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A Solid House of Cards

  • Ole Ravn
  • Ole Skovsmose
Chapter
Part of the History of Mathematics Education book series (HME)

Abstract

This chapter investigates the logicist programme that became influential in the first part of the 20th century. Logicism pursues the idea of reducing mathematics to the secure foundation that has been established in logic. It suggests how mathematical concepts can be defined in terms of logical concepts, and how mathematical theorems can be derived from logical proposition.

The chapter considers Frege’s critique of important philosophic conceptions of mathematics, before it presents his elaboration of the logicist programme as provided in his Begriffsschrift. Then follows a presentation of Russell and Whitehead’s continued detailed elaboration in Principia Mathematica. This elaboration was initiated by Russell’s discovery of a paradox that apparently destroyed the solidity of the logical foundation of mathematics in the format suggested by Frege. Russell communicated his discovery of the paradox to Frege, who got deeply chocked. While Frege gave up solving the paradox and paralysed in his logical endeavours, Russell moved on and tried to locate a solution that could save the logicist programme. In his own time, Frege was an unknown German mathematician with a taste for logic. His work remained unknown to the wider public until Russell discovered and developed his ideas. Later, it became evident that Frege was the most important person in the development of modern logic.

Keywords

Begriffsschrift Logicism Mathematica Mathematics as logic Paradox Principia 

References

  1. Dummett, M. (1981a). Frege: Philosophy of language. London, UK: Duckworth.Google Scholar
  2. Dummett, M. (1981b). The interpretation of Frege’s philosophy. London, UK: Duckworth.Google Scholar
  3. Dummett, M. (1991). Frege: Philosophy of mathematics. London, UK: Duckworth.Google Scholar
  4. Frege, G. (1893/1903). Grundgesetze der Arithmetik I-II. Jena, Germany: H. Poble.Google Scholar
  5. Frege, G. (1967). Begriffsschrift: A formula language, modelled upon that of arithmetic, for pure thought. In J. van Hiejenoort (Ed.), From Frege to Gödel: A source book in mathematical logic, 1879–1931 (pp. 1–82). Cambridge, MA: Harvard University Press.Google Scholar
  6. Frege, G. (1969). Über Sinn und Bedeutung. In G. Frege (Ed.), Funktion, Begriff, Bedeutung: Fünf logische Studien (pp. 40–65). Göttingen, Germany: Vandenhoech & Ruprecht.Google Scholar
  7. Frege, G. (1978). Die Grundlagen der Arithmetik/the foundations of arithmetic. Oxford, UK: Blackwell.Google Scholar
  8. Hersh, R. (1998). What is mathematics, really? London, UK: Vintage.Google Scholar
  9. Hume, D. (2000). A treatise of human nature. Oxford, UK: Oxford University Press.Google Scholar
  10. Ishiguro, H. (1972). Leibniz’s philosophy of logic and language. London, UK: Duckworth.Google Scholar
  11. Kneale, W., & Kneale, M. (1962). The development of logic. Oxford, UK: Clarendon Press.Google Scholar
  12. Leibniz, G. W. (1966). Logical papers: A selection. (Translated and edited with an introduction by G. H. R. Parkinson). Oxford, UK: Clarendon Press.Google Scholar
  13. Monk, R. (1996). Bertrand Russell: The spirit of solitude. London, UK: Jonathan Cape.Google Scholar
  14. Monk, R. (2000). Bertrand Russell: The ghost of madness. London, UK: Jonathan Cape.Google Scholar
  15. Moorehead, C. (1992). Bertrand Russell: A life. London, UK: Sinclair-Stevenson.Google Scholar
  16. Russell, B. (1978). Autobiography. London, UK: Unwin Paperbacks.Google Scholar
  17. Russell, B. (1992). The principles of mathematics. London, UK: Routledge.Google Scholar
  18. Russell, B. (1993). Introduction to mathematical philosophy. London, UK: Routledge.Google Scholar
  19. Whitehead, A., & Russell, B. (1910–1913). Principia mathematica I-III. Cambridge, UK: Cambridge University Press.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Ole Ravn
    • 1
  • Ole Skovsmose
    • 1
    • 2
  1. 1.Department of Learning and PhilosophyAalborg UniversityAalborgDenmark
  2. 2.Department of Mathematics EducationState University of São Paulo, (Universidade Estadual Paulista, Unesp)São PauloBrazil

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