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Abstract
In the first part, we remind Fréchet’s testimonies about the approach of his early work (1904-1906) in the emerging fields of Functional Analysis and General Analysis, in relation to his idea of introducing a topological structure in an abstract space. In the second part, we highlight the influence that had on this idea, the algebraic point of view of the time of extending the Cantorian notions to an abstract space with a finite group structure. Fréchet took advantage of techniques such as the “composition mode” between the elements of the space, to axiomatize operations and structures of the “class ” with sequential convergence, the “class ” with neighborhood system, the “class ” with “écart” (metric). Then, new historical data are used to reaffirm the proximity of the philosophical conceptions underlying these investigations with the ideas of Leibniz, specifically with regard to the method of “analysis of principles”. The third part examines the contribution of Hausdorff of 1912 and 1914 to the establishment of the neighborhood axiomatics for the topology of an abstract space. Taking into account the observations of Weyl and Bourbaki that Hausdorff drew on Hilbert, we examine the system of axioms for the neighborhoods of the plane introduced by Hilbert in two 1902 papers devoted to the problem of the space continuity. As regards Fréchet’s influence in Hausdorff, we explore the connections of Hausdorff’s “topological space” based on neighborhoods with the notions of metric, sequential convergence and neighborhoods, proposed years earlier by Fréchet. From the beginning, Hausdorff argued that the topology of the separable space had the characteristics of generality and formal rigor that allowed it to adapt to applications better than others. It is shown that all this was consistent with the ideals of simplicity, unity and economy of thought Hausdorff had acquired in his early philosophical works. © 2017. Acad. Colomb. Cienc. Ex. Fis. Nat.References
Alexandroff, P. (1961). Elementary Concepts of Topology. Dover: New York. (Traducción de Einfachste Grundbegriffe der Topologie. Springer, 1932).
Alexandroff, P. S, Fedorchuk, V. V. (1978). The Main Aspects in the Development of Set-Theoretical Topology. Russian Math. Surveys. 33: 1-53.
Alexandroff, P. S., Urysohn, P. S. (1929). Mémoire sur les espaces topologiques compacts. Ver. Kon. Akad. Wetensch. Amsterdam. 14: 1-96.
Anacona, M. P. (2017). La teoría de conjuntos y los números reales en el estructuralismo de Bourbaki. Tesis Doctoral. Cádiz: Universidad de Cádiz.
Arboleda, L. C. (1979). Les debuts de l’École topologique soviétique: notes sur les lettres de Paul S. Alexandrov et Paul S. Urysohn à Maurice Fréchet. Archive for History of Exact Sciencies. 32: 73-89.
Arboleda, L. C. (1980a). Contribution à l’étude des premières recherches topologiques d’après l’oeuvre et la correspondance de Maurice Fréchet. Tesis de Doctorado. École d’Hautes Études en Sciences Sociales. Paris.
Arboleda, L. C. (1980b). Las primeras investigaciones sobre los espacios topológicos. Sociedad Colombiana de Matemáticas. X Coloquio Colombiano de Matemáticas, Paipa.
Arboleda, L. C. (1982). Consideraciones metodológicas sobre el aporte de M. Fréchet a la topología general. Actualités mathématiques: actes du VIe Congrès du regroupement des mathématiciens d’expression latine. Paris: Gauthier-Villars; 493- 499.
Arboleda, L. C. (1984). Sobre los fundamentos de la teoría de los espacios compactos. Asclepio, Madrid. 35: 123-157.
Arboleda, L. C. (2007). Modalidades constructivas y objetivación del cuerpo de los reales. Revista Brasileira de Historia da Matemática, Especial n°1: 215-230.
Arboleda, L. C., Recalde, L. C. (2003). Fréchet and the Logic of the Constitution of Abstract Spaces from Concrete Reality. Synthese. 134: 245-272.
Arboleda, L. C. (2012). Objetos Matemáticos y Prácticas Constitutivas: La Génesis de la Topología de Vecindades. Notae Philosophicae Scientiae Formalis. 1: 32-44.
Arhangel’skii, A. V. y Dranishnikov, A. N. (1997). P. S. Alexandroff and Topology: an introductory note. Topology and its Applications, 80: 1-6.
Aull, C. E., Lowen, R. (eds.) (1997). Handbooks of the History of General Topology. Vol.1. Dordrecht: Springer.
Bourbaki, N. (1971). Topologie Générale. Vol. 1, chapitres 1 à 4. Paris: Hermann.
Bourbaki, N. (1974). Éléments d’histoire des mathématiques. Paris: Hermann.
Cartan, H. (1937a). Théorie des filtres. C. R. Acad. Sci. Paris. 205:595-598.
Cartan, H. (1937b). Filtres et ultrafiltres. C. R. Acad. Sci. Paris. 206: 777-779.
De Risi, V. (ed.) (2015). Mathematizing Space. The Objects of Geometry from Antiquity to the Early Modern Age. Heidelberg: Birkhäuser.
Dieudonné, J. (1981). History of functional analysis. Amsterdam: North-Holland.
Dugac, P. (1984). Histoire des espaces complets. Revue d’Histoire des Sciences. 37: 3-28.
Dugac, P. (2003). Histoire de l’Analyse. Autour de la notion de limite et de ses voisinages. Paris: Vuibert.
Epple, M., Herrlich, H., Husek, M., Preuβ, G., Purkert, W., Scholz, E. (2002). Zum Begriff des topologischen Raumes. En: Felix Hausdorff: Gesammelte Werke, Bd. II. Berlin:Springer.
Epple, M. (2006): Felix Hausdorff’s Considered Empiricism. En J. Ferreirós, J. Gray (Ed) (2006): The Architecture of Modern Mathematics (263-289). New York: Oxford University Press.
Fréchet, M. (1904). Généralisation d’un théorème de Weierstrass. C. R. Acad. Sci. Paris. 139: 848-850.
Fréchet, M. (1906). Sur quelques points du calcul fontionnel. Rendiconti Circolo Mat. Palermo. 22: 1-74.
Fréchet, M. (1928). Les Espaces Abstraits et leur théorie considérée comme introduction à l’Analyse Générale. Paris: Gauthier-Villars.
Fréchet, M. (1933). Notice sur les travaux scientifiques de Maurice Fréchet. Paris: Hermann.
Fréchet, M. (1941). L’analyse générale et la question des fondements. En: Gonseth, F. (ed). (1941). Les entretiens de Zurich sur les fondements et la méthode des sciences mathématiques. Zurich: Leemann. p. 53-81.
Fréchet, M. et Halbwachs, M. (1924). Les calcul de probabilités à la portée de tous. Paris: Dunod.
Hadamard, J. (1912). Le calcul fonctionnel. Enseignement mathématique. 14: 5-18.
Halbwachs, M. (1906). Leibniz. Collection Les Philosophes. Paris: Delaplane. Segunda edición aumentada: Paris, Mellotée, 1928. Tercera edición: Paris, Mellotté, 1950.
Hausdorff, F. (1914). Grundzüge der Mengenlehre. Leipzig: Veit.
Hilbert, D. (1897). Zum Gedächtnis an Karl Weierstrass. En D.
Hilbert (1970). Gesammelte Abhandlungen (vol. 3). Berlin:Springer-Verlag.
Hilbert, D. (1899). Grundlagen der Geometrie. Leipzig: Teubner.
Hilbert, D. (1902): Über die Grundlagen der Geometrie. Göttinger Nachrichten. 4: 233-241.
Hilbert, D. (1902a). The Foundations of Geometry. Chicago: Open Court.
Hilbert, D. (1903): Über die Grundlagen der Geometrie. Math. Annalen. 56: 381-422.
Hirsch, G. (1978). Topologie. En J. Dieudonné. (Ed). Abrégé d’Histoire des mathématiques. 2: 211-266. Paris: Hermann. James, I. M. (Ed) (1999). History of Topology. Amsterdam: Elsevier.
Katz, V., Parshall, K. H. (2014). Taming the Unknown: A History of Algebra from Antiquity to the Early Twentieth Century. Princeton: Princeton University Press.
Koestsier, T. Y Van Mill, J. (1999). By their fruits ye shall know them: Some remarks on the interaction of general topology with other areas of mathematics. En I. M. James (1999):199-239.
Majer, U. (2006). The relation of Logic and Intuition in Kant’s Philosophy of Science, particularly Geometry. Western Ontario series in Philosophy of Science. 70: 47-66.
Manheim, J. (1964). The Genesis of point set Topology. Rinehart: Oxford.
Purkert, W. (2008). The Double Life of Felix Hausdorff/Paul Mongré. Math. Intellingencer. 30: 36-50.
Pier, J.-P. (1980). Historique de la notion de compacité. HistoriaMathematica. 7: 425-443.
Rodríguez, L. (2015). Frigyes Riesz and the emergence of general topology. The roots of ‘topological space’ in geometry. Archive for History of Exact Sciences, 69: 55-102.
Scholz, E. (1996). Logische Ordnungen im Chaos: Hausdorffs frühe Beiträge zur Mengenlehre. En E. Brieskorn (Ed) (1966). Felix Hausdorff zum Gedächtnis, Vol. 1, Aspekte seines Werkes (107-134). Wiesbaden: Vieweg.
Séguier, J. A. de (1904). Théorie des groupes finis. Éléments de la théorie des groupes abstraits. Paris: Gauthier-Villars. Tarrés Freixenet, J. (1994). La topología general desde sus comienzos hasta Hausdorff. Historia de la matemática en el siglo XIX (2ª. Parte). Madrid: Real Academia de Ciencias Exactas, Físicas y Naturales.
Taylor, A. E. (1982). A Study of Maurice Fréchet: I. His Early Work on Point Set Theory and the Theory of Functionals. Archive for History of Exact Sciences. 27: 233-295.
Taylor, A. E. (1985). A Study of Maurice Fréchet: II. Mainly about his Work on General topology, 1909-1928. Archive for History of Exact Sciences. 34: 279-380.
Weil, A. (1937). Sur les espaces à structure uniforme et sur la topologie générale. Publications de l’Institut de mathématiques de l’Université de Strasbourg. Paris: Hermann.
Weyl, H. (1949). David Hilbert and his Mathematical Work. Bulletin of American Mathematical Society. 50: 612-654.
Zalamea, F. (2009). Filosofía sintética de las matemáticas contemporáneas. Bogotá: Universidad Nacional de Colombia.
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