Henri poincare scientific biography of rory
Henri Poincaré
French mathematician, physicist and engineer (–)
For ships with this name, see French ship Henri Poincaré.
Jules Henri Poincaré (, ; French:[ɑ̃ʁipwɛ̃kaʁe]ⓘ;[1] 29 April &#;&#; 17 July ) 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",[2] since he excelled in all fields of the discipline as it existed during his lifetime. He has further been called "the Gauss of modern mathematics".[3] Due to his success in science, influence and philosophy, he has been called "the philosopher par excellence of modern science."[4]
As a mathematician and physicist, he made many original fundamental contributions to pure and applied mathematics, mathematical physics, and celestial mechanics.[5] 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.
Poincaré is regarded as the creator of the field of algebraic topology, and is further credited with introducing automorphic forms. He also made important contributions to algebraic geometry, number theory, complex analysis and Lie theory.[6] He famously introduced the concept of the Poincaré recurrence theorem, which states that a state will eventually return arbitrarily close to its initial state after a sufficiently long time, which has far-reaching consequences.[7] Early in the 20th century he formulated the Poincaré conjecture, which became, over time, one of the famous unsolved problems in mathematics.
It was eventually solved in – by Grigori Perelman. Poincaré popularized the use of non-Euclidean geometry in mathematics as well.[8]
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 Thus he obtained perfect invariance of all of Maxwell's equations, an important step in the formulation of the theory of special relativity, for which he is also credited with laying down the foundations for,[9] further writing foundational papers in [10] He first proposed gravitational waves (ondes gravifiques) emanating from a body and propagating at the speed of light as being required by the Lorentz transformations, doing so in [11] In , he wrote an influential paper which provided a mathematical argument for quantum mechanics.[12][13] Poincaré also laid the seeds of the discovery of radioactivity through his interest and study of X-rays, which influenced physicist Henri Becquerel, who then discovered the phenomena.[14] The Poincaré group used in physics and mathematics was named after him, after he introduced the notion of the group.[15]
Poincaré was considered the dominant figure in mathematics and theoretical physics during his time, and was the most respected mathematician of his time, being described as "the living brain of the rational sciences" by mathematician Paul Painlevé.[16] Philosopher Karl Popper regarded Poincaré as the greatest philosopher of science of all time,[17] with Poincaré also originating the conventionalist view in science.[18] Poincaré was a public intellectual in his time, and personally, he believed in political equality for all, while wary of the influence of anti-intellectual positions that the Catholic Church held at the time.[19] He served as the president of the French Academy of Sciences (), the president of Société astronomique de France (–), and twice the president of Société mathématique de France (, ).
Life
Poincaré was born on 29 April in Cité Ducale neighborhood, Nancy, Meurthe-et-Moselle, into an influential French family.[20] His father Léon Poincaré (–) was a professor of medicine at the University of Nancy.[21] His younger sister Aline married the spiritual philosopher Émile Boutroux.
Another notable member of Henri's family was his cousin, Raymond Poincaré, a fellow member of the Académie française, who was President of France from to , and three-time Prime Minister of France between and [22]
Education
During his childhood he was seriously ill for a time with diphtheria and received special instruction from his mother, Eugénie Launois (–).
In , 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".[23] Poor eyesight and a tendency towards absentmindedness may explain these difficulties.[24] He graduated from the Lycée in with a baccalauréat in both letters and sciences.
During the Franco-Prussian War of , he served alongside his father in the Ambulance Corps.
Poincaré entered the École Polytechnique as the top qualifier in and graduated in 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 From November to June 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 [25]
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 in which 18 miners died.
Henri poincare scientific biography of rory anderson His view and some later, more extreme versions of it came to be known as " conventionalism ". Please see your browser settings for this feature. Sur la dynamique de l' electron. HenriHe carried out the official investigation into the accident.
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. He graduated from the University of Paris in
First scientific achievements
After receiving his degree, Poincaré began teaching as junior lecturer in mathematics at the University of Caen in Normandy (in December ).
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 Poulain d'Andecy (–), granddaughter of Isidore Geoffroy Saint-Hilaire and great-granddaughter of Étienne Geoffroy Saint-Hilaire and on 20 April , they married.[26] Together they had four children: Jeanne (born ), Yvonne (born ), Henriette (born ), and Léon (born ).
Poincaré immediately established himself among the greatest mathematicians of Europe, attracting the attention of many prominent mathematicians. In 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 to , he taught mathematical analysis in the École Polytechnique.
In –, 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 career in the mining administration to mathematics. He worked at the Ministry of Public Services as an engineer in charge of northern railway development from to He eventually became chief engineer of the Corps des Mines in and inspector general in
Beginning in and for the rest of his career, he taught at the University of Paris (the Sorbonne).
Henri poincare scientific biography of rory and ryan Monthly Notices of the Royal Astronomical Society. As Miller notes in [ 21 ] :- Incredibly, he could work through page after page of detailed calculations, be it of the most abstract mathematical sort or pure number calculations, as he often did in physics, hardly ever crossing anything out. Buletin Valentin Isac. Vocations, IV Paris,He was initially appointed as the maître de conférences d'analyse (associate professor of analysis).[27] Eventually, he held the chairs of Physical and Experimental Mechanics, Mathematical Physics and Theory of Probability,[28] and Celestial Mechanics and Astronomy.
In , at the young age of 32, Poincaré was elected to the French Academy of Sciences.
He became its president in , and was elected to the Académie française on 5 March
In , 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.)
In , Poincaré joined the French Bureau des Longitudes, which engaged him in the synchronisation of time around the world.
In Poincaré backed an unsuccessful proposal for the decimalisation of circular measure, and hence time and longitude.[29] 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 , he intervened in the trials of Alfred Dreyfus, attacking the spurious scientific claims regarding evidence brought against Dreyfus.
Poincaré was the President of the Société Astronomique de France (SAF), the French astronomical society, from to [30]
Students
Poincaré had two notable doctoral students at the University of Paris, Louis Bachelier () and Dimitrie Pompeiu ().[31]
Death
In , Poincaré underwent surgery for a prostate problem and subsequently died from an embolism on 17 July , in Paris.
He was 58 years of age. He is buried in the Poincaré family vault in the Cemetery of Montparnasse, Paris, in section 16 close to the gate Rue Émile-Richard.
A former French Minister of Education, Claude Allègre, proposed in that Poincaré be reburied in the Panthéon in Paris, which is reserved for French citizens of the highest honour.[32]
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 mechanics, theory of relativity and physical cosmology.
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 , 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.
Henri poincare scientific biography of rory This will only follow from mathematical reasoning containing content and not just formal argument. Bell further noted his powerful memory, one that was even superior to Leonhard Euler 's, stating that: [ 85 ]. However, when the memoir was about to be published in Acta Mathematica , Phragmen , who was editing the memoir for publication, found an error. Rather he believed that formal arguments may reveal the mistakes of intuition and logical argument is the only means to confirm insights.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[34] and the book by Barrow-Green[35]).
The version finally printed[36] 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 and was generalised to the case of n&#;>&#;3 bodies by Qiudong Wang in the s. The series solutions have very slow convergence. It would take millions of terms to determine the motion of the particles for even very short intervals of time, so they are unusable in numerical work.[34]
Work on relativity
Main articles: Lorentz ether theory and History of special relativity
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 Lorentz had introduced an auxiliary quantity (without physical interpretation) called "local time" [37] 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).[38] 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 (), 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.[39] Based on these assumptions he discussed in 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.[40]
Principle of relativity and Lorentz transformations
Further information: History of Lorentz transformations
In Poincaré described hyperbolic geometry in terms of the hyperboloid model, formulating transformations leaving invariant the Lorentz interval, which makes them mathematically equivalent to the Lorentz transformations in 2+1 dimensions.[41][42] In addition, Poincaré's other models of hyperbolic geometry (Poincaré disk model, Poincaré half-plane model) as well as the Beltrami–Klein model can be related to the relativistic velocity space (see Gyrovector space).
In Poincaré developed a mathematical theory of light including polarization. His vision of the action of polarizers and retarders, acting on a sphere representing polarized states, is called the Poincaré sphere.[43] It was shown that the Poincaré sphere possesses an underlying Lorentzian symmetry, by which it can be used as a geometrical representation of Lorentz transformations and velocity additions.[44]
He discussed the "principle of relative motion" in two papers in [40][45] and named it the principle of relativity in , according to which no physical experiment can discriminate between a state of uniform motion and a state of rest.[46] In Poincaré wrote to Lorentz about Lorentz's paper of , 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.[47] 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.[48] Poincaré later delivered a paper at the meeting of the Academy of Sciences in Paris on 5 June in which these issues were addressed.
In the published version of that he wrote:[49]
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 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.[50] Poincaré expressed a lack of interest in a four-dimensional reformulation of his new mechanics in , because in his opinion the translation of physics into the language of four-dimensional geometry would entail too much effort for limited profit.[51] So it was Hermann Minkowski who worked out the consequences of this notion in [51][52]
Mass–energy relation
Like others before, Poincaré () 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.[40] 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.
Henri poincare scientific biography of rory end: Intelligencer 13 1 , 19 - He seemed to have made a good recovery, and was about to drive out for the first time this morning. Consequences Time dilation Length contraction Relativistic mass Mass—energy equivalence Relativity of simultaneity Relativistic Doppler effect Thomas precession Relativistic disk Bell's spaceship paradox Ehrenfest paradox. Toulouse's characterisation [ edit ].
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 ().[46] He rejected[53] the possibility that energy carries mass and criticized his own 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.
In the above quote he refers to the Hertz assumption of total aether entrainment that was falsified by the Fizeau experiment but that experiment does indeed show that that light is partially "carried along" with a substance. Finally in [54] he revisits the problem and ends with abandoning the principle of reaction altogether in favor of supporting a solution based in the inertia of aether itself.
But we have seen above that Fizeau's experiment does not permit of our retaining the theory of Hertz; it is necessary therefore to adopt the theory of Lorentz, and consequently to renounce the principle of reaction.
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 Marie Curie.
It was Albert Einstein's concept of mass–energy equivalence () that a body losing energy as radiation or heat was losing mass of amount m&#;=&#;E/c2 that resolved[55] Poincaré's paradox, without using any compensating mechanism within the ether.[56] 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 were mathematically equivalent.[57]
Gravitational waves
In Poincaré first proposed gravitational waves (ondes gravifiques) emanating from a body and propagating at the speed of light.
He wrote:
It has become important to examine this hypothesis more closely and in particular to ask in what ways it would require us to modify the laws of gravitation. That is what I have tried to determine; at first I was led to assume that the propagation of gravitation is not instantaneous, but happens with the speed of light.[58][49]
Poincaré and Einstein
Einstein's first paper on relativity was published three months after Poincaré's short paper,[49] but before Poincaré's longer version.[50] 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 () 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 , when Einstein considered Vaihinger's general outlook to be close to his own and Poincaré's to be close to Vaihinger's.[59] In public, Einstein acknowledged Poincaré posthumously in the text of a lecture in titled "Geometrie und Erfahrung (Geometry and Experience)" 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 ".[60]
Assessments on Poincaré and relativity
Further information: History of special relativity and Relativity priority dispute
Poincaré's work in the development of special relativity is well recognised,[55] though most historians stress that despite many similarities with Einstein's work, the two had very different research agendas and interpretations of the work.[61] 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.[62][63][64][65][66]
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.[67]
Algebra and number theory
Poincaré introduced group theory to physics, and was the first to study the group of Lorentz transformations.[68][69] He also made major contributions to the theory of discrete groups and their representations.
Topology
The subject is clearly defined by Felix Klein in his "Erlangen Program" (): 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.
Henri poincare scientific biography of rory and dean L'oeuvre de H. Other [ edit ]. Basel: Springer. Oscar II, King of Sweden and Norway, initiated a mathematical competition in to celebrate his sixtieth birthday inHis first article on this topic appeared in
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.[71]
Astronomy and celestial mechanics
Poincaré published two now classical monographs, "New Methods of Celestial Mechanics" (–) and "Lectures on Celestial Mechanics" (–).
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 (), 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.[72]
These monographs include an idea of Poincaré, which later became the basis 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 ().[73]
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" (–).[74] 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.[75]
- The singular points of the integral curves
Saddle
Focus
Center
Node
Character
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 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 (an intuitive), arguing that this is demonstrated by the fact that he worked so often by visual representation.
Jacques Hadamard wrote that Poincaré's research demonstrated marvelous clarity[76] 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 interesting not only 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 ().[77][78] In it, he discussed Poincaré's regular schedule:
- He worked during the same times each day in short periods of time. He undertook mathematical research for four hours a day, between 10&#;a.m. and noon then again from 5&#;p.m.
to 7&#;p.m.. He would read articles in journals later in the evening.
- His normal work habit was to solve a problem completely in his head, then commit the completed problem to paper.
- He was ambidextrous and nearsighted.
- His ability to visualise what he heard proved particularly useful when he attended lectures, since his eyesight was so poor that he could not see properly what the lecturer wrote on the blackboard.
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., ).
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 ()