HENRI POINCARÉ
The scientist does not study nature because it is useful;
he studies it because he delights in it, and he delights in
it because it is beautiful. If nature were not beautiful,
it would not be worth knowing, and if nature were not worth
knowing, life would not be worth living. Of course I do
not here speak of that beauty that strikes the senses,
the beauty of qualities and of appearances; not that I
undervalue such beauty, far from it, but it has nothing to
do with science; I mean that profounder beauty which comes
from the harmonious order of the parts, and which a pure
intelligence can grasp. This it is which gives body, a
structure so to speak, to the iridescent appearances which
flatter our senses, and without this support the beauty
of these fugitive dreams would be only imperfect, because
it would be vague and always fleeting. On the contrary,
intellectual beauty is sufficient unto itself, and it is for
its sake, more perhaps than for the future good of humanity,
that the scientist devotes himself to long and difficult
labors.
It is, therefore, the quest of this special beauty, the
sense of the harmony of the cosmos, which makes us choose
the facts most fitting to contribute to this harmony, just
as an artist chooses from among the features of his model
those which perfect the picture and give it character
and life. And we need not fear that this instinctive and
unavowed prepossession will turn the scientist aside
from the search for the true. One may dream a harmonious
world, but how far the real world will leave it behind!
The greatest artists that ever lived, the Greeks, made
their heavens; how shabby it is beside the true heavens,
ours!--Poincaré's "The Value of Science," p. 8.
Such language as this is extremely disconcerting to those who hold the
popular notion of science and scientists; regarding science as a vague
impending mass of solid fact, immutable, inexorable, threatening the
extinction of all such things as art, sentiment, poetry, and religion,
only to be diverted by a determination to remain ignorant of it;
regarding men of science as mere calculating machines, mechanically
grinding out logical grist for utilitarian purposes. Mathematical
astronomy is surely one of the sciences, the most rigid, remote,
and recondite of the sciences. Yet here is the leading mathematical
astronomer of the age talking about it as though it were one of the
fine arts, a thing of beauty that the artist creates for his own
delight in the making of it and shapes in accordance with his own ideas
of what is harmonious.
Now we cannot throw out of consideration M. Poincaré's opinion, on
the ground that he did not know what he was talking about. A man
who has made as much science as he has ought to know how science is
made, and what for. To most of us nature--or to avoid hurting our own
feelings let us rather say, opportunity--has denied the privilege of
knowing this by experience. Consequently M. Poincaré is an especially
interesting man to study, for he has been willing to tell us not only
what a man of science is, but also how it feels to be one. No other
contemporary of equal eminence has been so frank and accommodating in
the self-revelation of his methods or so willing to submit himself
as a subject of observation. We are admitted to the laboratory of a
mathematician, and we can watch the mechanism of scientific thought in
action.
So far as he is concerned, he has repudiated the idea that science is
purely utilitarian in the most emphatic language. August Comte said
that it would be idle to seek to know the composition of the sun, since
this knowledge would be of no use to sociology. Against such a charge
of uselessness Poincaré eloquently defended his science by showing the
practical value of astronomy even from Comte's point of view, but in
conclusion asserted his own opinion very plainly:
Was I wrong in saying that it is astronomy which has made us
a soul capable of comprehending nature; that under heavens
always overcast and starless, the earth itself would have
been for us eternally unintelligible; that we should there
have seen only caprice and disorder; and that, not knowing
the world, we should never have been able to subdue it? What
science could have been more useful? And in thus speaking
I put myself at the point of view of those who only value
practical applications. Certainly, this point of view is not
mine; as for me, on the contrary, if I admire the conquests
of industry, it is, above all, because they free us from
material cares, they will one day give to all the leisure
to contemplate nature. I do not say: Science is useful,
because it teaches us to construct machines. I say: Machines
are useful, because in working for us, they will some day
leave us more time to make science. But finally it is worth
remarking that between the two points of view there is no
antagonism, and that man having pursued a disinterested aim,
all else has been added unto him.--"Value of Science", p. 88.
It is this insistence upon the æsthetic value of science that caused
him to shrink from being called a "pragmatist", although those who
accept that name have always laid unusual stress upon the æsthetic
factor in thinking. But in his theory of knowledge Poincaré is
decidedly pragmatic, and no one has given a clear exposition or
stronger expression to the practical mode of thought by which the
natural sciences have made their progress and which is now being
extended to the fields of metaphysics, religion, ethics, and sociology.
Poincaré's favorite word is "convenient" (_commode_). Theories are
strictly speaking not to be classed as true or false. They are merely
more or less convenient. For example:
Masses are coefficients it is convenient to introduce
into calculations. We could reconstruct all mechanics
by attributing different values to all the masses. This
new mechanics would not be in contradiction either with
experience or with the general principles of dynamics.
Only the equations of this new mechanics would be _less
simple_.--"Science and Hypothesis", p. 76.
We have not a direct intuition of simultaneity, nor of
the equality of two durations. If we think we have this
intuition, this is an illusion. We replace it by the aid of
certain rules which we apply almost always without taking
count of them. But what is the nature of these rules? No
general rule, no rigorous rule; a multitude of little
rules applicable to each particular case. These rules
are not imposed upon us, and we might amuse ourselves by
inventing others; but they could not be cast aside without
greatly complicating the laws of physics, mathematics, and
astronomy. We therefore choose these rules, not because
they are true, but because they are most convenient, and we
may recapitulate them as follows: "The simultaneity of two
events or the order of their succession, the equality of two
durations, are to be so defined that the enunciation of the
natural laws may be as simple as possible; in other words,
all these rules, all these definitions, are only the fruit
of an unconscious opportunism."--"Value of Science", p. 35.
Time should be so defined that the equations of mechanics
may be as simple as possible. In other words, there is not
one way of measuring time more true than another. That
which is generally adopted is only more _convenient_. Of
two watches, we have no right to say that one goes true,
the other wrong: we can only say that it is advantageous
to conform to the indications of the first.--"Value of
Science", p. 30.
Behold then the rule we follow and the only one we can
follow: when a phenomenon appears to us as the cause of
another, we regard it as anterior. It is therefore by cause
we define time.--"Value of Science", p. 32.
Experience does not prove to us that space has three
dimensions. It only proves to us that it is convenient to
attribute three dimensions to it.--"Value of Science", p.
69.
It has often been observed that if all the bodies in the
universe were dilated simultaneously and in the same
proportion we should have no means of perceiving it, since
all our measuring instruments would grow at the same time
as the objects themselves which they serve to measure. The
world, after this dilatation, would continue on its course
without anything apprising us of so considerable an event.
--"Value of Science", p. 39.
But Poincaré goes farther and shows not only that two such worlds of
different sizes would be absolutely indistinguishable, but that they
would be equally indistinguishable if they were distorted in any manner
so long as they corresponded with each other point by point. This
conception of the relativity of space may be thought a little hard to
grasp, but M. Poincaré is kind enough to suggest a way by which any one
may see it for himself if he has ten cents to admit him to one of those
hilarious resorts where life-size concave and convex mirrors are to be
seen.[1] You may think yourself a gentleman of proper figure, that is
to say, somewhat portly, and you look upon the tall slim shape that
confronts you in the cylindrical mirror as absurdly misshapen. But you
would find it difficult to convince him of his deformity. His legs, as
well as yours, fulfill the requirement that Lincoln laid down as their
proper length; that is, they reach from the body to the ground. If you
touch your chin with your thumb and your brow with your forefinger, so
does he. It occurs to you that here is a case where your knowledge of
geometry would, if ever, prove useful, but when you appeal to it, you
will find that the geometry of his queer-looking world is just as good
as yours; in fact, is just the same. You get a foot rule and measure
yourself; 70 inches high, 14 inches in diameter at the equator, ratio
5:2. But meanwhile the mirror man is also measuring himself, and his
dimensions come out exactly the same as yours, 70 and 14 and 5:2, for
when he holds the rule perpendicular it lengthens and when horizontal
it shrinks. Lines that in your world are straight are curved in his,
but you cannot prove it to him, for when he lays his straightedge
against these curves of his, behold it immediately bends to correspond.
By this time, finding it so difficult to prove to the mirror man that
you are right and he is wrong, it occurs to you that perhaps he isn't,
that he may have just as much reason as you for believing that his is
the normal, well-proportioned world, and yours the distorted image of
it. Since, then, you have no way of perceiving the absolute length,
direction, or curvature of a line, your space may be as irregularly
curved and twisted as it looks to be in the funniest of the mirrors,
and you would not know it. Now the principle of the pragmatist is that
anything that does not make any difference to anything else is not
real. The reason why we have not been able to discover any differences
between the mirror space and our space, each considered by itself, is
because there is none. Or to return to the language of Poincaré, "space
is in reality amorphous and the things that are in it alone give it a
form." Why do we say that space has three dimensions instead of two or
four or more? Why do we stick to an old fogy like Euclid when Riemann
and Lobachevski proffer us new and equally self-consistent systems of
geometry wherein parallels may meet or part? Because:
by natural selection our mind has _adapted_ itself to
the conditions of the external world. It has adopted the
geometry _most advantageous_ to the species or, in other
words, _the most convenient_. Geometry is not true, it is
advantageous.
Such language may pass without notice in university halls, for all
scientists are more or less clearly conscious of the provisional and
practical nature of the hypotheses and conventions they employ. But
to the outside world it sounds startling. To some it seemed that the
foundations of the universe were being undermined. Others saw in it a
confession of what Brunetière had called "the bankruptcy of science"
and openly rejoiced over the discomfiture of the enemy of the Church.
Now Poincaré had chanced to use in discussing the relativity of motion
the following illustration:
Absolute space, that is to say, the mark to which it would
be necessary to refer the earth to know whether it really
moves, has no objective existence. Hence this affirmation
"the earth turns round" has no meaning, since it can be
verified by no experiment; since such an experiment not only
could not be either realized or dreamed by the boldest Jules
Verne but cannot be conceived of without contradiction. Or
rather these two propositions: "The earth turns round" and
"it is more convenient to suppose the earth turns round"
have the same meaning; there is nothing more in the one than
in the other.--"Science and Hypothesis", p. 85.
This remark was at once seized upon by the Catholic apologists, and
the Galileo case, once closed by the voice of Rome, was reopened
for the admission of this new evidence. If the Ptolemaic and the
Copernican theories are equally true, and the choice between them is
merely a matter of expediency, was not the Holy Inquisition justified
in upholding the established theory in the interests of religion
and morality? Monsignor Bolo, an eminent and sagacious theologian,
announced in _Le Matin_ of February 20, 1908, that M. Poincaré, the
greatest mathematician of the century, says that Galileo was wrong
in his obstinacy. To this Poincaré replied in the whispered words of
Galileo:
"E pur si muove, Monseigneur"?
In a later discussion of the point, he explains that what he said about
the rotation of the earth could be equally well applied to any other
accepted hypothesis, even the very existence of an external world, for
"these two propositions, 'the external world exists' or 'it is more
convenient to suppose that it exists' have one and the same meaning."
The Copernican theory is the preferable because it has a richer, more
profound content, since if we assume the earth is stationary we have to
invent other explanations for the flattening at the poles, the rotation
of Foucault's pendulum, the trade winds, etc., while the hypothesis of
a revolving earth brings all these together as the effects of a single
cause.
M. Le Roy, a Catholic pragmatist and a disciple of Bergson's, goes
much further than Poincaré in regard to the human element in science,
holding that science is merely a rule of action and can teach us
nothing of truth, for its laws are only artificial conventions. This
view Poincaré considered to be dangerously near to absolute nominalism
and skepticism, and in his controversy with Le Roy[2] he showed that
the scientist does not "create facts as Le Roy said, but merely the
language in which he enunciates them." Of the contingence upon which Le
Roy and Boutroux insist, Poincaré would admit only that scientific laws
can never be more than approximate and probable. Even in astronomy,
where the single and simple law of gravitation is involved, neither
absolute certainty nor absolute accuracy can be attained. Therefore we
cannot safely say that at a particular time Saturn will be at a certain
point in the heavens. We must limit ourselves to the prediction that
"Saturn will _probably_ be _near_" such a point.
[Illustration: Poincaré]
In an address before the International Philosophical Congress at
Bologna in April, 1910, Professor Poincaré discussed again the question
of whether the laws of nature may not change. He admitted that there
is not a sole law that we can enunciate with the certainty that it
has always been true in the past. Nevertheless, he concluded, there
is nothing to hinder the man of science from keeping his faith in
the principle of immutability, since no law can descend to the level
of a secondary and limited law without being replaced by another law
more general and more comprehensive. He considered in particular the
possibility that in the remote past the fundamental laws of mechanics
would not hold, for since the energy of the world has been continually
dissipating in the form of heat there must have been a time when bodies
moved faster than they do now. But according to the recent theories
of matter, no body can travel faster than light, and with velocities
approaching that of light its mass is no longer constant but increases
with its velocity. This, of course, would play havoc with all of
Newton's laws, which then we should have to regard as limited in their
scope to such ordinary conditions and moderate motion as we see about
us now.
But even at present we can hardly regard them with the same implicit
confidence as formerly. Take, for example, Newton's law that action and
reaction are equal and opposite. When a ball is fired from a cannon,
the cannon recoils at the same time and with the same momentum that the
ball goes forward. But suppose instead of a cannon we have a lamp with
a reflector sending a beam of light into space. It has been deduced
mathematically and proved experimentally that light exerts a minute
but measurable pressure on an object which it strikes. The reflector
therefore recoils like the cannon, but where is the ball if light is an
immaterial wave motion? To be sure, if the ray of light strikes some
planet out in space, it would give it an impulse equal and opposite to
that originally imparted to the reflector on our earth. But what if the
light goes on through vacant space and never hits anything at all? A
law that may have to wait several thousand years for its validation and
may even fail of it altogether is not what the layman has in mind when
he thinks of immutable and infrangible laws governing the universe.
But it is rather important just now that the layman gets to understand
what the scientist means when he talks of laws, theories, and
hypotheses. For we are in the midst of a stupendous revolution in
science. Our nicely arranged nineteenth century cosmos seems to be
dissolving into chaos again. We have seen the elements melt with
fervent heat and we can no longer rely upon the uniformity of atomic
weights. The laws of the conservation of matter and energy, which
were the guiding stars of research to the last generation, are
becoming dimmed. The old-fashioned ether, in its time a useful but
never entirely satisfactory contrivance, for it had to be patched up
repeatedly with divers new properties to enable it to bear the various
duties thrust upon it, seems no longer competent to stand the strain
and may have to be sent to the scientific scrap-heap at any moment. We
hear physicists of supposed sanity assert that all bodies contract in
the direction of their motion and that their weight varies with their
speed and the direction in which they are going. We read of "atoms of
light," and of corpuscles of electricity which, though they are but
a thousandth part of the hydrogen atom, are caught and counted and
weighed one by one.
Now what puzzles the lay mind is the calmness with which the scientists
survey this crash of worlds and shock of systems. They do not have the
mien of exposed impostors. They are not, like the augurs of decadent
Rome, unable to meet without laughing in each other's faces. They do
not resent the overthrow of their former idols. They have no fear
of heretics, consequently no hatred for them. They regard all this
iconoclasm with a mild curiosity quite in contrast to their intense and
personal interest in science generally. It is hard to get out a quorum
at the Association for the Advancement of Science to hear a discussion
of the principle of relativity with all its revolutionary consequences.
Compare this apparent indifference to the fate of fundamental
principles in scientific circles with what would happen in a
Presbyterian assembly if it should be proposed to eliminate
predestination from the Westminster Confession or in an Episcopal
convocation if the Virgin Birth were denied; with what would happen
in a stockholders' meeting if doubt were expressed as to the rights
of capital, or in a socialist convention if the class conflict were
questioned. Now the existence of the ether has the same importance to
scientific thought that predestination has to theological or capitalism
to economic thought. Its refutation or modification would be quite as
upsetting to faith and practice. Yet scientists are men; they have red
blood in their veins, and it not infrequently shows in their cheeks
when they debate something that seems to them worth while. Pure theory
rarely seems to them worth while because it is recognized as pure
conventionality and convenience.
The scientific man, especially the scientific investigator, holds his
theories with a light hand, but keeps a firm grip on his facts. This is
just the opposite of the lay attitude toward science. If the layman is
interested in knowing the speed of light, it is because he thinks that
he learns from it that all space is filled with a rigid elastic solid,
at which he cannot but wonder. The scientist is interested in the ether
because it helps him in his calculation of the speed of light.
A lecturer on wireless telegraphy will use in the course of the hour
two or three more or less contradictory conceptions of electricity.
If afterward you call his attention to the inconsistency and ask him
which is right and which is wrong, you will not get a very satisfactory
answer. He does not know and obviously does not care. You insist upon
his telling you which theory he personally believes in. He really
had not thought of "believing" in any of them. If he uses white
chalk on the blackboard in preference to red, it is not because he
denies the existence of red chalk and its occasional usefulness. So,
too, the astronomer will speak of the sun's rising and in the next
breath of the earth's turning toward the sun, quite innocent of his
inconsistency. The botanist alludes to a certain flower as a poppy and
again as Eschscholtzia. He means the same thing but is using different
languages; in the first case English, in the second case I don't know
what.
It is eminently desirable that people should have faith in science, but
in order to do that they must have the same sort of faith in it that
the scientist has. Otherwise they will regard it as a lot of ingenious
fancies which are proved false by each succeeding generation. Science
is moulting just now and looks queer. The public ought to understand
clearly that the process means growth and not disease. There is another
reason now for the popularization of the scientific mode of thought. It
is beginning to be applied where entirely different conceptions have so
far prevailed--to art, ethics, religion, sociology, and the like. This
is already arousing a great commotion and will cause more before the
process is complete. It will, for example, involve the rewriting and to
a large extent the reinvestigation of history. Poincaré has hinted at
this in a passage which seems to me of very great significance:
Carlyle has somewhere said something like this: "Nothing but
facts are of importance. John Lackland passed by here. Here
is something that is admirable. Here is a reality for which
I would give all the theories in the world." Carlyle was a
fellow countryman of Bacon, but Bacon would not have said
that. That is the language of the historian. The physicist
would say rather: "John Lackland passed by here. That
makes no difference to me for he never will pass this way
again."--"Science and Hypothesis", p. 102.
The aim of science is prevision, and I believe that this will
eventually be recognized as the true aim of all knowledge. The
historian, or let me say rather the antiquarian, for the historian may
have the scientific temperament, values facts for their rarity. The
scientist values facts for their commonness. A unique fact, if there
be such, would have no possible interest to him. The antiquarian goes
about looking for things, facts, or furniture, which have been of
importance in the past. The scientist is looking only for things that
will be of importance in the future.
According to Poincaré, the proper choice of facts is the first duty of
the scientist. He must be able to pick out the significant and reject
all the rest. "Invention consists in avoiding the constructing of
useless combinations and in constructing the useful combinations which
are in infinite minority. To invent is to discern, to choose." It is
most desirable to bring together elements far distant from one another.
Such unions are mostly sterile, but when this is not the case, they are
the most fruitful of all. The successful scientist does not, like a
shopper, look over one by one all available samples and pick out what
he wants. Life is too short. The unsuitable ideas do not even present
themselves to his mind. It is as if he were an examiner of second
resort who only concerns himself with the candidates who have passed
the first test. This preliminary sifting and sorting process is done
largely by the unconscious mind, as Poincaré shows by telling how he
came to make his first mathematical discoveries:
For a fortnight I labored to demonstrate that there could
exist no function analogous to those that I have since
called the fuchsian functions.[3] I was then very ignorant.
Every day I seated myself at my work table and spent an
hour or two there, trying a great many combinations, but I
arrived at no result. One night when, contrary to my custom,
I had taken black coffee and I could not sleep, ideas
surged up in crowds. I felt them as they struck against one
another until two of them stuck together, so to speak, to
form a stable combination. By morning I had established the
existence of a class of fuchsian functions, those which are
derived from the hyper-geometric series. I had merely to put
the results in shape, which only took a few hours.--"Science
et Méthode", p. 52.
After working out the deductions from this discovery, he went on a
geological excursion of the School of Mines. The distractions of travel
took his mind from his mathematical labor. But at Constance, just as
he was stepping into an omnibus for some excursion, the idea occurred
to him, without any connection with his previous thoughts, that his
fuchsian functions were identical in their transformations with
those of the non-Euclidian geometry. He took his seat in the omnibus
and continued his conversation, feeling absolutely certain of his
discovery, which he worked out at his leisure on his return to his home
at Caen.
He next devoted himself to the study of arithmetical questions,
without reaching any results of importance and without suspecting that
this subject could have the slightest connection with his earlier
researches. Disgusted at his lack of success, he went to pass some days
at the seashore, where he was occupied with other things. One day as he
was walking on the cliff, the thought came to him, brief, sudden, and
certain as usual, that he had been employing the same transformations
in his arithmetical and geometrical work.
He thereupon went back to Caen and undertook the systematic application
of his theory. But he was stopped by an insurmountable obstacle, and
while in this perplexity he was called away to his military service
at Mont-Valérien, where he had no time for mathematics. One day while
walking on the street, the solution of the difficulty appeared to him
in a flash. He did not try to think it out at the time, but after his
release from the army, he completed his memoir without trouble.
These fascinating glimpses into the soul of a mathematician will remind
the reader of many other instances of such subconscious assistance
on record and doubtless of personal experiences as well. We think of
Alfred Russel Wallace at Ternate, his brain inflamed with tropical
fever, seized with the sudden inspiration of the theory of natural
selection, the key to the biological problems which had perplexed him
for so many months. How fortunate that his clerical opponents did not
know of this and so could not dismiss evolution as the dream of a
diseased imagination. But as James says in his "Varieties of Religious
Experience", we have no right to discountenance unwelcome theories as
feverish fancies, since for all we know 102° may be a more favorable
temperature for truth to germinate and sprout in than the ordinary
bloodheat of 98°.
We are reminded, too, of Kekulé of Bonn puzzling over the constitution
of benzene, trying in vain to satisfy six carbon atoms with six
hydrogen atoms when they wanted fourteen. In the evening as he sat
by the fire, his wearied brain refused to rest, and he seemed to see
the four-handed carbon imps dancing with their one-armed hydrogen
partners on the floor. Suddenly six of them joined hands in a ring
and the problem was solved. Since then the benzene sextet has been
dancing through hundreds of volumes and has added millions annually
to the wealth of Germany. Professor Hilprecht of the University of
Pennsylvania has told how a Chaldean priest, custodian of the "Temple
Library", appeared to him in a dream and showed him how to put together
the fragments of a cuneiform inscription which he had for a long time
been striving in vain to translate.
Then there was Stevenson in Samoa, writing for dear life, but not
failing to give credit to his "brownies" for doing a large part of his
work for him. But the brownies do not work unbidden, and they will not
make bricks without straw. Poincaré insists upon the necessity of the
preliminary period of conscious effort without which these subliminal
inspirations never come and the subsequent period of verification,
development, and application, without which they are fruitless. Such
ideas came to him most often in the evening or morning when he was in
bed and half awake. He did not regard the operations of his unconscious
mind as merely mechanical. On the contrary, it is distinguished by the
power of choice, selecting and presenting to the conscious ego only
those combinations that seem profitable and important. This choice
is made, in Poincaré's opinion, under the guidance of the artistic
instinct.
The usual combinations are precisely the most beautiful; I
mean those which can best charm that special sensibility
which all mathematicians recognize but at which the profane
are tempted to smile. Among the numerous combinations
which the subliminal self has blindly formed, almost all
are without interest and without utility. For that reason
they have no action upon the æsthetic sensibility and never
come into consciousness. Only those that are harmonious and
consequently both useful and beautiful are capable of moving
that special sensibility of the geometrician of which I
spoke, and which, once excited, calls our attention to them
and so gives them the chance to become conscious.--"Science
et Méthode", p. 58.
Poincaré, if we may believe what he says on this point, was a poor
chess player and absolutely incapable of adding up a column of figures
correctly. But the reader should beware of the common fallacy of
reversing a proposition of this kind and assuming that if he, too,
makes mistakes in addition he has the mind of a great mathematician.
Poincaré's memory was, however, exceptionally good, especially for
figures and formulas. On returning from a walk he was able to recall
the numbers of the carriages he had met. When he was in the Polytechnic
School he followed the courses in mathematics without taking a note and
without looking at the syllabus provided by the professor. He was a
rapid mental calculator, using auditive imagery rather than visual. He
associated colors with the sound of words.[4]
In this connection may be quoted an anecdote told by M. Jules
Sageret:[5] At a _conférence_ in the Superior School of Telegraphy the
director called upon him to discuss a very difficult problem in the
propagation of the electric current. Poincaré complied and solved the
problem without taking any time in preparation. After the _conférence_
the director felicitated him on the solution. "Yes," said Poincaré, "I
found the value of _x_, but is it in kilograms or kilometers?"
Poincaré did not find it profitable to work more than two hours at a
time. His custom was to stay at his desk from ten o'clock to noon and
from five to seven in the afternoon, never working in the evening after
dinner. He drank wine at meals, but never smoked. He went to bed at ten
and rose at seven, but did not sleep soundly.
He was a blond, five feet five inches in height and weighed 154 pounds.
His head was unusually large, especially in breadth. His eyes were
myopic and unsteady. He stood stoopingly with his wrinkled forehead
upturned. He spoke somewhat slowly and with a distraught air, as though
he were thinking of something else, even though he might be at the time
interested and keenly observant. He talked English and German readily
and read Latin and Italian. He was fond of music, especially Wagner.
Of the absent-mindedness that had been characteristic of him from
youth, many stories are told. Like most mathematicians he was fond
of walking while thinking, his fingers opening and closing in an
unconscious gesture. One day on his return from a walk he was surprised
to find that he was carrying a wicker cage, new and happily empty. He
could not imagine how he had got it, but retracing his steps he found
upon the sidewalk the stock of the basket maker whom he had innocently
despoiled.
When as an engineering student he made a trip to Austria, his mother
was afraid he would drop his portfolio sometime without noticing it.
So, realizing doubtless that his memory was auditory, she sewed little
bells on it. The plan was successful. His mother found on his return
that he had brought back in his valise not only the portfolio but
also an Austrian bed sheet neatly folded, which, some morning, he had
mistaken for his night clothes.
These and similar anecdotes were told by M. Frédéric. Masson when he
welcomed M. Poincaré into the Académie Française, January 28, 1909,[6]
and it must have been a trifle embarrassing to the new member to listen
to such a minute analysis of his life and character addressed to him
in the second person. How deftly the director of the Academy mingled
eulogy and raillery may be seen from a quotation:
"You did not delay revealing your vocation and will be justly cited as
the most precocious of infant prodigies. You were nine months old when
for the first time as night came your eyes were directed toward the
sky. You saw there a star light up. You persistently pointed it out to
your mother, who was also your nurse. Then you discovered another with
some astonishment, and your reason cried '_Enco lo la bas_!' A third,
a fourth, more cries of joy and equal enthusiasm. You had to be put
to bed because you became so excited discovering stars. That evening
was your first contact with the infinite, and you had inaugurated your
courses in astronomy, the youngest professor known."
Henri Poincaré was born April 29, 1854, at Nancy, where his ancestors
had long been established. His grandfather was a pharmacist and his
father a physician of more than usual scholarship. The name, he said,
was originally Pontcaré, for, one can imagine a square bridge but
not a square point. But the philologists who took the question up
discovered in the register of the university a student named "Petrus
Pugniquadrati" in 1403 and "Jehan Poing-quarré" in 1418, so the name
Poincaré meant "clenched-fist." His cousin, Raymond Poincaré, son of a
distinguished engineer, has long been one of the most prominent figures
in the political world, a member of the Academy, senator, minister, and
is now president of the French republic.
In the Nancy _lycée_ he led all his classes and showed a special
aptitude for history and literature. At the age of thirteen he
composed a five-act tragedy in verse, and since he was a Lorrainer,
the heroine was of course Jeanne d'Arc. But as soon as he caught sight
of a geometry, his true vocation became apparent. His instructor ran
to his home and announced to his mother: "Madame, your son will be a
mathematician."
Passing through the Polytechnic School he entered the National School
of Mines, and for a few years after graduation he served as engineer
in the Government departments of mines and railroads. At the age of
twenty-seven he was called to a chair of mathematics in the University
of Paris, where he remained, also filling the positions of Professor
of Astronomy in the Polytechnic School and Professor of Theoretical
Electricity in the Professional School of Posts and Telegraphs. He was
received into the Academy of Sciences at the early age of thirty-two,
and at the time of his election to the French Academy he had been
honored by election to membership by thirty-five foreign academies.
He took his seat in the Académie française very appropriately as
the successor of Sully-Prudhomme, who likewise was an engineer by
profession and a philosopher by temperament. To Poincaré as well as to
Sully-Prudhomme, science appealed to the æsthetic sense as a thing of
beauty and an inspiration to the imagination.
He married at the age of twenty-seven and had four children, three
daughters and a son. His younger sister is the wife of the philosopher,
Émile Boutroux, well known in this country from the lectures he gave at
Harvard and Princeton.
Poincaré was influential in introducing improved methods in teaching
mathematics, promoting the use of natural and dynamical methods
instead of the abstract and static methods of Euclid and Legendre.
He was skeptical in regard to religion and indifferent to politics.
When called upon to contribute to a symposium on the old question
of the scholar in politics,[7] he responded that savants like all
citizens ought to interest themselves in the affairs of the country.
But politics has become a profession, and a savant who entered into it
would have to devote half his time to public business if he would be
useful and the other half to his constituents if he wished to keep his
seat, so he would have no time for science.
When asked for his opinion on woman suffrage,[8] he replied as follows:
I see no theoretical reason for refusing the political
suffrage to women, married or not. They pay taxes the same
as men, and they contribute their sons, so it is even
heavier upon them than upon men. Perhaps woman suffrage is
the sole means of combating alcoholism. I fear only the
clerical influence over women.
Of the achievements that have given M. Poincaré his world-wide fame
I am not competent to speak. Readers who would know the significance
and value of his work on fuchsian, hyper-fuchsian, theta-fuchsian,
abelian, and elliptical functions must go further for the information.
I can only quote the opinions of those most competent to express an
opinion as to his contributions to science. In 1905 he received the
Bolyai Prize of ten thousand crowns, which is awarded by the Hungarian
Academy of Sciences every five years for the best work in mathematics
done during that period. The official report by Gustave Rados begins as
follows:
"Henri Poincaré is incontestably the first and most powerful
investigator of the present time in the domain of mathematics and
mathematical physics. His strongly marked individuality permits us to
recognize in him a savant endowed with intuition, who knows how to draw
from the exhaustless well of geometrical and mechanical intuitions the
elements and the origins of his profound and penetrating researches,
yet using besides the most admirable logical power in working out
his conceptions. In addition to his brilliant inventive genius we
must recognize in him an ability for the finest and most fruitful
generalizations of mathematical relations, which has often enabled him
to push back, far beyond the point where others have hitherto been
stopped, the limits of our knowledge in different branches of pure
and applied mathematics. This was shown already in his first work on
automorphic functions with which he began the series of his brilliant
publications which must be classed with the greatest mathematical
discoveries of all time."
In this country Poincaré has become known largely through the efforts
of Professor George Bruce Halsted of the State Normal School of
Greeley, Colorado, who has translated his philosophical works and has
for many years been indefatigable in spreading the new gospel of the
non-Euclidian geometry. Professor Halsted has at my request kindly
contributed the following account of one of Poincaré's astronomical
triumphs and of the visit that Professor Sylvester of Johns Hopkins
paid to Poincaré many years ago:
"The kernel of Poincaré's power lies in an oracle Sylvester often
quoted from Hesiod: Only the genius knows how much more the part is
than the whole. He penetrates at once the divine simplicity of the
perfectly general case, and thence descends, as from Olympus, to the
special concrete earthly particulars. Thus his memoir of 1885, which
Sir George Darwin says came to him as a revelation, on a rotating fluid
mass, and his book 'Les Méthodes nouvelles de la Mécanique céleste,'
1892-1899, were ready with prevision when the shocking special case
occurred of Phoebe, ninth satellite of Saturn, discovered in 1900,
afterward found, incredible as it seemed, to be revolving in the
direction contrary to that of all the others. It follows that Saturn
himself originally rotated in the reverse direction. Again, on February
29, 1908, was found an eighth satellite of Jupiter, Jviii, revolving
round Jove in the shocking Phoebe retrograde direction. Zeus must
have turned over. All the planets have turned over, and some are now
making another somersault. Moreover, Jviii does not even revolve in a
closed orbit; its path is an open twister of unreturning turns.
"For Poincaré the inexhaustible source, the lamp of Aladdin, has ever
been the non-Euclidian geometry. In him the Bolyai-Lobachevski-Riemann
germ flowers fair.
"Personally Poincaré is the most lovable of men. At our very first
meeting I realized that I had already been intimately associated with
him for two years in the person of Sylvester. I told him the story of
Sylvester's discovery of him, and he showed me how vividly and tenderly
reconnaissant he was toward the great old master.
"Midsummer, and up a stuffy Paris stairway labors a giant gnome, beard
on enormous chest, fortunately no neck, for no neck could upbear such
a monstrous head, bald but for the inverted halo of hair collaring its
juncture with the broad shoulders; small inefficient hands holding big
hat and damp handkerchief; breath puffing with the heat and exertion.
It is Sylvester, self-driven to seek out the source of new creations
strangely akin to his own. At the sought door, open, he pauses, seized
by doubt, the person within is so young, so slight, so dazed. Can this
be the new incarnation of the eternal world-genius of geometry? But
the aloof sensitiveness of the face, the broad sphericity of the head
reassure him. This is Henri Poincaré. And so the old King finds the
True Prince, who in turn finds himself at last truly comprehended,
anointed to the succession, and given high heart to establish his
dominion."
The sudden death of Henri Poincaré, July 18, 1912, at the age of
fifty-eight, shocked the scientific world. This marvelous thinking
machine was stopped, this repository of the exact sciences was lost
to the world, by the trifling accident of a clot of blood catching
in one of the valves of the heart. He had gone to the hospital for a
minor operation which was apparently successful. Ten days later he was
pronounced well enough to leave and was dressing when he was struck
down.
The funeral service at the church of Saint-Jacques-du-Haut-Pas was
attended by a remarkable assemblage of men of science and letters,
government officials, and representatives of foreign countries. At
the Montparnasse cemetery orations were delivered by M. Guist'hau,
Minister of Public Instruction, Jules Claretie of the Académie
française, M. Appell, dean of the Faculty of Sciences, M. Bigourdan of
the Observatory, Paul Painlevé of the Academy of Sciences, and General
Cornille, Commandant of the École polytechnique. M. Painlevé said of
him:
"The life of Henri Poincaré was one intense and uninterrupted
meditation, that despotic and pitiless meditation which bows the
shoulders and bends the head, which absorbs the vital influx of one's
being and too soon uses up the body it possesses.
"Henri Poincaré was not only a great creator in the positive sciences;
he was a great philosopher and a great writer. Certain of his aphorisms
remind one of Pascal: 'Thought is only a flash between two long nights,
but this flash is everything.' His style followed the movement of his
thought; brief and arresting formulas, often paradoxical when isolated,
joined by hasty explanations which discard the easy details and say
only the essential. This is why superficial critics have accused him
of being 'incoherent'; the truth being that without some previous
scientific education, such logical movement is difficult to follow.
Mice cannot keep step with a lion.
"It is likewise a lack of comprehension of his philosophy as a whole
that has led certain commentators to think they found a transcendental
skepticism in his critical studies of the principles of science. Must
he not have had faith in science who has written 'The search for truth
ought to be the aim of our activity; it is the sole aim worthy of it'?
His philosophy of the rational science will live as long as his own
discoveries. The totality of the mathematical sciences seemed to him
like a gigantic measuring instrument, harmoniously adjusted and well
adapted for the evaluation of the phenomena of the universe. There
remains one trait of his character that I cannot pass over in silence;
that is his admirable intellectual sincerity. He gave himself, he gave
to all, so far as words permit, the whole of his thought and even the
mechanism of his thought. In his last publication, appearing only a few
days before his death and dealing with the problem of the stability of
our universe, he excused himself for giving out such incomplete results:
'It would seem under these conditions that I ought to
abstain from all publication until I had solved the problem,
but after the fruitless efforts I have made for months, it
appeared to me wisest to let the problem ripen while I let
it alone for some years. That would have been very well if
I had been sure of taking it up again some day, _but at my
age I could not be sure of that_. Besides, the importance of
the subject is too great, and the results already obtained
are on the whole too considerable for me to be content to
leave them altogether useless. I hope that the geometricians
who will interest themselves in this problem and who will
doubtless be more fortunate than I will be able to get
something out of it and make use of it in finding the path
they should pursue.'
"What words can be added to this scientific testament, so simple and
so noble, of a life altogether consecrated, without faltering even to
the last hour, to the search for truth? For the first time in half a
century this unparalleled brain has found repose."
Poincaré, as we have seen, was awake to the wider aspects of science.
He was interested in its effects upon human life and conduct, although
he himself was engaged in one of its most remote and abstract branches.
Shortly before his death he discussed a question which nowadays
arouses intense interest, the question of what effect the advance
and popularization of science will have on ethics. Will science in
destroying superstitions, in changing utterly the traditional way of
regarding the universe and man, undermine the morality which forms
the foundation of our civilization? This question Poincaré answers in
the negative. He believes that our moral instincts lie too deep to
be affected by such a revolution in thought, but on the other hand
he does not think, as some do, that science will ever be able of
itself to provide the moral imperative. A few paragraphs from this
essay, published posthumously in "Last Thoughts", may well serve as a
conclusion to this sketch of his philosophy:
There can be no scientific morality; but no more can there
be immoral science. And the reason is simple; it is a
reason--how shall I say it?--purely grammatical.
If the premises of a syllogism are both in the indicative,
the conclusion likewise will be in the indicative. For
the conclusion to be put in the imperative, it would be
necessary that at least one of the premises should itself
be in the imperative. Now, the principles of science,
the postulates of geometry, are and can be only in the
indicative; still in this same mood are the experimental
verities, and at the foundation of the sciences there
is, there can be, nothing else. Hence, the most subtle
dialectician may juggle with these principles as he will,
combine them, frame them up one upon another; all he will
get from them will be in the indicative. He will never
obtain a proposition which shall say: do this, or don't
do that; that is to say, a proposition which confirms or
contradicts morality....
Some therefore think that science will be destructive; they
fear the ruin it will make and dread lest, where it shall
have passed, society can no longer survive.
Is there not in these fears a sort of internal
contradiction? If it is scientifically proved that such
or such a custom, regarded as indispensable to the very
existence of human society, had not in reality the
importance attributed to it and deceived us only by its
venerable antiquity, if that be proved, admitting this proof
to be possible, will the moral life of humanity be shaken?
One of two things, either this custom is useful, and then a
reasonable science cannot prove that it is not; or else it
is useless and we should not regret it. From the moment that
we place at the foundation of our syllogisms one of those
generous emotions which engender morality, it is still this
emotion, and consequently it is still morality which we must
find at the end of our whole chain of reasonings, if this
has been conducted in accordance with the rules of logic.
What is in danger of perishing is the non-essential, that
which was merely an accident in our moral life; the sole
important thing cannot fail to be found in the conclusions
since it is in the premises....
Science, right or wrong, is deterministic; everywhere it
penetrates it introduces determinism. So long as it is
only a question of physics or even of biology, this is
unimportant. The domain of conscience remains inviolate.
What will happen when morality in turn shall become the
object of science?
Is all despair, or if some day morality should accommodate
itself to determinism, could it so adapt itself without
dying from the effects? So profound a metaphysical
revolution would doubtless have much less influence upon
morals than we think. It is of course understood that
penal repression is not in question. What is called crime
or punishment, would be called sickness or prophylaxis,
but society would retain intact its right, which is not to
punish, but simply the right of self-defense. What is more
serious is that the idea of merit or demerit would have to
disappear or be transformed. But we should continue to love
the good man, as we love all that is beautiful; we should
no longer have the right to hate the vicious man, who would
then inspire only disgust; but is hate necessary? Enough
that we do not cease to hate vice.
Apart from that, all would go on as in the past. Instinct is
stronger than all metaphysics, and even though one should
have laid it bare, even if one should understand the secret
of its force, its power would not thereby be weakened. Is
gravitation less irresistible since Newton? The moral forces
which guide us would continue to guide us.[9]
* * * * *
HOW TO READ POINCARÉ
A complete analytical bibliography of Poincaré's writings up to
1909 will be found in Ernest Lebon's "Henri Poincaré" (Paris:
Gaultier-Villars), which contains the biographical address of M.
Frédéric Masson on his admission to the French Academy and other
eulogies. The list comprises 436 articles and books classified
as follows: Mathematical analysis, 146; analytical and celestial
mechanics, 85; mathematic physics, 78; scientific philosophy, 51;
necrology, 17; miscellaneous, 59; an astonishing output for thirty
years' work, considering the amount and difficulty of the labor
involved in some of the contributions.
The mathematical works of Poincaré are too difficult for the layman
and indeed for many professional mathematicians. But there are five
volumes of general interest published by Flammarion, Paris: "La Science
et l'Hypothèse", "La Valeur de la Science", "Science et Méthode",
"Savants et Ecrivains", and "Dernières Pensées." The first of these has
had a wide popularity, having been translated into English, German,
Spanish, Hungarian, and Japanese. The English translation of "Science
and Hypothesis", by Professor George Bruce Halsted (New York: Science
Press), which appeared in 1905, is introduced by an interesting
criticism of Poincaré's philosophy by Professor Josiah Royce, of
Harvard. Two years later "The Value of Science" was published in this
country (Science Press). "Science et Méthode", though it contains
some matter of more general interest than the others, particularly
his account of the rôle played by unconscious mind in mathematical
invention and his explanation of the newer conceptions of physics,
has not yet appeared in English. The fourth volume, "Savants et
Ecrivains", is an evidence of Poincaré's good will rather than his
literary talents, as it consists of perfunctory addresses on deceased
Academicians, the most extensive being that on Sully-Prudhomme, whose
chair he holds. The fifth, published after his death, contains the
essay on "Science and Morality" from which I have quoted, as well as
interesting discussions of recent science and philosophy. The volume
entitled "Foundations of Science" (published by the Science Press,
New York) contains "Science and Hypothesis", "Value of Science", and
"Science and Method" with the introduction by Professor Royce.
From either of the two volumes, "Science and Hypothesis" or "The Value
of Science", one can get an idea of Poincaré's philosophy, which is of
importance because it is not merely the philosophy of an individual
but the point of view of most men of science nowadays, though rarely
so definitely recognized or clearly expressed. Both books consist
of a somewhat heterogeneous collection of studies on the method and
logic of the mathematical and physical sciences, containing much that
the general reader will have to skip because of its use of unfamiliar
terms, but it will not be safe for him to skip any whole pages without
looking them over carefully, for he is likely to find brilliant and
suggestive sentences embedded in the most unpromising material.
Separate articles by Poincaré, forming chapters from the
above-mentioned volumes, are accessible in American periodicals. "The
Future of Mathematics" in _Monist_, Vol. XX, pp. 76-92; also in the
1909 Smithsonian Report, which is in every public library. "The Choice
of Facts" in _Monist_, Vol. XIX, pp. 231-239. "The Principles of
Mathematical Physics" in the report of the St. Louis Congress of Arts
and Sciences, Vol. I, pp. 604-624, and in _Monist_, Vol. XV, pp. 1-24.
"The Bolyai Prize" (Report on the Work of Hilbert) in _Science_, May
19 and 26, 1911. "Mathematical Creations" in _Monist_, Vol. XX, pp.
321-335. "The Value of Science" was first published complete in the
_Popular Science Monthly_, September, 1906, and later; "Relativity of
Space", "The New Logics", and "Chance" in the _Monist_, 1913.
For biographical details besides the references already given in
footnotes, see Nordmann's article on Poincaré in Smithsonian Report,
1912; Darbou's eulogy in _Le Temps_, December 15, 1913; and articles in
_Revue du Mois_, February 10, 1913; _La Revue de Paris_, February 15,
1913; _The Nation,_ September 12, 1912.
[1] "Science et Méthode," p. 101.
[2] Part III of "The Value of Science."
[3] M. Poincaré, in relating these experiences for their psychological
interest, was kind enough to say that the non-mathematical reader
need not be frightened at these barbarous names, for it is not at all
necessary for him to know what they mean.
[4] Dr. Toulouse has devoted a volume of his series of
medico-psychological studies of men of genius to observations on the
memory, reaction time, mode of thinking, habits, and physiological
constitution of Henri Poincaré (Paris: Flammarion).
[5] _Revue des Idées_, 1909, p. 488.
[6] Masson's address may be found in Le Bon's bibliography; also in
_Popular Science Monthly_. An entertaining account of Poincaré's
reception into the Academy was written for _Le Figaro_ by André
Beaunier and translated for the Boston _Transcript_.
[7] _Revue bleue_, June 4, 1907, p. 708.
[8] _La Revue_, 1910.
[9] Translated by Professor Halsted from "Science and Morals" in
_Dernières Pensées_.