DOI: 10.4324/9780415249126-Y061-1
Version: v1,  Published online: 1998
Retrieved September 21, 2020, from

References and further reading

  • Carnap, R. (1931) ‘Die logizistische Grundlegung der Mathematik’, Erkenntnis 2: 91–105; trans. E. Putnam and G. Massey, ‘The Logicist Foundations of Mathematics’, in P. Benacerraf and H. Putnam (eds) Philosophy of Mathematics: Selected Readings, Cambridge: Cambridge University Press, 2nd edn, 1983, 41–52.

    (A classic account of logicism. Referred to in §5.)

  • Carnap, R. (1939) Foundations of Logic and Mathematics, Chicago, IL: University of Chicago Press.

    (A lucid account of Carnap’s view of the relation of logic to the analysis of language, and of how mathematics – a part of the purely logical aspect of language – acquires relevance for the formulation of propositions of empirical science.)

  • Dedekind, R. (1888) Was sind und was sollen die Zahlen?, Braunschweig: Vieweg; repr. in Gesammelte mathematische Werke (Collected Mathematical Works), ed. R. Fricke, E. Noether and Ö. Ore, Braunschweig: Vieweg, 1932, vol. 3, 335–390; trans. W.W. Beman (1901), ‘The Nature and Meaning of Numbers’, in Essays on the Theory of Numbers, New York: Dover, 1963.

    (Dedekind’s theory of the natural numbers. Referred to in §§2–4.)

  • Frege, G. (1879) Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, Halle: Nebert; repr. Hildesheim: Olms, 1964; trans. S. Bauer-Mengelberg, ‘ Begriffsschrift, a Formula Language, Modelled Upon That of Arithmetic, for Pure Thought’, in van Heijenoort (1967), 5–82.

    (Includes the first formulation of Frege’s system of logic. Referred to in §3.)

  • Frege, G. (1884) Die Grundlagen der Arithmetik: eine logisch-mathematische Untersuchung über den Begriff der Zahl, Breslau: Koebner; repr. and trans. J.L. Austin, The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number, Evanston, IL: Northwestern University Press, 2nd edn, 1980.

    (An extended philosophical essay which culminates in Frege’s definitions of number and finite number, and his analysis of mathematical induction; referred to in §3.)

  • Frege, G. (1893, 1903) Grundgesetze der Arithmetik: begriffsschriftlich abgeleitet, Jena: Pohle, 2 vols; repr. Hildesheim: Olms, 1966; part 1 of vol. 1 trans. M. Furth, Basic Laws of Arithmetic: An Exposition of the System, Berkeley, CA: University of California Press, 1964.

    (The first two volumes of Frege’s projected complete systematic treatment of arithmetic, including the theory of the real numbers, on an explicitly logical foundation. In consequence of Russell’s discovery of a contradiction in the system, the third volume was never written. Referred to in §§1, 3.)

  • Frege, G. (1924–5) ‘Erkenntnisquellen der Mathematik und der mathematischen Naturwissenschaften’, ‘Zahlen und Arithmetik’ and ‘Versuch einer neuen Begründung für die Arithmetik’, in Nachgelassene Schriften und Wissenschaftlicher Briefwechsel, vol. 1, ed. H. Hermes, F. Kambartel and F. Kaulbach, Hamburg: Meiner, 1969; trans. P. Long and R. White, ‘Sources of Knowledge of Mathematics and the Mathematical Natural Sciences’, ‘Numbers and Arithmetic’ and ‘A New Attempt at a Foundation for Arithmetic’, in Posthumous Writings, ed. H. Hermes, F. Kambartel and F. Kaulbach, Chicago, IL: University of Chicago Press, 1979.

    (Three very late brief manuscripts, giving Frege’s final thoughts on the foundations of mathematics. Referred to in §§1, 5.)

  • Frege, G. (1980) Philosophical and Mathematical Correspondence, ed. B. McGuinness, Chicago, IL: University of Chicago Press, 34–48.

    (Frege’s correspondence with Hilbert on the latter’s Foundations of Geometry. Referred to in §4.)

  • Gödel, K. (1930) ‘Die Vollständigkeit der Axiome des logischen Funktionenkalküls’, Monatshefte für Mathematik und Physik 37: 349–360; trans. S. Bauer-Mengelberg, ‘The Completeness of the Axioms of the Functional Calculus of Logic’, in van Heijenoort (1967), 582–591.

    (Gödel’s proof of the completeness of first-order logic; referred to in §5.)

  • Gödel, K. (1931) ‘Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I’, Monatshefte für Mathematik und Physik 38: 173–198; trans. J. van Heijenoort, ‘On Formally Undecidable Propositions of Principia Mathematica and Related Systems’, in van Heijenoort (1967), 592–617.

    (Gödel’s celebrated incompleteness theorems, the first of which is referred to in §5.)

  • Heijenoort, J. van (1967) From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, Cambridge, MA: Harvard University Press.

    (Includes translations of several of the works discussed in this article, with introductions.)

  • Löwenheim, L. (1915) ‘Über Möglichkeiten im Relativkalkül’, Mathematische Annalen 76: 447–470; trans. S. Bauer-Mengelberg, ‘On Possibilities in the Calculus of Relatives’, in van Heijenoort (1967), 232–251.

    (Löwenheim’s original proof – difficult, and couched in now unfamiliar notation – of the Löwenheim–Skolem theorem; van Heijenoort’s introduction is very helpful. Referred to in §5.)

  • Ramsey, F.P. (1925) The Foundations of Mathematics, Proceedings of the London Mathematical Society, series 2, 25(5): 338–384; repr. in The Foundations of Mathematics and Other Logical Essays, ed. R.B. Braithwaite, London: Routledge, 1931.

    (Referred to in §5. The posthumous collection, of both previously published and unpublished works, in which this essay appears includes important discussions of the theory of types, the axiom of choice and (a point not discussed in the text above) Russell and Whitehead’s axiom of infinity – needed in their version of a logicist theory in order to obtain a complete system of arithmetic.)

  • Russell, B.A.W. (1903) The Principles of Mathematics, Cambridge: Cambridge University Press; 2nd edn, London: Allen & Unwin, 1937; repr. London: Routledge, 1992.

    (A historically important work, of substantial difficulty; quoted in §1.)

  • Skolem, T. (1920) ‘Logisch-kombinatorische Untersuchungen über die Erfüllbarkeit oder Beweisbarkeit mathematischer Sätze nebst einem Theorem über dichte Mengen’, Videnskapsselskapets skrifter, I. Matematisk-naturvidenskabelig klasse 4; §1 trans. S. Bauer-Mengelberg, ‘Logico-Combinatorial Investigations in the Satisfiability or Provability of Mathematical Propositions: A Simplified Proof of a Theorem by L. Löwenheim and Generalizations of the Theorem’, in van Heijenoort (1967), 254–263.

    (Skolem’s more satisfactory proof, and strengthening, of Löwenheim’s theorem on countable models; referred to in §5.)

  • Skolem, T. (1923) ‘Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre’, in Matematikerkongressen i Helsingfors den 4–7 Juli 1922, Den femte skandinaviska matematikerkongressen, Redogörelse, Helsinki: Akademiska Bokhandeln, 217–232; trans. S. Bauer-Mengelberg, ‘Some Remarks on Axiomatized Set Theory’, in van Heijenoort (1967), 291–301.

    (A brief, remarkably clear and very deep discussion of a series of issues concerning the foundations of mathematics. Referred to in §5.)

  • Stein, H. (1988) ‘Logos, Logic, and Logistiké: Some Philosophical Remarks on the Nineteenth-Century Transformation of Mathematics’, in W. Aspray and P. Kitcher (eds) History and Philosophy of Modern Mathematics, Minneapolis, MN: University of Minnesota Press, 238–259.

    (Includes a discussion of some of the nineteenth-century developments that form the mathematical background to the logicist thesis referred to in §2; and gives an indication of the views of Riemann and Dedekind on geometry, referred to in §4.)

  • Whitehead, A.N. and Russell, B.A.W. (1910–13) Principia Mathematica, Cambridge: Cambridge University Press, 3 vols; 2nd edn, 1925–7.

    (Referred to in §4.)

  • Zermelo, E. (1904) ‘Beweis, daß jede Menge wohlgeordnet werden kann’, Mathematische Annalen 59: 514–516; trans. S. Bauer-Mengelberg, ‘Proof that Every Set can be Well-Ordered’, in van Heijenoort (1967), 139–141.

    (The article that brought prominently to the attention of mathematicians the axiom of choice – here ‘the assumption that coverings exist’; referred to in §4.)

Citing this article:
Stein, Howard. Bibliography. Logicism, 1998, doi:10.4324/9780415249126-Y061-1. Routledge Encyclopedia of Philosophy, Taylor and Francis,
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