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Poincaré’s geometric worldview and philosophy. (English) Zbl 07265533
Dani, S. G. (ed.) et al., Geometry in history. Cham: Springer (ISBN 978-3-030-13608-6/hbk; 978-3-030-13611-6/pbk; 978-3-030-13609-3/ebook). 435-450 (2019).
Henri Poincaré is one of the pioneers in non-Euclidean geometry and topology. The author gives first short insights into Poincaré’s work on hyperbolic geometry and its connections to his research in function theory and differential equations. He then goes briefly into Poincaré’s topological papers referring to more detailed analysis by K. S. Sarkaria [in: History of topology. Amsterdam: Elsevier. 123–167 (1999; Zbl 0959.54002)], and being very short on Poincaré’s famous conjecture which was recently resolved. The shorter second part of the paper is devoted to Poincaré’s philosophical work for which the author mainly refers to Poincaré’s semi-popular books of the first decade of the 20th century. The author builds on work by the philosopher L. Althusser (1974) on “spontaneous philosophies of scientists”. While Althusser looks at these philosophies rather critical, the author interprets Poincaré’s “spontaneous philosophy” in a rather positive sense. He underlines that Poincaré’s “geometric conventionalism” which “depends on his research in geometry and topology” was way beyond the level of understanding of contemporary philosophers such as Husserl, Lenin, and Russell. Several topics discussed in this article are also included in biographies of Poincaré such as the one by J. Gray [Henri Poincaré. A scientific biography. Princeton, NJ: Princeton University Press (2013; Zbl 1263.01002)], which is not cited.
For the entire collection see [Zbl 1426.01005].
01A55 History of mathematics in the 19th century
01A60 History of mathematics in the 20th century
01A70 Biographies, obituaries, personalia, bibliographies
30-03 History of functions of a complex variable
51-03 History of geometry
54-03 History of general topology
57-03 History of manifolds and cell complexes
00A30 Philosophy of mathematics
Biographic References:
Poincaré, Henri
Full Text: DOI
[1] L. Althusser, Philosophie et philosophie spontanée des savants, François Maspero, Paris, 1974.
[2] L. Althusser, Lenine et la philosophie, François Maspero, Paris, 1972.
[3] E. C. Banks, Kant, Herbart and Riemann, Kant-Studien, Philosophische Zitschrift, 96 (2005) 208-234.
[4] E. Beltrami, Teoria fondamentale degli spazii di curvatura costante, Annali di Matematica Pura ed Applicata, 2 (1868) 232-255. · JFM 01.0208.03
[5] V. N. Berestovskii, Poincaré conjecture and related statements, in Geometry in History, edited by S.G. Dani, A. Papadopoulos, Springer, Cham, 2019. https://doi.org/10.1007/978-3-030-13609-3_17 · Zbl 1149.57311
[6] E. Betti, Sopra gli spazi de un numero qualunque di dimensioni, Annali di Matematica (2) 4, 140-158. · JFM 03.0301.01
[7] M. Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982), 357-453. · Zbl 0528.57011
[8] G. Frege, Die Grundlagen der Arithmetik, eine logisch mathematische Untersuchung über den Begriff der Zahl, Koebner, Breslau, 1884. · Zbl 0654.03005
[9] G. Frege, Kleine Schriften ; herausgegeben von Ignacio Angelell, Georg Olms, Hildesheim, 1967.
[10] J. F. Fries, Neue oder anthropologische Kritik der Vernunft, Bande 1-3, 1828-1831, Winter, Heidelberg.
[11] W. Goldman, Flat affine, projective and conformal structures on manifolds: a historical perspective, in Geometry in History, edited by S.G. Dani, A. Papadopoulos, Springer, Cham, 2019. https://doi.org/10.1007/978-3-030-13609-3_14
[12] J. Gray, Plato’s ghost: the modernist transformation of mathematics, Princeton University Press, Princeton NJ, 2008. · Zbl 1166.00005
[13] F. Hausdorff, Grundzüge der Mengenlehre, Veit, Leipzig, 1914. · JFM 45.0123.01
[14] P. Heegaard, Forstudier til en topologisk teori for de algebraiske Fladers Sammenhäng, Thesis; Traduction française: Sur l’Analysis Situs, Bull. Soc. Math. France, 44 (1916), 161-242. · JFM 46.0835.02
[15] J. F. Herbart, Allgemeine Metaphysik, nebst den Anfängen der philosophischen Naturlehre (2 Teile), A.W. Unzer, Königsberg, T.1 1828, T. 2, 1829.
[16] E. Husserl, Husserliana : gesammelte Werke Bd. 21, Studien zur Arithmetik und Geometrie : Texte aus dem Nachlass (1886-1901), Nijhoff, Hague, 1983. · Zbl 0522.01017
[17] I. Kant, Critik der reinen Vernunft, Johann Friedrich Hartknoch, Riga 1781, 2 Aufl 1787.
[18] F. Klein, Ueber die sogenannte Nicht-Euklidische Geometrie, Math. Ann., 4, 573-625 (1871). · JFM 03.0231.02
[19] V. Lénine, Matérialisme et empirio-criticisme, Editions sociales Paris et Editions du progrès Moscou, (traduit du russe).
[20] N. Lobatchevsky, Pangeometry, edited by A. Papadopoulos, EMS Heritage of European Mathematics, 4, 2010.
[21] J. Morgan and G. Tian, Ricci Flow and the Poincare Conjecture, Clay Mathematics Monographs, 3. American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2007. xlii+521. · Zbl 1179.57045
[22] M. Morse, Calculus of Variations in the Large. Colloquium Publications Volume: 18; 1934; 368 pp; American Mathematical Society, Providence, R.I..
[23] K. Ohshika, The origin of the notion of manifold: from Riemann’s Habilitationsvortrag onward, in “From Riemann to differential geometry and relativity” edited by L. Ji, A. Papadopoulos and S. Yamada, Springer 2017. · Zbl 1385.01007
[24] A. Papadopoulos, On the work of Euler and his followers on spherical geometry, Gaṇita Bhāratı̄ 36 (2014), no. 1, 53-108. · Zbl 1329.01027
[25] A. Papadopoulos, Euler, la géométrie sphérique et le calcul des variations, in “Leonhard Euler : Mathématicien, physicien et théoricien de la musique”, CNRS Éditions, Paris, 349-392, 2015. · Zbl 1327.01024
[26] E. Picard, Sur une extension aux fonctions de deux variables du problème de Riemann relatif aux fonctions hypergéométriques. Annales scientifiques de l’École Normale Supérieure, Sér. 2, 10 (1881), pp. 305-322. · JFM 13.0389.01
[27] V. Poénaru, A glimpse into the problems of the fourth dimension, in this volume.
[28] H. Poincaré, Théorie des groupes Fuchsiens, Acta Math., 1, 1-62, (1882). · JFM 14.0338.01
[29] H. Poincaré, Sur l’Analysis Situs, Comptes rendus hebdomadaires de l’Académie des sciences de Paris, 115 (1892), 189-192.
[30] H. Poincaré, Analysis situs, Journal de l’École Polytechnique, t. 1, pp.1-121 (1895). · JFM 26.0541.07
[31] H. Poincaré, Complément à l’Analysis situs, premier complément, Rendiconti del Circolo Matematico di Palermo, 13, 285-343 (1899). · JFM 30.0435.02
[32] H. Poincaré, Second complément à l’Analysis situs, Proceedings of the London Mathematical Society, 32, 277-308 (1900). · JFM 31.0477.10
[33] H. Poincaré, Sur certaines surfaces algébriques ; troisième complément à l’Analysis Situs, Bulletin de la Société Mathématique de France, 30, 49-70 (1902). · JFM 33.0499.12
[34] H. Poincaré, Sur les cycles des surfaces algébriques. Quatrième complément à l’Analysis Situs, Journal de Mathématiques, 8, 169-214 (1902). · JFM 33.0500.01
[35] H. Poincaré, Cinquième complément à l’Analysis Situs, Rendiconti del Circolo matematico di Palermo, 18, 45-110 (1904). · JFM 35.0504.13
[36] H. Poincaré, La science et l’hypothèse, Flammarion, Paris, 1902. · JFM 34.0080.12
[37] H. Poincaré, La valeur de la science, Flammarion, Paris,1905. · JFM 33.0071.03
[38] H. Poincaré, Science et méthode, Flammarion, Paris, 1908 · JFM 39.0095.03
[39] H. Poincaré, Dernières pensées, Flammarion, Paris, 1913.
[40] B. Riemann, Über die Hypothesen, welche der Geometrie zu Grunde liegen, Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 13, (1867). · JFM 47.0770.01
[41] L. Rollet, Henri Poincaré, des Mathématiques à la philosophie, Étude du parcours intellectuel, social et politique d’ un mathématicien au début du siècle, Thèse, 1999, Université de Nancy.
[42] B. Russell, An essay on the foundations of geometry, Cambridge University Press, Cambridge 1897. · JFM 28.0413.01
[43] K. S. Sarkaria, The topological work of Henri Poincaré, in “History of Topology”, Edited by I.M. James, North-Holland, 123-167. · Zbl 0959.54002
[44] E. Scholtz, The Concept of Manifold, 1850-1950, in “History of Topology”, Edited by I.M. James, North-Holland, 25-64.
[45] J. Shipley, Frege on the foundation of geometry in intuition, Journal for the History of Analytical Philosophy, 3-6, (2015).
[46] S. Smale, Generalized Poincaré’s conjecture in dimensions greater than four, Ann. of Math. (2) 74 1961 391-406. · Zbl 0099.39202
[47] W. · Zbl 0496.57005
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