Background

Jules Henri Poincaré was born on April 29, 1854, in Nancy, Meurthe-et-Moselle. His father Leon Poincaré was a professor of medicine at the University of Nancy and his mother was Eugénie Launois. His adored younger sister Aline married the spiritual philosopher Emile Boutroux. Another notable member of Henri's family was his cousin, Raymond Poincaré, who would serve as President of France from 1913 to 1920, and who was a fellow member of the Académie française.

Education

Henri attended elementary school and the lycée in Nancy (now renamed the Lycée Henri Poincaré in his honor) in 1862 and graduated in 1871. Then he entered the École Polytechnique in Paris at the age of 18. There he demonstrated his brilliance in mathematics and also his phenomenal memory. Although his eyesight was poor, he never took notes in class, and after reading a book he could recall the page on which any statement occurred. Strangely enough, at this time Poincaré seems not to have fathomed his own mathematical power, for in 1875 he entered the School of Mines with the intention of becoming an engineer and 3 years later he qualified as a mining engineer.

At the same time, Poincaré was preparing for his Doctorate in Science in mathematics under the supervision of Charles Hermite. His doctoral thesis was in the field of differential equations. It was named Sur les propriétés des fonctions définies par les équations aux différences partielles. He realized that they could be used to model the behavior of multiple bodies in free motion within the solar system. Poincaré graduated from the University of Paris in 1879.

Career

Poincaré immediately established himself among the greatest mathematicians of Europe, attracting the attention of many prominent mathematicians. In 1881 he was invited to take a teaching position at the Faculty of Sciences of the University of Paris; he accepted the invitation. In 1881-1882, Poincaré created a new branch of mathematics: qualitative theory of differential equations. He showed how it is possible to derive the most important information about the behavior of a family of solutions without having to solve the equation (since this may not always be possible). He successfully used this approach to problems in celestial mechanics and mathematical physics. During the years of 1883 to 1897, he taught mathematical analysis in École Polytechnique.

From 1881 to 1885, Poincaré was in charge of the development of the northern railway. Eventually, he would become Chief Engineer of the Corps de Mines in 1893 and Inspector General in 1910. Meanwhile, in spite of Poincare’s busy schedule, his academic output did not diminish, leading to his election to the chair of Mathematical Physics and Theory of Probability at the University of Paris in 1886. Later, he also held the chairs of Physical and Experimental Mechanics and Celestial Mechanics and Astronomy.

Poincaré summarized his new mathematical methods in astronomy in Les Méthodes Nouvelles de la mécanique céleste, 3 volumes (1892, 1893, and 1899; “The New Methods of Celestial Mechanics”). He was led by this work to contemplate mathematical spaces (now called manifolds) in which the position of a point is determined by several coordinates. Very little was known about such manifolds, and, although the German mathematician Bernhard Riemann had hinted at them a generation or earlier, few had taken the hint. Poincaré took up the task and looked for ways in which such manifolds could be distinguished, thus opening up the whole subject of topology, then known as analysis situs.

In 1887, he entered a competition held by the King of Sweden for finding a solution to the "three-body problem." Although he could not solve it, the judges found his work to be “of such importance that its publication will inaugurate a new era in the history of celestial mechanics.”

In 1887, at the young age of 32, Poincaré was elected to the French Academy of Sciences. He became its president in 1906 and was elected to the Académie française on 5th March 1908. In 1887, Henri won Oscar II, King of Sweden’s mathematical competition for a resolution of the three-body problem concerning the free motion of multiple orbiting bodies.

In 1892, he published his first volume of "Les Méthodes Nouvelles de la mécanique céleste," one of his major works on celestial mechanics. He later published two more volumes on the subject, publishing the third volume in 1899.

In 1893, Poincaré joined the French Bureau des Longitudes, which engaged him in the synchronization of time around the world. Also in the same year, he started working on the theory of analytic functions of several complex variables, writing his first paper on the subject. In 1897 he backed an unsuccessful proposal for the decimalization of circular measure, and hence time and longitude. It was this post that led him to consider the question of establishing international time zones and the synchronization of time between bodies in relative motion.

Poincaré’s main achievement in mathematical physics was his magisterial treatment of the electromagnetic theories of Hermann von Helmholtz, Heinrich Hertz, and Hendrik Lorentz. His interest in this topic which, he showed seemed to contradict Newton’s laws of mechanics led him to write a paper in 1905 on the motion of the electron. This paper, and others of his at this time, came close to anticipating Albert Einstein’s discovery of the theory of special relativity. But he never took the decisive step of reformulating traditional concepts of space and time into space-time, which was Einstein’s most profound achievement. Attempts were made to obtain a Nobel Prize in physics for Poincaré, but his work was too theoretical and insufficiently experimental for some tastes.

In 1894, he started working on homotopy theory, which reduces topological questions to algebra by associating various groups of algebraic invariants with topological spaces. In his paper, he introduced the fundamental or the first group, distinguishing different types of 2-dimensional surfaces.

About 1900 Poincaré acquired the habit of writing up accounts of his work in the form of essays and lectures for the general public. Published as La Science et l’hypothèse (1903; Science and Hypothesis), La Valeur de la science (1905; The Value of Science), and Science et méthode (1908; Science and Method), these essays form the core of his reputation as a philosopher of mathematics and science. His most famous claim in this connection is that much of science is a matter of convention. He came to this view on thinking about the nature of space: Was it Euclidean or non-Euclidean? He argued that one could never tell because one could not logically separate the physics involved from mathematics, so any choice would be a matter of convention. Poincaré suggested that one would naturally choose to work with the easier hypothesis.

In 1895, Poincaré published "Analysis Situs," a seminal work on mathematics. Later, he published five supplementary papers on the subject, thus developing an early systematic treatment of topology and revolutionizing the subject by using algebraic structures, a work he completed by 1905. During this period, he also started writing scientific articles meant for laymen, trying to popularize the subject among them. These works, which conveyed the meaning and importance of science and mathematics in general terms, proved that he was equally gifted in literature.

In 1899, and again more successfully in 1904, Poincaré intervened in the trials of Alfred Dreyfus. He attacked the spurious scientific claims of some of the evidence brought against Dreyfus, who was a Jewish officer in the French army charged with treason by colleagues.

He was dismayed by Georg Cantor’s theory of transfinite numbers and referred to it as a “disease” from which mathematics would eventually be cured. Poincaré said, “There is no actual infinite; the Cantorians have forgotten this, and that is why they have fallen into contradiction.”

His often imprecise mathematical exposition, masked by a delightful prose style, was alien to the generation in the 1930s who modernized French mathematics under the collective pseudonym of Nicolas Bourbaki, and they proved to be a powerful force. His philosophy of mathematics lacked the technical aspect and profundity of developments inspired by the German mathematician David Hilbert’s work. However, its diversity and fecundity have begun to prove attractive again in a world that sets more store by applicable mathematics and less by systematic theory.

Poincaré did not win the Nobel Prize. This is because he worked mainly on theory and did not make any specific invention or discovery. Yet, between 1904 and 1912, he received a total number of 51 nominations. Among them, 34 nominations were for the 1910 Nobel Prize only.

Achievements

• 

  Henri Poincaré's greatest achievement was not only in becoming one of the most inventive, versatile, and productive mathematicians of all time, but also a leading physicist who almost won a Nobel Prize for physics and a prominent philosopher of science whose fresh and surprising essays are still in print a century later. The first in-depth and comprehensive look at his many accomplishments, Henri Poincaré explores all the fields that Poincaré touched, the debates sparked by his original investigations, and how his discoveries still contribute to society today.
  
  Poincaré's influence was wide-ranging and permanent. His novel interpretation of non-Euclidean geometry challenged contemporary ideas about space, stirred heated discussion, and led to flourishing research. And Poincaré's reformulation of celestial mechanics and discovery of chaotic motion started the modern theory of dynamical systems.
  
  In physics, his insights on the Lorentz group preceded Einstein's, and he was the first to indicate that space and time might be fundamentally atomic. Poincaré the public intellectual did not shy away from scientific controversy, and he defended mathematics against the attacks of logicians such as Bertrand Russell, opposed the views of Catholic apologists, and served as an expert witness in probability for the notorious Dreyfus case that polarized France.
  
  The generality and originality of his methods enabled him to master and then break new ground in mathematical physics, celestial mechanics, and nearly every branch of pure mathematics. It was Poincaré's style to emphasize qualitative solutions rather than quantitative recipes. He published several papers on the behavior of solutions and on the properties of integral curves.
  
  Poincaré was elected to the Academy of Sciences in 1887, and he became president of that body in 1906. Two years later he was elected to the literary section of the French Institute, in recognition of his popular works on the philosophy and methods of science, which were widely read in France and translated into six languages.
  
  In 1889 Poincaré won a prize from the King of Sweden for a question posed by Weierstrass on the stability of the solar system, that is, the three-body (or n-body) problem in classical mechanics. Despite a mathematical error that he discovered at the last moment (after questions raised by the Swedish mathematician Lars Edvard Phragmén) and frantically corrected, the work was important for its use of topology and as a founding document in chaos theory, for Poincaré showed that in general, the stability of such systems cannot be demonstrated. It is also in this context that he proved his famous recurrence theorem.
  
  Poincaré is also famous for his 1904 conjecture concerning the topology of three-dimensional spheres which remained one of the major unsolved problems in mathematics until the Russian mathematician Grigori Perelman succeeded in demonstrating it nearly one hundred years later. Poincaré lectured on contemporary mathematical physics for years and was very much abreast of current developments.
  
  Also, he was responsible for formulating the Poincaré conjecture, which was one of the most famous unsolved problems in mathematics until it was solved in 2002–2003 by Grigori Perelman. In his research on the three-body problem, Poincaré became the first person to discover a chaotic deterministic system that laid the foundations of modern chaos theory.
  
  The Institut Henri Poincaré (mathematics and theoretical physics center), Poincaré Prize (Mathematical Physics International Prize), Annales Henri Poincaré (Scientific Journal), Poincaré Seminar (nicknamed "Bourbaphy"), the crater Poincaré on the Moon, Asteroid 2021 Poincaré.
  
  In mathematics and physics, numerous ideas have been named after him. Among them are Poincaré Complex, Poincaré conjunction, Poincaré disk model, Poincaré duality, Poincaré group, Poincaré symmetry, Poincaré-Einstein synchronization, Poincaré-Bendixson theorem, Poincaré-Birkhoff theorem, and Poincaré-Birkhoff-Witt theorem, etc.
  
  "La Science et l’Hypothès" (Science and Hypothesis), first published in 1901, is one of Henri Poincaré’s most popular works. In this book, he talked about mathematics, physics, space, and nature in non-technical terms.

Works

More photos

• book

  • Memoire Sur Les Courbes Definies Par Une Equation Differentielle 1881
  • Thermodynamique 1893
  • The Value of Science: Essential Writings of Henri Poincare 1902
  • The Foundations of Science: Science and Hypothesis, The Value of Science, Science and Method 1913
  • Science and hypothesis 1901
  • Works of Henri Poincaré 1900
  • Science and Method 1913
  • The Measure of Time 1906
  • Last Thoughts 1909
  • Mathematics And Science: Last Essays 1913
  • La Dynamique de l’électron 1913
  • Le Continu mathématique 1901
  • Papers on Topology: Analysis Situs and Its Five Supplements 1904
  • L'oeuvre de Henri Poincaré 1913
  • Papers on Fuchsian Functions 1892
  • Figures D'équilibre D'une Masse Fluide 1900
  • Electricite Et Optique: Cour de Physique Mathematique 1890
  • Les oscillations électriques 1894

Religion

Henri Poincaré was raised in the Roman Catholic faith but later left the religion. He became a freethinker, believing in the search for truth and was said to be an atheist.

Views

In the foundations of mathematics, Poincaré argued for conventionalism, against formalism, against logicism, and against Cantor's treating his new infinite sets as being independent of human thinking. Poincaré stressed the essential role of intuition in a proper constructive foundation for mathematics. He believed that logic was a system of analytic truths, whereas arithmetic was synthetic and a priori, in Kant's sense of these terms. Mathematicians can use the methods of logic to check a proof, but they must use intuition to create a proof, he believed.

He thought that mathematical logic was barren, and when he heard that antinomies had crept into the logistic system of Bertrand Russell and Alfred North Whitehead he could barely conceal his glee.

Poincaré made clear the importance of paying attention to the invariance of laws of physics under different transformations and was the first to present the Lorentz transformations in their modern symmetrical form. Poincaré discovered the remaining relativistic velocity transformations and recorded them in a letter to Dutch physicist Hendrik Lorentz (1853–1928) in 1905. Thus he obtained perfect invariance of all of Maxwell's equations, an important step in the formulation of the theory of special relativity.

Poincaré had philosophical views opposite to those of Bertrand Russell and Gottlob Frege, who believed that mathematics was a branch 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."

Poincaré believed that arithmetic is a synthetic science. He argued that Peano's axioms cannot be proven non-circularly with the principle of induction, therefore concluding that arithmetic is a priori synthetic and not analytic. Poincaré then went on to say that mathematics cannot be deduced from logic since it is not analytic. His views were similar to those of Immanuel Kant. He strongly opposed Cantorian set theory, objecting to its use of impredicative definitions.

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. Poincaré held that convention plays an important role in physics. His view (and some later, more extreme versions of it) came to be known as "conventionalism." Poincaré believed that Newton's first law was not empirical but is a conventional framework assumption for mechanics.

He also believed that the geometry of physical space is conventional. He considered examples in which either the geometry of the physical fields or gradients of temperature can be changed, either describing space as non-Euclidean measured by rigid rulers or as a Euclidean space where the rulers are expanded or shrunk by a variable heat distribution. However, Poincaré thought that we were so accustomed to Euclidean geometry that we would prefer to change the physical laws to save Euclidean geometry rather than shift to non-Euclidean physical geometry.

Quotations: "Facts do not speak."

"The facts of science and, à fortiori, its laws are the artificial work of the scientist; science, therefore, can teach us nothing of the truth; it can only serve us as rule of action."

"Mathematics has a threefold purpose. It must provide an instrument for the study of nature. But this is not all: it has a philosophical purpose, and, I daresay, an aesthetic purpose."

"Principles are conventions and definitions in disguise."

"For a long time the objects that mathematicians dealt with were mostly ill-defined; one believed one knew them, but one represented them with the senses and imagination, but one had but a rough picture and not a precise idea on which reasoning could take hold."

"Astronomy is useful because it raises us above ourselves; it is useful because it is grand (it is useful because it is beautiful)… It shows us how small is man’s body, how great his mind since his intelligence can embrace the whole of this dazzling immensity, where his body is only an obscure point, and enjoy its silent harmony."

"Let a drop of wine fall into a glass of water; whatever be the law that governs the internal movement of the liquid, we will soon see it tint itself uniformly pink and from that moment on, however, we may agitate the vessel, it appears that the wine and water can separate no more. All this, Maxwell and Boltzmann have explained, but the one who saw it in the cleanest way, in a book that is too little read because it is difficult to read, is Gibbs, in his Principles of Statistical Mechanics."

"When the physicists ask us for the solution of a problem, it is not drudgery that they impose on us, on the contrary, it is us who owe them thanks."

"A scientist worthy of the name, above all a mathematician, experiences in his work the same impression as an artist; his pleasure is as great and of the same Nature."

"A very small cause which escapes our notice determines a considerable effect that we cannot fail to see, and then we say that the effect is due to chance. If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of that same universe at a succeeding moment."

"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…. The 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."

"All that we can hope from these inspirations, which are the fruits of unconscious work, is to obtain points of departure for such calculations. As for the calculations themselves, they must be made in the second period of conscious work which follows the inspiration, and in which the results of the inspiration are verified and the consequences deduced."

"All the scientist creates in a fact is the language in which he enunciates it. If he predicts a fact, he will employ this language, and for all those who can speak and understand it, his prediction is free from ambiguity. Moreover, this prediction once made, it evidently does not depend upon him whether it is fulfilled or not."

"Astronomy has not only taught us that there are laws, but that from these laws there is no escape, that with them there is no possible compromise."

"Before a complex of sensations becomes a recollection placeable in time, it has ceased to be actual. We must lose our awareness of its infinite complexity, or it is still actual... It is only after a memory has lost all life that it can be classed in time, just as only dissected flowers find their way into the herbarium of a botanist."

"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."

"Chance ... must be something more than the name we give to our ignorance."

"Consider now the Milky Way. Here also we see an innumerable dust, only the grains of this dust are no longer atoms but stars; these grains also move with great velocities, they act at a distance one upon another, but this action is so slight at great distances that their trajectories are rectilineal; nevertheless, from time to time, two of them may come near enough together to be deviated from their course, like a comet that passed too close to Jupiter. In a word, in the eyes of a giant, to whom our Suns were what our atoms are to us, the Milky Way would only look like a bubble of gas."

"Does the harmony the human intelligence thinks it discovers in nature exist outside of this intelligence? No, beyond doubt, a reality completely independent of the mind which conceives it, sees or feels it, is an impossibility."

"Every definition implies an axiom since it asserts the existence of the object defined. The definition then will not be justified, from the purely logical point of view, until we have ‘proved’ that it involves no contradiction either in its terms or with the truths previously admitted."

"Governments and parliaments must find that astronomy is one of the sciences which cost most dear: the least instrument costs hundreds of thousands of dollars, the least observatory costs millions; each eclipse carries with it supplementary appropriations. And all that for stars which are so far away, which are complete strangers to our electoral contests, and in all probability will never take any part in them. It must be that our politicians have retained a remnant of idealism, a vague instinct for what is grand; truly, I think they have been calumniated; they should be encouraged and shown that this instinct does not deceive them, that they are not dupes of that idealism."

"How is it that there are so many minds that are incapable of understanding mathematics?... the skeleton of our understanding,... and actually they are the majority... We have here a problem that is not easy of solution, but yet must engage the attention of all who wish to devote themselves to education."

"I then began to study arithmetical questions without any great apparent result, and without suspecting that they could have the least connexion with my previous researches. Disgusted at my want of success, I went away to spend a few days at the seaside and thought of entirely different things. One day, as I was walking on the cliff, the idea came to me, again with the same characteristics of conciseness, suddenness, and immediate certainty, that arithmetical transformations of indefinite ternary quadratic forms are identical with those of non-Euclidian geometry."

"I think, and I am not the only one who does, that it is important never to introduce any conception which may not be completely defined by a finite number of words. Whatever may be the remedy adopted, we can promise ourselves the joy of the physician called in to follow a beautiful pathological case."

"If all the parts of the universe are interchained in a certain measure, any one phenomenon will not be the effect of a single cause, but the resultant of causes infinitely numerous."

"If we ought not to fear mortal truth, still less should we dread scientific truth. In the first place, it can not conflict with ethics? But if science is feared, it is above all because it can give no happiness? Man, then, can not be happy through science but today he can much less be happy without it."

"If we wish to foresee the future of mathematics, our proper course is to study the history and present condition of the science."

"If we work, it is less to obtain those positive results the common people think are our only interest, than to feel that aesthetic emotion and communicate it to those able to experience it."

"In addition to this, it (mathematics) provides its disciples with pleasures similar to painting and music. They admire the delicate harmony of the numbers and the forms; they marvel when a new discovery opens up to them an unexpected vista, and does the joy that they feel not have an aesthetic character even if the senses are not involved at all?… For this reason, I do not hesitate to say that mathematics deserves to be cultivated for its own sake, and I mean the theories which cannot be applied to physics just as much as the others."

"It is a misfortune for a science to be born too late when the means of observation have become too perfect. That is what is happening at this moment with respect to physical chemistry; the founders are hampered in their general grasp by third and fourth decimal places."

"What, in fact, is mathematical discovery? It does not consist in making new combinations with mathematical entities that are already known. That can be done by anyone, and the combinations that could be so formed would be infinite in number, and the greater part of them would be absolutely devoid of interest. Discovery consists precisely in not constructing useless combinations, but in constructing those that are useful, which are an infinitely small minority. Discovery is discernment, selection."

Membership

Poincaré was appointed a member of the Académie Française in 1909 and elected a foreign member of the Royal Society in 1894. He was a member of the Société Astronomique de France (SAF), the French astronomical society, from 1901 to 1903, and of the American Philosophical Society from 1899. He was also a foreign member of the Royal Society of Edinburgh from 1895 and a foreign member of the Royal Netherlands Academy of Arts and Sciences (1897).

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  Royal Society of London , United Kingdom

  1894 - 1912

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  American Philosophical Society , United States

  1899 - 1912

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  Royal Society of Edinburgh , United Kingdom

  1895 - 1912

Personality

Even during his childhood Henri Poincaré was a very intelligent boy with a strong memory, he had to read a book only once to be able to recall on which page any specific line could be found. Although music was not his strong point, he loved music and acting, staging plays with his cousins.

Physical Characteristics: During his childhood, Henri Poincaré was seriously ill for a time with diphtheria. He also had weak eyes, but despite the fact that his eyesight was poor, he never took notes in class, and after reading a book he could recall the page on which any statement occurred.

Quotes from others about the person

• "Poincaré was a vigorous opponent of the theory that all mathematics can be rewritten in terms of the most elementary notions of classical logic; something more than logic, he believed, makes mathematics what it is." - Eric Temple Bell

Interests

• reading, music, acting, theater

Connections

Poincaré met his future wife named Louise Poulin d'Andesi (Jeanne Marie Louise Poulain d'Andécy) and on 20 April 1881, they were married. Together they had four children: Jeanne (born 1887), Yvonne (born 1889), Henriette (born 1891), and Léon (born 1893).

Father:

Charles Émile Léon Poincaré

1828–1892, Charles Émile Léon Poincaré father was a professor of Hygiene in the School of Medicine at the University of Nancy.

Mother:

Marie Pierrette Eugénie Launois

1830-1897

Wife:

Jeanne Marie LouisePoulain d'Andécy

1857-1934

Daughter:

Yvonne Poincaré

1889-1939

Daughter:

Henriette Burnier (Poincaré)

1890-1970

Daughter:

Jeanne Eugénie Pauline Daum (Poincaré)

1887-1975

Cousin:

Raymond Poincaré

Sister:

Aline Catherine Eugénie Boutroux (Poincaré)

Son:

Léon Maurice Poincaré

1893-1972

teacher:

Charles Hermite

After Poincaré entered the École Polytechnique as the top qualifier in 1873 and graduated in 1875, there he studied mathematics as a student of Charles Hermite.

associate:

Louis Bachelier

Louis Bachelier used to be one of Poincaré's students.

associate:

Dimitrie Pompeiu

References

• Jules Henri Poincaré
  2017
• The Value of Science
  2001
• Dictionary of Scientific Biography Volume 11
  1975
• Men of Mathematics
  1986
• Henri Poincare: A Scientific Biography
  2012
• Poincare and the Three Body Problem
  1996
• Henri Poincaré: Impatient Genius
  2012
• All Cry Chaos
  2011
• Mathematical Heritage of Henri Poincare
  1983
• Einstein's Clocks and Poincare's Maps
  2004
• La correspondance entre Henri Poincaré et les physiciens, chimistes et ingénieurs
  2008
• Henri Poincaré, le problème des trois corps
  2016
• Poincare's Prize: The Hundred-Year Quest to Solve One of Math's Greatest Puzzles
  2008
• Henri Poincaré, l'oeuvre philosophique
  2016
• Henri Poincaré
  2013
• Henri Poincaré
  1914
• Henri Poincaré Biographie, Bibliographie Analytique des Écrits
  2011
• Henri Poincaré
  2017
• Henri Poincaré:A Biography Through the Daily Papers
  2013
• Henri Poincaré; Biographie, Bibliographie Analytique des écrits
  2009
• Poincare's Philosophy: From Conventionalism to Phenomenology
  2001