Archive for History of Exact Sciences

, Volume 34, Issue 1–2, pp 141–192 | Cite as

“The True” in Gottlob Frege's “Über die Grundlagen der Geometrie”

  • William Boos
Article

Abstract

In this essay, I examine the metaphysical and metalogical ramifications of Gottlob Frege's controversy with David Hilbert and Alwin Korselt, over Hilbert'sGrundlagen der Geometrie. These ramifications include

(1) Korselt's original appeals to general metatheoretic Deutungen (interpretations);

(2) Hilbert's puzzling belief that whatever is consistent in some sense ‘exists;’ and

(3) Frege's ‘semantic monist’ conviction that theoretical sense and reference (mathematical and other) ‘must’ be “eindeutig lösbar” (“uniquely solvable”).

My principal conclusions are

(4) that Frege's position in (3) represented a pervasively dogmatic presumption that his newly discovered quantification theory ‘must’ have a propositional metatheory (‘the True’; ‘the False’); and

(5) that this needless assumption adversely affected not only his polemic against the moderate semantic relativism of Hilbert and Korselt, but also his reception of type-theoretic ideas, and greatly facilitated his vulnerability to the sort of self-referential inconsistency Russell discovered in Grundgesetz V.

These conclusions also seem to me to provide a conceptual framework for several of Frege's other arguments and reactions which might seem more particular and disparate. These include

(6) his arbitrary restrictions on the range of second-order quantification, which undercut his own tentative attempts to give accounts of independence and semantic consequence;

(7) his uncharacteristic hesitation, even dismay, at the prospect that such accounts might eventuate in a genuinely quantificational metamathematics, whose Gegenstände (objects) might themselves be Gedanken (thoughts); and, perhaps most revealingly

(8) his otherwise quite enigmatic, quasi-stoic doctrine that ‘genuine’ formal deduction ‘must’ be from premises that are ‘true.’

A deep reluctance to pluralize or iterate ‘the’ transition from theory to meta-theory would also be consonant, of course, with Frege's vigorous insistence that there can be only one level each of linguistic Begriffe (concepts) and Gegenstände (objects). With hindsight, such an assumption may seem more gratuitous in the philosophy of language (where it contributed, I would argue, to Wittgenstein's famous transition to ‘the mystical’ in 6.45 and 6.522 of the Tractatus); but its more implausible implications in this wider context seemed to emerge more slowly.

In the mathematical test-case discussed here, however, such strains were immediately and painfully apparent; the first models of hyperbolic geometry were described some thirty years before Frege drafted his polemic against Hilbert's pioneering exposition. It is my hope that a careful study of Frege's lines of argument in this relatively straightforward mathematical controversy may suggest other, parallel approaches to the richer and more ambiguous problems of his philosophy of language.

Keywords

Conceptual Framework Quantification Theory Careful Study Undercut Semantic Relativism 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Bartlett, James. “Funktion und Gegenstand. Eine Untersuchung in der Logik von Gottlob Frege.” Ph. D. dissertation, University of Munich, 1961.Google Scholar
  2. Boos, William, “Limits of Inquiry”, Erkenntnis 20 (1983), 150–194.Google Scholar
  3. Cantor, Georg. Gesammelte Abhandlungen. Berlin: Springer-Verlag, 1933; reprint Hildesheim: Olms Verlag, 1966.Google Scholar
  4. Dedekind, Richard. Was sind und was sollen die Zahlen. 10th ed. Braunschweig: Friedrich Vieweg und Sohn, 1969.Google Scholar
  5. Dummett, Michael. “Frege on the Consistency of Mathematical Theories,” in [Schirn 1975], Studies on Frege, edited by Matthias Schirn, pp. 229–42.Google Scholar
  6. Frege, Gottlob. “Kritische Beleuchtung einiger Punkte in E. Schröders Vorlesungen über die Algebra der Logik,” Archiv für Systematische Philosophie 1, pp. 433–56. Reprinted in [Frege 1960].Google Scholar
  7. Frege, Gottlob. “Über die Grundlagen der Geometrie,” I und II. Jahresberichte der Deutschen Mathematikvereinigung, XII (1903): 319–329, 368–75.Google Scholar
  8. Frege, Gottlob. “Über die Grundlagen der Geometrie, I, II, and III, Jahresberichte der Deutschen Mathematikvereinigung, XV (1906): 293–309, 377–403, and 423–430.Google Scholar
  9. Frege, Gottlob. Funktion, Begriff, Bedeutung, Fünf Logische Studien, ed. Günther Patzig. Göttingen: Vandenhoeck und Ruprecht, 1962.Google Scholar
  10. Frege, Gottlob. Grundgesetze der Arithmetik. Band I. Jena: Phole Verlag, 1893; Band II. Jena: Pohle Verlag, 1903; Olms Verlag, 1962.Google Scholar
  11. Frege, Gottlob. Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle: L. Nebert, 1879: reprint Begriffsschrift und andere Aufsätze, edited by Ignacio Angelelli. Hildesheim: Olms Verlag, 1964.Google Scholar
  12. Frege, Gottlob. Logische Untersuchungen, ed. Günther Patzig. Göttingen: Vandenoeck und Ruprecht, 1966.Google Scholar
  13. Frege, Gottlob. Kleine Schriften, ed. Ignacio Angelelli. Hildesheim: Olms Verlag, 1967.Google Scholar
  14. Frege, Gottlob. Nachgelassene Schriften, ed. Hans Hermes, Friedrich Kambartel und Friedrich Kaulbach. Hamburg: Felix Meiner Verlag, 1969.Google Scholar
  15. Frege, Gottlob. Die Grundlagen der Arithmetik. Breslau: Verlag von Wilhelm Koebner, 1884; reprint translated J. L. Austin, Oxford: Basil Blackwell, 1974.Google Scholar
  16. Frege, Gottlob. Wissenschaftlicher Briefwechsel, ed. Gottfried Gabriel et al. Hamburg: Felix Meiner Verlag, 1976.Google Scholar
  17. Frege, Gottlob. Die Grundlagen der Arithmetik. Breslau: Verlag von Wilhelm Koebner, 1884; reprint Hildesheim: Olms Verlag, 1977.Google Scholar
  18. Freudenthal, Hans. “The Main Trends in the Foundations of Geometry in the 19th Century”, in [Nagel, Suppes, Tarski], Logic, Methodology and Philosophy of Science, ed. Ernest Nagel, Patrick Suppes & Alfred Tarski. Stanford: Stanford University Press, 1962, pp. 613–21.Google Scholar
  19. Goldfarb, Warren. “Logic in the Twenties: the Nature of the Quantifier.” Journal of Symbolic Logic, 44, 3 (1979): 351–368.Google Scholar
  20. Henkin, Leon. “Completeness in the Theory of Types.” Journal of Symbolic Logic, 15 (1950): 81–91.Google Scholar
  21. Henkin, Leon, Suppes, Patrick & Tarski, Alfred. The Axiomatic Method. Amsterdam: North Holland Publishing Company, 1958.Google Scholar
  22. Kambartel, Friedrich. “Frege und die axiomatische Methode. Zur Kritik mathematikhistorischer Legitimationsversuche der formalistischen Ideologie,” in [Schirn 1975], Studies in Frege, edited by Matthias Schirn, pp. 215–28.Google Scholar
  23. Hilbert, David. “Mathematische Probleme. Vortrag, gehalten auf dem internationalen Mathematiker-Kongress zu Paris 1900,” Archiv der Mathematik und Physik, 3rd series, 1 (1901), pp. 44–63, 213–37; reprinted in [Hilbert 1935], Hilbert, David, Gesammelte Abhandlungen, Berlin: Springer Verlag, 1935, pp. 290–329.Google Scholar
  24. Hilbert, David. Grundlagen der Geometrie, 2nd ed. Leipzig: Teubner Verlag, 1903.Google Scholar
  25. Hilbert, David. “Über die Grundlagen der Logik und der Arithmetik.” Verhandlungen des Dritten Internationalen Mathematiker-Kongresses in Heidelberg vom 8. bis 13. August 1904. Leipzig: Teubner Verlag, 1905.Google Scholar
  26. König, Julius. “Zum Kontinuum-Problem.” Mathematische Annalen, 60 (1905): 177–80, in [van Heijenoort 1967], From Frege to Gödel, edited by Jan van Heigenoort, pp. 145–49.Google Scholar
  27. Korselt, Alwin. “Über die Grundlagen der Geometrie.” Jahresberichte der Deutschen Mathematikervereinigung, 12 (1903): 402–407.Google Scholar
  28. Korselt, Alwin. “Über die Grundlagen der Mathematik.” Jahresberichte der Deutschen Mathematikervereinigung, 14 (1905): 365–389.Google Scholar
  29. Korselt, Alwin. “Paradoxien in der Mengenlehre.” Jahresberichte der Deutschen Mathematikervereinigung, 15 (1906): 215–19.Google Scholar
  30. Korselt, Alwin. “Über die Logik der Geometrie.” Jahresberichte der Deutschen Mathematikervereinigung, 17 (1908): 98–124.Google Scholar
  31. Largeault, Jean. Logique et philosophie chez Frege. Publications de la Faculté des Lettres et Sciences de Paris-Sorbonne, Série “Recherches”, tome 50. Louvain: Editions Nauwelaerts, 1970.Google Scholar
  32. Nagel, Ernest, Patrick Suppes & Alfred Tarski, eds. Logic, Methodology and Philosophy of Sciences. Stanford: Standford University Press, 1962.Google Scholar
  33. Pascal, Blaise. “De l'espirit géométrique,” in Oeuvres Complètes edited Louis Lafuma. Paris: Editions du Seuil, 1963, pp. 348–55 (written before 1658, first published in full in 1844).Google Scholar
  34. Reinhardt, William. “Remarks on Reflection Principles, Large Cardinals, and Elementary Embeddings,” in Thomas Jech, ed.. Axiomatic Set Theory, Proceedings of Symposia in Pure Mathematics, vol. 13, 2. Providence: American Mathematical Society, pp. 189–205.Google Scholar
  35. Resnick, Michael. “The Frege-Hilbert Controversy.” Philosophy and Phenomenological Research, 34 (1973–74): 386–403.Google Scholar
  36. Schirn, Matthias, ed. Studies in Frege. Stuttgart/Bad Constalt: Fromann-Holzboog-Verlag, 1975.Google Scholar
  37. Schröder, Ernst, Vorlesungen über die Algebra der Logik (Exakte Logik), Leipzig: B. G. Teubner Verlag, 1890.Google Scholar
  38. Skolem, Thoralf, “Über die Nichtcharakterisierbarkeit mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen, Fundamenta Mathematicae 23 (1934), pp. 150–61.Google Scholar
  39. Sluga, Hans. Gottlob Frege. London: Routledge and Kegan Paul, 1980.Google Scholar
  40. Thiel, Christian. Sinn und Bedeutung in der Logik Gottlob Freges. Meisenheim am Glan: Verlag Anton Hain, 1965.Google Scholar
  41. Von Kutschera, Franz. Die Antinomien der Logik. Freiburg im Breisgau: Karl Alber Verlag, 1965.Google Scholar

Copyright information

© Springer-Verlag GmbH & Co. KG 1985

Authors and Affiliations

  • William Boos
    • 1
  1. 1.Department of PhilosophyUniversity of New MexicoAlbuquerque

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