The Ancient versus the Modern Continuum

The Ancient versus the Modern Continuum

Tags:

• Aristotle,
• Cantor,
• continuum,
• Dedekind,
• foundations of mathematics,
• mathematical continuity,
• mathematical practice,
• philosophy of mathematics

Notes:

• Aristotle. Categories. In Categories and De Interpretatione, edited by J. L. Ackrill. New York: Oxford University Press, 1963.
  Aristotle. On Indivisible Lines. In The Complete Works of Aristotle. Vol. 2. The Revised Oxford Translation, edited by Jonathan Barnes. Princeton: Princeton University Press, 1995.
  Aristotle. Physica. In The Works of Aristotle. Vol. 2. Physica. De Caelo. De Generatione et Corruptione, edited by W. D. Ross. London: Oxford University Press, 1930.
  Bell, John L. The Continuous and the Infinitesimals in Mathematics and Philosophy. Milano: Polimetrica, 2005.
  Bostock, David. “Aristotle on Continuity in Physics VI.” In Aristotle’s Physics: A Collection of Essays, edited by Lindsay Judson, 179-212. New York: Oxford University Press, 2003.
  Cantor, Georg. “Foundations of a General Theory of Manifolds: a Mathematico-philosophical Investigation into the Theory of the Infinite.” In From Kant to Hilbert: A Source Book in the Foundations of Mathematics. Vol. 2, edited by William Ewald, 878-920. New York: Oxford University Press, 2005.
  Cantor, Georg. Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, edited by Ernst Zermelo. Berlin: J. Springer, 1932.
  Cantor, Georg. Grundlagen einer allgemeinen Mannigfaltigkeitslehere: ein mathematisch-philosophischer Versuch des Unendilchen. Lepizig: Teubner, 1883.
  Cantor, Georg. “Letter to Dedekind of 15 September 1890.” In From Kant to Hilbert: A Source Book in the Foundations of Mathematics. Vol. 2, edited by William Ewald, 872-874. New York: Oxford University Press, 2005.
  Cantor, Georg. “Letter to Dedekind of 29 November 1873.” In From Kant to Hilbert: A Source Book in the Foundations of Mathematics. Vol. 2, edited by William Ewald, 844. New York: Oxford University Press, 2005.
  Cantor, Georg. “On a Property of the Set of Real Algebraic Numbers.” In From Kant to Hilbert: A Source Book in the Foundations of Mathematics. Vol. 2, edited by William Ewald, 839-843. New York: Oxford University Press, 2005.
  Cantor, Georg. “Über die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen.” Mathematische Annalen 5, (1872): 123-132.
  Dauben, Joseph Warren. Georg Cantor: His Mathematics and Philosophy of the Infinite. Princeton: Princeton University Press, 1990.
  Dedekind, Richard. “Continuity and Irrational Numbers.” In Essays in the Theory of Numbers. Continuity and Irrational Numbers. The Nature and Meaning of Numbers. Translated by Wooster Woodruff Beman. New York: Dover, 1963.
  Dedekind, Richard. “From On the Theory of Algebraic Integers.” In From Kant to Hilbert: A Source Book in the Foundations of Mathematics. Vol. 2, edited by William Ewald, 779-787. New York: Oxford University Press, 2005.
  Dedekind, Richard. La création des nombres. Translated by Hourya Benis Sinaceur. Paris: Vrin, 2008.
  Dedekind, Richard. “On the Introduction of New Functions in Mathematics.” In From Kant to Hilbert: A Source Book in the Foundations of Mathematics. Vol. 2, edited by William Ewald, 754-762. New York: Oxford University Press, 2005.
  Dedekind, Richard. ¿Qué son y para qué sirven los números? Y otros escritos sobre los fundamentos de las matemáticas. Translated by José Ferreirós. Madrid: Alianza Editorial, 2014.
  Dedekind, Richard. “Sur la théorie des nombres entiers algébriques.” Bulletin des Sciences mathématiques et astronomiques 11, no. 1 (1876): 278-288.
  Dedekind, Richard. “Sur la théorie des nombres entiers algébriques,” Bulletin des Sciences mathématiques et astronomiques 2e série 1, no. 1 (1877): 17-248.
  Dedekind, Richard. “The Nature and Meaning of Numbers.” In Essays in the Theory of Numbers. Continuity and Irrational Numbers. The Nature and Meaning of Numbers. Translated by Wooster Woodruff Beman. New York: Dover, 1963.
  Dedekind, Richard. “Über die Einführung neuer Funktionen in der Mathematik.” In Gesammelte mathematische Werke. Vol. 3, edited by Robert Fricke, Emmy Noether and Öystein Ore, 428-438. Braunschweig: F. Vieweg und Sohn, 1932.
  Dedekind, Richard and Hourya Benis Sinaceur, La Création des Nombres, Paris, Vrin, 2008.
  Dugac, Pierre. “Elements d’analyse de Karl Wierstrass.” Archive for History of Exact Sciences 10, (1973): 41-176.
  Ferreirós, José. Labyrinth of Thought. A History of Set Theory and Its Role in Modern Mathematics. Basel-Boston-Berlin: Springer, 2000.
  Ferreirós, José. Mathematical Knowledge and the Interplay of Practices. Princeton: Princeton University Press, 2016.
  Guthrie, William Keith Chambers. A History of Greek Philosophy. Vol. 2. The Presocratic Tradition from Parmenides to Democritus. Cambridge: Cambridge University Press, 1980.
  Heath, Thomas. A History of Greek Mathematics. Vol. 1. From Thales to Euclid. London: Oxford University Press, 1921.
  Heath, Thomas. Mathematics in Aristotle. Oxford: Oxford University Press, 1970.
  Kirck, Geoffrey Stephen and Raven, John Earle. The Presocratic Philosophers: A Critical History with a Selection of Texts. New York: Cambridge University Press, 1977.
  Kossak, Ernest. Die Elemente der Arithmetik. Programmabhanlung des Werderschen Gymnasiums in Berlin. Berlin: Nicloai’sche Verlagsbuchhandlung, 1872.
  Lipschitz, Rudolph. “Carta a Dedekind 6 de agosto 1876.” In ¿Qué son y para qué sirven los números? Y otros escritos sobre los fundamentos de las matemáticas. Translated by José Ferreirós. Madrid: Alianza Editorial, 2014.
  Newstead, Anne G. J. “Aristotle and Modern Mathematical Theories of the Continuum.” In Aristotle and Contemporary Science. Vol. 2, edited by Jagdish Hattiangadi, David M. Johnson and Demetra Sfendoni-Mentzou, 113-129. Frankfurt: Peter Lang, 2001.
  Russell, Bertrand. The Principles of Mathematics. New York: W. W. Norton & Company, 1996.
  Trifogli, Cecilia. Oxford Physics in the Twentieth Century (ca. 1250-1270): Motion, Infinity, Place and Time. Leiden: Brill, 2000.
  Tweddle, J. Christopher. “Weierstrass’s Construction of the Irrational Numbers.” Mathematische Semesterberichte 58, no. 1 (2011): 47-58.
  White, Michael J. “On Continuity: Aristotle versus Topology?” History and Philosophy of Logic  9 (1988): 1-12.