Abstract
Frege claims that mathematical theories are collections of thoughts, and that scientific continuity turns on thought-identity. This essay explores the difficulties posed for this conception of mathematics by the conceptual development canonically involved in mathematical progress. The central difficulties are (i) that mathematical development often involves sufficient conceptual progress that mature versions of theories do not involve easily-recognizable synonyms of their earlier versions, and (ii) that the introduction of new elements in the domains of mathematical theories would seem to conflict with Frege’s view that the original theories involved determinate reference. It is argued here that the difficulties apparently posed to Frege’s central views stem from an overly-simple view of Frege’s understanding of mathematical objects and of reference. The positive view recommended is one on which Frege’s view of mathematical theories is largely consistent with, and helps make sense of, the phenomenon of theoretical unity across conceptual development.
Versions of this essay were presented at the 2014 “Frege@Stirling” workshop at Stirling University, and at the 2014 Logic Colloquium in Vienna. Many thanks to the organizers and audience members, especially to Philip Ebert, Bob Hale, and Rob Trueman for helpful comments.
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Notes
- 1.
- 2.
See Burge (1979).
- 3.
See Burge (1984).
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- 5.
“Thus it is shown that our eight primitive names have a reference and thereby that the same applies to all names correctly formed out of them. However, not only a reference but also a sense belongs to all names correctly formed from our signs. Every such name of a truth-value expresses a sense, a thought.” [Frege (1893) §32].
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This account of Frege’s project as one of conceptual analysis is argued for in Blanchette (2012a).
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The logical equivalence here requires the faulty principle about extensions, assumed by Frege in Grundlagen, that the extension of F = the extension of G iff ∀x (Fx iff Gx).
References
Blanchette, P. (2012a). Frege’s conception of logic. New York: Oxford University Press.
Blanchette, P. (2012b). Frege on shared belief and total functions. The Journal of Philosophy CIX, 1/2, 9–39.
Burge, T. (1979). Sinning against frege. The Philosophical Review, 88, 398–432. (Reprinted in Burge (2005) 213–239).
Burge, T. (1984). Frege on extensions of concepts, from 1884 to 1903. The Philosophical Review, 93, 3–34. Reprinted in Burge (2005) 273–298.
Burge, T. (2005). Truth, thought, reason: Essays on Frege. Oxford: Clarendon Press.
Frege, G. (1884). Die Grundlagen der Arithmetik. Breslau: William Koebner. [English edition: Frege, G (1978). The foundations of arithmetic]. (J. L. Austin, Trans.). Evanston, Ill.: Northwestern University Press.
Frege, G. (1891). Funktion und Begriff. Jena: Hermann Pohle. Reprinted in Frege (1967): 125–142. [English edition: Frege, G. (1984a). Function and concept. Frege (1984): 137–156] (P. Geach, Trans.).
Frege, G. (1892a). Über Sinn und Bedeutung. Zeitschrift für Philosophie und Philosophische Kritik, 100, 25–50. Reprinted in Frege (1967): 143–162. [English edition: Frege, G. (1892). On Sense and Reference. Frege (1984): 157–177] (M. Black, Trans.).
Frege, G. (1892b). Über Begriff und Gegenstand. Vierteljahrsschrift für wissenschaftliche Philosophie, 16, 192–205. Reprinted in Frege, G. (1967): 167–178. [English edition: Frege, G. (1984b). On Concept and Object. Frege (1984): 182–194] (P. Geach, Trans.).
Frege, G. (1893). Grundgesetze der Arithmetik (Vol. I). Jena: Hermann Pohle. [English edition: Frege, G. (2013). Basic laws of arithmetic (P. Ebert, & M. Rossberg, Trans.). Oxford: Oxford University Press.
Frege, G. (1903). Grundgesetze der Arithmetik (Vol. II). Jena: Hermann Pohle. [English edition: Frege, G. (2013). Basic laws of arithmetic (P. Ebert, & M. Rossberg, Trans.). Oxford: Oxford University Press.
Frege, G. (1967). Kleine Schriften. In I. Angelelli (Ed.), Olms: Hildesheim.
Frege, G. (1979). Posthumous writings. In H. Hermes, F. Kambartel, & F. Kaulbach (Ed.), (Long & White, Trans.). Chicago: University of Chicago Press.
Frege, G. (1983). Nachgelassene Schriften. In H. Hermes, Friedrich Kambartel, & F. Kaulbach (Eds.) Hamburg: Felix Meiner Verlag.
Frege, G. (1984). Collected papers on mathematics, logic, and philosophy. B. McGuinness (Ed.), Oxford: Blackwell.
Peano, G. (1898). Riposta. Rivista di Matematica, 6, 60–61.
Reck, E., & Awodey, S. (Eds.). (2004). Frege’s lectures on logic: Carnap’s student notes, 1910–1914. Chicago: Open Court.
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Blanchette, P. (2016). Frege on Mathematical Progress. In: Costreie, S. (eds) Early Analytic Philosophy - New Perspectives on the Tradition. The Western Ontario Series in Philosophy of Science, vol 80. Springer, Cham. https://doi.org/10.1007/978-3-319-24214-9_1
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