# Henri Poincaré

Jules Henri Poincaré was born on April $29^{th}$ 1854 in Cité Ducale[BA] a in Nancy, a city in France. He was the son of Dr. Léon Poincaré (1828-1892) who was a professor at the University of Nancy in the faculty of medicine.[SJ] His mother, Eugénie Launois (1830-1897) was described as a “gifted mother”[EB] who gave special instruction to her son. She was 24 and his father 26 years of age when Henri was born[MT]. Two years after the birth of Henri they gave birth to his sister Aline.[EB]

In 1862 Henri entered the Lycée of Nancy which is today, called in his honor, the Lycée Henri Poincaré. In fact the University of Nancy is also named in his honor. He graduated from the Lycée in 1871 with a bachelors in letters and sciences. Henri was the top of class in almost all subjects, he did not have much success in music and was described as “ at best” in any physical activities.[MT] This could be blamed on his poor eyesight and absentmindedness.[BC] Later in 1873, Poincaré entered l’Ecole Polytechnique where he performed better in mathematics than all the other students. He published his first paper at 20 years of age, titled Démonstration nouvelle des propriétés de l’indicatrice d’une surface.[BR] He graduated from the institution in 1876. The same year he decided to attend l’Ecole des Mines and graduated in 1879 with a degree in mining engineering.[SJ] After his graduation he was appointed as an ordinary engineer in charge of the mining services in Vesoul. At the same time he was preparing for his doctorate in sciences (not surprisingly), in mathematics under the supervision of Charles Hermite. Some of Charles Hermite’s most famous contributions to mathematics are: Hermite’s polynomials, Hermite’s differential equation, Hermite’s formula of interpolation and Hermitian matrices.[MT] Poincaré, as expected graduated from the University of Paris in 1879, with a thesis relating to differential equations. He then became a teacher at the University of Caen, where he taught analysis. He remained there until 1881. He then was appointed as the “maître de conférences d’analyse”[SJ] (professor in charge of analysis conferences) at the University of Paris. Also in that same year he married Miss Poulain d’Andecy. Together they had four : Jeanne born in 1887, Yvonne born in 1889, Henriette born in 1891, and finally Léon born in 1893. He had now returned to work at the Ministry of Public Services as an engineer. He was responsible for the development of the northern railway. He held that position from 1881 to 1885. This was the last job he held in administration for the government of France. In 1893 he was awarded the title of head engineer in charge of the mines. After that his career awards and position continuously escalated in greatness and quantity. He died two years before the war on July $17^{th}$ 1912 of an embolism at the age of 58. Interestingly, at the beginning of World War I, his cousin Raymond Poincaré was the president of the French Republic.

Poincaré’s work habits have been compared to a bee flying from flower to flower. Poincaré was interested in the way his mind worked, he studied his habits. He gave a talk about his observations in 1908 at the Institute of General Psychology in Paris. He linked his way of thinking to how he made several discoveries. His mental organization was not only interesting to him but also to Toulouse, a psychologist of the Psychology Laboratory of the School of Higher Studies in Paris. Toulouse wrote a book called Henri Poincaré which was published in 1910. He discussed Poincaré’s schedule: he worked during the same times each day in short of time. He never spent a long time on a problem since he believed that the subconscious would continue working on the problem while he worked on another problem. Toulouse also noted that Poincaré also had an exceptional memory. In addition he stated that most mathematicians worked from principle already established while Poincaré was the that started from basic principle each time.[MT] His method of thinking is well summarized as:

Habitué à négliger les détails et à ne regarder que les cimes, il passait de l’une à l’autre avec une promptitude surprenante et les faits qu’il découvrait se groupant d’eux-mêmes autour de leur étaient instantanémant et automatiquement classé dans sa mémoire. (He neglected details and jumped from idea to idea, the facts gathered from each idea would then come together and solve the problem) [BA]

The mathematician Darboux claimed he was “un intuitif”(intuitive)[BA], arguing that this is demonstrated by the fact that he worked so often by visual . He did not care about being rigorous and disliked . He believed that logic was not a way to invent but a way to ideas and that logic ideas.

Poincaré had the opposite philosophical views of Bertrand Russell and Gottlob Fredge who believed that mathematics were a of logic. Poincaré strongly disagreed, claiming that intuition was the life of mathematics. Poincaré gives an interesting point of view in his book Science and Hypothesis:

For a superficial observer, scientific truth is beyond the possibility of doubt; the logic of science is infallible, and if the scientists are sometimes mistaken, this is only from their mistaking its rule. [PHSH]

Poincaré believed that arithmetic is a synthetic science. He argued that Peano’s axioms cannot be proven non-circularly with the principle of induction.[MM] Therefore concluding that arithmetic is a priori synthetic and not analytic. Poincaré then went on to say that mathematics can not be a deduced from logic since it is not analytic. It is important to note that even today Poincaré has not been proven wrong in his argumentation. His views were the same as those of Kant[KD]. However Poincaré did not share Kantian views in all branches of philosophy and mathematics. For example in geometry Poincaré believed that the structure of non-Euclidean space can be known analytically. He wrote 3 books that made his philosophies known: Science and Hypothesis, The Value of Science and Science and Method.

Poincaré’s first area of interest in mathematics was the Fuchsian function that he named after the mathematician Lazarus Fuch because Fuch was known for being a good teacher and done alot of research in differential equations and in the of functions. The functions did not keep the fuchsian and are today called automorphic. Poincaré actually developed the concept of those functions as part of his doctoral thesis.[MT] An automorphic function is a function $f(z)$ where $z\in\mathbb{C}$ which is analytic under its domain and which is invariant under a denumerable infinite group of linear fractional transformations, they are the generalizations of trigonometric functions and elliptic functions. Below Poincaré explains how he discovered Fuchsian functions:

[PHSM]

This is a clear indication Henri Poincaré brilliance. Poincaré communicated a lot with Klein another mathematician working on Fuchsian functions. They were able to discuss and further the theory of automorphic(Fuchsian) functions. Apparently Klein became jealous of Poincaré’s high opinion of Fuch’s work and ended their relationship on bad .

Poincaré contributed to the field of algebraic topology and published Analysis situs in 1895 which was the first systematic look at topology. He acquired most of his knowledge from his work on differential equations. He also formulated the Poincaré conjecture, one of the great unsolved mathematics problems. It is currently one of the “Millennium Prize Problems”. The problem is stated as:

Consider a compact 3-dimensional manifold V without boundary. Is it possible that the fundamental group V could be trivial, even though V is not homeomorphic to the 3-dimensional sphere? [CMI]

The problem has been attacked by many mathematicians such as Henry Whitehead in 1934, but without success. Later in the 50’s and 60’s progress was made and it was discovered that for higher-dimension manifolds the problem was easier. (Theorems have been stated for those higher dimensions by Stephe Smale, John Stallings, Andrew Wallace, and many more) [CMI] Poincaré also studied homotopy theory, which is the study of topology reduced to various groups that are algebraically invariant.[MT] He introduced the fundamental group in a paper in 1894, and later stated his infamous conjecture. He also did work in analytic functions, algebraic geometry, and Diophantine problems where he made important contributions not unlike most of the areas he studied in.

In 1887, Oscar II, King of Sweden and Norway held a competition to celebrate his sixtieth birthday and to promote higher learning.[EE] The King wanted a contest that would be of interest so he decided to hold a mathematics competition. Poincaré entered the competition submitting a memoir on the three problem which he describes as:

Le but final de la Mécanique céleste est de résoudre cette grande question de savoir si la loi de Newton explique à elle seule tous les phénomènes astronomiques; le seul moyen d’y parvenir est de faire des observation aussi précises que possible et de les comparer ensuite aux résultats du calcul. (The goal of celestial mechanics is to answer the great question of whether Newtonian mechanics explains all astronomical phenomenons. The only way this can be proven is by taking the most precise observation and comparing it to the theoretical calculations.) [PHMC]

Poincaré did in fact win the competition. In his memoir he described new mathematical ideas such as homoclinic points. The memoir was about to be published in Acta Mathematica when an error was found by the editor. This error in fact led to the discovery of chaos theory. The memoir was published later in 1890.[MT] In addition Poincaré proved that the determinism and predictability were disjoint problems. He also found that the solution of the three body problem would change drastically with small change on the initial conditions. This area of research was neglected until 1963 when Edward Lorenz discovered the famous a chaotic deterministic system using a of the atmosphere.[MM]

Henri Poincaré and Albert Einstein had an interesting relationship concerning their work on relativity (one might actually describe it as a lack of a relationship). Their interaction begins in 1905 when Poincaré published his first paper on relativity. The topic of the paper was “partly kinematic, partly dynamic”[PA] which included the correction of Lorentz’s proof related to the Lorentz transformation (actually named by Poincaré). About a month later Einstein published his first paper on relativity. Both continued publishing work about relativity, but neither of them would reference each others work. Not only did Einstein not reference Poincaré’s work but claimed never to have read it(and it is not known if he eventually read Poincaré papers)[PA]. On one occasion Einstein referenced Poincaré aknowledging his work on relativity in the text of a lecture in 1921 called ‘Geometrie und Erahrung’.[PA] Although later in Einstein’s life, he would comment on Poincaré as being one of the pioneers of relativity. Before Einstein’s death, Einstein said:

Lorentz had already recognized that the transformation named after him is essential for the analysis of Maxwell’s equations, and Poincaré deepened this insight still further…[PA]

Poincaré made many contributions to different fields of applied mathematics such as: celestial mechanics, fluid mechanics, optics, electricity, telegraphy, capillarity, elasticity, thermodynamics, potential theory, quantum theory, theory of relativity and cosmology. In the field of differential equations Poincaré has given many results that are critical for the qualitative theory of differential equations, for example the Poincaré sphere and the Poincaré .

It is that intuition that led him to discover and study so many areas of science. Poincaré is considered to be the next universalist after Gauss. After Gauss’s death in 1855 people generally believed that there would be no one else that could master all branches of mathematics. However they were wrong because Poincaré took all areas of mathematics as “his province”[BC].

## 1 Appendix

### 1.1 Awards:

• In 1901 was awarded the http://www.royalsoc.ac.uk/Royal Society Sylvester Medal.

• The 1911 http://www.astrosociety.org/Astronomical Society of the Pacific, http://www.phys-astro.sonoma.edu/BruceMedalists/Poincare/index.htmlBruce Medalist.

### 1.2 Fellowships:

• In 1894 was elected fellow of http://www.royalsoc.ac.uk/The Royal Society.

• In 1985 was elected fellow of http://www.royalsoced.org.uk/The Royal Society of Edinburgh.

## References

• EE The 1911 Edition Encyclopædia: Oscar II of Sweden and Norway, [online], http://63.1911encyclopedia.org/http://63.1911encyclopedia.org/
• BA Belliver, André: Henri Poincaré ou la vocation souveraine, Gallimard, 1956.
• BR Bour P-E., Rebuschi M.: Serveur W3 des Archives H. Poincaré [online] http://www.univ-nancy2.fr/ACERHP/http://www.univ-nancy2.fr/ACERHP/
• BC Boyer B. Carl: A History of Mathematics: Henri Poincaré, John Wiley & Sons, inc., Toronto, 1968.
• CMI Clay Mathematics Institute: Millennium Prize Problems, 2000, [online] http://www.claymath.org/prizeproblems/http://www.claymath.org/prizeproblems/.
• EB Encyclopædia Britannica: Biography of Jules Henri Poincaré.
• MM Murz, Mauro: Jules Henri Poincaré [Internet Encyclopedia of Philosophy], [online] http://www.utm.edu/research/iep/p/poincare.htmhttp://www.utm.edu/research/iep/p/poincare.htm, 2001.
• KD Kolak, Daniel: Lovers of Wisdom (second edition), Wadsworth, Belmont, 2001.
• MT O’Connor, J. John & Robertson, F. Edmund: The MacTutor History of Mathematics Archive, [online] http://www-gap.dcs.st-and.ac.uk/ history/http://www-gap.dcs.st-and.ac.uk/ history/, 2002.
• OHP Oeuvres de Henri Poincaré: Tome XI, Gauthier-Villard, Paris, 1956.
• PA Pais, Abraham: Subtle is the Lord…, Oxford University Press, New York, 1982.
• PHSM Poincaré, Henri: Science and Method; The Foundations of Science, The Science Press, Lancaster, 1946.
• PHSH Poincaré, Henri: Science and Hypothesis; The Foundations of Science, The Science Press, Lancaster, 1946.
• PHMC Poincaré, Henri: Les méthodes nouvelles de la mécanique celeste, Dover Publications, Inc. New York, 1957.
• SJ Sageret, Jules: Henri Poincaré, Mercvre de France, Paris, 1911.
• WHP Wikipedia: Henri, Poincaré. [online] http://en.wikipedia.org/wiki/Henri_Poincarehttp://en.wikipedia.org/wiki/Henri_Poincare, 2004.