ABC of MathematiciansAuthor(s): Michael HoltSource: Mathematics in School, Vol. 4, No. 1 (Jan., 1975), pp. 8-9Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30211306 .

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Poincard was without doubt the leading mathematician of the last quarter of the nineteenth century. Asked by an army man to name the greatest man France had produced in modern times, Bertrand Russell replied unhesitatingly: "Poincare." "What! That man?" spluttered the British Blimp thinking Russell referred to Raymond Poincard, the President of the French Republic. Realizing what was in his mind, Russell explained: "I was thinking of Raymond's cousin, Henri Poincare." With his vast output, Poincare was the last man who was able to regard all branches of mathematics as his kingdom. All this he could justly boast. And he found a place in the hall of literary fame, too; for he was a master of exposition, his works being widely translated, and was accorded France's highest literary honour being made a member of the Academy. He also wrote a very penetrating study of the psychology of mathematics, now at the heart of much modern educational theory.

Jules Henri Poincare was born on 29 April 1854 at Nancy, Lorraine, In France. His father Leon was a celebrated physician; his mother's identity is coyly concealed by Poincare's official biographer, Gaston Darboux, the noted mathematician; apparently her family name was Lannois but was not well enough connected for Darboux to quote it. Whatever her social background she lavished every care on her brilliant son who as a result developed rapidly. He was ambidextrous and this was to play a significant part in his intellectual life for it left him extremely clumsy. So much so that he could not even draw simple figures for doing Euclid. Perhaps there was more than a hint of self-justification in his later remark about topology, to which he contributed so much: Propositions of topology, he said, "would remain true if the figures were copied by an inexpert draughtsman who should grossly change all the proportions and replace the straight line by lines more or less sinuous."

At 5 the young Poincare caught diphtheria which left him for 9 months with a paralyzed larynx. He could read in- credibly fast and had, like Lord Macauley and Euler, an almost perfect photographic memory. But because of his poor eye- sight he often could not see the blackboard at school and so learnt things by ear; most mathematicians work by eye. How- ever, at primary school he was brilliant, not in mathematics but in natural history; indeed he retained his love of animals all his life. At the age of nine, E. T. Bell tells us, Poincare showed the first promise of his literary talent. His teacher declared that a short essay he had handed in "was 'a little masterpiece'. But he also advised his pupil to be more conventional-stupider-if he

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wished to make a good impression on the school examiners" At the age of 15 the maths bug caught him. His friends

noted the first signs in him of what was later to be his famous absent-mindedness: he often forgot meals. All his mathematics was done in his head as he paced restlessly about; he would write out his results when all his working was done. When he was at his peak, a distinguished Finnish mathematician came to visit him in "his airy perch in the Rue Gay-Lussac". The maid announced the arrival of the guest but Poincare was mathematiz- ing, pacing his room. His visitor waited in an anteroom for three hours, at the end of which Poincare stuck his buli-like head through the curtains and bellowed "Vouz me derangez beacoup" ("You're upsetting me"). The poor man fled, never to gain his interview he had come so far for. An example perhaps not so much of Poincare's absent-mindedness as of his bloody- mindedness.

In 1870 the Franco-Prussian war broke out. Poincard, 16, was too young to see active service. Nevertheless he saw bloodshed first hand. His grandparents' home, which the Poincares used to visit, was pillaged by Prussians. The shock to the young Poincare made him a French patriot for life. He learnt German, however, in order to know how the war was progressing, for he had to live in German-occupied Nancy.

Poincare took his first degree in 1871 at the age of 17. He almost failed in, of all subjects, mathematics. He fell down on the proof of the simple formula for a geometric progression (g.p.):

S=1+q+1+4++... and on and on for an infinite number of terms; the common ratio is

,. Use the same trick as Poincare should have used in

proving the general formula by algebra. That is, work out AS. Then subtract from S. (Answer next time.)

However, he took his next hurdle with no trouble: he gained first prize in mathematics when he sat his entrance exam to the School of Forestry. Fellow students recalled that to almost any mathematical problem his reply "came like an arrow". This flashing intuition stayed with him all his life.

At the end of the first year he moved to the Ecole Poly- technique where his mathematics was brilliant, his Physical Education abysmal, and as for his drawing, his examiners had to give him "'zero" So poor was his draughtsmanship that it affected his geometry and so lost him first place in his final examinations. He left the Ecole at 21 in 1875, to go to the School of Mines. Here he became an engineer, not perhaps the most obvious choice for one who could not draw! In his spare

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time he wrote up a solution to a general problem in differential equations. He presented this as a thesis for a doctorate to the Faculty of Sciences in 1878. His doctorate was gained on brilliance although it was noted there were gaps and errors in his reasoning. He was as courageous physically as he was impetuous mentally. After a mining disaster at Caen, Poincare went down the mine with the crew to rescue survivors. On 1 December 1879 he became a professional mathematician when he was made Professor of Mathematical Analysis at Caen.

At 27 he was promoted to the University of Paris where he took charge of-inappropriately one might think-mechanics and experimental physics. Soon he was to be the undisputed master of French mathematics. His output was vast: 500 papers on new mathematics, 30 books on mathematics, physics and astronomy, not to mention sheaves of popular essays. His greatest work was his book on celestial mechanics, a worthy rival to Laplace's work on the same subject (see Laplace). Poincare also came close to solving a celebrated insoluble pro- blem-the three-body problem-how, for instance, the Sun, 'Earth and Moon attract one another and move in unison. His

work on rotating fluid bodies had enormous importance in cosmogony, the science of how the universe came into being.

For teachers his most intriguing work must be his interest in mathematical creation. Before summarizing his thoughts it is worth quoting an anecdote relating wheh Poincare, as a mature mathematician, took one of Binet's early "intelligence tests". Binet, the reader will recall, was the "father" of IQ tests. Poincare's results seemed to show that far from being the most intelligent man in France (it seems to be an unavoidable fact that to do maths you have to be intelligent) he had the mind of a young child! History does not relate what Binet's reactions were. (I personally heard the same story told about Paul Dienes, another distinguished mathematician and undoubtedly intelli- gent, the father of Zoltan Dienes the mathematician educa- tionist.) For Poincare's thoughts on mathematical creation I quote from an essay he delivered in the first years of this century as a lecture before the Psychological Society in Paris. It is one of the most celebrated attempts to describe what goes on in the mathematician's brain. Since then there have been many fur- ther attempts which I have recounted elsewhere*.

Poincare felt that mathematical creation "should intensely interest the psychologist ... so that in studying the procedure of geometric thought we may hope to reach what is most essential in man's mind ...."

"How does it happen," he goes on, "there are people who do not understand mathematics?... That not every one can retain a demonstration once learned may... pass. But that not every one can understand mathematical reasoning when ex- plained 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?... Need we add that mathematicians themselves are not infallible?"

"As for myself. I must confess, I am absolutely incapable even of adding without mistakes... My memory is not bad, but it would be insufficient to make a good chess-player. Why then does it not fail me in a difficult piece of mathematical reasoning?"

He goes on to describe the various levels of mathematical thinking, with creativity at the top, "In fact, what is mathe- matical creation? It does not consist in making new combina- tions with mathematical entitles already known. Anyone could do that," (!) "but the combinations so made would be infinite in number and most of them without interest." Here Poincare

puts his finger on the fallacy underlying creation by computer: the computer cannot, unless programmed, tell dross from gold, and so far no-one has programmed for creative selection.

In an oft-quoted passage, Poincare tells how he wrote his memoir on Fuchsian functions: how every day, for fifteen days "I seated myself at my work table, stayed for an hour or two. *Mathematics in a Changing World, with D. T. E. Marjoram, Heinemann Educational Books, London 1973.

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 next morning I had established the existence of a class of Fuchsian functions ... I had only to write out the results, which took but a few hours."

Poincare then outlines a recipe for creative work. First ponder (as he did for fifteen days), then let the unconscious get to work and the solution will appear as if by magic. To support this he tells, how one day when on a geological expedition,

with- out a thought of mathematics in his head, he was at Coutances about to board a bus. "At the moment when I put my foot on the step the idea came to me, without my former thoughts seeming to have paved the way for it..." (He had thh identity of his Fuchsian functions with those of non-Euclidean geometry

(see Lobachewski for a brief description of this kind of geometry). "I did not verify the idea; I should not have had the time, as, upon taking my seat in the omnibus, I went on with a conversation already commenced." On his return to Caen he verified the result at his leisure: the point being, as with the crowd of ideas rising in his mind, he had mathematical ideas galore and the technique to sort the good ones from the bad.

Later in the essay he stresses the operation of the subliminal

self, of a "delicate intuition" and of "emotional sensibility":

"It may be surprising to see emotional sensibility invoked a propos of mathematical demonstrations which ... can only interest intellect. This would be to forget the feeling of mathematical beauty, of the harmony of numbers and form, of geometric elegance."

He saw the future elements of a mathematician's ideas as combining liked hooked atoms. With the mind at rest they are hooked on the wall.., but "during unconscious work, they unhook themselves and flash in every direction... as would a swarm of gnats. Then their mutual impact may produce new combinations." What is the role of the preliminary unconscious work? Poincare asks: "To unhook these atoms from the wall

and put them in swing." Then it is that the will .comes into play; for "the only combinations that have a chance of forming are those chosen by our will." On another occasion he showed the vital need for a theory in science, akin to combinations of ideas: "A collection of facts is no more a science than a heap

of stones is a house." His life was busy, tranquil and loaded with honours. In 1906,

at the age of fifty-two, he achieved the highest distinction possible to a French scientist, the Presidency of the Academy of Sciences. Yet he remained, if at times bloodyminded, truly humble and unaffectedly simple. His marriage was a happy one. He had a son and three daughters in whose company he took great pleasure. One of his passions was symphonic music.

In 1908 illness struck him down. He was prevented from attending the International Mathematical Congress at Rome in which he was to present his paper on The Future ofMathematical Physics. Italian surgeons operated on his prostate gland and apparently cured him. But in 1911 he began to fear he might not live long. On 9 December he wrote to the editor of a mathematical journal asking him to break custom and accept an unfinished memoir: "... at my age, I may not be able to solve it," but he felt it might put researchers on "a new and un- expected path" too full of promise to pass up. He had spent two years fruitlessly trying to solve what has become known as "Poincard's last theorem"; it is about the possibility of solving the three-body problem. The proof that had evaded Poincare was given shortly after the publication of his memoir by a young American mathematician, George David Birkhoff.

The next spring Poincare fell ill again; he underwent a second operation on 9 July. It was successful. But on 17 July he died of a blood clot while he was dressing. He died, at 59, at the height of powers.

Answer to Pascal problem.

"iMlystic hexagon": Points of intersection of pairs of opposite sides of hexagon always lie on a straight line. [

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