Jules
Henri Poincaré was born on 29 April 1854 in Cité
Ducale neighborhood, Nancy, Meurthe-et-Moselle into an influential family.
His father Leon Poincaré (1828–1892) was a professor of medicine at the University
of Nancy. His cousin, Raymond Poincaré, was the President of France, 1913 to
1920. He was raised in the Roman Catholic faith, however, he rejected later
rejected Christianity and became an atheist. Poincare was a French mathematician,
theoretical physicist, engineer, and a science philosopher. He is described as The
Last Universalist in mathematics as he excelled in all fields of the
discipline existing during his lifetime. He made numerous original fundamental
contributions to pure and applied mathematics, mathematical physics, and celestial
mechanics. He was responsible for formulating the Poincaré conjecture, which
was one of the most famous unsolved problems in mathematics until it was solved
in 2002–2003. Poincaré was the first person to discover a chaotic deterministic
system through his works on three-body problem which laid the foundations of
modern chaos theory. He is also considered to be one of the founders of the
field of topology.
Poincaré elucidated
the importance to the invariance of laws of physics under different
transformations (there are overall 14 such transformations collectively known
as the Poincare Group or the fill Lorentz Group). Checking for such invariance forms
the basis of establishing new physical laws. He was the first to present the Lorentz
transformations in their modern symmetrical form and obtained perfect
invariance of all of Maxwell's equations, an important step in the formulation
of the theory of special relativity.
In the following
article on Mathematical Creation,
Poincare has beautifully elicited the mechanism behind the scientific
creativity and the role of conscious and unconscious self, trying to explain
those sudden moments of eureka, that
we find behind every scientific breakthrough throughout the history of science,
through his own mathematical endeavors.
* * * * *
The genesis of
mathematical creation is a problem which should intensely interest the psychologist.
It is the activity in which the human mind seems to take least from the outside
world, in which it acts or seems to act only of itself and on itself, so that
in studying the procedure of geometric thought we may hope to reach what is
most essential in man’s mind.
This has long
been appreciated, and some time back the journal called L’enseignement mathématique,
edited by Laisant and Fehr, began an investigation of the mental habits and
methods of work of different mathematicians. I had finished the main outlines
of this article when the results of that inquiry were published, so I have
hardly been able to utilize them and shall confine myself to saying that the
majority of witnesses confirm my conclusions; I do not say all, for when the
appeal is to universal suffrage unanimity is not to be hoped.
A first fact
should surprise us, or rather would surprise us if we were not so used to it.
How does it happen there are people who do not understand mathematics? If
mathematics invokes only the rules of logic, such as are accepted by all normal
minds; if its evidence is based on principles common to all men, and that none
could deny without being mad, how does it come about that so many persons are
here refractory?
That not every
one can invent is nowise mysterious. That not every one can retain a demonstration
once learned may also pass. But that not every one can understand mathematical
reasoning when explained appears very surprising when we think of it. And yet
those who can follow this reasoning only with difficulty are in the majority:
that is undeniable, and will surely not be gainsaid by the experience of
secondary-school teachers.
And further: how is error possible in
mathematics? A sane mind should not be guilty of a logical fallacy, and yet
there are very fine minds who do not trip in brief reasoning such as occurs in
the ordinary doings of life, and who are incapable of following or repeating without
error the mathematical demonstrations which are longer, but which after all are
only an accumulation of brief reasonings wholly analogous to those they make so
easily. Need we add that mathematicians themselves are not infallible?
The answer seems to me evident. Imagine
a long series of syllogisms, and that the conclusions of the first serve as
premises of the following: we shall be able to catch each of these syllogisms,
and it is not in passing from premises to conclusion that we are in danger of
deceiving ourselves. But between the moment in which we first meet a
proposition as conclusion of one syllogism, and that in which we reencounter it
as premise of another syllogism occasionally some time will elapse, several
links of the chain will have unrolled; so it may happen that we have forgotten
it, or worse, that we have forgotten its meaning. So it may happen that we
replace it by a slightly different proposition, or that, while retaining the
same enunciation, we attribute to it a slightly different meaning, and thus it is
that we are exposed to error.
 |
| Marie Curie and Henri Poicare at the Solvay conference, 1911 |
Often the mathematician uses a rule.
Naturally he begins by demonstrating this rule; and at the time when this proof
is fresh in his memory he understands perfectly its meaning and its bearing,
and he is in no danger of changing it. But subsequently he trusts his memory
and afterward only applies it in a mechanical way; and then if his memory fails
him, he may apply it all wrong. Thus it is, to take a simple example, that we
sometimes make slips in calculation because we have forgotten our
multiplication table.
According to this, the special aptitude
for mathematics would be due only to a very sure memory or to a prodigious
force of attention. It would be a power like that of the whistplayer who
remembers the cards played; or, to go up a step, like that of the chess-player who
can visualize a great number of combinations and hold them in his memory. Every
good mathematician ought to be a good chess player, and inversely; likewise he
should be a good computer. Of course that sometimes happens; thus Gauss was at
the same time a geometer of genius and a very precocious and accurate computer.
But there are exceptions; or rather I
err; I can not call them exceptions without the exception being more than the
rule. Gauss it is, on the contrary, who was an exception. As for myself, I must
confess, I am absolutely incapable even of adding without mistakes. In the same
way, I should be but a poor chess-player; I would perceive that by a certain
play I should expose myself to a certain danger; I would pass in review several
other plays, rejecting them for other reasons, and then finally I should make
the move first examined, having meantime forgotten the danger I had foreseen.
In a word, my memory is not bad, but it
would be insufficient to make me a good chessplayer. Why then does it not fail
me in a difficult piece of mathematical reasoning where most chess-players
would lose themselves? Evidently because it is guided by the general march of
the reasoning. A mathematical demonstration is not a simple juxtaposition of syllogisms,
it is syllogisms placed in a certain order, and the order in which these
elements are placed is much more important than the elements themselves. If I
have the feeling, the intuition, so to speak, of this order, so as to perceive
at a glance the reasoning as a whole, I need no longer fear lest I forget one
of the elements, for each of them will take its allotted place in the array,
and that without any effort of memory on my part.
It seems to me then, in repeating a
reasoning learned, that I could have invented it. This is often only an
illusion; but even then, even if I am not so gifted as to create it by myself, I
myself re-invent it in so far as I repeat it.
We know that this feeling, this
intuition of mathematical order, that makes us divine hidden harmonies and
relations, can not be possessed by every one. Some will not have either this
delicate feeling so difficult to define, or a strength of memory and attention beyond
the ordinary, and then they will be absolutely incapable of understanding
higher mathematics. Such are the majority. Others will have this feeling only
in a slight degree, but they will be gifted with an uncommon memory and a great
power of attention. They will learn by heart the details one after another;
they can understand mathematics and sometimes make applications, but they
cannot create. Others, finally, will possess in a less or greater degree the
special intuition referred to, and then not only can they understand mathematics
even if their memory is nothing extraordinary, but they may become creators and
try to invent with more or less success according as this intuition is more or less
developed in them.
In fact, what is mathematical creation?
It does not consist in making new combinations with mathematical entities
already known. Any one could do that, but the combinations so made would be
infinite in number and most of them absolutely without interest. To create
consists precisely in not making useless combinations and in making those which
are useful and which are only a small minority. Invention is discernment,
choice.
How to make this choice I have before
explained; the mathematical facts worthy of being studied are those which, by
their analogy with other facts, are capable of leading us to the knowledge of a
mathematical law just as experimental facts lead us to the knowledge of a physical
law. They are those which reveal to us unsuspected kinship between other facts,
long known, but wrongly believed to be strangers to one another.
Among chosen combinations the most
fertile will often be those formed of elements drawn from domains which are far
apart. Not that I mean as sufficing for invention the bringing together of
objects as disparate as possible; most combinations so formed would be entirely
sterile. But certain among them, very rare, are the most fruitful of all.
To invent, I have said, is to choose;
but the word is perhaps not wholly exact. It makes one think of a purchaser
before whom are displayed a large number of samples, and who examines them, one
after the other, to make a choice. Here the samples would be so numerous that a
whole lifetime would not suffice to examine them. This is not the actual state
of things. The sterile combinations do not even present themselves to the mind
of the inventor. Never in the field of his consciousness do combinations appear
that are not really useful, except some that he rejects but which have to some
extent the characteristics of useful combinations. All goes on as if the
inventor were an examiner for the second degree who would only have to question
the candidates who had passed a previous examination.
But what I have hitherto said is what
may be observed or inferred in reading the writings of the geometers, reading
reflectively.
It is time to penetrate deeper and to
see what goes on in the very soul of the mathematician. For this, I believe, I
can do best by recalling memories of my own. But I shall limit myself to
telling how I wrote my first memoir on Fuchsian functions. I beg the reader’s
pardon; I am about to use some technical expressions, but they need not
frighten him, for he is not obliged to understand them. I shall say, for
example, 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.
For fifteen days I strove to prove that
there could not be any functions like those I have since called Fuchsian
functions. I was then very ignorant; every day I seated myself at my work
table, stayed an hour or two, tried a great number of combinations and reached
no results. One evening, 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. But the next morning I had
established the existence of a class of Fuchsian functions, those which come
from the hypergeometric series; I had only to write out the results, which took
but a few hours.
Then I wanted to represent these
functions by a quotient of two series; this idea was perfectly conscious and
deliberate, the analogy with elliptic functions guided me. I asked myself what
properties these series must have if they existed, and I succeeded without difficulty
in forming the series I have called theta-Fuchsian.
Just at this time I left Caen, where I
was then living, to go on a geological excursion under the auspices of the
school of mines. The changes of 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-Euclidean 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 commenced, but I felt a perfect certainty. On my return to Caen, for
conscience’ sake I verified the result at my leisure.
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Then I turned my attention to the study
of some arithmetic questions apparently without much success and without a
suspicion of any connection with my preceding researches. Disgusted with my
failure, I went to spend a few days at the seaside, and thought of something
else. One morning, walking on the bluff, the idea came to me, with just the same
characteristics of brevity, suddenness and immediate certainty, that the
arithmetic transformations of indeterminate ternary quadratic forms were
identical with those of non-Euclidean geometry.
Returned to Caen, I meditated on this
result and deduced the consequences. The example of quadratic forms showed me
that there were Fuchsian groups other than those corresponding to the
hypergeometric series; I saw that I could apply to them the theory of theta-Fuchsian
series and that consequently there existed Fuchsian functions other than those from
the hypergeometric series, the ones I then knew. Naturally I set myself to form
all these functions. I made a systematic attack upon them and carried all the
outworks, one after another. 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. All
this work was perfectly conscious.
Thereupon I left for Mont-Valérien,
where I was to go through my military service; so I was very differently
occupied. One day, going along the street, the solution of the difficulty which
had stopped me suddenly appeared to me. I did not try to go deep into it immediately,
and only after my service did I again take up the question. I had all the elements
and had only to arrange them and put them together. So I wrote out my final memoir
at a single stroke and without difficulty.
I shall limit myself to this single
example; it is useless to multiply them. In regard to my other researches I
would have to say analogous things, and the observations of other mathematicians
given in L’enseignement mathématique would only confirm them.
Most striking at first is this
appearance of sudden illumination, a manifest sign of long, unconscious prior
work. The rôle of this unconscious work in mathematical invention appears to me
incontestable, and traces of it would be found in other cases where it is less evident.
Often when one works at a hard question, nothing good is accomplished at the first
attack. Then one takes a rest, longer or shorter, and sits down anew to the
work. During the first half-hour, as before, nothing is found, and then all of
a sudden the decisive idea presents itself to the mind. It might be said that
the conscious work has been more fruitful because it has been interrupted and
the rest has given back to the mind its force and freshness. But it is more
probable that this rest has been filled out with unconscious work and that the
result of this work has afterwards revealed itself to the geometer just as in
the cases I have cited; only the revelation, instead of coming during a walk or
a journey, has happened during a period of conscious work, but independently of
this work which plays at most a role of excitant, as if it were the goad
stimulating the results already reached during rest, but remaining unconscious,
to assume the conscious form.
There is another remark to be made about
the conditions of this unconscious work: it is possible, and of a certainty it
is only fruitful, if it is on the one hand preceded and on the other hand
followed by a period of conscious work. These sudden inspirations (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 agoing the unconscious machine
and without them it would not have moved and would have produced nothing.
The need for the second period of
conscious work, after the inspiration, is still easier to understand. It is
necessary to put in shape the results of this inspiration, to deduce from them
the immediate consequences, to arrange them, to word the demonstrations, but above
all is verification necessary. I have spoken of the feeling of absolute
certitude accompanying the inspiration; in the cases cited this feeling was no
deceiver, nor is it usually. But do not think this a rule without exception;
often this feeling deceives us without being any the less vivid, and we only
find it out when we seek to put on foot the demonstration. I have especially
noticed this fact in regard to ideas coming to me in the morning or evening in
bed while in a semi-hypnagogic state.
Such are the realities; now for the
thoughts they force upon us. The unconscious, or, as we say, the subliminal
self plays an important rôle in mathematical creation; this follows from what
we have said. But usually the subliminal self is considered as purely
automatic. Now we have seen that mathematical work is not simply mechanical,
that it could not be done by a machine, however perfect. It is not merely a
question of applying rules, of making the most combinations possible according
to certain fixed laws. The combinations so obtained would be exceedingly
numerous, useless and cumbersome. The true work of the inventor consists in
choosing among these combinations so as to eliminate the useless ones or rather
to avoid the trouble of making them, and the rules which must guide this choice
are extremely fine and delicate. It is almost impossible to state them
precisely; they are felt rather than formulated. Under these conditions, how
imagine a sieve capable of applying them mechanically?
A first hypothesis now presents itself;
the subliminal self is in no way inferior to the conscious self; it 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? You recognize the full
importance of this question. Boutroux in a recent lecture has shown how it came
up on a very different occasion, and what consequences would follow an
affirmative answer. (See also, by the same author, Science et Religion, pp.
313 ff.)
Is this affirmative answer forced upon
us by the facts I have just given? I confess that, for my part, I should hate
to accept it. Reexamine the facts then and see if they are not compatible with
another explanation.
It is certain that the combinations
which present themselves to the mind in a sort of sudden illumination, after an
unconscious working somewhat prolonged, are generally useful and fertile
combinations, which seem the result of a first impression. Does it follow that
the subliminal self, having divined by a delicate intuition that these
combinations would be useful, has formed only these, or has it rather formed
many others which were lacking in interest and have remained unconscious?
In this second way of looking at it, all
the combinations would be formed in consequence of the automatism of the
subliminal self, but only the interesting ones would break into the domain of
consciousness. And this is still very mysterious. What is the cause that, among
the thousand products of our unconscious activity, some are called to pass the threshold,
while others remain below? Is it a simple chance which confers this privilege?
Evidently not; among all the stimuli of
our senses, for example, only the most intense fix our attention, unless it has
been drawn to them by other causes. More generally 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 invoked à 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 harmony 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.
Now, what are the mathematic entities to
which we attribute this character of beauty and elegance, and which are capable
of developing in us a sort of esthetic emotion? They are those whose elements
are harmoniously disposed so that the mind without effort can embrace their
totality while realizing the details. This harmony is at once a satisfaction of
our esthetic needs and an aid to the mind, sustaining and guiding. And at the
same time, in putting under our eyes a well-ordered whole, it makes us foresee
a mathematical law. Now, we have said above, the only mathematical facts worthy
of fixing our attention and capable of being useful are those which can teach
us a mathematical law. So that we reach the following conclusion: The useful
combinations are precisely the most beautiful, I mean those best able to charm
this special sensibility that all mathematicians know, but of which the profane
are so ignorant as often to be tempted to smile at it.
What happens then? Among the great
numbers of combinations blindly formed by the subliminal self, almost all are
without interest and without utility; but just for that reason they are also
without effect upon the esthetic sensibility. Consciousness will never know them;
only certain ones are harmonious, and, consequently, at once useful and
beautiful. They will be capable of touching this special sensibility of the
geometer of which I have just spoken, and which, once aroused, will call our
attention to them, and thus give them occasion to become conscious.
This is only a hypothesis, and yet here
is an observation which may confirm it: when a sudden illumination seizes upon
the mind of the mathematician, it usually happens that it does not deceive him,
but it also sometimes happens, as I have said, that it does not stand the test
of verification; well, we almost always notice that this false idea, had it
been true, would have gratified our natural feeling for mathematical elegance.
Thus it is this special esthetic
sensibility which plays the rôle of the delicate sieve of which I spoke, and
that sufficiently explains why the one lacking it will never be a real creator.
Yet all the difficulties have not disappeared. The conscious self is narrowly
limited, and as for the subliminal self we know not its limitations, and this
is why we are not too reluctant in supposing that it has been able in a short
time to make more different combinations than the whole life of a conscious
being could encompass. Yet these limitations exist. Is it likely that it is
able to form all the possible combinations, whose number would frighten the
imagination? Nevertheless that would seem necessary, because if it produces
only a small part of these combinations, and if it makes them at random, there
would be small chance that the good, the one we should choose, would be found
among them.
Perhaps we ought to seek the explanation
in that preliminary period of conscious work which always precedes all fruitful
unconscious labor. Permit me a rough comparison. Figure the future elements of
our combinations as 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 meeting, and consequently without any combination between them.
On the other hand, during a period of
apparent rest and unconscious work, certain of them are detached from the wall
and put in motion. They flash in every direction through the space (I was about
to say the room) where they are enclosed, as would, for example, a swarm of
gnats or, if you prefer a more learned comparison, like the molecules of gas in
the kinematic theory of gases. Then their mutual impacts may produce new
combinations. What is the rôle of the preliminary conscious work? It is
evidently to mobilize certain of these atoms, to unhook them from the wall and
put them in swing. We think we have done no good, because we have moved these
elements a thousand different ways in seeking to assemble them, and have found
no satisfactory aggregate. But, after this shaking up imposed upon them by our
will, these atoms do not return to their primitive rest. They freely continue
their dance.
Now, our will did not choose them at
random; it pursued a perfectly determined aim. The mobilized atoms are
therefore not any atoms whatsoever; they are those from which we might
reasonably expect the desired solution. Then 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. Again I beg pardon, my
comparison is very rough, but I scarcely know how otherwise to make my thought
understood.
However it may be, the only combinations
that have a chance of forming are those where at least one of the elements is
one of those atoms freely chosen by our will. Now, it is evidently among these
that is found what I called the good combination. Perhaps this is a way
of lessening the paradoxical in the original hypothesis.
Another observation. It never happens
that the unconscious work gives us the result of a somewhat long calculation all
made, where we have only to apply fixed rules. We might think the wholly
automatic subliminal self particularly apt for this sort of work, which is in a
way exclusively mechanical. It seems that thinking in the evening upon the
factors of a multiplication we might hope to find the product ready made upon
our awakening, or again that an algebraic calculation, for example a
verification, would be made unconsciously. Nothing of the sort, as observation
proves. All one may hope from these inspirations, fruits of unconscious work,
is a point of departure for such calculations. As for the calculations
themselves, they must be made in the second period of the conscious work, that
which follows the inspiration, that in which one verifies the results of this inspiration
and deduces their consequences. The rules of these calculations are strict and complicated.
They require discipline, attention, will and therefore consciousness. In the subliminal
self, on the contrary, reigns what I should call liberty, if we might give this
name to the simple absence of discipline and to the disorder born of chance.
Only, this disorder itself permits unexpected combinations.
I shall make a last remark; when above I
made certain personal observations, I spoke of a night of excitement when I
worked in spite of myself. Such cases are frequent, and it is not necessary
that the abnormal cerebral activity be caused by a physical excitant as in that
I mentioned. 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
distinguishes the two mechanisms or, if you wish, the working methods of the
two egos. And the psychologic observations I have been able, thus, to make seem
to me to confirm in their general outlines the views I have given.
Surely they have need of it, for they
are and remain in spite of all very hypothetical: the interest of the questions
is so great that I do not repent of having submitted them to the reader.
* * * * *
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