Ortega y Gasset on Georg Cantor’s Theory of Transfinite Numbers

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Abstract

Ortega y Gasset is known for his philosophy of life and his effort to propose an alternative to both realism and idealism. The goal of this article is to focus on an unfamiliar aspect of his thought. The focus will be given to Ortega’s interpretation of the advancements in modern mathematics in general and Cantor’s theory of transfinite numbers in particular. The main argument is that Ortega acknowledged the historical importance of the Cantor’s Set Theory, analyzed it and articulated a response to it. In his writings he referred many times to the advancements in modern mathematics and argued that mathematics should be based on the intuition of counting. In response to Cantor’s mathematics Ortega presented what he defined as an ‘absolute positivism’. In this theory he did not mean to naturalize cognition or to follow the guidelines of the Comte’s positivism, on the contrary. His aim was to present an alternative to Cantor’s mathematics by claiming that mathematicians are allowed to deal only with objects that are immediately present and observable to intuition. Ortega argued that the infinite set cannot be present to the intuition and therefore there is no use to differentiate between cardinals of different infinite sets.

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  • Beck M. and Geoghegan R. 2011 The Art of Proof. Basic Training for Deeper Mathematics New York Springer.

  • Cantor G. 2015 “Contributions to the Founding of the Theory of Transfinite Numbers in Contributions to the Founding of the Theory of Transfinite Numbers. Translated by Philip E.B. Jourdain Dover Publications New York.

  • Dauben J.W. 1989 Georg Cantor: His Mathematics and Philosophy of the Infinite Princeton University Press New Jersey.

  • Einstein A. 2007 ‘Geometry and Experience’ An expanded form of Address to the Prussian Academy of Sciences in Berlin of January 27th 1921. In: The Essential Einstein. His Greatest Works. Edited with commentary by Stephen Hawking (Penguin Books)

  • Harold E. 1988 Kronecker’s Place in History. In: “History and Philosophy of Modern Mathematics” W. Aspray and P. Kitcher eds. Minnesota Studies in the Philosophy of Science vol. 11 Univ. of Minn. Press.

  • Gillies D.A. 1980 “Brouwer’s Philosophy of Mathematics. Review Article”. In: “Erkenntnis 15 105-126.

  • Gillman L. 2012 “Two Classical Surprises Concerning the Axiom of Choice and the Continuum Hypothesis” In: The Mathematical Association of America (Monthly 109).

  • Jech T. 2006. Set Theory The Third Millennium Edition Revised and Expended Berlin Springer.

  • Morón A. 1968 El sistema de Ortega y Gasset Madrid Ediciones Alcalá.

  • Natorp P. 1912 Kant und die Marburger Schule in Kant Studien 17 (1-3): 193-221 Wurzburg Kabitsch.

  • Ortega y Gasset J. 1983 En torno a Galileo (1933) Obras completas. Tomo V Alianza Editorial Madrid.

  • Ortega y Gasset 1987 Cartas de un joven español Madrid El Arquero.

  • Ortega y Gasset 1992a La idea de principio en Leibniz y la evolución de la teoría deductiva Alianza Editorial Madrid.

  • Ortega y Gasset 1992b La idea de principio en Leibniz y la evolución de la teoría deductiva Alianza Editorial Madrid.

  • Ortega y Gasset 1995 Qué es filosofía? (Introducción: Ignacio Sánchez Cámara) Editorial Espasa Calpe Madrid (1929).

  • Ortega y Gasset 2004 “Vicistudes en las ciencies” (1930) in Meditacion de la tecnica y otros ensyaos sobre ciencia y filosofía Alianza Editorial Madrid.

  • Reichenbach H. The Philosophy of Space and Time. Translated by Maria Reichenbach and John Freund with Introduction and remarks by Rudolf Carnap New York 2014.

  • Russell B. 1917 “Mathematics and the Metaphysicians” In: Mysticism and Logic London George Allen.

  • Russell B. 2009 “Mathematics and Logic” in The Basic Writings of Bertrand Russell London New York Routledge.

  • Russell B. 2010 Introduction to Mathematical Philosophy London George Allen & Unwin.

  • Sánchez C. 1995 Introducción a Qué es filosofía (in Qué es filosofía) Espasa Calpe Madrid.

  • San Martín J. 1994 Ensayos sobre Ortega Universidad Nacional de Educación a Distancia Madrid.

  • Thiele R. 2005 “Georg Cantor (1845-1918)” In: Mathematics and the Divine Amsterdam Elsevier pp. 525-547.

  • Trudeau R.J. 1993 Introduction to Graph Theory New York. Dover Publications.

  • Winskel G. 2010 Set Theory for Computer Sciencehttps://www.cl.cam.ac.uk/~gw104/STfCS2010.pdf. (Accessed 6 February 2016).

  • Zamora Bonilla J. 2005 El impulse orteguiano a la ciencia espanola Editorial Biblioteca Nueva Madrid.

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