Poincaré made clear the importance of paying attention to theinvariance of laws of physics under different transformations, and was the first to present theLorentz transformations in their modern symmetrical form. Poincaré discovered the remainingrelativistic velocity transformations and recorded them in a letter toHendrik Lorentz in 1905. Thus he obtained perfect invariance of all ofMaxwell's equations, an important step in the formulation of the theory ofspecial relativity, for which he is also credited with laying down the foundations,[10] further writing foundational papers in 1905.[11] He first proposedgravitational waves (ondes gravifiques) emanating from a body and propagating at the speed of light as being required by the Lorentz transformations, doing so in 1905.[12] In 1912, he wrote an influential paper which provided a mathematical argument forquantum mechanics.[13][14] Poincaré also laid the seeds of the discovery ofradioactivity through his interest and study ofX-rays, which influenced physicistHenri Becquerel, who then discovered the phenomena.[15] ThePoincaré group used in physics andmathematics was named after him, after he introduced the notion of the group.[16]
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 withdiphtheria and received special instruction from his mother, Eugénie Launois (1830–1897).
In 1862, Henri entered the Lycée inNancy (now renamed theLycée Henri-Poincaré [fr] in his honour, along withHenri 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 theconcours 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".[24] Poor eyesight and a tendency towards absentmindedness may explain these difficulties.[25] He graduated from the Lycée in 1871 with abaccalauréat in both letters and sciences.
Poincaré entered theÉcole Polytechnique as the top qualifier in 1873 and graduated in 1875. There he studied mathematics as a student ofCharles 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 theÉcole des Mines, while continuing the study of mathematics in addition to themining engineering syllabus, and received the degree of ordinary mining engineer in March 1879.[26]
As a graduate of the École des Mines, he joined theCorps des Mines as an inspector for theVesoul region in northeast France. He was on the scene of a mining disaster atMagny in August 1879 in which 18 miners died. He carried out the official investigation into the accident.
At the same time, Poincaré was preparing for hisDoctorate in Science in mathematics under the supervision of Charles Hermite. His doctoral thesis was in the field ofdifferential equations. It was namedSur 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 theSolar System. He graduated from theUniversity of Paris in 1879.
After receiving his degree, Poincaré began teaching as juniorlecturer in mathematics at theUniversity of Caen in Normandy (in December 1879). At the same time he published his first major article concerning the treatment of a class ofautomorphic functions.
There, inCaen, he met his future wife, Louise Poulain d'Andecy (1857–1934), granddaughter ofIsidore Geoffroy Saint-Hilaire and great-granddaughter ofÉtienne Geoffroy Saint-Hilaire and on 20 April 1881, they married.[27] 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 theUniversity of Paris; he accepted the invitation. During the years 1883 to 1897, he taughtmathematical analysis in theÉcole 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 incelestial mechanics andmathematical physics.
He never fully abandoned his career in the mining administration to mathematics. He worked at theMinistry of Public Services as an engineer in charge of northern railway development from 1881 to 1885. He eventually became chief engineer of theCorps des Mines in 1893 and inspector general in 1910.
Beginning in 1881 and for the rest of his career, he taught at theUniversity of Paris (theSorbonne). He was initially appointed as themaître de conférences d'analyse (associate professor of analysis).[28] Eventually, he held the chairs of Physical and Experimental Mechanics, Mathematical Physics and Theory of Probability,[29] and Celestial Mechanics and Astronomy.
In 1893, Poincaré joined the FrenchBureau des Longitudes, which engaged him in thesynchronisation of time around the world. In 1897 Poincaré backed an unsuccessful proposal for thedecimalisation of circular measure, and hence time andlongitude.[31] 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. (Seework on relativity section below.)
In 1904, he intervened in thetrials ofAlfred Dreyfus, attacking the spurious scientific claims regarding evidence brought against Dreyfus.
In 1912, Poincaré underwent surgery for aprostate problem and subsequently died from anembolism on 17 July 1912, in Paris. He was 58 years of age. He is buried in the Poincaré family vault in theCemetery of Montparnasse, Paris, in section 16, close to the Rue Émile-Richard.
A former French Minister of Education,Claude Allègre, proposed in 2004 that Poincaré be reburied in thePanthéon in Paris, which is reserved for French citizens of the highest honour.[34]
The problem of finding the general solution to the motion of more than two orbiting bodies in theSolar System had eluded mathematicians sinceNewton's time. This was known originally as the three-body problem and later then-body problem, wheren is any number of more than two orbiting bodies. Then-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 byGösta 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 toNewton'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 seriesconverges 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 distinguishedKarl 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[36] and the book byBarrow-Green[37]). The version finally printed[38] contained many important ideas which led to thetheory of chaos. The problem as stated originally was finally solved byKarl F. Sundman forn = 3 in 1912 and was generalised to the case ofn > 3 bodies byQiudong Wang in the 1990s. 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.[36]
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 theoristHendrik 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"[39] and introduced the hypothesis oflength contraction to explain the failure of optical and electrical experiments to detect motion relative to the aether (seeMichelson–Morley experiment).[40] 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. InThe 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 apostulate to give physical theories the simplest form.[41]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.[42]
Principle of relativity and Lorentz transformations
In 1892 Poincaré developed amathematical theory oflight includingpolarization. His vision of the action of polarizers and retarders, acting on a sphere representing polarized states, is called thePoincaré sphere.[45] 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.[46]
He discussed the "principle of relative motion" in two papers in 1900[42][47] and named it theprinciple of relativity in 1904, according to which no physical experiment can discriminate between a state of uniform motion and a state of rest.[48]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.[49] 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.[50] 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:[51]
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 isinvariant. 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 offour-vectors.[52] 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.[53] So it wasHermann Minkowski who worked out the consequences of this notion in 1907.[53][54]
Likeothers before, Poincaré (1900) discovered a relation betweenmass andelectromagnetic energy. While studying the conflict between theaction/reaction principle andLorentz ether theory, he tried to determine whether thecenter of gravity still moves with a uniform velocity when electromagnetic fields are included.[42] 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 fictitiousfluid (fluide fictif) with a mass density ofE/c2. If thecenter of mass frame is defined by both the mass of matterand 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 aparadox when changing frames: if a Hertzian oscillator radiates in a certain direction, it will suffer arecoil from the inertia of the fictitious fluid. Poincaré performed aLorentz boost (to orderv/c) to the frame of the moving source. He noted that energy conservation holds in both frames, but that thelaw of conservation of momentum is violated. This would allowperpetual motion, a notion which he abhorred. The laws of nature would have to be different in theframes 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 theether.
Poincaré himself came back to this topic in his St. Louis lecture (1904).[48] He rejected[55] 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 theFizeau experiment but that experiment does indeed show that light is partially "carried along" with a substance. Finally in 1908[56] 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 andKaufmann's experiments on the mass of fast moving electrons and (2) the non-conservation of energy in the radium experiments ofMarie Curie.
It wasAlbert Einstein's concept ofmass–energy equivalence (1905) that a body losing energy as radiation or heat was losing mass of amountm = E/c2 that resolved[57] Poincaré's paradox, without using any compensating mechanism within the ether.[58] 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.[59]
In 1905 Poincaré first proposedgravitational 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.[60][51]
Einstein's first paper on relativity was published three months after Poincaré's short paper,[51] but before Poincaré's longer version.[52] 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 onspecial relativity. However, Einstein expressed sympathy with Poincaré's outlook obliquely in a letter toHans 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.[61] In public, Einstein acknowledged Poincaré posthumously in the text of a lecture in 1921 titled "Geometrie und Erfahrung (Geometry and Experience)" in connection withnon-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 ....".[62]
Poincaré's work in the development of special relativity is well recognised,[57] though most historians stress that despite many similarities with Einstein's work, the two had very different research agendas and interpretations of the work.[63] 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.[64][65][66][67][68]
While this is the view of most historians, a minority go much further, such asE. T. Whittaker, who held that Poincaré and Lorentz were the true discoverers of relativity.[69]
Poincaré introducedgroup theory to physics, and was the first to study the group ofLorentz transformations.[70][71] He also made major contributions to the theory of discrete groups and their representations.
The subject is clearly defined byFelix Klein in his "Erlangen Program" (1872): the geometry invariants of arbitrary continuous transformation, a kind of geometry. The term "topology" was introduced, as suggested byJohann Benedict Listing, instead of previously used "Analysis situs". Some important concepts were introduced byEnrico Betti andBernhard 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.[72]
His research ingeometry led to the abstract topological definition ofhomotopy andhomology. He also first introduced the basic concepts and invariants of combinatorial topology, such asBetti numbers and thefundamental group. Poincaré proved a formula relating the number of edges,vertices and faces ofn-dimensionalpolyhedron (theEuler–Poincaré theorem) and gave the first precise formulation of the intuitive notion of dimension.[73]
Title page to volume I ofLes Méthodes Nouvelles de la Mécanique Céleste (1892)
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 ofalgebraic andtranscendental functions through unambiguous coordinates and velocities of the bodies. His work in this area was the first major achievement in celestial mechanics sinceIsaac Newton.[74]
These monographs include an idea of Poincaré, which later became the basis for mathematical "chaos theory" (see, in particular, thePoincaré recurrence theorem) and the general theory ofdynamical systems. Poincaré authored important works onastronomy for theequilibrium figures of a gravitating rotating fluid. He introduced the important concept ofbifurcation 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).[75]
After defending his doctoral thesis on the study of singular points of the system ofdifferential equations, Poincaré wrote a series of memoirs under the title "On curves defined by differential equations" (1881–1882).[76] 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 alimit cycle and theloop 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 thefinite-difference equations, he created a new direction – theasymptotic analysis of the solutions. He applied all these achievements to study practical problems ofmathematical physics andcelestial mechanics, and the methods used were the basis of its topological works.[77]
Photographic portrait of H. Poincaré by Henri Manuel
Poincaré's work habits have been compared to abee flying from flower to flower. Poincaré was interested in the way hismind worked; he studied his habits and gave a talk about his observations in 1908 at the Institute of General Psychology inParis. He linked his way ofthinking to how he made several discoveries.
The mathematicianDarboux claimed he wasun intuitif (anintuitive), 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[78] 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.
Poincaré's mental organisation was interesting not only to Poincaré himself but also toÉdouard Toulouse, apsychologist of the Psychology Laboratory of the School of Higher Studies in Paris. Toulouse wrote a book entitledHenri Poincaré (1910).[79][80] 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.
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.[81]
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).
Poincaré is credited with laying the foundations ofspecial relativity,[11][10] with somearguing that he should be credited with its creation.[82] He is said to have "dominated the mathematics and the theoretical physics of his time", and that "he was without a doubt the most admiredmathematician while he was alive, and he remains today one of the world's most emblematic scientific figures."[83] Poincaré is regarded as a "universal specialist", as he refinedcelestial mechanics, he progressed nearly all parts of mathematics of his time, including creating new subjects, is a father of special relativity, participated in all the great debates of his time in physics, was a major actor in the greatepistemological debates of his day in relation tophilosophy of science, and Poincaré was the one who investigated the 1879Magny shaft firedamp explosion as an engineer.[83] Due to the breadth of his research, Poincaré was the only member to be elected to every section of theFrench Academy of Sciences of the time, those being geometry, mechanics, physics, astronomy and navigation.[84]
PhysicistHenri Becquerel nominated Poincaré for aNobel Prize in 1904, as Becquerel took note that "Poincaré's mathematical and philosophical genius surveyed all of physics and was among those that contributed most to human progress by giving researchers a solid basis for their journeys into the unknown."[85] After his death, he was praised by many intellectual figures of his time, as the authorMarie Bonaparte wrote to his widowed wife Louise that "He was – as you know better than anyone – not only the greatest thinker, the most powerful genius of our time – but also a deep and incomparable heart; and having been close to him remains the precious memory of a whole life."[86]
MathematicianE.T. Bell titled Poincaré as "The Last Universalist", and noted his prowess in many fields, stating that:[87]
Poincaré was the last man to take practically all mathematics, both pure and applied, as his province ... few mathematicians have had the breadth of philosophical vision that Poincaré had and none is his superior in the gift of clear exposition.
When philosopher and mathematicianBertrand Russell was asked who was the greatest man thatFrance had produced in modern times, he instantly replied "Poincaré".[87] Bell noted that if Poincaré had been as strong in practical science as he was in theoretical, he might have "made a fourth with the incomparable three,Archimedes,Newton, andGauss."[88]
Bell further noted his powerful memory, one that was even superior toLeonhard Euler's, stating that:[88]
His principal diversion was reading, where his unusual talents first showed up. A book once read - at incredible speed - became a permanent possession, and he could always state the page and line where a particular thing occurred. He retained this powerful memory all his life. This rare faculty, which Poincaré shared with Euler who had it in a lesser degree, might be called visual or spatial memory. In temporal memory - the ability to recall with uncanny precision a sequence of events long passed — he was also unusually strong.
Bell notes the terrible eyesight of Poincaré, he almost completely remembered formulas and theorems by ear, and "unable to see the board distinctly when he became a student of advanced mathematics, he sat back and listened, following and remembering perfectly without taking notes - an easy feat for him, but one incomprehensible to most mathematicians."[88]
The fact that renownedtheoretical physicists like Poincaré,Boltzmann orGibbs were not awarded theNobel Prize is seen as evidence that the Nobel committee had more regard for experimentation than theory.[95][96] 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.[92]
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.
However, Poincaré did not shareKantian views in all branches of philosophy and mathematics. For example, in geometry, Poincaré believed that the structure ofnon-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".[98] Poincaré believed thatNewton's first law was not empirical but is a conventional framework assumption formechanics (Gargani, 2012).[99] He also believed that the geometry ofphysical space is conventional. He considered examples in which either the geometry of the physical fields orgradients of temperature can be changed, either describing a space as non-Euclidean measured by rigid rulers, or as aEuclidean space where the rulers are expanded or shrunk by avariable heat distribution. However, Poincaré thought that we were so accustomed toEuclidean geometry that we would prefer to change the physical laws to save Euclidean geometry rather than shift to non-Euclidean physical geometry.[100]
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.[102]
Poincaré's two stages—random combinations followed by selection—became the basis forDaniel Dennett's two-stage model offree will.[103]
Poincaré, Henri (1902–1908).The Foundations of Science. New York: Science Press.; reprinted in 1921; this book includes the English translations of Science and Hypothesis (1902), The Value of Science (1905), Science and Method (1908).
1890.Poincaré, Henri (2017).The three-body problem and the equations of dynamics: Poincaré's foundational work on dynamical systems theory. Translated by Popp, Bruce D. Cham, Switzerland: Springer International Publishing.ISBN978-3-319-52898-4.
1892–99.New Methods of Celestial Mechanics, 3 vols. English trans., 1967.ISBN1-56396-117-2.
1905. "The Capture Hypothesis of J. J. See", The Monist, Vol. XV.
Ewald, William B., ed., 1996.From Kant to Hilbert: A Source Book in the Foundations of Mathematics, 2 vols. Oxford Univ. Press. Contains the following works by Poincaré:
1894, "On the Nature of Mathematical Reasoning", 972–981.
1898, "On the Foundations of Geometry", 982–1011.
1900, "Intuition and Logic in Mathematics", 1012–1020.
1905–06, "Mathematics and Logic, I–III", 1021–1070.
1910, "On Transfinite Numbers", 1071–1074.
1905. "The Principles of Mathematical Physics",The Monist, Vol. XV.
1910. "The Future of Mathematics",The Monist, Vol. XX.
1910. "Mathematical Creation",The Monist, Vol. XX.
Other:
1904.Maxwell's Theory and Wireless Telegraphy, New York, McGraw Publishing Company.
1905. "The New Logics",The Monist, Vol. XV.
1905. "The Latest Efforts of the Logisticians",The Monist, Vol. XV.
Poincaré's recurrence theorem: certain systems will, after a sufficiently long but finite time, return to a state very close to the initial state.
Poincaré–Bendixson theorem: a statement about the long-term behaviour of orbits of continuous dynamical systems on the plane, cylinder, or two-sphere.
Poincaré–Hopf theorem: a generalization of the hairy-ball theorem, which states that there is no smooth vector field on a sphere having no sources or sinks.
Poincaré separation theorem: gives the upper and lower bounds of eigenvalues of a real symmetric matrix B'AB that can be considered as the orthogonal projection of a larger real symmetric matrix A onto a linear subspace spanned by the columns of B.
Poincaré–Birkhoff theorem: every area-preserving, orientation-preserving homeomorphism of an annulus that rotates the two boundaries in opposite directions has at least two fixed points.
^Heinzmann, Gerhard; Stump, David (2024)."Henri Poincaré". In Zalta, Edward N.; Nodelman, Uri (eds.).The Stanford Encyclopedia of Philosophy (Summer 2024 ed.). Metaphysics Research Lab, Stanford University. Retrieved11 March 2025.
^Irons, F. E. (August 2001). "Poincaré's 1911–12 proof of quantum discontinuity interpreted as applying to atoms".American Journal of Physics.69 (8):879–884.Bibcode:2001AmJPh..69..879I.doi:10.1119/1.1356056.
^Poincaré, J. Henri (2017).The three-body problem and the equations of dynamics: Poincaré's foundational work on dynamical systems theory. Popp, Bruce D. (Translator). Cham, Switzerland: Springer International Publishing.ISBN9783319528984.OCLC987302273.
^Poincaré, H. (2007)."38.3, Poincaré to H. A. Lorentz, May 1905". In Walter, S. A. (ed.).La correspondance entre Henri Poincaré et les physiciens, chimistes, et ingénieurs. Basel: Birkhäuser. pp. 255–257.
^Poincaré, H. (2007)."38.4, Poincaré to H. A. Lorentz, May 1905". In Walter, S. A. (ed.).La correspondance entre Henri Poincaré et les physiciens, chimistes, et ingénieurs. Basel: Birkhäuser. pp. 257–258.
^abWalter, Scott (2007). "Breaking in the 4-Vectors: The Four-Dimensional Movement in Gravitation, 1905–1910".The Genesis of General Relativity. Vol. 3. Dordrecht: Springer Netherlands. pp. 1118–1178.doi:10.1007/978-1-4020-4000-9_18.ISBN978-1-4020-3999-7.
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^"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."
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