Now for a discovery of a diametrically opposite kind, where intuition
plays the dominant part. The extracts which follow are from a celebrated
lecture by Henri Poincaré at the Societé de Psychologie
in Paris, and concern one of his best-known mathematical discoveries:
the theory of the so-called 'Fuchsian functions'. To reassure the reader
I hasten to quote from Poincaré's own introductory remarks:
I beg your pardon; I am about to use some technical expressions, but
they need not frighten you for you are 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. . . .
And now follows one of the most lucid introspective accounts of the
Eureka act by a great scientist:
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. By 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 the 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-Fuchsia.
Just at this time I left Caen, where I was then living, to go on a
geologic 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.
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 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 hyper-geometric
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 . . .
Most striking at first is this appearance of sudden illumination,
a manifest sign of long, unconscious prior work. The role of
this unconscious work in mathematical invention appears to me
incontestable. . . . [10]
Similar experiences have been reported by other mathematicians. They seem
to be the rule rather than the exception. One of them is Jacques Hadamard:
[11]
. . . 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 awakened by
an external noise, a solution 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.
A few more examples. André Marie Ampère (1775-1836),
after whom the unit of electric current is named, a genius of childlike
simplicity, recorded in his diary the circumstances of his first
mathematical discovery:
On April 27, 1802, he tells us, I gave a shout of joy . . . It was
seven years ago I proposed to myself a problem which I have not been
able to solve directly, but for which I had found by chance a solution,
and knew that it was correct, without being able to prove it. The matter
often returned to my mind and I had sought twenty times unsuccessfully
for this solution. For some days I had carried the idea about with me
continually. At last, I do not know how, I found it, together
with a large number of curious and new considerations concerning the
theory of probability. As I think there are very few mathematicians
in France who could solve this problem in less time, I have no doubt
that its publication in a pamphlet of twenty pages is a good method
for obtaining a chair of mathematics in a college. [12]
The memoir did in fact get him a professorship at the Lycée in Lyon.
It was called
Considerations of the Mathematical Theory of Games of
Chance
, and demonstrated, among other things, that habitual gamblers
are, in the long run, bound to lose.
Another great mathematician, Karl Friedrich Gauss, described in a letter
to a friend how he finally proved a theorem on which he had worked
unsuccessfully for four years:
At last two days ago I succeeded, not by dint of painful effort but so
to speak by the grace of God. As a sudden flash of light, the enigma
was solved. . . . For my part I am unable to name the nature of the
thread which connected what I previously knew with that which made my
success possible. [13]
On another occasion Gauss is reported to have said: 'I have had my
solutions for a long time, but I do not yet know how I am to arrive
at them.' Paraphrasing him, Polya -- a contemporary mathematician --
remarks: 'Wlien you have satisfied yourself that the theorem is true,
you start proving it.' [14]
We have seen quite a few cats being let out of the bag -- the mathematical
mind, which is supposed to have such a dry, logical, rational texture. As
a last example in this chapter I shall quote the dramatic case of
Friedrich August von Kekulé, Professor of Chemistry in Ghent, who,
one afternoon in 1865, fell asleep and dreamt what was probably the most
important dream in history since Joseph's seven fat and seven lean cows:
I turned my chair to the fire and dozed, he relates. Again the atoms
were gambolling before my eyes. This time the smaller groups kept
modestly in the background. My mental eye, rendered more acute by
repeated visions of this kind, could now distinguish larger structures,
of manifold conformation; long rows, sometimes more closely fitted
together; all twining and twisting in snakelike motion. But look! What
was that? One of the snakes had seized hold of its own tail, and the
form whirled mockingly before my eyes. As if by a flash of lightning
I awoke . . . Let us learn to dream, gentlemen. [15]
The serpent biting its own tail gave Kekulé the clue to a discovery
which has been called 'the most brilliant piece of prediction to be found
in the whole range of organic chemistry' and which, in fact, is one of
the cornerstones of modern science. Put in a somewhat simplified manner,
it consisted in the revolutionary proposal that the molecules of certain
important organic compounds are not open structures but closed chains or
'rings' -- like the snake swallowing its tail.
Summary
When life presents us with a problem it will be attacked in accordance
with the code of rules which enabled us to deal with similar problems
in the past. These rules of the game range from manipulating sticks
to operating with ideas, verbal concepts, visual forms, mathematical
entities. When the same task is encountered under relatively unchanging
conditions in a monotonous environment, the responses will become
stereotyped, flexible skills will degenerate into rigid patterns,
and the person will more and more resemble an automaton, governed by
fixed habits, whose actions and ideas move in narrow grooves. He may be
compared to an engine-driver who must drive his train along fixed rails
according to a fixed timetable.
Vice versa, a changing, variable environment will tend to create flexible
bchaviour -- patterns with a high degree of adaptability to circumstances
-- the driver of a motor-car has more degrees of freedom than the
engine-driver. But novelty can be carried to a point -- by life or in the
laboratory -- where the situation still resembles
in some respects
other situations encountered in the past, yet contains new features or
complexities which make it impossible to solve the problem by the same
rules of the game which were applied to those past situations. When
this happens we say that the situation is
blocked
-- though the
subject may realize this fact only after a series of hopeless tales,
or never at all. To squeeze the last drop out of the metaphor: the
motorist is heading for a frontier to which all approaches are barred,
and all his skill as a driver will not help him -- short of turning his
car into a helicopter, that is, playing a different kind of game.
A blocked situation increases the stress of the frustrated drive. What
happens next is much the same in the chimparizee's as in Archimedes's
case. When all hopeful attempts at solving the problem by traditional
methods have been exhausted, thought runs around in circles in the
blocked matrix like rats in a cage. Next, the matrix of organized,
purposeful behaviour itself seems to go to pieces, and random trials
make their appearance, accompanied by tantrums and attacks of despair --
or by the distracted absent-mindedness of the creative obsession. That
absent-mindedness is, of course, in fact single-mindedness; for at this
stage -- the 'period of incubation' -- the whole personality, down to
the unverbalized and unconscious layers, has become saturated with the
problem, so that on some level of the mind it remains active, even while
attention is occupied in a quite different field -- such as looking at a
tree in the chimpanzee's case, or watching the rise of the water-level;
until either chance or intuition provides a link to a quite different
matrix, which bears down vertically, so to speak, on the problem blocked
in its old horizontal context, and the two previously separate matrices
fuse. But for that fusion to take place a condition must be fulfilled
which I called 'ripeness'.
Concerning the psychology of the creative act itself, I have mentioned
the following, interrelated aspects of it: the displacement of attention
to something not previously noted, which was irrelevant in the old and
is relevant in the new context; the discovery of hidden analogies as a
result of the former; the bringing into consciousness of tacit axioms and
habits of thought which were implied in the code and taken for granted;
the uncovering of what has always been there.
This leads to the paradox that the more original a discovery the more
obvious it seems afterwards. The creative act is not an act of creation
in the sense of the Old Testament. It does not create something out
of nothing; it uncovers, selects, re-shuffles, combines, synthesizes
already existing facts, ideas, faculties, skills. The more familiar the
parts, the more striking the new whole. Man's knowledge of the changes
of the tides and the phases of the moon is as old as his observation
that apples fall to earth in the ripeness of time. Yet the combination
of these and other equally familiar data in Newton's theory of gravity
changed mankind's outlook on the world.
'It is obvious', says Hadamard, 'that invention or discovery, be it in
mathematics or anywhere else, takes place by combining ideas. . . .
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
.'
The 'ripeness' of a culture for a new synthesis is reflected in the
recurrent phenomenon of multiple discovery, and in the emergence
of similar forms of art, handicrafts, and social institutions in
diverse cultures. But when the situation is ripe for a given type of
discovery it still needs the intuitive power of an exceptional mind,
and sometimes a favourable chance event, to bring it from potential into
actual existence. On the other hand, some discoveries represent striking
tours de force by individuals who seem to be so far ahead of their time
that their contemporaries are unable to understand them.
Thus at one end of the scale we have discoveries which seem to be due
to more or less conscious, logical reasoning, and at the other end
sudden insights which seem to emerge spontaneously from the depth of
the unconscious. The same polarity of logic and intuition will be found
to prevail in the methods and techniques of artistic creation. It is
summed up by two opposite pronouncements: Bernard Shaw's 'Ninety per cent
perspiration, ten per cent inspiration', on the one hand, Picasso's 'I do
not seek -- I find' (
je ne cherche pas, je trouve
), on the other.
VI
THREE ILLUSTRATIONS
Before proceeding further, let me return for a moment to the basic,
bisociative pattern of the creative synthesis: the sudden interlocking
of two previously unrelated skills, or matrices of thought. I shall give
three somewhat more detailed examples which display this pattern from
various angles: Gutenberg's invention of printing with movable types;
Kepler's synthesis of astronomy and physics; Darwin's theory of evolution
by natural selection.
1. The Printing Press
At the dawn of the fifteenth century printing was no longer a novelty
in Europe. Printing from wooden blocks on vellum, silk, and cloth
apparently started in the twelfth century, and printing on paper was
widely practised in the second half of the fourteenth. The blocks were
engraved in relief with pictures or text or both, then thoroughly wetted
with a brown distemper-like substance; a sheet of damp paper was laid
on the block and the back of the paper was rubbed with a so-called
frotton
-- a dabber or burnisher -- until an impression of the
carved relief was transferred to it. Each sheet could be printed on
only one side by this method, but the blank backs of the sheets could
be pasted together md then gathered into quires and bound in the same
manner as manuscript-books. These 'block books' or xylographs circulated
already in considerable numbers during Gutenberg's youth.
He was born in 1398 at Mainz and was really called Gensfleisch,
meaning gooseflesh, but preferred to adopt the name of his mother's
birthplace. The story of his life is obscure, highlighted by a succession
of lawsuits against money-lenders and other printers; his claim to
priority is the subject of a century-old controversy. But there exists
a series of letters to a correspondent, Frère Cordelier, which
has an authentic ring and gives a graphic description of the manner in
which Gutenberg arrived at his invention. [1] Whether others, such as
Costa of Haarlem, made the same invention at the same time or before
Gutenberg is, from our point of view, irrelevant.
Oddly enough, the starting point of Gutenberg's invention was not the
block-books -- he does not seem to have been acquainted with them --
but playing-cards. In his first letter to Cordelier he wrote:
For a month my head has been working; a Minerva, fully armed, must
issue from my brain. . . . You have seen, as I have, playing-cards
and pictures of saints. . . . These cards and pictures are engraved on
small pieces of wood, and below the pictures there are words and entire
lines also engraved. . . . A thick ink is applied to the engraving; and
upon this a leaf of paper, slightly damp, is placed; then this wood,
this ink, this paper is rubbed and rubbed until the back of the paper
is polished. This paper is then taken off and you see on it the picture
just as if the design had been traced upon it, and the words as if they
had been written; the ink applied to the engraving has become attached
to the paper, attracted by its softness and by its moisture. . . .
Well, what has been done for a few words, for a few lines, I must
succeed in doing for large pages of writing, for large leaves covered
entirely on both sides, for whole books, for the first of all books,
the Bible. . . .
How? It is useless to think of engraving on pieces of wood the whole
thirteen hundred pages. . . .
What am I to do? I do not know: but I know what I want to do: I wish to
manifold the Bible, I wish to have the copies ready for the pilgrimage
to Aix la Chapelle.
Here, then, we have matrix or skill No. 1: the printing from wood-blocks
by means of rubbing.
In the letters which follow we see him desperately searching for a
simpler method to replace the laborious carving of letters in wood:
Every coin begins with a punch. The punch is a little rod of steel,
one end of which is engraved with the shape of one letter, several
letters, all the signs which are seen in relief on a coin. The punch is
moistened and driven into a piece of steel, which becomes the 'hollow'
or 'stamp'. It is into these coin-stamps, moistened in their turn,
that are placed the little discs of gold, to be converted into coins,
by a powerful blow.
This is the first intimation of the method of type-casting. It leads
Gutenberg, by way of analogy, to the
seal
: 'When you apply to
the vellum or paper the seal of your community, everything has been
said, everything is done, everything is there. Do you not see that you
can repeat as many times as necessary the seal covered with signs and
characters?'
Yet all this is insufficient. He may cast letters in the form of coins,
or seals, instead of engraving the wood, yet they will never make a
clear print by the clumsy rubbing method; so long as his search remains
confined to this one and only traditional method of making an 'imprint',
the problem remains blocked. To solve it, an entirely different kind
of skill must be brought in. He tries this and that; he thinks of
everything under the sun: it is the period of incubation. When the
favourable opportunity at last offers itself he is ready for it:
I took part in the wine harvest. I watched the wine flowing, and going
back from the effect to the cause, I studied the power of this press
which nothing can resist. . . .
At this moment it occurs to him that the same, steady pressure might be
applied by a seal or coin -- preferably of lead, which is easy to cast --
on paper, and that owing to the pressure, the lead would leave a trace
on the paper -- Eureka!
. . . A simple substitution which is a ray of light. . . . To work
then! God has revealed to me the secret that I demanded of Him. . . .
I have had a large quantity of lead brought to my home and that is the
pen with which I shall write.
'The ray of light, was the bisociation of wine-press and seal -- which, added together, become the letter-press. The wine-press has been lifted out of its context, the mushy pulp, the flowing red liquid, the jolly revelry -- as Sultan's branch was wrenched out of the context of the tree -- and connected with the stamping of vellum with a seal. From now
onward these separate skills, which previously had been as different
as the butcher's, the baker's, and the candlestick-maker's, will appear
integrated in a single, complex matrix:
One must strike, cast, make a form like the seal of your community;
a mould such as that used for casting your pewter cups; letters in
relief like those on your coins, and the punch for producing them like
your foot when it multiplies its print. There is the Bible!
2. Gravity and the Holy Ghost
'If I have been able to see farther than others,' said Newton, 'it
was because I stood on the shoulders of giants.' One of the giants was
Johannes Kepler (1471-1530) whose three laws of planetary motion provided
the foundation on which the Newtonian universe was built. They were the
first 'natural laws' in the modern sense: precise, verifiable statements
expressed in mathematical terms; at the same time, they represent the
first attempt at a synthesis of astronomy and physics which, during the
preceding two thousand years, had developed on separate lines.