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Jules Henri Poincaré (29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and a philosopher of science. He is often described as a polymath, and in mathematics as The Last Universalist, since he excelled in all fields of the discipline as it existed during his lifetime. As a mathematician and physicist, he made many original fundamental contributions to pure and applied mathematics, mathematical physics and celestial mechanics. He was responsible for formulating the Poincaré conjecture, one of the most famous unsolved problems in mathematics, until it was solved in 2002–3. 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. Poincaré made clear 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. The Poincaré group used in physics and mathematics was named after him. Poincaré was born on 29 April 1854 in Cité Ducale neighborhood, Nancy, Meurthe - et - Moselle into an influential family. His father Leon Poincaré (1828 – 1892) was a professor of medicine at the University of Nancy. His adored younger sister Aline married the spiritual philosopher Emile Boutroux. Another notable member of Jules' family was his cousin, Raymond Poincaré, who would become the President of France, 1913 to 1920, and a fellow member of the Académie française. During his childhood he was seriously ill for a time with diphtheria and received special instruction from his mother, Eugénie Launois (1830 – 1897). In 1862, Henri entered the Lycée in Nancy (now renamed the Lycée Henri Poincaré in his honor, along with the University of 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". However, poor eyesight and a tendency towards absentmindedness may explain these difficulties. 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. 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. He graduated in 1875 or 1876. He went on to study at the École des Mines, continuing to study mathematics in addition to the mining engineering syllabus and received the degree of ordinary engineer in March 1879. 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 sciences 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 différences. 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 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. Soon after, he was offered a post as junior lecturer in mathematics at Caen University, but 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). Eventually, he held the chairs of Physical and Experimental Mechanics, Mathematical Physics and Theory of Probability, and Celestial Mechanics and Astronomy. Also in that same year, Poincaré married Miss Poulain d'Andecy. Together they had four children: Jeanne (born 1887), Yvonne (born 1889), Henriette (born 1891), and Léon (born 1893). 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 in 1909. 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. In 1893, Poincaré joined the French Bureau des Longitudes, which engaged him in the synchronization of time around the world. In 1897 Poincaré backed an unsuccessful proposal for the decimalization of circular measure, and hence time and longitude. It was this post which led him to consider the question of establishing international time zones and the synchronization of time between bodies in relative motion. 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 fellow colleagues. 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, has recently proposed that Poincaré be reburied in the Panthéon in Paris, which is reserved for French citizens only of the highest honor. Poincaré had two notable doctoral students at the University of Paris, Louis Bachelier (1900) and Dimitrie Pompeiu (1905). 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 popularizer of mathematics and physics and wrote several books for the lay public. Among the specific topics he contributed to are the following:
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 nineteenth century. Indeed in 1887, in honor 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:
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). The version finally printed 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 generalized to the case of n > 3 bodies by Qiudong Wang in the 1990s. 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 synchronized. 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"
and introduced the hypothesis of length contraction to
explain the failure of optical and electrical experiments
to detect motion relative to the aether (Michelson –
Morley experiment).
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. Based on these assumptions he
discussed in 1900 Lorentz's "wonderful invention" of local
time and remarked that it arose when moving clocks are
synchronized by exchanging light signals assumed to travel
with the same speed in both directions in a moving frame. He discussed the "principle of relative motion" in two papers in 1900 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. 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. 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 gave what is now known as the relativistic velocity - addition law. 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:
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. Poincaré expressed a disinterest 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.
So it was Hermann Minkowski who worked out the
consequences of this notion in 1907. 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. 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). This time (and later also in 1908) he rejected the possibility that energy carries mass and criticized the ether solution to compensate the above mentioned problems:
He also discussed two other unexplained effects: (1) non - conservation of mass implied by Lorentz's variable mass γm, 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 Poincaré's paradox, without using any compensating mechanism within the ether. 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. Einstein's first paper on relativity was published three months after Poincaré's short paper, but before Poincaré's longer version. It relied on the principle of relativity to derive the Lorentz transformations and used a similar clock synchronization procedure (Einstein synchronisation) that Poincaré (1900) had described, but was remarkable in that it contained no references at all. Poincaré never acknowledged Einstein's work on special relativity. Einstein acknowledged Poincaré 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 recognized that the transformation named after him is essential for the analysis of Maxwell's equations, and Poincaré deepened this insight still further ...." Poincaré's
work in the development of special relativity is well
recognized, though most
historians stress that despite many similarities with
Einstein's work, the two had very different research
agendas and interpretations of the work. 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 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. 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. 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. He
believed that logic was not a way to invent but a way to
structure ideas and that logic limits ideas. Poincaré's mental organization was not only interesting to Poincaré himself but also to Toulouse, a psychologist of the Psychology Laboratory of the School of Higher Studies in Paris. Toulouse wrote a book entitled Henri Poincaré (1910). 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. His method of thinking is well summarized as: 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. Poincaré said, "There is no actual infinite; the Cantorians have forgotten this, and that is why they have fallen into contradiction." Poincaré saw mathematical work in economics and finance as peripheral. In 1900 Poincaré commented on Louis Bachelier's thesis "The Theory of Speculation", saying: "M. Bachelier has evidenced an original and precise mind [but] the subject is somewhat remote from those our other candidates are in the habit of treating." Bachelier's work explained what was then the French government's pricing options on French Bonds and anticipated many of the pricing theories in financial markets used today. 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:
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 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. 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. Although he most often spoke of a deterministic universe, Poincaré said that the subconscious generation of new possibilities involves chance.
Poincaré's two stages — random combinations followed by
selection — became the basis for Daniel Dennett's two -
stage model of free will. Charles Émile Picard FRS
(24 July 1856 – 11 December 1941) was a French
mathematician. He was elected the fifteenth member to
occupy seat 1 of the Académie
Française in 1924. Picard's mathematical papers, textbooks, and many popular writings exhibit an extraordinary range of interests, as well as an impressive mastery of the mathematics of his time. Modern students of complex variables are probably familiar with two of his named theorems. His lesser theorem states that every non constant entire function takes every value in the complex plane, with perhaps one exception. His greater theorem states that an analytic function with an essential singularity takes every value infinitely often, with perhaps one exception, in any neighborhood of the singularity. He also made important contributions in the theory of differential equations, including work on Picard – Vessiot theory, Painlevé transcendents and his introduction of a kind of symmetry group for a linear differential equation, the Picard group. In connection with his work on function theory, he was one of the first mathematicians to use the emerging ideas of algebraic topology. In addition to his path breaking theoretical work, Picard also made important contributions to applied mathematics, including the theories of telegraphy and elasticity. His collected papers run to four volumes. Like his contemporary, Henri Poincaré, Picard was much concerned with the training of mathematics, physics, and engineering students. He wrote a classic textbook on analysis and one of the first textbooks on the theory of relativity. Picard's popular writings include biographies of many leading French mathematicians, including his father in law, Charles Hermite. |