenth, the German mathe-
matician Kummer, in order to attack that question of the
**last theorem of Fermat," was obliged to introduce a new
and audacious conception, the "ideals," a grand idea which
entirely revolutionized algebra. As we just said, even that
powerful tool given to mathematical thought allows, as
yet, only a partial proof of the mysterious theorem.

Riemann (18&6-1866). Bernhard Riemann, whose ex-
traordinary intuitive power we have already mentioned,
has especially renovated our knowledge of the distribution
of prime numbers, also one of the most mysterious ques-
tions in mathematics. 2 He has taught us to deduce results

iThe use of considerations of that kind has been attempted by the
most prominent masters beginning with Abel during the last two cen-
turies. Every significant gain which can be obtained in that direction
seems to have been reached, and those gains are quite limited ones, The
French Academy of Sciences in Paris yearly receives several papers on
that subject, most of which are absurd, while a few reproduce known
results of Abel or others.

2 Both those instances concerning Fermat and Riemann relate to arith-
metic. Indeed, arithmetic, which is the first study hi elementary teaching,
is one of the most difficult, if not the most difficult branch of mathematics,
when one tries to penetrate it more deeply. Essential gains are generally
obtained, as happens hi our examples, on an arithmetical question by
reconducting it to higher algebra or to the infinitesimal calculus.

It must be observed that the example of this discovery of Riemann



118 PARADOXICAL CASES OF INTUITION

in that line from considerations borrowed from the integral
calculus : more precisely, from the study of a certain quan-
tity, a function of a variable s which may assume not only
real, but also imaginary values. He proved some important
properties of that function, but pointed out several
as important ones without giving the proof. At the death
of Riemann, a note was found among his papers, saying
'These properties of (s) (the function in question) are
deduced from an ｅxpression of it which, however, I did not
succeed in simplifying enough to publish it."

We still have not the slightest idea of what the expres-
sion could be. As to the properties he simply enunciated,
some thirty years elapsed before I was able to prove all of
them but one. The question concerning that last one re-
mains unsolved as yet, though, by an immense labor pur-
sued throughout this last half century, some highly inter-
esting discoveries in that direction have been achieved. It
seems more and more probable, but still not at all certain,
that the "Riemann hypothesis" is true. Of course, all these
complements could be brought to Riemann's publication
only by the help of facts which were completely unknown
in his time; and, for one of the properties enunciated by
him, it is hardly conceivable how he can have found it with-
out using some of these general principles, no mention of
which is made in his paper.

Galois (1811-1831). Most striking is the personality
of Evariste Galois whose tragic life, abruptly ended in his

again illustrates the difference between two aspects of intuition which
Poincare' believed to be identical. In general, Riemann's intuition, as
Poiocar^ observes, is highly geometrical; but this is not the case for his
memoir on prime numbers, the one in which that intuition is the most
powerful and mysterious: in that memoir, there is no important role of
geometrical elements.



PARADOXICAL CASES OF INTUITION 119

early youth, brought to science one of the most capital
monuments we know of. Galois' passionate nature was cap-
tivated by mathematical science from the moment he be-
came acquainted with Legendre's geometry. However, he
was violently dominated by another overpowering feeling,
enthusiastic devotion to republican and liberal ideas, for
which he fought in a passionate and sometimes very im-
prudent way. Nevertheless, the death he met with at the
age of twenty did not occur in that struggle, but in an
absurd duel.

Galois spent the night before that duel in revising his
notes on his discoveries. First, the manuscript which had
been rejected by the Academy of Sciences as being unintel-
ligible (one may not wonder that such highly intuitive
minds are very obscure) ; then, in a letter directed to a
friend, scanty and hurried mention of other beautiful
views, with the same words hastily and repeatedly inscribed
in the margin "I have no time." Indeed, few hours remained
to him before going where death awaited him.

All those profound ideas were at first forgotten and it
was only after fifteen years that, with admiration, scientists
became aware of the memoir which the Academy had re-
jected. It signifies a total transformation of higher alge-
bra, projecting a full light on what had been only glimpsed
thus far by the greatest mathematicians, and, at the same
time, connecting that algebraic problem with others in
quite different branches of science.

But what especially belongs to our subject is one point
in the letter written by Galois to his friend and enunciating
a theorem on the "periods" of a certain kind of integrals.
Now, this theorem, which is clear for us, could not have
been understood bv scientists living at the time of Galois :



120 PARADOXICAL CASES OF INTUITION

these "periods' 5 had no meaning in the state of science of
that day ; they acquired one only by means of some prin-
ciples in the theory of functions, today classical, but which
were not found before something like a quarter of a cen-
tury after the death of Galois. It must be admitted, there-
fore: (1) that Galois must have conceived these principles
in some way; (2) that they must have been unconscious
in his mind, since he makes no allusion to them, though
they by themselves represent a significant discovery.

The case of Galois deserves some attention in connection
with our former distinction. In some ways he reminds us of
Hermite. He is, like him, a thoroughly analytical mathe-
matician, though he came to his first and enthusiastic
vision of science by the geometiy of Legendre. one of his
early essays while a schoolboy was of a geometrical nature,
but it was the only one. A curious thing is that Galois'
teacher in mathematics in the high school, Mr. Richard,
who had the merit of discovering at once his extraordinary
abilities, was also, fifteen years later, the teacher of Her-
mite ; this, however, cannot be regarded otherwise than as
a mere coincidence, since the genius of such men is evidently
a gift of nature, independent of any teaching.

On the other hand, Galois, who was evidently highly in-
tuitive according to our definition (A) , does not appear as
such in terms of the definition (B). In the proof of the
general theorem which affords a definitive solution to the
main problem of algebra, there is no trace of "scattered
ideas," no combination of apparently heterogeneous prin-
ciples : his thought is, so to speak, an intensive and not an
extensive one; and I should be inclined to say as much of
the discoveries contained in his posthumous letter (the
letter written during the night just before his fatal duel),



PARADOXICAL CASES OF INTUITION 121

though the stream of thought cannot be so surely charac-
terized on such a sequel of simply and briefly enunciated
results. This does not exclude the occasional possibility of
a connection between the aspects (A) and (B) of intui-
tion; but in Galois' case, they appear to be independent
of each other.

From the second point of view, it is clear that Galois
profoundly differs from Hermite, whose discovery con-
cerning quadratic forms is a typical example of "thinking
aside."

A Case in the Work of Poincare. It seems to have been
unnoticed that something similar occurs in Poincare's
Methodes Nouvelles de la Mecanique Celeste. In his Vol-
ume HI (see p. 261), he has to deal with the calculus of
variations and he uses a sufficient condition for a minimum,
equivalent to the one which results from Weierstrass's
method (see above, p. Ill) . But he does not give a proof
of that condition : he speaks of it as a known fact. Now,
as we have said, Weierstrass's method was not published
at the time when that volume of the Methodes NouveUes
was written. Moreover, he does not make any mention of
Weierstrass's discovery, which he should have necessarily
done if he had received any private communication of it.
Above al! 5 it must be added that the condition is formulated
in a form slightly different (though basically equivalent)
from the one which is classically known as resulting from
Weierstrass's method. Must we think that Weierstrass's
argument or an analogous one was found by Poincare
and remained unconscious in his mind? 8

The case appears a still stranger one If we notice that at the same
page (p. 261) of bis third volume, just a few lines before, PoincarS writes:
"Cette recherche se rattache a la difficile question de la variation seconde."



122 PARADOXICAL CASES OF INTUITION

Historical Comparisons. In such cases, we must admit
that some parts of the mental process develop so deeply
in the unconscious that some parts of it, even important
ones, remain hidden from our conscious self. We come very
near the phenomena of dual personality such as were ob-
served by psychologists of the nineteenth century.

Even intermediaries seem to have existed between the
two kinds of phenomena. I think of Socrates' ideas being
suggested to him by a familiar demon or also of the nymph
Egeria whom Numa Pompilius used to consult frequently.

An analogous example can possibly be spoken of in the
mathematical field. It is Cardan, who is not only the in-
ventor of a well-known joint which is an essential part of
automobiles, but who has also fundamentally transformed
mathematical science by the invention of imaginaries. Let
us recall what an imaginary quantity is. The rules of alge-
bra show that the square of any number, whether positive
or negative, is a positive number : therefore, to speak of the
square root of a negative number is mere absurdity. Now,
Cardan deliberately commits that absurdity and begins to
calculate on such "imaginary" quantities.

One would describe this as pure madness.; and yet the
whole development of algebra and analysis would have been

Now, in Weierstrass's theory that is, in our present view of the calculus
of variations there is no question of the second variation, which is fully
left out of consideration.

Therefore, there is a curious contradiction in that page of the Zftthodes
Novoellt*. The allusion to the "variation seconde" is that of a man having
no idea of the new theory. on the contrary, Poincar proves to have been
foDy aware of it when he enunciated his condition (A) (his form of Weier-
strass's condition): nobody thought of anything of that kind in the older
ideas on the subject; one only knew of the classic (but less adequate)
"Legendre condition."

Should oae think of the case of Poincar6 as of a kind of dual per-
sonality?



PARADOXICAL CASES OF INTUITION 123

impossible without that fundament which, of course, was,
in the nineteenth century, established on solid and rigorous
bases. It has been written that the shortest and best way
between two truths of the real domain often passes through
the imaginary one.

We have mentioned Cardan's case with Socrates* and
Numa Pompilius 9 , because he too is reported by some of
his biographers to have received suggestions from a mys-
terious voice at certain periods of his life. However, testi-
monies on that point do not agree at least in details.



IX. THE GENERAL DIRECTION
OF RESEARCH



BEFORE trying to discover anything or to solve a deter-
minate problem, there arises the question: what shall we
try to discover? What problem shall we try to solve?

Two Conceptions of Invention. Claparede, in his intro-
ductory lecture before the above-mentioned meeting at the
Centre de Synthese, observes that there are two kinds of
invention : one consists, a goal being given, in finding the
means to reach it, so that the mind goes from the goal to the
means, from th'e question to the solution ; the other consists,
on the contrary, in discovering a fact, then imagining
what it could be useful for, so that, this time, mind goes
from the means to the goal ; the answer appears to us before
the question.

Now, paradoxical as it seems, that second kind of inven-
tion is the more general one and becomes more and more
so as science advances. Practical application is found by
not looking for it, and one can say that the whole progress
of civilization rests on that principle. When the Greeks,
some four centuries B.C., considered the ellipse i.e., the
curve generated by the points M in a plane such that the
sum MF -{~ -k^' f their distances from two given points
F, F' be a constant and found many remarkable proper-
ties of it, they did not think and could not think of any
possible use for such discoveries. However, without these
studies, Kepler could not have discovered, two thousand
years later, the laws of motion of planets, and Newton
could not have discovered universal attraction.



GENERAL DIRECTION OF RESEARCH 125

Even results which are more strictly practical obey the
same rule. Balloons, in earlier days, were filled with hydro-
gen or lighting gas, which constituted a serious danger of
fire. At the present time, we are able to fill balloons with
incombustible gas. This progress has been possible for two
reasons: in the first place, because one has succeeded in
knowing which substances exist in the atmosphere of the
sun and which do not; secondly, because research was
started, by Lord Rayleigh and Ramsay among others, in
order to determine the density of nitrogen exactly to the
1/10,000, instead of the precision of 1/1,000 which was
known previously.

Both are subjects which were investigated and eluci-
dated without foreseeing any possible applications.

We must add, however, that, conversely, application is
useful and eventually essential to theory by the very fact
that it opens new questions for the latter. one could say
that application's constant relation to theory is the same
as that of the leaf to the tree: one supports the other, but
the former feeds the latter. Not to mention several im-
portant physical examples, the first mathematical founda-
tion in Greek science, geometry, was suggested by practical
necessity, as can be seen by its very name, which means
**land-measuring."

But this example is exceptional in the sense that prac-
tical questions are most often solved by means of existing
theories: practical applications of purely scientific dis-
coveries, important as they may be, are generally remote
in time (though, in recent years, this delay may be consid-
erably shortened, as happened in the case of radio teleg-
raphy, which occurred a few years after the discovery of
Hertzian waves) . It seldom happens that important mathe-



126 GENERAL DIRECTION OF RESEARCH

matical researches are directly undertaken in view of a
given practical use : they are inspired b j the desire which
is the common motive of every scientific work, the desire
to know and to understand. Therefore, between the two
kinds of invention we have just distinguished from each
other, mathematicians are accustomed only to the second
one.

The Choice of Subjects. But setting aside practical ap-
plications, which generally, if they exist, lie far away in
time, mathematical discoveries can be more or less rich in
theoretical consequences. Even these are most often un-
known to us, as fully unknown as incombustible balloons
were to the men who, for the first time, discovered the
chemical composition of the atmosphere of the sun.

Then, how are we to select subjects of research? This
delicate choice is one of the most important things in re-
search; according to it we form, generally in a reliable
manner, our judgment of the value of a scientist.

Upon it we base even our judgment of research students.
Students have often consulted me for subjects of research ;
when asked for such guidance, I have given it willingly,
but I must confess that provisionally, of course I have
Been inclined to classify the man as second rate. In a dif-
ferent field, such was the opinion of our great Indianist
Sylvain Levi, who told me that, on being asked such a
question, he was tempted to reply : Now, my young friend,
you have attended our courses for, say, three or four years
and you have never perceived that there is something want-
ing further investigation?

But how is that important and difficult choice to be
directed? The answer is hardly doubtful: it is the same
which Poincare gave us concerning the means of discovery,



GENERAL DIRECTION OF RESEARCH 127

the same for the "drive" as for the "mechanism." The guide
we must confide in is that sense of scientific beauty, that
special esthetic sensibility, the importance of which he has
pointed out.

As Renan also curiously notices, 1 there is a scientific
taste just as there is a literary or artistic one; and that
taste, according to individuals, may be more or less sure.

Concerning the fruitfulness of the future result, about
which, strictly speaking, we most often do not know any-
thing in advance, that sense of beaufcy can inform us and
I cannot see anything else allowing us to foresee. At least,
contesting that would seem to me to be a mere question of
words. Without knowing anything further, we feel that
such a direction of investigation is worth following; we
feel that the question in itself deserves interest, that its
solution will be of some value for science, whether it permits
further applications or not. Everybody is free to call or
not to call that a feeling of beauty. This is undoubtedly
the way the Greek geometers thought when they investi-
gated the ellipse, because there is no other conceivable way.

As to applications, though completely unforeseen, they
do most often arise later on, if our original feeling has
been a right one. I shall report one or two personal in-
stances, apologizing for that repeated intervention of my
own example on which, of course, I am especially informed.

When I presented my doctor's thesis for examination,
Hermite observed that it would be most useful to find ap-
plications. At that time, I had none available. Now, be-
tween the time my manuscript was handed in and the day
when the thesis was sustained, I became aware of an impor-
tant question (the one we have spoken of at p. 118 in con-

i L'Avenir de la Science, p. 115.



128 GENERAL DIRECTION OF RESEARCH

nection with Biemann) which had been proposed by the
French Academy of Sciences as a prize subject; and pre-
cisely, the results in my thesis gave the solution of that
question. I had been uniquely led by my feeling of the in-
terest of the problem and it led me in the right way.

A few years later, having, in a further study of the same
kind of questions, obtained a very simple result 2 which
seemed to me an elegant one, I communicated it to my
friend, the physicist Duhem. He asked to what it applied.
When I answered that so far I had not thought of that,
Duhem, who was a remarkable artist as well as a promi-
nent physicist, compared me to a painter who would begin
by painting a landscape without leaving his studio and
only then start on a walk to find in nature some landscape
suiting his picture. This argument seemed to be correct,
but, as a matter of fact, I was right in not worrying about
applications : they did come afterwards.

Some years before (1893), I had been attracted by a
question in algebra (on determinants). When solving it, I
had no suspicion of any definite use it might have, only
feeling that it deserved interest; then in 1900 appeared
Fredholm's theory, 3 for which the result obtained in 1893
happens to be essential.

Most surprising I should say bewildering facts of
that kind are connected with the extraordinary march of
contemporary physics. In 1913, Elie Cartan, one of the
first among French mathematicians, thought of a remark-
able class of analytic and geometric transformations in

2 For technicians, the "composition theorem."

* This is the theory which, as said in Section IV, I failed to discover.
It has been a consolation for nay self-esteem to have brought a necessary
link to Fredholm's arguments.



GENEEAL DIRECTION OF RESEARCH 129

relation to the theory of groups. No reason was seen, at
that time, for special consideration of those transforma-
tions except just their esthetic character. Then, some fif-
teen years later, experiments revealed to physicists some
extraordinary phenomena concerning electrons, which they
could only understand by the help of Cartan's ideas of
1913.

But hardly any more typical instance in that line can
be set forth than modern functional calculus. When Jean
Bernoulli, in the eighteenth century, asked for the curve
along which a small heavy body would go down from a
point A to a point B in the shortest possible time, he was
necessarily tempted by the beauty of that problem, so dif-
ferent from what had been attacked hitherto though evi-
dently offering an analogy with those already treated by
infinitesimal calculus. That beauty alone could tempt him.
The consequences which "calculus of variations" i.e., the
theory of problems of that kind would carry for the im-
provement of mechanics, at the end of the eighteenth cen-
tury and the beginning of the nineteenth, could not be
suspected in his time.

Much more surprising is the fate of the extension given
to that initial conception in the last part of the nineteenth
century, chiefly under the powerful impulse of Volterra.
Why was the great Italian geometer led to operate on func-
tions as infinitesimal calculus had operated on numbers,
that is to consider a function as a continuously variable
element? only because he realized that this was a harmoni-
ous way of completing the architecture of the mathemati-
cal building, just as the architect sees that the building will
be better poised by the addition of a new wing. one could
already imagine that, as explained in Section III, such a



ISO GENERAL DIRECTION OF EESEARCH

harmonious creation could be of help for solving problems
concerning functions considered in the previous fashion;
but that "functional," as we called the new conception,
could be in direct relation with reality could not be thought
of otherwise than as mere absurdity. Functionals seemed
to be an essentially and completely abstract creation of
mathematicians.

Now, precisely the absurd has happened. Hardly intelli-
gible and conceivable as it seems, in the ideas of contempo-
rary physicists (in the recent theory of "wave mechan-
ics") , the new notion, the treatment of which is accessible
only to students already familiar with very advanced cal-
culus, is absolutely necessary for the mathematical repre-
sentation of any physical phenomenon. Any observable
element, such as a pressure, a speed, etc., which one used to
define by a number, can no longer be considered as such,
but is mathematically represented by a functional !

These examples are a sufficient answer to Wallas's doubt
on the value of the sense of beauty as a "drive" for discov-
ery. on the contrary, in our mathematical field, it seems to
be almost the only useful one.

We again see how direction in thought implies affective
elements, such being especially the case as concerns that
continuity of attention, that faithfulness of the mind to its
object, the importance of which we have already pointed
out in Section IV.*

In the present stage, as in inspiration, choice is directed
by the sense of beauty; but, this time we refer to it con-

* In a question of inversive geometry (see Section IV), I had under-
estimated the beauty of the question and failed to devote to it a sufficient
continuity of attention.



GENERAL DIRECTION OF RESEARCH 182

sciously, while it works in the unconscious to give us in-
spiration.

Direction of Inventive Work and Desire of Originality.
May other reasons influence the direction of research?

As Dr. de Saussure rightly observes, the intervention
of emotional causes is often possible (he gives me typical
examples in the life of Freud, the creator of psycho-
analysis) . However, this chances to be less the case as con-
cerns mathematics, on account of the abstract character of
that science where, according to Bertrand Russell's cele-
brated word : **We never know what we are talking about,
nor whether what we are saying is true."

Dr. de Saussure has also raised the question whether
creative workers could not be moved by a less laudable kind
of passion, deriving from human vanity : the desire of doing
something unlike others.

It seems to me that something of this kind is possible in
art or literature. More exactly, any question of vanity
being set aside, not being similar to others is a requisite
which the artist (or similarly, the literary man) must con-
sider in itself. Of course, this observation does not apply
to the really great ones: for instance, we have seen by
Mozart's letter (p. 17) that he did not have to think of
being original. But did not such a necessity have its part
in the founding of some schools of painters ; or in the works
where some literary men try to interpret in a paradoxical
way the actions or psychology of known personalities?
One may ask that question.

We might see some connection between this and a cer-
tain number of known cases where poets or other artists
have produced works in abnormal states (for instance,



182 GENERAL DIRECTION OF RESEARCH

Coleridge in a state of laudanum-sleep). Wallas, 5 who re-
ports such examples, considers that a slight degree of "dis-
sociation of mind" may be useful for the artist {t who wishes
to break with his own habits of thought and vision and
those of his school." Also, it is not unusual to hear of poeti-
cal works composed in dreams, while we have seen that this
is very rare, if not doubtful, in mathematical production.

The case of the scientist who, as has been said in the be-
ginning, is a servant and not a master, is indeed a different
one. Any result, the solution of any problem he knows of,
makes new problems arise before him. As a matter of fact,
I can hardly think of more than two or three memoirs
which I would describe as bizarre rather than as truly
original.

Nevertheless, the scientist may be and often is discour-
aged from studying such and such a problem not by the
knowledge that it has been solved, but by the fear that It
has been solved without his knowing it, a fact which would
render his work useless ; or and this is more disinterested
on his part it is natural for him to be attracted by a ques-
tion not devoid of importance in itself on account of its
having been overlooked until then. Such has often been my
case; I even add that, after having started a certain set
of questions and seeing that several other authors had be-
gun to follow that same line, I happened to drop it and in-
vestigate something else. I have been told by physicists that
some of the prominent men in contemporary physics often
act in the same way.



Art of Tkovgkt, pp. 206-210. That very great masters do not
Deed to strive for originality is interpreted by Wallas in saying that, for
tbem, "at the moment of production, a harmony is attained between an
intense activity of the whole nervous system, higher and lower alike, and
tfae conscious wiH. w



FINAL REMARKS 133

Also, we see clearly how mistaken Souriau was, certainly
because he did not direct inquiries among scholars, when he
spoke of them as being desirous of some great discovery "in
order to attract public attention" or, "to get an agreeable
and independent appointment." We can admit that motives
of that kind occasionally influence the life of some of us
when tempted to slacken our work as did the classic word :
**Thou sleepest, Brutus." It is possible that Ampere was
doing more than answering Julie Ampere's urgent anxieties
when he wrote her that publishing one of his discoveries
would be a good means of securing a professorship in a
lycee. But it was not that which made him discover; nor
could I conceive of a scientific man who would be led to
discover chiefly in that way. Scholars with minds of that
sort could get only poor results ; whether it be in the choice
of questions or in their treatment, a man without some love
of science could not be successful, because he would be un-
able to choose. 6

FINAL REMARKS

I have tried to report and interpret observations, per-
sonal or gathered from other scholars engaged in the work
of invention. There remain many other important aspects

Points of view substantially analogous to ours in this section are the
object of G. H. Hardy's recent and suggestive little book A Mathema-
tician's Apology. Although he does not get to a complete definition of the
beauty or, as he calls it, "seriousness" of a mathematical question or
result, on which, of course, esthetic feeling must intervene, he gives a
very delicate and acute analysis of the conditions suitable for the ap-
proach to such a definition.

He also discusses the motives which may influence the desire of re-
search. He sees three chiefly important ones, the first of them being, of
course, the desire to know the truth. For the reason given in the text, I
should insist more than he does on the predominant character and even
necessity of that first one.



IS* FINAL REMAKES

of the subject, especially "objective" ones, which we have
already had occasion to mention. Such are the possible re-
lations between inventive thought and bodily phenomena.
Ideas more or less analogous to those of Gall would deserve
to be pursued. But how could this be done ? It would require
somebody more qualified than I am better acquainted
with the physiology of the brain. However, here we meet
with the difficulty which we mentioned in the beginning;
while mathematicians have not sufficient knowledge of
neurology, neurologists cannot be expected to penetrate
deeply (as would be necessary) into mathematical studies.
Will it ever happen that mathematicians will know enough
of that subject of the physiology of the brain and that neu-
rophysiologists know enough of mathematical discovery
for efficient cooperation to be possible?

Similarly, I could not venture to say anything about the
social and historical influences which surely act on inven-
tion as they do on everything else, I do not know much
about the mechanism of that influence ; and the question is
whether anybody does. Such attempts as Taine's in his
Philosophic de FArt, although their principle bears the
mark of genius, are certainly premature and very hypo-
thetical in their conclusions. Indeed, the difficulties of such
an attempt are obvious : not only is there the fact that no
experiment is possible, but even (genius apart) men with
notable inventive powers are too rare to allow an extensive
application of comparative methods, so that Taine's ques-
tion and our own are among the most difficult even between
those of an historical nature. Social influences govern
mathematical development in the same unconscious and
rather mysterious way as they do literary or artistic ones.
There may certainly be something right in Klein's idea of



FINAL REMARKS 135

an intervention of Galton's heredity theories as concerns
intuitive and logical qualities of the mind (and the same
might be said of mathematical aptitude in general and of
the way various minds use concrete representations) ; but
it is quite unlikely that things are as simple as was imagined
by the school of Taine. It is certainly not a fortuitous thing
that, at the time of the Renaissance and especially in Italy,
there were so many extraordinary men of every kind, a
Benvenuto Cellini and a Leonardo da Vinci as well as a
Galilei ; but it is more doubtful that the reasons for such a
marvellous phenomenon are those supposed by Taine, 1

Things may eventually be clearer when, instead of gen-
eral cases, we consider some individual ones. Saying that,
I think of the case of Cardan, who lived at that same time
and who, in fact, was one of the most extraordinary char-
acters of that extraordinary time. It could be naturally
expected that that discovery of imaginaries which seems
nearer to madness than to logic and which, in fact, has il-
luminated the whole mathematical science, would come
from such a man whose adventurous life was not always
commendable from the moral point of view, and who from
childhood suffered from fantastic hallucinations to such an
extent that he was chosen by Lombroso as a typical ex-
ample in the chapter "Genius and Insanity" of his book
on The Man of Genius.

If we do not recur to such special cases, the exceptional

i Is the similitude in the evolution of ideas among Greek philosophers
and among thinkers of the Middle Ages, as concerns words and
wordless thought, more than a fortuitous coincidence, and would it mean
a general law in the evolution of thought? Of course, one should not dare
to make a positive assertion on the basis of only two instances. If proved,
the fact would be a rather significant one. A study on that question m
Arabian philosophy (especially in the Spanish period) or hi Asiatic
philosophies could be of interest



136 FINAL REMARKS

character of the phenomena which we have considered
creates an obstacle to study as soon as one leaves aside the
data supplied by introspection. But, on the other hand, one
may wonder whether such processes cannot help us to eluci-
date those which go on in other psychological realms : for
instance, as we have seen, those examined in Section VI may
have some features in common with the role of images as
considered by Taine or with problems raised by the Gestalt
theory. In conformity with a rule which seems applicable
to every science of observation (that it even applies in
mathematics appears from the fact noticed in Section VIII,
p. 117 footnote), it is the exceptional phenomenon which
is likely to explain the usual one ; and, consequently, what-
ever we can observe that has to do with invention or even, as
in this study, this or that kind of invention, is capable of
throwing light on psychology in general.



APPENDIX I

AN INQUIRY INTO THE WORKING METHODS
OF MATHEMATICIANS

Translated from L'Enteignement Mathematique,
Vol. IV, 1902 and Vol. VI, 1904



1. At what time, as well as you can remember, and under what
circumstances did you begin to be interested in mathematical
sciences? *Have you inherited your liking for mathematical sci-
ences? Were any of your immediate ancestors or members of
your family (brothers, sisters, uncles, cousins, etc.) particularly
good at mathematics? Was their influence or example to any
extent responsible for your propensity for mathematics?

2. Toward what branches of mathematical science did you feel
especially attracted?

3. Are you more interested in mathematical science per se or
in its applications to natural phenomena?

4. Have you a distinct recollection of your manner of working
while you were pursuing your studies, when the goal was rather
to assimilate the results of others than to indulge in personal re-
search? Have you any interesting information to offer on that
point?

5. After having completed the regular course of mathematical
studies (which, for instance, corresponds to the program of the
Licence mathematique or of two Licences 1 or of the Aggregation,*
in what direction did you consider it expedient to continue your
studies? Did you endeavor, in the first place, to obtain a general
and extensive knowledge of several parts of science before writing
or publishing anything of consequence? Did you, on the contrary,
at first try to penetrate rather deeply into a special subject, study-

* Items preceded by an asterisk appeared in Vol. VI.

1 French grade corresponding to B.A. and M*A. degrees.

2 A grade, or rather a competition, required for teaching in high
schools.



138 APPENDIX I

ing almost exclusively what was strictly requisite for that pur-
pose, and only afterwards extending your studies little by little ?
If you have used other methods, can you indicate them briefly?
Which one do you prefer?

6. Among the truths which you have discovered, have you
attempted to determine the genesis of those you consider the most
valuable ?

7. What, in your estimation, is the role played by chance or
inspiration in mathematical discoveries? Is this role always as
great as it appears to be?

8. Have you noticed that, occasionally, discoveries or solutions
on a subject entirely foreign to the one you are dealing with
occur to you and that these relate to previous unsuccessful re-
search efforts of yours?

*8b. Have you ever worked in your sleep or have you found
in dreams the answers to problems? Or, when you waken in the
morning, do solutions which you had vainly sought the night be-
fore, or even days before, or quite unexpected discoveries, present
themselves ready-made to your mind?

9. Would you say that your principal discoveries have been
the result of deliberate endeavor in a definite direction, or have
they arisen, so to speak, spontaneously in your mind?

10. When you have arrived at a conclusion about something
you are investigating with a view to the publication of your find-
ings, do you immediately write down the part of your work to
which that discovery applies; or do you let your conclusions
accumulate in the form of notes and begin the redaction of the
work only when its contents are important enough?

11. Generally speaking, how much importance do you attach
to reading for mathematical research ? What advice in this respect
would you give to a young mathematician who has had the usual
classical education?

12. Before beginning a piece of research work, do you first
attempt to assimilate what has already been written on that sub-
ject?

IS. Or do you prefer to leave your mind free to work unbiased
and do you only afterwards verify by reading about the subject



APPENDIX I 189

so as to ascertain just what is your personal contribution to the
conclusions reached?

14. When you take np a question, do yon try to make as general
a study as possible of the more or less specific problems which
occur to you? Do yon usually prefer, first to study special cases
or a more inclusive one, and then to generalize progressively?

15. As far as method is concerned, do yon make any distinction
between invention and redacting?

16. Does it seem to you that your habits of work are appreci-
ably the same as they were before you had completed yonr stud-
ies?

17. In your principal research studies, have yon followed the
same line of thought steadily and uninterruptedly to the end,
or have you laid it aside at times and subsequently taken it np
again?

18. What is, in your opinion, the minimum number of hours
during the day, the week, or the year, which a mathematician
who has other demands on his time should devote to mathematics
so as to study profitably certain branches of these same mathe-
matics? Do you believe that one should, if one can, study a little
every day, say for one hour at the very least?

19. Do artistic and literary occupations, especially those of
music and poetry, seem to yon likely to hamper mathematical
invention, or do you think they help it by giving the mind tem-
porary rest?

19a. What are your favorite hobbies, pursuits, or chief inter-
ests, aside from mathematics, or in your leisure time b. Do
metaphysical, ethical, or religious questions attract or repel you?

20. If you are absorbed by professional duties, how do you
fit these in with your personal studies?

21. What counsels, in brief, would you offer to a young man
studying mathematics? *b. to a young mathematician who has
finished the usual course of study and desires to follow a scien-
tific career?

QUESTIONS ABOUT DAILY HABITS

22. Do you believe that it is beneficial to a mathematician to
observe a few special rules of hygiene such as diet, regular meals,
time for rest, etc,?



140 APPENDIX I

23. What do you consider the normal amount of sleep neces-
sary?

24. Would you say that a mathematician's work should be in-
terrupted by other occupations or by physical exercises which
are suited to the individual's age and strength?

25. Or, on the contrary, do you think one should devote the
whole day to one's work and not allow anything to interfere with
it; and, when it is finished, take several days of complete rest?
*b. Do you experience definite periods of inspiration and enthusi-
asm succeeded by periods of depression and incapacity for work ?

c. Have you noticed whether these intervals alternate regularly
and, if so, how many days, approximately, does the period of
activity last and also the period of inertia? d. Do physical or
meteorological conditions (i.e., temperature, light, darkness, the
season of the year, etc.) exert an appreciable influence on your
ability to work?

26. What physical exercises do you do, or have you done as
relaxation from mental work? Which do you prefer?

27. Would you rather work in the morning or in the evening?

28. If you take a vacation, do you spend it in studying mathe-
matics (if so, to what extent?) or do you devote the entire time
to rest and relaxation ?

Final remarks. *Of course, there may be many other details
which it would be useful to learn by an inquiry: 29a. Does one
work better standing, seated or lying down; b, at the blackboard
or on paper ; c. to what extent is one disturbed by outside noises ;

d. can one pursue a problem while walking or in a train; e, how
do stimulants or sedatives (tobacco, coffee, alcohol, etc.) affect
the quality and quantity of one's work?

*3Q. It would be very helpful for the purpose of psychological
investigation to know what internal or mental images, what kind
of "internal word" mathematicians make use of; whether they
are motor, auditory, visual, or mixed, depending on the subject
which they are studying.

If any persons who have been well acquainted with defunct
mathematicians are able to furnish answers to any of the preced-
ing questions, we ask them instantly to be kind enough to do so.



APPENDIX I 141

In this way they will make an important contribution to the his-
tory and development of mathematical science.

Added by the writer. The final question SO corresponds to onr
discussion of Section VI, and it would be especially important
to get further answers on it. Such answers ought to be of two
different kinds, corresponding respectively to ordinary thought
and to research effort.

Moreover, question 30 should be usefully supplemented by

3 la. Especially in research thought, do the mental pictures
or internal words present themselves in the full consciousness or
in the fringe-consciousness (such as denned in Wallas's Art of
Thought, pp. 51, 95 or under the name "antechamber of conscious-
ness" in Galton's Inquiries into Human Faculty, p. 203 of the
edition of 1883 ; p. 146 of the edition of 1910) ?

3lb. The same question is asked concerning the arguments
which these mental pictures or words may symbolize. 5

3 only a few mathematicians, until now, have answered questions Sla,
and 31b, especially as concerns topological arguments such as the proof
of Jordan's theorem (see Section VII, p. 103). For all of them without
any exception, it is the geometrical aspect of the argument which directly
appears in the full consciousness. one or two of them immediately feel
the possibility of arithmetizlng any link of it and are even able to find
that arithmetization (so that it must be present in their fringe-con-
sciousness) ; for others, it would require more or less effort.



APPENDIX II

A TESTIMONIAL FROM PROFESSOR EINSTEIN



Concerning the subject of the above study and, especially,
matters treated in Section VI, the writer has received several
answers to the questions which he has asked. All of them were
valuable to him, but one is more important than any other, not only
because of the personality of its author, but also as dealing with
the question in a quite circumstantial and thorough manner. We
owe it to the great scientist Albert Einstein, and it reads as
follows: 1

MY DEAR COLLEAGUE:

In the following, I am trying to answer in brief your questions
as well as I am able. I am not satisfied myself with those answers
and I am willing to answer more questions if you believe this
could be of any advantage for the very interesting and difficult
work you have undertaken.

(A) The words or the language, as they are written or spoken,
do not seem to play any role in my mechanism of thought. The
psychical entities which seem to serve as elements in thought are
certain signs and more or less clear images which can be "volun-
tarily" reproduced and combined.

There is, of course, a certain connection between those elements
and relevant logical concepts. It is also clear that the desire to
arrive finally at logically connected concepts is the emotional basis
of tM rather vague play with the above mentioned elements. But
taken from a psychological viewpoint, this combinatory play
seems to be the essential feature in productive thought before
there is any connection with logical construction in words or other
kinds of signs which can be communicated to others.

i Questions (A), (B), (C) correspond to number 80 of the ques-
tionnaire issued by L'Enseign&m&nt Mathdrrtatique (see Appendix I).

I hare asked question (D) on the psychological type, not in research
bet in usual tKonght.

Question (E) corresponds to onr number 31.



APPENDIX II 148

(B) The above mentioned elements are, in my case, of visual
and some of muscular type. Conventional words or other signs
have to be sought for laboriously only in a secondary stage, when
the mentioned associative play is sufficiently established and can
be reproduced at will.

(C) According to what has been said, the play with the men-
tioned elements is aimed to be analogous to certain logical con-
nections one is searching for.

(D) Visual and motor. In a stage when words intervene at
all, they are, in my case, purely auditive, but they interfere only
in a secondary stage as already mentioned*

(E) It seems to me that what you call full consciousness is a
limit case which can never be fully accomplished. This seems to
me connected with the fact called the narrowness of consciousness
(Enge des Bewusstseins).

Remark: Professor Max Wertheimer has tried to investigate
the distinction between mere associating or combining of repro-
ducible elements and between understanding (organisches Be-
greifen) ; I cannot judge how far his psychological analysis
catches the essential point. 2

With kind regards . . ,

ALBERT EINSTEIN

2 As can be seen, phenomena in Professor Einstein's mind arc substan-
tially analogous to those mentioned in Section VI, with, as natural,
special features in several details. A more important and remarkable
difference concerns question (E), i.e^ the role of fringe- or full con-
sciousness. Professor Einstein refers to the "narrowness of conscious-
ness": a subject which we should have spoken of in our Section II if
we had not been afraid of being carried too far afield, and a treatment
of which will be found in William James's Ptyckology, Chap. XIII,
pp. 217 ff.

It would be interesting to compare Max Wertheimer's ideas (con-
nected with the Gestaltist school) not only with our Section VI, but with
the first part of Section VII.



APPENDIX III

THE INVENTION OF INFINITESIMAL CALCULUS



In addition to the case of Pascal (p. 53), other similar occur-
rences have happened in the history of science. one of the most
striking instances in modern times is the invention of Infinitesimal
Calculus.

Heraelitos's profound idea that everything ought to be con-
sidered in its "devenir," i.e. in its continuous transformation, had
not been understood during Antiquity. only in the fourteenth
century A.D. did one of the greatest medieval thinkers, Nicole
Oresme, notice that the rate of increase or decrease of a quantity
is slowest in the neighborhood of a maximum or mini'mrim. The
bearing of such a remark was not perceived by anybody, including
Oresme himself, to whom it did not appear that such a funda-
mental idea had to be developed.

Three centuries later the same principle was enunciated by
Johannes Kepler, but Kepler went no further than Oresme had
already gone before him ; the discovery went only halfway.

Then, in Fennat's hands, the principle received mathematical
ｅxpression. In several instances, considering one quantity in terms
of another (such as time), Fermat used a mathematical operation
which gave zero as the result if the quantity in question was a
maximum or minimum. Moreover, the same method allowed him to
find tangents to several curves considered by his contemporaries.

The operation performed by Fermat is precisely what we now
call differentiation. Does this mean, as many are inclined to think,
that he invented the Differential Calculus? In one sense we must
answer "yes/' for we see him applying his method to various prob-
lems, and even pointing out that the method could be applied to
similar ones. 1 But in another sense we must say "no," for the
method he used appeared to nobody in his time, not even to him-

1 1 am indebted for this information to the kindness of Professor
Sergescn.



APPENDIX III 145

self, as a general rtde for solving a whole class of problems, or as
a new conception the properties of which deserved further investi-
gation. Adapting an ｅxpression of Poincare, we can say that
things are more or less discovered, not discovered outright from
complete obscurity to complete revelation. one step consists in ac-
quiring the idea of a principle ; another, if not several others, in
giving a precise form to that idea and driving it far enough to be
able to make it a starting point for further researches. In the
present case, this was the work of Newton and Leibniz.

But the Differential Calculus is not the whole Infinitesimal
Calculus. There is a second branch, the Integral Calculus, the
fundamental operation of which is the valuation of plane curvi-
linear areas ; and this implies a discovery which lay deep and had
been entirely unsuspected, viz. the fact that integration is nothing
else than the converse of differentiation. Who made that essential
and difficult discovery? In the development of differentiation
something of it was in the making. Torricelli and Fennat (per-
haps even, though doubtful, Descartes) used methods which
looked like the application of the principle and were indeed rather
near it, but with a fundamental difference. I should say the same
even of Barrow, Newton's master, in his Lectiones Geometrical,
although the essential content of the n 11 of his zth lecture is
really equivalent to that principle. 2 Besides Descartes, who did
not let the connection appear clearly, both Torricelli and Fermat
treated special cases whose properties concealed the general prin-
ciple; while for Barrow the true meaning of the principle was
hidden by the meaning he gave to the notion of the tangent.

Thus Oresme, Kepler, and Fermat failed to discover the Dif-
ferential Calculus because they did not pursue their initial and
fruitful ideas.

We see how psychological considerations can be illustrated by
the history of science; and conversely, how they can help us to
understand correctly a question which, very often, can be con-
sidered as a badly set one: Who is the author of such or such a
discovery ?

2 It is not certain whether Barrow was not influenced by his immortal
disciple, whose genius he recognized.



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