Henri Poincaré

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Henri Poincaré
PSM V82 D416 Henri Poincare.png
Henri Poincaré
(photograph published in 1913)
Born(1854-04-29)29 April 1854
Died17 July 1912(1912-07-17) (aged 58)
Residence France
Nationality French
Other namesJules Henri Poincaré
EducationLycée Nancy (now Lycée Henri-Poincaré  [ fr ])
Alma mater
Known for
Awards
Scientific career
FieldsMathematics and physics
Institutions
Thesis Sur les propriétés des fonctions définies par les équations différences  (1879)
Doctoral advisor Charles Hermite
Doctoral students
Other notable students
Influences
Influenced
Signature
Henri Poincare Signature.svg
Notes
He was an uncle of Pierre Boutroux.

Jules Henri Poincaré ( /pwæ̃kɑːˈr/ ; [2] French:  [ɑ̃ʁi pwɛ̃kaʁe] ( Loudspeaker.svg listen ); [3] [4] 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The Last Universalist," [5] since he excelled in all fields of the discipline as it existed during his lifetime.

Mathematician person with an extensive knowledge of mathematics

A mathematician is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems.

Philosophy of science is a sub-field of philosophy concerned with the foundations, methods, and implications of science. The central questions of this study concern what qualifies as science, the reliability of scientific theories, and the ultimate purpose of science. This discipline overlaps with metaphysics, ontology, and epistemology, for example, when it explores the relationship between science and truth.

Polymath person whose expertise spans a significant number of different subject areas

A polymath is a person whose expertise spans a significant number of subject areas, known to draw on complex bodies of knowledge to solve specific problems.

Contents

As a mathematician and physicist, he made many original fundamental contributions to pure and applied mathematics, mathematical physics, and celestial mechanics. [6] 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 which laid the foundations of modern chaos theory. He is also considered to be one of the founders of the field of topology.

Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, but the pure mathematicians are not primarily motivated by such applications. Instead, the appeal is attributed to the intellectual challenge and esthetic beauty of working out the logical consequences of basic principles.

Applied mathematics Application of mathematical methods to other fields

Applied mathematics is the application of mathematical methods by different fields such as science, engineering, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models. In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics where abstract concepts are studied for their own sake. The activity of applied mathematics is thus intimately connected with research in pure mathematics.

Mathematical physics Application of mathematical methods to problems in physics

Mathematical physics refers to the development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines the field as "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories". It is a branch of applied mathematics, but deals with physical problems.

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 Hendrik Lorentz 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. In 1905, Poincaré first proposed gravitational waves (ondes gravifiques) emanating from a body and propagating at the speed of light as being required by the Lorentz transformations.

Hendrik Lorentz Dutch physicist

Hendrik Antoon Lorentz was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect. He also derived the transformation equations underpinning Albert Einstein's theory of special relativity.

Maxwells equations set of partial differential equations that describe how electric and magnetic fields are generated and altered by each other and by charges and currents

Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar etc. Maxwell's equations describe how electric and magnetic fields are generated by charges, currents, and changes of the fields. One important consequence of the equations is that they demonstrate how fluctuating electric and magnetic fields propagate at the speed of light. Known as electromagnetic radiation, these waves may occur at various wavelengths to produce a spectrum from radio waves to γ-rays. The equations are named after the physicist and mathematician James Clerk Maxwell, who between 1861 and 1862 published an early form of the equations that included the Lorentz force law. He also first used the equations to propose that light is an electromagnetic phenomenon.

Special relativity Theory of interwoven space and time by Albert Einstein

In physics, special relativity is the generally accepted and experimentally well-confirmed physical theory regarding the relationship between space and time. In Albert Einstein's original pedagogical treatment, it is based on two postulates:

  1. the laws of physics are invariant in all inertial systems ; and
  2. the speed of light in a vacuum is the same for all observers, regardless of the motion of the light source.

The Poincaré group used in physics and mathematics was named after him.

Poincaré group group of isometries of Minkowski spacetime

The Poincaré group, named after Henri Poincaré (1906), was first defined by Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group of fundamental importance in physics.

Life

Poincaré was born on 29 April 1854 in Cité Ducale neighborhood, Nancy, Meurthe-et-Moselle into an influential family. [7] His father Leon Poincaré (1828–1892) was a professor of medicine at the University of Nancy. [8] His younger sister Aline married the spiritual philosopher Emile Boutroux. Another notable member of Henri's family was his cousin, Raymond Poincaré, a fellow member of the Académie française, who would serve as President of France from 1913 to 1920. [9]

Raymond Poincaré French statesman and lawyer

Raymond Nicolas Landry Poincaré was a French statesman who served three times as 58th Prime Minister of France, and as President of France from 1913 to 1920. He was a conservative leader, primarily committed to political and social stability.

Académie française Pre-eminent council for the French language

The Académie française is the pre-eminent French council for matters pertaining to the French language. The Académie was officially established in 1635 by Cardinal Richelieu, the chief minister to King Louis XIII. Suppressed in 1793 during the French Revolution, it was restored as a division of the Institut de France in 1803 by Napoleon Bonaparte. It is the oldest of the five académies of the institute.

Education

Plaque on the birthplace of Henri Poincare at house number 117 on the Grande Rue in the city of Nancy. Henri Poincare maison natale Nancy plaque.jpg
Plaque on the birthplace of Henri Poincaré at house number 117 on the Grande Rue in the city of Nancy.

During his childhood he was seriously ill for a time with diphtheria and received special instruction from his mother, Eugénie Launois (1830–1897).

Diphtheria Infectious disease

Diphtheria is an infection caused by the bacterium Corynebacterium diphtheriae. Signs and symptoms may vary from mild to severe. They usually start two to five days after exposure. Symptoms often come on fairly gradually, beginning with a sore throat and fever. In severe cases, a grey or white patch develops in the throat. This can block the airway and create a barking cough as in croup. The neck may swell in part due to enlarged lymph nodes. A form of diphtheria that involves the skin, eyes, or genitals also exists. Complications may include myocarditis, inflammation of nerves, kidney problems, and bleeding problems due to low levels of platelets. Myocarditis may result in an abnormal heart rate and inflammation of the nerves may result in paralysis.

In 1862, Henri entered the Lycée in Nancy (now renamed the Lycée Henri-Poincaré  [ fr ] in his honour, along with Henri Poincaré University, also in Nancy). He spent eleven years at the Lycée and during this time he proved to be one of the top students in every topic he studied. He excelled in written composition. His mathematics teacher described him as a "monster of mathematics" and he won first prizes in the concours général, a competition between the top pupils from all the Lycées across France. His poorest subjects were music and physical education, where he was described as "average at best". [10] However, poor eyesight and a tendency towards absentmindedness may explain these difficulties. [11] He graduated from the Lycée in 1871 with a bachelor's degree in letters and sciences.

During the Franco-Prussian War of 1870, he served alongside his father in the Ambulance Corps.

Poincaré entered the École Polytechnique in 1873 and graduated in 1875. There he studied mathematics as a student of Charles Hermite, continuing to excel and publishing his first paper (Démonstration nouvelle des propriétés de l'indicatrice d'une surface) in 1874. From November 1875 to June 1878 he studied at the École des Mines, while continuing the study of mathematics in addition to the mining engineering syllabus, and received the degree of ordinary mining engineer in March 1879. [12]

As a graduate of the École des Mines, he joined the Corps des Mines as an inspector for the Vesoul region in northeast France. He was on the scene of a mining disaster at Magny in August 1879 in which 18 miners died. He carried out the official investigation into the accident in a characteristically thorough and humane way.

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. Poincaré devised a new way of studying the properties of these equations. He not only faced the question of determining the integral of such equations, but also was the first person to study their general geometric properties. He realised that they could be used to model the behaviour of multiple bodies in free motion within the solar system. Poincaré graduated from the University of Paris in 1879.

The young Henri Poincare Young Poincare.jpg
The young Henri Poincaré

First scientific achievements

After receiving his degree, Poincaré began teaching as junior lecturer in mathematics at the University of Caen in Normandy (in December 1879). At the same time he published his first major article concerning the treatment of a class of automorphic functions.

There, in Caen, he met his future wife, Louise Poulin d'Andesi (Louise Poulain d'Andecy) and on 20 April 1881, they married. Together they had four children: Jeanne (born 1887), Yvonne (born 1889), Henriette (born 1891), and Léon (born 1893).

Poincaré immediately established himself among the greatest mathematicians of Europe, attracting the attention of many prominent mathematicians. In 1881 Poincaré was invited to take a teaching position at the Faculty of Sciences of the University of Paris; he accepted the invitation. During the years of 1883 to 1897, he taught mathematical analysis in École Polytechnique.

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.

Career

He never fully abandoned his mining career to mathematics. He worked at the Ministry of Public Services as an engineer in charge of northern railway development from 1881 to 1885. He eventually became chief engineer of the Corps de Mines in 1893 and inspector general in 1910.

Beginning in 1881 and for the rest of his career, he taught at the University of Paris (the Sorbonne). He was initially appointed as the maître de conférences d'analyse (associate professor of analysis). [13] Eventually, he held the chairs of Physical and Experimental Mechanics, Mathematical Physics and Theory of Probability [14] , and Celestial Mechanics and Astronomy.

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 5 March 1908.

In 1887, he 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. (See three-body problem section below.)

The Poincare family grave at the Cimetiere du Montparnasse Poincare gravestone.jpg
The Poincaré family grave at the Cimetière du Montparnasse

In 1893, Poincaré joined the French Bureau des Longitudes, which engaged him in the synchronisation of time around the world. In 1897 Poincaré backed an unsuccessful proposal for the decimalisation of circular measure, and hence time and longitude. [15] It was this post which led him to consider the question of establishing international time zones and the synchronisation of time between bodies in relative motion. (See work on relativity section below.)

In 1899, and again more successfully in 1904, he 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.

Poincaré was the President of the Société Astronomique de France (SAF), the French astronomical society, from 1901 to 1903. [16]

Students

Poincaré had two notable doctoral students at the University of Paris, Louis Bachelier (1900) and Dimitrie Pompeiu (1905). [17]

Death

In 1912, Poincaré underwent surgery for a prostate problem and subsequently died from an embolism on 17 July 1912, in Paris. He was 58 years of age. He is buried in the Poincaré family vault in the Cemetery of Montparnasse, Paris.

A former French Minister of Education, Claude Allègre, proposed in 2004 that Poincaré be reburied in the Panthéon in Paris, which is reserved for French citizens only of the highest honour. [18]

Work

Summary

Poincaré made many contributions to different fields of pure and applied mathematics such as: celestial mechanics, fluid mechanics, optics, electricity, telegraphy, capillarity, elasticity, thermodynamics, potential theory, quantum theory, theory of relativity and physical cosmology.

He was also a populariser of mathematics and physics and wrote several books for the lay public.

Among the specific topics he contributed to are the following:

Three-body problem

The problem of finding the general solution to the motion of more than two orbiting bodies in the solar system had eluded mathematicians since Newton's time. This was known originally as the three-body problem and later the n-body problem, where n is any number of more than two orbiting bodies. The n-body solution was considered very important and challenging at the close of the 19th century. Indeed, in 1887, in honour of his 60th birthday, Oscar II, King of Sweden, advised by Gösta Mittag-Leffler, established a prize for anyone who could find the solution to the problem. The announcement was quite specific:

Given a system of arbitrarily many mass points that attract each according to Newton's law, under the assumption that no two points ever collide, try to find a representation of the coordinates of each point as a series in a variable that is some known function of time and for all of whose values the series converges uniformly.

In case the problem could not be solved, any other important contribution to classical mechanics would then be considered to be prizeworthy. The prize was finally awarded to Poincaré, even though he did not solve the original problem. One of the judges, the distinguished Karl Weierstrass, said, "This work cannot indeed be considered as furnishing the complete solution of the question proposed, but that it is nevertheless of such importance that its publication will inaugurate a new era in the history of celestial mechanics." (The first version of his contribution even contained a serious error; for details see the article by Diacu [21] and the book by Barrow-Green [22] ). The version finally printed [23] contained many important ideas which led to the theory of chaos. The problem as stated originally was finally solved by Karl F. Sundman for n = 3 in 1912 and was generalised to the case of n > 3 bodies by Qiudong Wang in the 1990s.

Work on relativity

Marie Curie and Poincare talk at the 1911 Solvay Conference Curie and Poincare 1911 Solvay.jpg
Marie Curie and Poincaré talk at the 1911 Solvay Conference

Local time

Poincaré's work at the Bureau des Longitudes on establishing international time zones led him to consider how clocks at rest on the Earth, which would be moving at different speeds relative to absolute space (or the "luminiferous aether"), could be synchronised. At the same time Dutch theorist Hendrik Lorentz was developing Maxwell's theory into a theory of the motion of charged particles ("electrons" or "ions"), and their interaction with radiation. In 1895 Lorentz had introduced an auxiliary quantity (without physical interpretation) called "local time" [24] and introduced the hypothesis of length contraction to explain the failure of optical and electrical experiments to detect motion relative to the aether (see Michelson–Morley experiment). [25] Poincaré was a constant interpreter (and sometimes friendly critic) of Lorentz's theory. Poincaré as a philosopher was interested in the "deeper meaning". Thus he interpreted Lorentz's theory and in so doing he came up with many insights that are now associated with special relativity. In The Measure of Time (1898), Poincaré said, " A little reflection is sufficient to understand that all these affirmations have by themselves no meaning. They can have one only as the result of a convention." He also argued that scientists have to set the constancy of the speed of light as a postulate to give physical theories the simplest form. [26] Based on these assumptions he discussed in 1900 Lorentz's "wonderful invention" of local time and remarked that it arose when moving clocks are synchronised by exchanging light signals assumed to travel with the same speed in both directions in a moving frame. [27]

Principle of relativity and Lorentz transformations

In 1881 Poincaré described hyperbolic geometry in terms of Weierstrass coordinates of the hyperboloid model. There, he formulated transformations leaving invariant the Lorentz interval , which makes them mathematically equivalent to the Lorentz transformations in 2+1 dimensions. [28] [29]

He discussed the "principle of relative motion" in two papers in 1900 [27] [30] and named it the principle of relativity in 1904, according to which no physical experiment can discriminate between a state of uniform motion and a state of rest. [31] In 1905 Poincaré wrote to Lorentz about Lorentz's paper of 1904, which Poincaré described as a "paper of supreme importance." In this letter he pointed out an error Lorentz had made when he had applied his transformation to one of Maxwell's equations, that for charge-occupied space, and also questioned the time dilation factor given by Lorentz. [32] In a second letter to Lorentz, Poincaré gave his own reason why Lorentz's time dilation factor was indeed correct after all—it was necessary to make the Lorentz transformation form a group—and he gave what is now known as the relativistic velocity-addition law. [33] Poincaré later delivered a paper at the meeting of the Academy of Sciences in Paris on 5 June 1905 in which these issues were addressed. In the published version of that he wrote: [34]

The essential point, established by Lorentz, is that the equations of the electromagnetic field are not altered by a certain transformation (which I will call by the name of Lorentz) of the form:

and showed that the arbitrary function must be unity for all (Lorentz had set by a different argument) to make the transformations form a group. In an enlarged version of the paper that appeared in 1906 Poincaré pointed out that the combination is invariant. He noted that a Lorentz transformation is merely a rotation in four-dimensional space about the origin by introducing as a fourth imaginary coordinate, and he used an early form of four-vectors. [35] Poincaré expressed a lack of interest in a four-dimensional reformulation of his new mechanics in 1907, because in his opinion the translation of physics into the language of four-dimensional geometry would entail too much effort for limited profit. [36] So it was Hermann Minkowski who worked out the consequences of this notion in 1907.

Mass–energy relation

Like others before, Poincaré (1900) discovered a relation between mass and electromagnetic energy. While studying the conflict between the action/reaction principle and Lorentz ether theory, he tried to determine whether the center of gravity still moves with a uniform velocity when electromagnetic fields are included. [27] He noticed that the action/reaction principle does not hold for matter alone, but that the electromagnetic field has its own momentum. Poincaré concluded that the electromagnetic field energy of an electromagnetic wave behaves like a fictitious fluid (fluide fictif) with a mass density of E/c2. If the center of mass frame is defined by both the mass of matter and the mass of the fictitious fluid, and if the fictitious fluid is indestructible—it's neither created or destroyed—then the motion of the center of mass frame remains uniform. But electromagnetic energy can be converted into other forms of energy. So Poincaré assumed that there exists a non-electric energy fluid at each point of space, into which electromagnetic energy can be transformed and which also carries a mass proportional to the energy. In this way, the motion of the center of mass remains uniform. Poincaré said that one should not be too surprised by these assumptions, since they are only mathematical fictions.

However, Poincaré's resolution led to a paradox when changing frames: if a Hertzian oscillator radiates in a certain direction, it will suffer a recoil from the inertia of the fictitious fluid. Poincaré performed a Lorentz boost (to order v/c) to the frame of the moving source. He noted that energy conservation holds in both frames, but that the law of conservation of momentum is violated. This would allow perpetual motion, a notion which he abhorred. The laws of nature would have to be different in the frames of reference, and the relativity principle would not hold. Therefore, he argued that also in this case there has to be another compensating mechanism in the ether.

Poincaré himself came back to this topic in his St. Louis lecture (1904). [31] This time (and later also in 1908) he rejected [37] the possibility that energy carries mass and criticized the ether solution to compensate the above-mentioned problems:

The apparatus will recoil as if it were a cannon and the projected energy a ball, and that contradicts the principle of Newton, since our present projectile has no mass; it is not matter, it is energy. [..] Shall we say that the space which separates the oscillator from the receiver and which the disturbance must traverse in passing from one to the other, is not empty, but is filled not only with ether, but with air, or even in inter-planetary space with some subtile, yet ponderable fluid; that this matter receives the shock, as does the receiver, at the moment the energy reaches it, and recoils, when the disturbance leaves it? That would save Newton's principle, but it is not true. If the energy during its propagation remained always attached to some material substratum, this matter would carry the light along with it and Fizeau has shown, at least for the air, that there is nothing of the kind. Michelson and Morley have since confirmed this. We might also suppose that the motions of matter proper were exactly compensated by those of the ether; but that would lead us to the same considerations as those made a moment ago. The principle, if thus interpreted, could explain anything, since whatever the visible motions we could imagine hypothetical motions to compensate them. But if it can explain anything, it will allow us to foretell nothing; it will not allow us to choose between the various possible hypotheses, since it explains everything in advance. It therefore becomes useless.

He also discussed two other unexplained effects: (1) non-conservation of mass implied by Lorentz's variable mass , Abraham's theory of variable mass and Kaufmann's experiments on the mass of fast moving electrons and (2) the non-conservation of energy in the radium experiments of Madame Curie.

It was Albert Einstein's concept of mass–energy equivalence (1905) that a body losing energy as radiation or heat was losing mass of amount m = E/c2 that resolved [38] Poincaré's paradox, without using any compensating mechanism within the ether. [39] The Hertzian oscillator loses mass in the emission process, and momentum is conserved in any frame. However, concerning Poincaré's solution of the Center of Gravity problem, Einstein noted that Poincaré's formulation and his own from 1906 were mathematically equivalent. [40]

Gravitational waves

In 1905 Henri Poincaré first proposed gravitational waves (ondes gravifiques) emanating from a body and propagating at the speed of light. [34] "Il importait d'examiner cette hypothèse de plus près et en particulier de rechercher quelles modifications elle nous obligerait à apporter aux lois de la gravitation. C'est ce que j'ai cherché à déterminer; j'ai été d'abord conduit à supposer que la propagation de la gravitation n'est pas instantanée, mais se fait avec la vitesse de la lumière."

Poincaré and Einstein

Einstein's first paper on relativity was published three months after Poincaré's short paper, [34] but before Poincaré's longer version. [35] Einstein relied on the principle of relativity to derive the Lorentz transformations and used a similar clock synchronisation procedure (Einstein synchronisation) to the one that Poincaré (1900) had described, but Einstein's paper was remarkable in that it contained no references at all. Poincaré never acknowledged Einstein's work on special relativity. However, Einstein expressed sympathy with Poincaré's outlook obliquely in a letter to Hans Vaihinger on 3 May 1919, when Einstein considered Vaihinger's general outlook to be close to his own and Poincaré's to be close to Vaihinger's. [41] In public, Einstein acknowledged Poincaré posthumously in the text of a lecture in 1921 called Geometrie und Erfahrung in connection with non-Euclidean geometry, but not in connection with special relativity. A few years before his death, Einstein commented on Poincaré as being one of the pioneers of relativity, saying "Lorentz had already recognised that the transformation named after him is essential for the analysis of Maxwell's equations, and Poincaré deepened this insight still further ...." [42]

Assessments on Poincaré and relativity

Poincaré's work in the development of special relativity is well recognised, [38] though most historians stress that despite many similarities with Einstein's work, the two had very different research agendas and interpretations of the work. [43] Poincaré developed a similar physical interpretation of local time and noticed the connection to signal velocity, but contrary to Einstein he continued to use the ether-concept in his papers and argued that clocks at rest in the ether show the "true" time, and moving clocks show the local time. So Poincaré tried to keep the relativity principle in accordance with classical concepts, while Einstein developed a mathematically equivalent kinematics based on the new physical concepts of the relativity of space and time. [44] [45] [46] [47] [48]

While this is the view of most historians, a minority go much further, such as E. T. Whittaker, who held that Poincaré and Lorentz were the true discoverers of relativity. [49]

Algebra and number theory

Poincaré introduced group theory to physics, and was the first to study the group of Lorentz transformations. [50] He also made major contributions to the theory of discrete groups and their representations.

Topological transformation of the torus into a mug Mug and Torus morph.gif
Topological transformation of the torus into a mug

Topology

The subject is clearly defined by Felix Klein in his "Erlangen Program" (1872): the geometry invariants of arbitrary continuous transformation, a kind of geometry. The term "topology" was introduced, as suggested by Johann Benedict Listing, instead of previously used "Analysis situs". Some important concepts were introduced by Enrico Betti and Bernhard Riemann. But the foundation of this science, for a space of any dimension, was created by Poincaré. His first article on this topic appeared in 1894. [51]

His research in geometry led to the abstract topological definition of homotopy and homology. He also first introduced the basic concepts and invariants of combinatorial topology, such as Betti numbers and the fundamental group. Poincaré proved a formula relating the number of edges, vertices and faces of n-dimensional polyhedron (the Euler–Poincaré theorem) and gave the first precise formulation of the intuitive notion of dimension. [52]

Astronomy and celestial mechanics

Chaotic motion in three-body problem (computer simulation). N-body problem (3).gif
Chaotic motion in three-body problem (computer simulation).

Poincaré published two now classical monographs, "New Methods of Celestial Mechanics" (1892–1899) and "Lectures on Celestial Mechanics" (1905–1910). In them, he successfully applied the results of their research to the problem of the motion of three bodies and studied in detail the behavior of solutions (frequency, stability, asymptotic, and so on). They introduced the small parameter method, fixed points, integral invariants, variational equations, the convergence of the asymptotic expansions. Generalizing a theory of Bruns (1887), Poincaré showed that the three-body problem is not integrable. In other words, the general solution of the three-body problem can not be expressed in terms of algebraic and transcendental functions through unambiguous coordinates and velocities of the bodies. His work in this area was the first major achievement in celestial mechanics since Isaac Newton. [53]

These monographs include an idea of Poincaré, which later became the base for mathematical "chaos theory" (see, in particular, the Poincaré recurrence theorem) and the general theory of dynamical systems. Poincaré authored important works on astronomy for the equilibrium figures of a gravitating rotating fluid. He introduced the important concept of bifurcation points and proved the existence of equilibrium figures such as the non-ellipsoids, including ring-shaped and pear-shaped figures, and their stability. For this discovery, Poincaré received the Gold Medal of the Royal Astronomical Society (1900). [54]

Differential equations and mathematical physics

After defending his doctoral thesis on the study of singular points of the system of differential equations, Poincaré wrote a series of memoirs under the title "On curves defined by differential equations" (1881–1882). [55] In these articles, he built a new branch of mathematics, called "qualitative theory of differential equations". Poincaré showed that even if the differential equation can not be solved in terms of known functions, yet from the very form of the equation, a wealth of information about the properties and behavior of the solutions can be found. In particular, Poincaré investigated the nature of the trajectories of the integral curves in the plane, gave a classification of singular points (saddle, focus, center, node), introduced the concept of a limit cycle and the loop index, and showed that the number of limit cycles is always finite, except for some special cases. Poincaré also developed a general theory of integral invariants and solutions of the variational equations. For the finite-difference equations, he created a new direction – the asymptotic analysis of the solutions. He applied all these achievements to study practical problems of mathematical physics and celestial mechanics, and the methods used were the basis of its topological works. [56] [57]

Character

Photographic portrait of H. Poincare by Henri Manuel Henri Poincare by H Manuel.jpg
Photographic portrait of H. Poincaré by Henri Manuel

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 and 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.

The mathematician Darboux claimed he was un intuitif (intuitive), arguing that this is demonstrated by the fact that he worked so often by visual representation. He did not care about being rigorous and disliked logic. [58] (Despite this opinion, Jacques Hadamard wrote that Poincaré's research demonstrated marvelous clarity [59] and Poincaré himself wrote that he believed that logic was not a way to invent but a way to structure ideas and that logic limits ideas.)

Toulouse's characterisation

Poincaré's mental organisation was not only interesting to Poincaré himself but also to Édouard Toulouse, a psychologist of the Psychology Laboratory of the School of Higher Studies in Paris. Toulouse wrote a book entitled Henri Poincaré (1910). [60] [61] In it, he discussed Poincaré's regular schedule:

These abilities were offset to some extent by his shortcomings:

In addition, Toulouse stated that most mathematicians worked from principles already established while Poincaré started from basic principles each time (O'Connor et al., 2002).

His method of thinking is well summarised 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 centre étaient instantanément et automatiquement classés dans sa mémoire. (Accustomed to neglecting details and to looking only at mountain tops, he went from one peak to another with surprising rapidity, and the facts he discovered, clustering around their center, were instantly and automatically pigeonholed in his memory.)

Belliver (1956)

Attitude towards transfinite numbers

Poincaré was dismayed by Georg Cantor's theory of transfinite numbers, and referred to it as a "disease" from which mathematics would eventually be cured. [62] Poincaré said, "There is no actual infinite; the Cantorians have forgotten this, and that is why they have fallen into contradiction." [63]

Honours

Awards

Named after him

Henri Poincaré did not receive the Nobel Prize in Physics, but he had influential advocates like Henri Becquerel or committee member Gösta Mittag-Leffler. [65] [66] The nomination archive reveals that Poincaré received a total of 51 nominations between 1904 and 1912, the year of his death. [67] Of the 58 nominations for the 1910 Nobel Prize, 34 named Poincaré. [67] Nominators included Nobel laureates Hendrik Lorentz and Pieter Zeeman (both of 1902), Marie Curie (of 1903), Albert Michelson (of 1907), Gabriel Lippmann (of 1908) and Guglielmo Marconi (of 1909). [67]

The fact that renowned theoretical physicists like Poincaré, Boltzmann or Gibbs were not awarded the Nobel Prize is seen as evidence that the Nobel committee had more regard for experimentation than theory. [68] [69] In Poincaré's case, several of those who nominated him pointed out that the greatest problem was to name a specific discovery, invention, or technique. [65]

Philosophy

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 (Murzi, 1998), 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 (Kolak, 2001, Folina 1992). He strongly opposed Cantorian set theory, objecting to its use of impredicative definitions[ citation needed ].

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 (Gargani, 2012). [70] 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 a 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 a non-Euclidean physical geometry. [71]

Free will

Poincaré's famous lectures before the Société de Psychologie in Paris (published as Science and Hypothesis , The Value of Science , and Science and Method) were cited by Jacques Hadamard as the source for the idea that creativity and invention consist of two mental stages, first random combinations of possible solutions to a problem, followed by a critical evaluation. [72]

Although he most often spoke of a deterministic universe, Poincaré said that the subconscious generation of new possibilities involves chance.

It is certain that the combinations which present themselves to the mind in a kind of sudden illumination after a somewhat prolonged period of unconscious work are generally useful and fruitful combinations... all the combinations are formed as a result of the automatic action of the subliminal ego, but those only which are interesting find their way into the field of consciousness... A few only are harmonious, and consequently at once useful and beautiful, and they will be capable of affecting the geometrician's special sensibility I have been speaking of; which, once aroused, will direct our attention upon them, and will thus give them the opportunity of becoming conscious... In the subliminal ego, on the contrary, there reigns what I would call liberty, if one could give this name to the mere absence of discipline and to disorder born of chance. [73]

Poincaré's two stages—random combinations followed by selection—became the basis for Daniel Dennett's two-stage model of free will. [74]

See also

Concepts

Theorems

Other

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References

Footnotes and primary sources

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  8. Sagaret, 1911
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  11. Carl, 1968
  12. F. Verhulst
  13. Sageret, 1911
  14. Laurent Mazliak. « Poincaré’s Odds ». In : Poincaré 1912-2012 : Poincaré Seminar 2012.  B. Duplantier et V. Rivasseau, Editors. T. 67. Progress in Mathematical Physics. Basel : Birkhäuser

  15. see Galison 2003
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Poincaré's writings in English translation

Popular writings on the philosophy of science:

On algebraic topology:

On celestial mechanics:

On the philosophy of mathematics:

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General references

Secondary sources to work on relativity

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