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A Concise History of Mathematics for Philosophers

Published online by Cambridge University Press:  06 June 2019

John Stillwell
Affiliation:
University of San Francisco

Summary

This Element aims to present an outline of mathematics and its history, with particular emphasis on events that shook up its philosophy. It ranges from the discovery of irrational numbers in ancient Greece to the nineteenth- and twentieth-century discoveries on the nature of infinity and proof. Recurring themes are intuition and logic, meaning and existence, and the discrete and the continuous. These themes have evolved under the influence of new mathematical discoveries and the story of their evolution is, to a large extent, the story of philosophy of mathematics.
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Online ISBN: 9781108610124
Publisher: Cambridge University Press
Print publication: 06 June 2019

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Bibliography

Beltrami, E. (1868). Teoria fondamentale degli spazii di curvatura costante.Annali di Matematica Pura ed Applicata, Ser. 2 2, 232–55. In his Opere Matematiche 1, 406–29, English translation in Stillwell (1996).Google Scholar
Bolzano, B. (1817). Rein analytischer Beweis des Lehrsatzes dass zwischen je zwey Werthen, die ein entgegengesetzes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege. Ostwald’s Klassiker 153. Repr., Leipzig: Engelmann, 1905. English translation in Russ (2004), 251–77.Google Scholar
Bolzano, B. (1851). Paradoxien der Unendlichen. Repr., Leipzig: Bei C. H. Reclam Sen. English translation in Russ (2004), 591678.Google Scholar
Bombelli, R. (1572). L’algebra. Prima edizione integrale. Introduzione di U. Forti. Prefazione di E. Bortolotti. Repr., Milan: Giangiacomo Feltrinelli Editore LXIII, 1966.Google Scholar
Boole, G. (1847). Mathematical Analysis of Logic. Repr., London: Basil Blackwell, 1948.Google Scholar
Cantor, G. (1874). Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen. Journal für reine und angewandte Mathematik 77, 258–62. In his Gesammelte Abhandlungen, 145–8. English translation by W. Ewald in Ewald (1996), 2:840–43.Google Scholar
Cantor, G. (1891). Über eine elementare Frage der Mannigfaltigkeitslehre. Jahresbericht deutschen Mathematiker-Vereinigung 1, 75–8. English translation by W. Ewald in Ewald (1996), 2:920–22.Google Scholar
Cardano, G. (1545). Ars magna. Translation in Richard Witmer,The Great Art or the Rules of Algebra, Cambridge, MA: MIT Press, 1968.Google Scholar
Church, A. (1936). An unsolvable problem in elementary number theory. American Journal of Mathematics 58, 345–63.Google Scholar
Cohen, P. (1963a). The independence of the continuum hypothesis I. Proceedings of the National Academy of Sciences of the United States of America 50, 1143–8.Google Scholar
Cohen, P. (1963b). The independence of the continuum hypothesis II. Proceedings of the National Academy of Sciences of the United States of America 51, 105–10.Google Scholar
Davis, M. (Ed.) (2004). The Undecidable. Mineola, NY: Dover Publications Inc. Corrected reprint of the 1965 original [Raven Press, Hawlett, NY].Google Scholar
Dedekind, R. (1888). Was sind und was sollen die Zahlen? Braunschweig: Vieweg und Sohn. English translation in Essays on the Theory of Numbers, New York: Dover, 1963.Google Scholar
Dehn, M. (1900). Über raumgleiche Polyeder. Göttingen Nachrichten 1900, 345–54.Google Scholar
Descartes, R. (1637). The Geometry of René Descartes (With a Facsimile of the First Edition, 1637). Translated by David Eugene Smith and Marcia L. Latham. Repr., New York: Dover, 1954.Google Scholar
Euler, L. (1748). Introductio in analysin infinitorum, I. Volume 8 of his Opera Omnia, series 1. English translation by John D. Blanton, Introduction to the Analysis of the Infinite, Book I, New York: Springer, 1988.Google Scholar
Ewald, W. (1996). From Kant to Hilbert: A Source Book in the Foundations of Mathematics. Vols. I and II. Oxford, UK: Clarendon Press.Google Scholar
Field, H. H. (1980). Science without Numbers. Princeton, NJ: Princeton University Press.Google Scholar
Frege, G. (1879). Begriffschrift. English translation in van Heijenoort (1967), 5–82.Google Scholar
Gauss, C. F. (1816). Demonstratio nova altera theorematis omnem functionem algebraicum rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse. Commentationes societas regiae scientiarum Gottingensis recentiores 3, 107–42. In his Werke 3, 31–56.Google Scholar
Gödel, K. (1930). Die Vollständigkeit der Axiome des logischen Funktionenkalküls. Monatshefte für Mathematik und Physik 37, 349–60. English translation in Gödel (1986), 103–23.Google Scholar
Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme. I. Monatshefte für Mathematik und Physik 38, 173–98. English translation in van Heijenoort (1967), 596–616.Google Scholar
Gödel, K. (1938). The consistency of the axiom of choice and the generalized continuum hypothesis. Proceedings of the National Academy of Sciences 25, 220–24.Google Scholar
Gödel, K. (1986). Collected Works. Vol. I. The Clarendon Press, Oxford University Press, New York. Publications 1929–1936, Edited and with a preface by Solomon Feferman.Google Scholar
Grassmann, H. (1844). Die lineale Ausdehnungslehre. Otto Wiegand, Leipzig. English translation in Grassmann (1995), 1312.Google Scholar
Grassmann, H. (1847). Geometrische Analyse geknüpft an die von Leibniz gefundene Geometrische Charakteristik. Weidmann’sche Buchhandlung, Leipzig. English translation in Grassmann (1995), 313414.Google Scholar
Grassmann, H. (1861). Lehrbuch der Arithmetic. Berlin: Enslin.Google Scholar
Grassmann, H. (1995). A New Branch of Mathematics. Chicago: Open Court. The Ausdehnungslehre of 1844 and other works, translated from the German with a note by Lloyd C. Kannenberg, and with a foreword by Albert C. Lewis.Google Scholar
Hamilton, W. R. (1835). Theory of conjugate functions, or algebraic couples. Communicated to the Royal Irish Academy, 1 June 1835. Mathematical Papers 3: 7696.Google Scholar
Heath, T. L. (1956). The Thirteen Books of Euclid’s Elements translated from the text of Heiberg. Vol. I: Introduction and Books I, II. Vol. II: Books III–IX. Vol. III: Books X–XIII and Appendix. Translated with introduction and commentary by Thomas L. Heath. 2nd edn. New York: Dover.Google Scholar
Hilbert, D. (1899). Grundlagen der Geometrie. Leipzig: Teubner. English translation in Foundations of Geometry, Chicago: Open Court, 1971.Google Scholar
Hilbert, D. (1902). Mathematical problems. Bulletin of the American Mathematical Society 8, 437–79. Translated by Frances Winston Newson.CrossRefGoogle Scholar
Huygens, C. (1659). Fourth part of a treatise on quadrature. Œuvres Complètes 14: 337.Google Scholar
Lambert, J. H. (1766). Die Theorie der Parallellinien. Magazin für reine und angewandte Mathematik 1786, 137–64, 325–58.Google Scholar
Minkowski, H. (1908). Raum und Zeit. Jahresbericht der Deutschen Mathematiker-Vereinigung 17, 7588.Google Scholar
Peano, G. (1888). Calcolo Geometrico secondo l’Ausdehnungslehre di H. Grassmann, preceduto dalle operazioni della logica deduttiva. Turin: Bocca. English translation in Peano (2000).Google Scholar
Peano, G. (1889). Arithmetices principia. Turin: Bocca.Google Scholar
Peano, G. (2000). Geometric Calculus. Boston, MA: Birkhäuser. According to the Ausdehnungslehre of H. Grassmann, translated from the Italian by Lloyd C. Kannenberg.Google Scholar
Poincaré, H. (1952). Science and Method. New York: Dover. Translated by Francis Maitland, with a preface by Bertrand Russell.Google Scholar
Post, E. L. (1941). Absolutely unsolvable problems and relatively undecidable propositions – an account of an anticipation. In Davis (2004), 338433.Google Scholar
Riemann, G. F. B. (1854). Über die Hypothesen, welche der Geometrie zu Grunde liegen. In Werke, 2nd edn., 272–87. English translation in Ewald (1996), 2:652–61.Google Scholar
Russ, S. (2004). The Mathematical Works of Bernard Bolzano. Oxford: Oxford University Press.CrossRefGoogle Scholar
Russell, B. (1903). The Principles of Mathematics. Cambridge: Cambridge University Press.Google Scholar
Saccheri, G. (1733). Euclid Vindicated from Every Blemish. Classic Texts in the Sciences. Repr., Cham, Switzerland: Birkhäuser/Springer, 2014. Dual Latin–English text, edited and annotated by Vincenzo De Risi translated from the Italian by G. B. Halsted and L. Allegri.Google Scholar
Schwarz, H. A. (1872). Über diejenigen Fälle, in welchen die Gaussische hypergeometrische Reihe eine algebraische Function ihres vierten Elementes darstellt. Journal für reine und angewandte Mathematik 75, 292335. In his Mathematische Abhandlungen 2, 211–259.Google Scholar
Skolem, T. (1922). Einige Bemerkungen zu axiomatischen Begründung der Mengenlehre. In Mathematikerkongressen i Helsingfors den 4–7 Juli 1922, Den femte skandinaviska matematikerkongressen, Redogörelse, 217–32. English translation in van Heijenoort (1967), 290–301.Google Scholar
Stillwell, J. (1996). Sources of Hyperbolic Geometry. Providence, RI: American Mathematical Society.Google Scholar
Turing, A. (1936). On computable numbers, with an application to the Entscheidungsproblem. Proceedings of the London Mathematical Society 42, 230–65.Google Scholar
van Heijenoort, J. (1967). From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. Cambridge, MA: Harvard University Press.Google Scholar
Vilenkin, N. Y. (1995). In Search of Infinity. Boston, MA: Birkhäuser. Translated from the Russian original by Abe Shenitzer with the editorial assistance of Hardy Grant and Stefan Mykytiuk.Google Scholar
von Koch, H. (1904). Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire. Archiv för Matematik, Astronomi och Fysik 1, 681704.Google Scholar
Whitehead, A. N., and Russell, B. (1910–13). Principia Mathematica. 3 vols. Cambridge: Cambridge University Press.Google Scholar
Zermelo, E. (1908). Untersuchungen über die Grundlagen der Mengenlehre I. Mathematische Annalen 65, 261–81. English translation in van Heijenoort (1967), 200–215.CrossRefGoogle Scholar

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A Concise History of Mathematics for Philosophers
  • John Stillwell, University of San Francisco
  • Online ISBN: 9781108610124
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A Concise History of Mathematics for Philosophers
  • John Stillwell, University of San Francisco
  • Online ISBN: 9781108610124
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A Concise History of Mathematics for Philosophers
  • John Stillwell, University of San Francisco
  • Online ISBN: 9781108610124
Available formats
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