# Jules Henri Poincaré

### Quick Info

Born
29 April 1854
Nancy, Lorraine, France
Died
17 July 1912
Paris, France

Summary
Henri Poincaré can be said to have been the originator of algebraic topology and of the theory of analytic functions of several complex variables.

### Biography

Henri Poincaré's father was Léon Poincaré and his mother was Eugénie Launois. They were 26 and 24 years of age, respectively, at the time of Henri's birth. Henri was born in Nancy where his father was Professor of Medicine at the University. Léon Poincaré's family produced other men of great distinction during Henri's lifetime. Raymond Poincaré, who was prime minister of France several times and president of the French Republic during World War I, was the elder son of Léon Poincaré's brother Antoine Poincaré. The second of Antoine Poincaré's sons, Lucien Poincaré, achieved high rank in university administration.

Henri was [2]:-
... ambidextrous and was nearsighted; during his childhood he had poor muscular coordination and was seriously ill for a time with diphtheria. He received special instruction from his gifted mother and excelled in written composition while still in elementary school.
In 1862 Henri entered the Lycée in Nancy (now renamed the Lycée Henri Poincaré in his honour). 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. Henri was described by his mathematics teacher 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.

Poincaré entered the École Polytechnique in 1873, graduating in 1875. He was well ahead of all the other students in mathematics but, perhaps not surprisingly given his poor coordination, performed no better than average in physical exercise and in art. Music was another of his interests but, although he enjoyed listening to it, his attempts to learn the piano while he was at the École Polytechnique were not successful. Poincaré read widely, beginning with popular science writings and progressing to more advanced texts. His memory was remarkable and he retained much from all the texts he read but not in the manner of learning by rote, rather by linking the ideas he was assimilating particularly in a visual way. 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 the symbols properly that his lecturers were writing on the blackboard.

After graduating from the École Polytechnique, Poincaré continued his studies at the École des Mines. His [21]:-
... meticulous notes taken on field trips while a student there exhibit a deep knowledge of the scientific and commercial methods of the mining industry; a subject that interested him throughout his life.
After completing his studies at the École des Mines Poincaré spent a short while as a mining engineer at Vesoul while completing his doctoral work. As a student of Charles Hermite, Poincaré received his doctorate in mathematics from the University of Paris in 1879. His thesis was on differential equations and the examiners were somewhat critical of the work. They praised the results near the beginning of the work but then reported that the (see for example [21]):-
... remainder of the thesis is a little confused and shows that the author was still unable to express his ideas in a clear and simple manner. Nevertheless, considering the great difficulty of the subject and the talent demonstrated, the faculty recommends that M Poincaré be granted the degree of Doctor with all privileges.
Immediately after receiving his doctorate, Poincaré was appointed to teach mathematical analysis at the University of Caen. Reports of his teaching at Caen were not wholly complimentary, referring to his sometimes disorganised lecturing style. He was to remain there for only two years before being appointed to a chair in the Faculty of Science in Paris in 1881. In 1886 Poincaré was nominated for the chair of mathematical physics and probability at the Sorbonne. The intervention and the support of Hermite was to ensure that Poincaré was appointed to the chair and he also was appointed to a chair at the École Polytechnique. In his lecture courses to students in Paris [2]:-
... changing his lectures every year, he would review optics, electricity, the equilibrium of fluid masses, the mathematics of electricity, astronomy, thermodynamics, light, and probability.
Poincaré held these chairs in Paris until his death at the early age of 58.

Before looking briefly at the many contributions that Poincaré made to mathematics and to other sciences, we should say a little about his way of thinking and working. He is considered as one of the great geniuses of all time and there are two very significant sources which study his thought processes. One is a lecture which Poincaré gave to l'Institute Général Psychologique in Paris in 1908 entitled Mathematical invention in which he looked at his own thought processes which led to his major mathematical discoveries. The other is the book [30] by Toulouse who was the director of the Psychology Laboratory of l'École des Hautes Études in Paris. Although published in 1910 the book recounts conversations with Poincaré and tests on him which Toulouse carried out in 1897.

In [30] Toulouse explains that Poincaré kept very precise working hours. He undertook mathematical research for four hours a day, between 10 am and noon then again from 5 pm to 7 pm. He would read articles in journals later in the evening. An interesting aspect of Poincaré's work is that he tended to develop his results from first principles. For many mathematicians there is a building process with more and more being built on top of the previous work. This was not the way that Poincaré worked and not only his research, but also his lectures and books, were all developed carefully from basics. Perhaps most remarkable of all is the description by Toulouse in [30] of how Poincaré went about writing a paper. Poincaré:-
... does not make an overall plan when he writes a paper. He will normally start without knowing where it will end. ... Starting is usually easy. Then the work seems to lead him on without him making a wilful effort. At that stage it is difficult to distract him. When he searches, he often writes a formula automatically to awaken some association of ideas. If beginning is painful, Poincaré does not persist but abandons the work.
Toulouse then goes on to describe how Poincaré expected the crucial ideas to come to him when he stopped concentrating on the problem:-
Poincaré proceeds by sudden blows, taking up and abandoning a subject. During intervals he assumes ... that his unconscious continues the work of reflection. Stopping the work is difficult if there is not a sufficiently strong distraction, especially when he judges that it is not complete ... For this reason Poincaré never does any important work in the evening in order not to trouble his sleep.
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.
Let us examine some of the discoveries that Poincaré made with this method of working. Poincaré was a scientist preoccupied by many aspects of mathematics, physics and philosophy, and he is often described as the last universalist in mathematics. He made contributions to numerous branches of mathematics, celestial mechanics, fluid mechanics, the special theory of relativity and the philosophy of science. Much of his research involved interactions between different mathematical topics and his broad understanding of the whole spectrum of knowledge allowed him to attack problems from many different angles.

Before the age of 30 he developed the concept of automorphic functions which are functions of one complex variable invariant under a group of transformations characterised algebraically by ratios of linear terms. The idea was to come in an indirect way from the work of his doctoral thesis on differential equations. His results applied only to restricted classes of functions and Poincaré wanted to generalise these results but, as a route towards this, he looked for a class functions where solutions did not exist. This led him to functions he named Fuchsian functions after Lazarus Fuchs but were later named automorphic functions. The crucial idea came to him as he was about to get onto a bus, as he relates in Science and Method (1908):-
At the moment when I put my foot on the step the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformation that I had used to define the Fuchsian functions were identical with those of non-euclidean geometry.
In a correspondence between Klein and Poincaré many deep ideas were exchanged and the development of the theory of automorphic functions greatly benefited. However, the two great mathematicians did not remain on good terms, Klein seeming to become upset by Poincaré's high opinions of Fuchs's work. Rowe examines this correspondence in [149].

Poincaré's Analysis situs , published in 1895, is an early systematic treatment of topology. He can be said to have been the originator of algebraic topology and, in 1901, he claimed that his researches in many different areas such as differential equations and multiple integrals had all led him to topology. For 40 years after Poincaré published the first of his six papers on algebraic topology in 1894, essentially all of the ideas and techniques in the subject were based on his work. The Poincaré conjecture remained as one of the most baffling and challenging unsolved problems in algebraic topology until it was settled by Grisha Perelman in 2002.

Homotopy theory reduces topological questions to algebra by associating with topological spaces various groups which are algebraic invariants. Poincaré introduced the fundamental group (or first homotopy group) in his paper of 1894 to distinguish different categories of 2-dimensional surfaces. He was able to show that any 2-dimensional surface having the same fundamental group as the 2-dimensional surface of a sphere is topologically equivalent to a sphere. He conjectured that this result held for 3-dimensional manifolds and this was later extended to higher dimensions. Surprisingly proofs are known for the equivalent of Poincaré's conjecture for all dimensions strictly greater than three. No complete classification scheme for 3-manifolds is known so there is no list of possible manifolds that can be checked to verify that they all have different homotopy groups.

Poincaré is also considered the originator of the theory of analytic functions of several complex variables. He began his contributions to this topic in 1883 with a paper in which he used the Dirichlet principle to prove that a meromorphic function of two complex variables is a quotient of two entire functions. He also worked in algebraic geometry making fundamental contributions in papers written in 1910-11. He examined algebraic curves on an algebraic surface $F(x, y, z) = 0$ and developed methods which enabled him to give easy proofs of deep results due to Émile Picard and Severi. He gave the first correct proof of a result stated by Castelnuovo, Enriques and Severi, these authors having suggested a false method of proof.

His first major contribution to number theory was made in 1901 with work on [1]:-
... the Diophantine problem of finding the points with rational coordinates on a curve $f (x, y) = 0$, where the coefficients of f are rational numbers.
In applied mathematics he studied optics, electricity, telegraphy, capillarity, elasticity, thermodynamics, potential theory, quantum theory, theory of relativity and cosmology. In the field of celestial mechanics he studied the three-body-problem, and the theories of light and of electromagnetic waves. He is acknowledged as a co-discoverer, with Albert Einstein and Hendrik Lorentz, of the special theory of relativity. We should describe in a little more detail Poincaré's important work on the 3-body problem.

Oscar II, King of Sweden and Norway, initiated a mathematical competition in 1887 to celebrate his sixtieth birthday in 1889. Poincaré was awarded the prize for a memoir he submitted on the 3-body problem in celestial mechanics. In this memoir Poincaré gave the first description of homoclinic points, gave the first mathematical description of chaotic motion, and was the first to make major use of the idea of invariant integrals. However, when the memoir was about to be published in Acta Mathematica, Phragmen, who was editing the memoir for publication, found an error. Poincaré realised that indeed he had made an error and Mittag-Leffler made strenuous efforts to prevent the publication of the incorrect version of the memoir. Between March 1887 and July 1890 Poincaré and Mittag-Leffler exchanged fifty letters mainly relating to the Birthday Competition, the first of these by Poincaré telling Mittag-Leffler that he intended to submit an entry, and of course the later of the 50 letters discuss the problem concerning the error. It is interesting that this error is now regarded as marking the birth of chaos theory. A revised version of Poincaré's memoir appeared in 1890.

Poincaré's other major works on celestial mechanics include Les Méthodes nouvelles de la mécanique céleste in three volumes published between 1892 and 1899 and Leçons de mecanique céleste (1905). In the first of these he aimed to completely characterise all motions of mechanical systems, invoking an analogy with fluid flow. He also showed that series expansions previously used in studying the 3-body problem were convergent, but not in general uniformly convergent, so putting in doubt the stability proofs of Lagrange and Laplace.

He also wrote many popular scientific articles at a time when science was not a popular topic with the general public in France. As Whitrow writes in [2]:-
After Poincaré achieved prominence as a mathematician, he turned his superb literary gifts to the challenge of describing for the general public the meaning and importance of science and mathematics.
Poincaré's popular works include Science and Hypothesis (1901), The Value of Science (1905), and Science and Method (1908). A quote from these writings is particularly relevant to this archive on the history of mathematics. In 1908 he wrote:-
The true method of foreseeing the future of mathematics is to study its history and its actual state.
Finally we look at Poincaré's contributions to the philosophy of mathematics and science. The first point to make is the way that Poincaré saw logic and intuition as playing a part in mathematical discovery. He wrote in Mathematical definitions in education (1904):-
It is by logic we prove, it is by intuition that we invent.
In a later article Poincaré emphasised the point again in the following way:-
Logic, therefore, remains barren unless fertilised by intuition.
McLarty [119] gives examples to show that Poincaré did not take the trouble to be rigorous. The success of his approach to mathematics lay in his passionate intuition. However intuition for Poincaré was not something he used when he could not find a logical proof. Rather he believed that formal arguments may reveal the mistakes of intuition and logical argument is the only means to confirm insights. Poincaré believed that formal proof alone cannot lead to knowledge. This will only follow from mathematical reasoning containing content and not just formal argument.

It is reasonable to ask what Poincaré meant by "intuition". This is not straightforward, since he saw it as something rather different in his work in physics to his work in mathematics. In physics he saw intuition as encapsulating mathematically what his senses told him of the world. But to explain what "intuition" was in mathematics, Poincaré fell back on saying it was the part which did not follow by logic:-
... to make geometry ... something other than pure logic is necessary. To describe this "something" we have no word other than intuition.
The same point is made again by Poincaré when he wrote a review of Hilbert's Foundations of geometry (1902):-
The logical point of view alone appears to interest [Hilbert]. Being given a sequence of propositions, he finds that all follow logically from the first. With the foundations of this first proposition, with its psychological origin, he does not concern himself.
We should not give the impression that the review was negative, however, for Poincaré was very positive about this work by Hilbert. In [181] Stump explores the meaning of intuition for Poincaré and the difference between its mathematically acceptable and unacceptable forms.

Poincaré believed that one could choose either euclidean or non-euclidean geometry as the geometry of physical space. He believed that because the two geometries were topologically equivalent then one could translate properties of one to the other, so neither is correct or false. for this reason he argued that euclidean geometry would always be preferred by physicists. This, however, has not proved to be correct and experimental evidence now shows clearly that physical space is not euclidean.

Poincaré was absolutely correct, however, in his criticism that those like Russell who wished to axiomatise mathematics; they were doomed to failure. The principle of mathematical induction, claimed Poincaré, cannot be logically deduced. He also claimed that arithmetic could never be proved consistent if one defined arithmetic by a system of axioms as Hilbert had done. These claims of Poincaré were eventually shown to be correct.

We should note that, despite his great influence on the mathematics of his time, Poincaré never founded his own school since he did not have any students. Although his contemporaries used his results they seldom used his techniques.

Poincaré achieved the highest honours for his contributions of true genius. He was elected to the Académie des Sciences in 1887 and in 1906 was elected President of the Academy. The breadth of his research led to him being the only member elected to every one of the five sections of the Academy, namely the geometry, mechanics, physics, geography and navigation sections. In 1908 he was elected to the Académie Francaise and was elected director in the year of his death. He was also made chevalier of the Légion d'Honneur and was honoured by a large number of learned societies around the world. He won numerous prizes, medals and awards.

Poincaré was only 58 years of age when he died [3]:-
M Henri Poincaré, although the majority of his friends were unaware of it, recently underwent an operation in a nursing home. He seemed to have made a good recovery, and was about to drive out for the first time this morning. He died suddenly while dressing.
His funeral was attended by many important people in science and politics [3]:-
The President of the Senate and most of the members of the Ministry were present, and there were delegations from the French Academy, the Académie des Sciences, the Sorbonne, and many other public institutions. The Prince of Monaco was present, the Bey of Tunis was represented by his two sons, and Prince Roland Bonaparte attended as President of the Paris Geographical Society. The Royal Society was represented by its secretary, Sir Joseph Larmor, and by the Astronomer Royal, Mr F W Dyson.
Let us end with a quotation from an address at the funeral:-
[M Poincaré was] a mathematician, geometer, philosopher, and man of letters, who was a kind of poet of the infinite, a kind of bard of science.

### References (show)

1. J Dieudonne, Biography in Dictionary of Scientific Biography (New York 1970-1990). See THIS LINK.
2. Biography in Encyclopaedia Britannica. http://www.britannica.com/biography/Henri-Poincare
3. P Appell, Henri Poincaré (Paris, 1925).
4. J Barrow-Green, Poincaré and the three body problem (London, 1997).
5. A Bellivier, Henri Poincaré, ou la vocation souveraine. Vocations, IV (Paris, 1956).
6. F E Browder, (ed.), The mathematical heritage of Henri Poincaré. Part 1 (Providence, R.I., 1983).
7. F E Browder, (ed.), The mathematical heritage of Henri Poincaré. Part 2 (Providence, R.I., 1983).
8. T Dantzig, Henri Poincaré : critic of crisis : reflections on his universe of discourse (New York, 1954).
9. P Dugac, Georg Cantor et Henri Poincaré (Rennes, 1983).
10. J Folina, Poincaré and the philosophy of mathematics (New York, 1992).
11. J Giedymin, Science and convention : Essays on Henri Poincaré's philosophy of science and the conventionalist tradition (Oxford, 1982).
12. J Gray, Linear differential equations and group theory from Riemann to Poincaré (Boston, MA, 1986).
13. J-L Greffe, G Heinzmann and K Lorenz (eds.), Henri Poincaré : science et philosophie (Paris, 1996).
14. J Hadamard and H Poincaré, Essai sur la psychologie de l'invention dans le domaine mathématique : L'invention mathématique (Sceaux, 1993).
15. G Heinzmann, Poincaré, Russell, Zermelo et Peano (Paris, 1986).
16. G Holton, The thematic origins of scientific thought : Kepler to Einstein (Cambridge, MA, 1974).
17. A A Logunov, On Henri Poincare's papers 'On the dynamics of an electron' (Russian) (Moscow, 1988).
18. F Maitland (trs.), Henri Poincaré, Science and method (New York).
19. A I Miller, Imagery in scientific thought : Creating 20th century physics (Boston, MA, 1984).
20. A I Miller, Insights of genius : Imagery and creativity in science and art (New York, 1996).
21. J J A Mooij, La philosophie des mathématiques de Henri Poincaré (Louvain, 1966).
22. P Nabonnand, (ed.) La correspondance entre Henri Poincaré et Gösta Mittag-Leffler (Basel, 1999).
23. A Pap, The a priori in physical theory (New York, 1968).
24. J-C Pont, La topologie algébrique des origines à Poincaré (Paris, 1974).
25. A Rey, La théorie de la physique chez les physiciens contemporains (Paris, 1907).
26. L A P Rougier, La philosophie géométrique de Henri Poincaré (Paris, 1920).
27. E Scholz, Geschichte des Mannigfaltigkeitsbegriffs von Riemann bis Poincaré (Boston, Mass., 1980).
28. R Torretti, Philosophy of geometry from Riemann to Poincaré (Dordrecht-Boston, Mass., 1984).
29. E Toulouse, Henri Poincaré (Paris, 1910).
30. A Tyapkin and A Shibanov, Poincaré (Bulgarian) (Sofia, 1984).
31. P S Aleksandrov, Poincaré and topology (Russian), Uspekhi Mat. Nauk 27 (1)(163) (1972), 147-158.
32. K G Andersson, Poincaré's discovery of homoclinic points, Arch. Hist. Exact Sci. 48 (2) (1994), 133-147.
33. M Atten, La nomination de H Poincaré à la chaire de physique mathématique et calcul des probabilités de la Sorbonne, Cahiers du séminaire d'histoire des mathématiques 9 (1988), 221-230.
34. M Atten, Poincaré et la tradition de la physique mathématique française, in Henri Poincaré : science et philosophie, Nancy, 1994 (Berlin, 1996), 35-44; 577-578.
35. S Bagce, Poincaré's philosophy of geometry and its relevance to his philosophy of science, in Henri Poincaré : science et philosophie, Nancy, 1994 (Berlin, 1996), 299-314; 582.
36. N L Balazs, The acceptability of physical theories : Poincaré versus Einstein, in General relativity, papers in honour of J L Synge (Oxford, 1972), 21-34.
37. H Barreau, Poincaré et l'espace-temps ou un conventionalisme insuffisant, in Henri Poincaré : science et philosophie, Nancy, 1994 (Berlin, 1996), 287-298; 581-582.
38. J Barrow-Green, Oscar II's prize competition and the error in Poincaré's memoir on the three body problem, Arch. Hist. Exact Sci. 48 (2) (1994), 107-131.
39. I G Bashmakova, Arithmetic of algebraic curves from Diophantus to Poincaré, Historia Math. 8 (4) (1981), 393-416.
40. I G Bashmakova, The arithmetic of algebraic curves from Diophantus to Poincaré (Russian), Istor.-Mat. Issled. Vyp. 20 (1975), 104-124; 379.
41. E T Bell, Men of Mathematics (New York, 1986), Chapter 28.
42. L C Biedenharn and Y Dothan, Poincaré's work on the magnetic monopole and its generalization in present day theoretical physics, in Differential topology-geometry and related fields, and their applications to the physical sciences and engineering (Leipzig, 1985), 39-50.
43. M Bognár and J Szenthe, The mathematical work of Henri Poincaré (Hungarian), Mat. Lapok 28 (4) (1977/80), 269-286.
44. A Borel, Henri Poincaré and special relativity, Enseign. Math. (2) 45 (3-4) (1999), 281-300.
45. M Brelot, Le balayage de Poincaré et l'épine de Lebesgue, in Proceedings of the 110th national congress of learned societies, Montpellier, 1985 (Paris, 1985), 141-151.
46. A Brenner, La nature des hypothèses physiques selon Poincaré, à la lumière de la controverse avec Duhem, in Henri Poincaré : science et philosophie, Nancy, 1994 (Berlin, 1996), 389-396; 584.
47. J Cassinet, La position d'Henri Poincaré par rapport à l'axiome du choix, à travers ses écrits et sa correspondance avec Zermelo (1905-1912), Hist. Philos. Logic 4 (2) (1983), 145-155.
48. M Castellet i Solanas, 150 years of topological genesis : from Euler to Poincaré (Catalan), in The development of mathematics in the nineteenth century (Barcelona, 1984), 195-209.
49. P G Cath, Jules Henri Poincaré (Nancy 1854-Paris 1912) (Dutch), Euclides, Groningen 30 (1954/55), 265-275.
50. A Châtelet, G Valiron, E LeRoy and E Borel, Hommage à Henri Poincaré, Congrès International de Philosophie des Sciences, Paris, 1949 Vol I (Paris, 1951), 37-64.
51. J Chazy, Henri Poincaré et la mécanique céleste, Bull. Astr. (2) 16 (1951), 145-160.
52. C S Chihara, Poincaré and logicism, in Henri Poincaré : science et philosophie, Nancy, 1994 (Berlin, 1996), 435-446; 585.
53. J-C Chirollet, Le continu mathématique 'du troisième ordre' chez Henri Poincaré, in La mathématique non standard (Paris, 1989), 83-116.
54. G Ciccotti and G Ferrari, Was Poincaré a herald of quantum theory?, European J. Phys. 4 (2) (1983), 110-116.
55. J da Silva, Poincaré's philosophy of mathematics (Portuguese), in The XIXth century : the birth of modern science, águas de Lindóia, 1991 (Campinas, 1992), 43-56.
56. A Dahan Dalmedico, Le difficile héritage de Henri Poincaré en systèmes dynamiques, in Henri Poincaré : science et philosophie, Nancy, 1994 (Berlin, 1996), 13-33; 577.
57. Yu A Danilov, Nonlinear dynamics : Poincaré and Mandel'shtam (Russian), in Nonlinear waves, 'Nauka' (Moscow, 1989), 5-15.
58. Yu A Danilov, Nonlinear dynamics : Poincaré and Mandelstam, in Nonlinear waves 1 (Berlin-New York, 1989), 2-13.
59. G Darboux, Eloge historique d'Henri Poincaré, Mémoires de l'Académie des sciences 52 (1914), 81-148.
60. O Darrigol, Henri Poincaré's criticism of fin de siècle electrodynamics, Stud. Hist. Philos. Sci. B Stud. Hist. Philos. Modern Phys. 26 (1) (1995), 1-44.
61. M A B Deakin, The development of the Laplace transform, 1737-1937. II. Poincaré to Doetsch, 1880-1937, Arch. Hist. Exact Sci. 26 (4) (1982), 351-381.
62. M Detlefsen, Poincaré against the logicians, Synthese 90 (3) (1992), 349-378.
63. M Detlefsen, Poincaré vs. Russell on the rôle of logic in mathematics, Philos. Math. (3) 1 (1) (1993), 24-49.
64. A Drago, Poincaré versus Peano and Hilbert about the mathematical principle of induction, in Henri Poincaré : science et philosophie, Nancy, 1994 (Berlin, 1996), 513-527; 586.
65. L M Druzkowski, Henri Poincaré - mathematician, physicist, astronomer and philosopher (Polish), Wiadom. Mat. 30 (1) (1993), 73-83.
66. P Dugac, Georg Cantor and Henri Poincaré, Bullettino Storia delle Scienze Matematiche 4 (1984), 65-96.
67. R Dugas, Henri Poincaré devant les principes de la mécanique, Revue Sci. 89 (1951), 75-82.
68. M Epple, Mathematical inventions: Poincaré on a 'Wittgensteinian' topic, in Henri Poincaré : science et philosophie, Nancy, 1994 (Berlin, 1996), 559-576; 587.
69. J Fiala, Henri Poincaré and the psychology of mathematics (Czech), Pokroky Mat. Fyz. Astronom. 22 (4) (1977), 205-217.
70. J Folina, Logic and intuition in Poincaré's philosophy of mathematics, in Henri Poincaré : science et philosophie, Nancy, 1994 (Berlin, 1996), 417-434; 584-585.
71. J Folina, Poincaré's conception of the objectivity of mathematics, Philos. Math. (3) 2 (3) (1994), 202-227.
72. M Friedman, Poincaré's conventionalism and the logical positivists, in Henri Poincaré : science et philosophie, Nancy, 1994 (Berlin, 1996), 333-344; 582-583.
73. E Giannetto, Henri Poincaré and the rise of special relativity, Hadronic J. Suppl. 10 (4) (1995), 365-433.
74. J Giedymin, Geometrical and physical conventionalism of Henri Poincaré in epistemological formulation, Stud. Hist. Philos. Sci. 22 (1) (1991), 1-22.
75. J Giedymin, On the origin and significance of Poincaré's conventionalism, Studies in Hist. and Philos. Sci. 8 (4) (1977), 271-301.
76. D S Gillman, Billiards and Poincaré : two unsolved problems (Spanish), Rev. Integr. Temas Mat. 3 (1) (1985), 7-13.
77. W Goldfarb, Poincaré against the Logicists, in History and philosophy of modern mathematics, Minneapolis, MN, 1985 (Minneapolis, MN, 1988), 61-81.
78. R L Gomes, Ruy Luis On the first centenary of the birth of Henri Poincaré (Portuguese), Gaz. Mat., Lisboa 15 (60-61) (1955), 1-3.
79. B S Gower, Henri Poincaré and Bruno de Finetti: conventions and scientific reasoning, Stud. Hist. Philos. Sci. B Stud. Hist. Philos. Modern Phys. 28 (4) (1997), 657-679.
80. J J Gray, Poincaré and Klein - groups and geometries, in 1830-1930 : a century of geometry, Paris, 1989 (Berlin, 1992), 35-44.
81. J J Gray, Did Poincaré say 'set theory is a disease'?, Math. Intelligencer 13 (1) (1991), 19-22.
82. J J Gray, Poincaré, Einstein, and the theory of special relativity, Math. Intelligencer 17 (1) (1995), 65-67, 75.
83. J J Gray, Les trois suppléments au Mémoire de Poincaré, écrit en 1880, sur les fonctions fuchsiennes et les équations différentielles, C. R. Acad. Sci. Paris Vie Académique 293 (8-12) (1981), suppl., 87-90.
84. J J Gray, Poincaré and electromagnetic theory, in Henri Poincaré : science et philosophie, Nancy, 1994 (Berlin, 1996), 193-208; 579.
85. J J Gray, Poincaré, topological dynamics, and the stability of the solar system, in The investigation of difficult things (Cambridge, 1992), 503-524.
86. J J Gray, The three supplements to Poincaré's prize essay of 1880 on Fuchsian functions and differential equations, Arch. Internat. Hist. Sci. 32 (109) (1982), 221-235.
87. P A Griffiths, Poincaré and algebraic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (2) (1982), 147-159.
88. O Gürel, Poincaré's bifurcation analysis, in Bifurcation theory and applications in scientific disciplines, New York, 1977 (New York, 1979), 5-26.
89. J Hadamard, Le centenaire de Henri Poincaré, Rev. Hist. Sci. Appl. 7 (1954), 101-108.
90. G Heinzmann, Helmholtz and Poincaré's considerations on the genesis of geometry, in 1830-1930 : a century of geometry, Paris, 1989 (Berlin, 1992), 245-249.
91. G Heinzmann, Poincaré et la philosophie des mathématiques, Cahiers du séminaire d'histoire des mathématiques 9 (1988), 99-121.
92. Henri Poincaré, La correspondance d'Henri Poincaré avec des mathématiciens de A à H, in Proceedings of the seminar on the history of mathematics 7 (Paris, 1986), 59-219.
93. Henri Poincaré, La correspondance d'Henri Poincaré avec des mathématiciens de J à Z, Cahiers du séminaire d'histoire des mathématiques 10 (1989), 83-229.
94. Henri Poincaré, Proc. Royal Soc. London 91 (1915), 5-16.
95. Henri Poincaré, Proc. London Math. Soc. 11 (1913), 41-48.
96. A Herreman, Le statut de la géométrie dans quelques textes sur l'homologie, de Poincaré aux années 1930, Rev. Histoire Math. 3 (2) (1997), 241-293.
97. P Hilton and J Pedersen, Descartes, Euler, Poincaré, Pólya-and polyhedra, Enseign. Math. (2) 27 (3-4) (1981), 327-343.
98. E Hlawka, Die Idee der 'willkürlichen' Funktionen von Poincaré im Laufe eines Jahrhunderts, Natur, Mathematik und Geschichte. Acta Hist. Leopold. No. 27 (1997), 189-200.
99. Igusa, Problems on abelian functions at the time of Poincaré and some at present, Bull. Amer. Math. Soc. (N.S.) 6 (2) (1982), 161-174.
100. L Indorato and G Masotto, Poincaré's role in the Crémieu-Pender controversy over electric convection, Ann. of Sci. 46 (2) (1989), 117-163.

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Written by J J O'Connor and E F Robertson
Last Update October 2003