Remarlzable Mathematicians

From Euler to von Neumann

loan James Mathematical Institute, Oxford

THE MATHEMATICAL ASSOCIATION OF AMERICA

• CAMBRIDGE UNIVERSITY PRESS

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HENRI POINCARE (1854-1912) The development of mathematics in the nineteenth century began under the inspiration of one giant; it ended under that of another. Poincare transformed many branches of mathematics and contributed greatly to mathematical physics, especially celestial mechanics, as well. Like Gauss a century earlier he always worked alone, had few students and founded no School, but unlike Gauss he published liberally and wrote and spoke about mathematics, science, and philosophy on many occasions.

The parents of Henri Poincare came from bourgeois families who had lived in Lorraine, especially the city of Nancy, for generations. His forbears included a number of distinguished scholars. One of his cousins, Raymond Poincare, was later to serve as Prime Minister of France and

as President of the French Republic during the First World War. Jules Henri Poincare, to give him his full name, was born in Nancy on April 2.9, 1854. His father Leon was a physician and professor of m edicine at

From Cantor to Hilbert

the University of Nancy. Henri and his younger sister Aline were at first educated by their mother; Poincare later traced his mathematical ability to his maternal grandmother. At the age of five he contracted diphtheria, which left him unable to speak for almost a year. This weakened his health and while it excluded him from the more boisterous childhood games it did not protect him from being bullied when, at the age of eight, he first went to school. Like Euler and Gauss before him, he was gifted with an exceptional memory, but his motor coordination was poor and that he was extremely myopic.

Poincare grew up in a comfortable academic environment. With ample time to read and study, he made rapid progress. At school his

unusual ability showed first in French composition, but by the end of his school career his awesome mathematical talent was already apparent. His classmate Appell remembered him as 'absorbed in his inner thoughts . . . when he spoke his eyes were filled with an expression of kindness, at the same time malicious and profound. I was struck by the way he talked in

Henri Poincare [1854-1912)

brief, jerky sentences, interspersed with long silences'. His schooling was

interrupted by the Franco-Prussian War, during which his home province of Alsace-Lorraine bore the brunt of the German invasion. Poincare accom­

panied his father on ambulance rounds at this time, becoming a fervent

French patriot as a result. He learned the German language during the war in order to read the news bulletins. Later in life he maintained friendly

relations with the German mathematicians.

At school Poincare carried off first prize in the Concours General for

elementary mathematics and came first after a brilliant performance in the

entrance examination for the Ecole Polytechnique in 1873. He also took the

entrance examination for the Ecole Normale Superieure and was placed fifth on the list of candidates, but since he was planning to become an engineer

he naturally chose the Polytechnique, unlike his near contemporaries Borel

and Hadamard.

At the Ecole Polytechnique, Poincare made rapid progress, although

clumsiness in drawing and experimental work cost him first place in the

final examination. He went on to the Ecole des Mines for the next four

years. As well as qualifying as a mining engineer he wrote a doctoral

thesis on the properties of functions defined by differential equations,

which was later published in the Journal de I' Ecole Polytechnique. After a

brief period working as a mining engineer he decided to devote himself

to mathematics and obtained a position at the University of Caen in

Normandy. It was at Caen that Poincare made his first important discovery:

the occurrence of non-euclidean geometry in the theory of automorphic

functions, and this is when he came into competition with Klein. Poincare's

own account of the circumstances of the discovery is of great interest.

He had been thinking about periodicity with respect to linear fractional transformations, after encountering functions with this property in the

work of Lazarus Fuchs. The functions in question arose from differential

equations, and Poincare had been struggling to understand them analyti­

cally when he was struck by an unexpected geometric inspiration. In his

own words:

For fifteen days I struggled to prove that no functions analogous to

those I have since called Fuchsian functions could exist; I was then

very ignorant. Every day I sat down at my work-table where I spent an

hour or two; I tried a great number of combinations but arrived at no

result. One evening, contrary to my custom, I took black coffee; I

could not go to sleep; ideas swarmed up in clouds; I sensed them

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From Cantor to Hilbert

clashing until, to put it so, a pair would hook together to form a stable

combination. By morning I had established the existence of a class of

Fuchsian functions, those derived from the hypergeometric series. I

had only to write up the results, which took me a few hours.

This experience of Poincare's, in which he seems to have been aware

of the workings of his own unconscious mind, is called extra-cognition

by psychologists. It seems to be a fairly rare phenomenon but instances

of the role of the unconscious in creative work are quite common. In

fact, the idea seems to be generally accepted that to 'sleep on a prob­

lem' which one has been thinking about will often result in a solution appearing as soon as one wakes up. Poincare, who was very interested in

such matters, continues by giving a remarkable illustration from his own

experience:

I then left Caen, where I was living at the time, to participate in a

geological trip organized by the School of Mines. The exigencies of travel made me forget my mathematical labours; reaching Coutances

we took a bus for some excursion or other. The instant I put my foot

on the step the idea came to me, apparently with nothing in my

previous thoughts having prepared me for it, that the transformations

I had used to define Fuchsian functions were identical to those of

non-Euclidean geometry. I did not make the verification; I should not have had the time, because once in the bus I resumed an interrupted

conversation; but I felt an instant and complete certainty. On

returning to Caen, I verified the result at my leisure to satisfy my

conscience.

It was this discovery which made him famous. As a rule, he seems to have made most of his discoveries while walking around. He continues:

I then undertook the study of certain arithmetical questions without much apparent success and without suspecting that such matters

could have the slightest connexion with my previous studies .

Disgusted at my lack of success, I went to spend a few days at the

seaside and thought of something else. One day, while walking along

the cliffs, the idea came to me, again with the same characteristics of

brevity, suddenness, and immediate certainty, that the transformations of indefinite ternary forms were identical with those

of non-Euclidean geometry.

Henri Poincare (1854-1912)

On returning to Caen, I reflected on this result and deduced its

consequences; the example of quadratic forms showed me that there

were Fuchsian groups other than those corresponding to

hypergeometric series; I saw that I could apply to them the theory of

thetafuchsian functions, and hence that there existed thetafuchsian

functions other than those derived from the hypergeometric series, the

only ones I had known up to then. Naturally I set myself the task of constructing all these functions . I conducted a systematic siege and,

one after another, carried out all the outworks; there was however one

which still held out and whose fall would bring about that of the

whole position. But all my efforts served only to make me better acquainted with the difficulty, which in itself was something.

At this point I left for Mont-Valerien, where I was to discharge my

military service. I had therefore very different preoccupations. One

day, while crossing the boulevard, the solution of the difficulty which had stopped me appeared to me all of a sudden. I did not seek to go

into it immediately, and it was only after my service that I resumed

the question. I had all the elements, and had only to assemble and

order them. So I wrote out my definitive memoir at one stroke and

with no difficulty.

The discovery of the underlying geometry (and topology, which soon

followed) put Fuchsian functions in a completely new light. For the next few years Poincare worked feverishly to develop these ideas, in friendly

competition with Klein. There were some reservations about his style

- undisciplined and lacking in rigour although very readable - but his brilliance was not contested.

Poincare returned to Paris in 1881, at the age of twenty-seven, and

in the same year married Jeanne Louise Marie Poulain d' Andecy, who had

a similar family background to his own: they had four children, a son and

three daughters. It must have been around this time that Sylvester visited

him at his 'airy perch' in the rue Gay-Lussac and was astonished when, after having toiled up three flights of narrow stairs leading to his study he

beheld a mere boy, 'so blond, so young', as the author of the deluge of papers

which had heralded the advent of a successor to Cauchy.

Sylvester got to know Poincare quite well. On one occasion Camille

Jordan had brought together several people, including Poincare, for a dinner in honour of Sylvester. When Poincare arrived, Sylvester monopolized his

From Cantor to Hilbert

attention without giving him time even to greet his hostess: 'I have a beau­tiful theorem to show you', said Sylvester, and proceeded to demonstrate it. From that moment Poincare said not a word; he ate his meal like an

automaton. After dinner Poincare recovered his awareness of the outside world and descended on Sylvester, exclaiming 'but your theorem is false! ' and proved it to him on the spot.

At first Poincare was just a lecturer in mathematical analysis at the Sorbonne but by 1886 he was professor of mathematical physics and calculus of probabilities, the chair he retained, winning higher and higher

honours, until the end of his life. In 1887, at thirty-two, he was elected to the Paris Academy as a result of his work on automorphic functions, and six years later to the Bureau des Longitudes. The following year he was elected to the Royal Society of London, the first of many such honours, and later became the first holder of the society's Sylvester medal.

A picture of the indefatigable Poincare at work can be found in a letter written by a nephew of his to Mittag-Leffler: 'In his peaceful study in Paris or in the shade of his garden in the Lozere, Henri Poincare would sit for hours every day in front of a pad of ruled paper, and one saw the sheets being covered, with a surprising regularity, in his delicate and angular hand­writing. Almost never an erasure, very rarely a hesitation. After some days a lengthy memoir will be finished, ready for the printer, and my uncle from then on was only interested in it as something in the past. He could scarcely be persuaded to cast a quick glance at the proof-sheets when they were sent to him by editors ... ' The letter goes on: 'It was often observed that Henri Poincare kept his thoughts to himself. Unlike certain other scientists, he did not believe that oral communication, the verbal exchange of ideas, could favour discovery ... my uncle regarded mathematical discovery as an idea which entirely excluded the possibility of collaboration. The intuition, by which discoveries are made, is a direct communion, without possible intermediaries, with the spirit and the truth.'

It was combinatorial topology, more than anything else, which domi­nated the later part of Poincare's working life. He virtually created the sub­

ject, organizing the miscellaneous results known previously, and setting the agenda for many years to come. His topological ideas not only breathed new life into complex analysis and mechanics, they amounted to the creation of a major new field. In publications between 1892 and 1904, he built up an arsenal of techniques and concepts around which topology developed. 'Everything I have done leads me to analysis situs', he wrote in 1901, 'I needed that science to pursue my studies of curves defined by higher order

Henri Poincare (1854-1912)

differential equations. I needed it for the study of non-uniform functions of two variables. I needed it for the study of the periods of multiple integrals and for the application of that study to the development of the perturbatrix

function. Then I entered analysis situs through an important problem in

group theory, research into discrete groups or finite groups contained in given continuous groups.'

Another of Poincare's great inventions was the qualitative theory of differential equations. He used this theory, which deals with such ques­tions as the long-term stability of a dynamical system, in his article 'Les methodes nouvelles de la mecanique celeste' (New methods in celestial mechanics), arguably the greatest advance in celestial mechanics since Newton's Principia. As we have seen, Poincare's memoir on the three-body problem was the winning entry in the international prize competition sponsored by the King of Sweden and Norway. It provided the foundation for Poincare's three-volume work with the same title, which contained the first mathematical description of chaotic behaviour in a dynamical system. He later produced another three-volume set, entitled Let;ons de mecanique celeste (Lectures on celestial mechanics), which covered much the same

ground but was less demanding technically.

Poincare was critical of the interest in pathological functions associ­ated with the school of Weierstrass: 'Logic sometimes makes monsters. For half a century we have seen a mass of bizarre functions which appear to be forced to resemble as little as possible honest functions which serve some purpose ... from the point of view of logic, these strange functions are the most generali on the other hand those which one meets without searching for them, and which follow simple laws appear as a particular case which does not amount to more than a small corner ... In former times when one invented a new function it was for a practical purposei today one invents them purposely to show up defects in the reasoning of our fathers and one will deduce from them only that.'

As he grew older Poincare devoted more and more attention to fun­damental questions about the nature of mathematics. He wrote a number of papers criticizing the logical and rational philosophies of Hilbert, Peano, and Russell. To some extent his work presaged some of the intuitionist

arguments of Brouwer. Courant, in a talk he gave at Yale, recalled an occasion when Poincare came to Gottingen shortly before his death to give talks on different topics: one was propagation of the electromagnetic wave around the earth and another was on the foundations of mathematics. It was a violent attack against Cantorism and against the use of the axiom

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244 From Cantor to Hilbert

of choice and the well-ordering principle. Poincare wanted to be polite (he could be devastatingly impolite if he tried) and he fulminated against the Cantor attitude and against the trend to do something in this direction. A

member of the audience described him, rather unkindly, as a dwarfish man with a slightly hunched back, a rough short beard, and very sad eyes.

Poincare's views on set theory seem to have shifted over the years. He first used the term Mengenlehre in the 1880s to describe early results and concepts he attributed to Cantor in what we would call point-set theory.

He was very enthusiastic about them, and took the opportunity in 1885, when secretary of the recently founded Societe Mathematique de France, to propose Cantor for membership. There is no reason to suppose that in later life he repented of his early enthusiasm for Mengenlehre, in the sense of point-set theory: it was axiomatic set theory that Poincare had been attacking at Gottingen.

Poincare was one of the last universal mathematicians, one who had a general grasp of all branches of mathematics. Like Euler he wrote fluently and copiously on every part of the subject, and in fact surpassed Euler in his popular writing. In 1908, when he was elected to the Academie

Frant;aise, the literary section of the Institut, it was for the quality of his expository work. The only other mathematicians to be so honoured were d'Alembert, Laplace, Fourier, and later Picard. He wrote several books on science and its philosophy, which were best sellers in the early part of the twentieth century. However for all his natural brilliance and formal training Poincare was apparently ignorant of much of the literature on

mathematics. One consequence of this was that each new subject he heard about drove his interests in a new direction. Thus when he heard about the work of Riemann and Weierstrass on Abelian functions he threw himself into Abelian functions. He characterized Riemann's method as 'above all a method of discovery' and that of Weierstrass and Dedekind as 'above all a method of demonstration'.

Intuition played an important part in Poincare's work and occasion­ally led him astray. For example in his early papers on analysis situs he did not pay sufficient attention to the phenomenon of torsion. And he did not seem to care very much if, for example, he used one definition of

manifold to prove one theorem and a quite different one to prove another, without showing the definitions to be equivalent. It must be admitted that the strength of Poincare's geometrical intuition sometimes led him to ignore the pedantic strictness of proofs. Here there is still another side: finding himself under constant influx of a set of ideas in the most diverse

Henri Poincare (1854- 1912)

fields of mathematics, Poincare did not have time to be rigorous, it was

said: he was often satisfied when his intuition gave him the confidence that the proof of such and such a theorem could be carried through to

complete logical rigour, and then assigned the completion of the proof to

others.

On the pure side of mathematics Poincare made fundamental con­

tributions to a wide range of subjects, particularly differential equations, the general themy of functions, and analysis situs. Much of this work

was more or less directly motivated by possible applications. Out of his

output of around 500 papers, about seventy dealt with problems closely related to physics, including light, electricity, capillarity, thermodynamics,

heat, elasticity, and telegraphy. He was particularly interested in theoretical physics. As early as 1899 he suggested that absolute motion does not exist:

the following year he also proposed the concept that nothing could travel

faster than light. These two concepts are of course central to Einstein's special theory of relativity, not announced until 1905, but there is no

indication that Poincare realized their full implications.

Poincare was often invited to address scientific conferences. For

example, in just one fortnight in the year 1900 he not only gave the

presidential address on the role of intuition and logic at the second Interna­

tional Congress of Mathematicians, but he also addressed the International

Congress of Philosophy on the principles of mechanics and the International Congress of Physics on the relation between experimental physics and

mathematical physics. In 1903 he crossed the Atlantic to lecture on the

present state and the future of mathematical physics at the International

Congress of Art and Science in StLouis, and took the opportunity to visit

other parts of the United States.

Already in 1908 at the fourth International Congress of Mathemati­cians in Rome those present were greatly concerned when Poincare needed

to return to Paris for medical treatment before giving his scheduled address

on the future of mathematics. In 1911 he took the unusual step of publishing

an unfinished paper, on periodic solutions of the three-body problem, being

afraid that he might not live to complete it. As we shall see, the proof of

Poincare's Last Geometrical Theorem was completed the following year by

the young American mathematician George Birkhoff. Henri Poincare died

unexpectedly on July 17, 1912 at the age of fifty-eight, due to a post-operative

embolism. The mathematical world was deeply shocked at the untimely

death of the greatest mathematician of his time, and one of the greatest of

all time.

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