Abstract
In this paper, I shall discuss several topics related to Frege’s paradigms of second-order abstraction principles and his logicism. The discussion includes a critical examination of some controversial views put forward mainly by Robin Jeshion, Tyler Burge, Crispin Wright, Richard Heck and John MacFarlane. In the introductory section, I try to shed light on the connection between logical abstraction and logical objects. The second section contains a critical appraisal of Frege’s notion of evidence and its interpretation by Jeshion, the introduction of the course-of-values operator and Frege’s attitude towards Axiom V, in the expression of which this operator occurs as the key primitive term. Axiom V says that the course-of-values of the function f is identical with the course-of-values of the function g if and only if f and g are coextensional. In the third section, I intend to show that in Die Grundlagen der Arithmetik (1884) Frege hardly could have construed Hume’s Principle (HP) as a primitive truth of logic and used it as an axiom governing the cardinality operator as a primitive sign. HP expresses that the number of Fs is identical with the number of Gs if and only if F and G are equinumerous. In the fourth section, I argue that Wright falls short of making a convincing case for the alleged analyticity of HP. In the final section, I canvass Heck’s arguments for his contention that Frege knew he could deduce the simplest laws of arithmetic from HP without invoking Axiom V. I argue that they do not carry conviction. I conclude this section by rejecting an interpretation concerning HP suggested by MacFarlane.
Similar content being viewed by others
References
I use the following abbreviations for references to Frege’s works:
BS: Begriffsschrift. Eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, L. Nebert, Halle a.S. 1879.
GGA: Grundgesetze der Arithmetik. Begriffsschriftlich abgeleitet, vol. I, H. Pohle, Jena 1893, vol. II, H. Pohle, Jena 1903.
GLA: Die Grundlagen der Arithmetik. Eine logisch mathematische Untersuchung über den Begriff der Zahl, W. Koebner, Breslau 1884.
KS: Kleine Schriften, ed. I. Angelelli, G. Olms, Hildesheim 1967.
NS: Nachgelassene Schriften, eds. H. Hermes, F. Kambartel and F. Kaulbach, F. Meiner, Hamburg 1969.
WB: Wissenschaftlicher Briefwechsel, eds. G. Gabriel, H. Hermes, F. Kambartel, C. Thiel, A. Veraart, F. Meiner, Hamburg 1976.
I refer by author and year of publication to the following works:
A. Antonelli R. May (2000) ArticleTitle‘Frege’s New Science’ Notre Dame Journal of Formal Logic 41 242–270
G. Boolos (1987a) ‘The Consistency of Frege’s Foundations of Arithmetic’ J. J. Thomson (Eds) On Being and Saying. Essays for Richard Cartwright MIT Press Cambridge, MA 3–20
Boolos, G.: 1987b, ‘Saving Frege from Contradiction’, Proceedings of the Aristotelian Society 1986–87, 137–151.
G. Boolos (1990) ‘The Standard of Equality of Numbers’ Boolos (Eds) Meaning and Method: Essays in Honor of Hilary Putnam Cambridge University Press Cambridge 261–277
G. Boolos (1995) ArticleTitle‘Frege’s Theorem and the Peano Postulates’ The Bulletin of Symbolic Logic 1 317–326
Boolos, G.: 1997, ‘Is Hume’s Principle Analytic?’, in Heck 1997a, pp. 245–261.
T. Burge (1984) ArticleTitle‘Frege on Extensions of Concepts, From 1884 to 1903’ The Philosophical Review 93 3–34
T. Burge (1998) ArticleTitle‘Frege on Knowing the Foundation’ Mind 107 305–347 Occurrence Handle10.1093/mind/107.426.305
T. Burge (2000) ‘Frege on Apriority’ P. Boghossian C. Peacocke (Eds) New Essays on the A Priori Clarendon Press Oxford 11–42
M. Dummett (1991) Frege. Philosophy of Mathematics Duckworth London
Fine K.: 1998, ‘The Limits of Abstraction’, in Schirn 1998, pp. 503–629.
B. Hale (1987) Abstract Objects Blackwell Oxford
B. Hale (2000) ArticleTitle‘Reals by Abstraction’ Philosophia Mathematica 8 100–123
B. Hale C. Wright (2001) The Reason’s Proper Study. Essays Towards a Neo-Fregean Philosophy of Mathematics Clarendon Press Oxford
Heck, R. G.: 1993, ‘The Development of Arithmetic in Frege’s Grundgesetze der Arithmetik’, The Journal of Symbolic Logic 58, 579–601; reprinted with minor revisions in W. Demopoulos (ed.), Frege’s Philosophy of Mathematics, Harvard University Press, Cambridge, MA., pp. 257–285.
R. G. Heck (1995) ‘Frege’s Principle’ J. Hintikka (Eds) From Dedekind to Gödel Kluwer Dordrecht 119–142
R. G. Heck (Eds) (1997a) Language, Thought, and Logic. Essays in Honour of Michael Dummett Clarendon Press Oxford
Heck, R. G.: 1997b, ‘The Julius Caesar Objection’, in Heck 1997a, pp. 273–308.
R. G. Heck (1997c) ArticleTitle‘Finitude and Hume’s Principle’ Journal of Philosophical Logic 26 589–617 Occurrence Handle10.1023/A:1004299720847
R. G. Heck (1998) ArticleTitle‘Grundgesetze der Arithmetik I §§29–32’ Notre Dame Journal of Formal Logic 38 437–474
Heck R. G.: forthcoming, ‘Frege and Semantics’, in T. Ricketts (ed.), The Cambridge Companion to Frege, Cambridge University Press, Cambridge.
H. Hodes (1984) ArticleTitle‘Logicism and the Ontological Commitments of Arithmetic’ Journal of Philosophy 81 123–149
R. Jeshion (2001) ArticleTitle‘Frege’s Notions of Self-Evidence’ Mind 110 937–976 Occurrence Handle10.1093/mind/110.440.937
Leibniz, G. W.: 1875–1890, in C. J. Gerhardt (ed.), Die philosophischen Schriften von G.W. Leibniz, Berlin; reprint Olms, Hildesheim 1960–1961.
Leibniz, G. W.: 1903, in L. Couturat (ed.), Opuscules et fragments inédits de Leibniz, Paris; reprint Olms, Hildesheim 1966.
J. MacFarlane (2002) ArticleTitle‘Frege, Kant, and the Logic in Logicism’ The Philosophical Review 111 25–63
Parsons, C.: 1997, ‘Wright on Abstraction and Set Theory’, in Heck 1997a, pp. 263–272.
M. Resnik (1986) ‘Frege’s Proof of Referentiality’ L. Haaparanta J. Hintikka (Eds) Frege Synthesized. Essays on the Philosophical and Foundational Work of Gottlob Frege D. Reidel Dordrecht, Boston 177–195
Russell B. (1903). The Principles of Mathematics. Cambridge; second edition, New York.
B. Russell (1919) Introduction to Mathematical Philosophy Redwood Books Trowbridge, Wiltsshire
M. Schirn (1989) ArticleTitle‘Frege on the Purpose and Fruitfulness of Definitions’ Logique et Analyse 125–126 61–80
M. Schirn (Eds) (1998) The Philosophy of Mathematics Today Clarendon Press Oxford
M. Schirn (2003) ArticleTitle‘Fregean Abstraction, Referential Indeterminacy and the Logical Foundations of Arithmetic’ Erkenntnis 59 203–232 Occurrence Handle10.1023/A:1024634404708
Thiel, C.: 1975, ‘Zur Inkonsistenz der Fregeschen Mengenlehre’, in C. Thiel (ed.), Frege und die moderne Grundlagenforschung, Anton Hain, Meisenheim am Glan, pp. 134–159.
P. Simons (1992) ArticleTitle‘Why Is There So Little Sense in Grundgesetze?’ Mind 101 753–766
J. Weiner (2002) ‘Section 31 Revisited: Frege’s Elucidations’ E. Reck (Eds) From Frege to Wittgenstein: Perspectives on Analytic Philosophy Oxford University Press Oxford 149–182
C. Wright (1983) Frege’s Conception of Numbers as Objects Aberdeen University Press Aberdeen
Wright, C.: 1997, ‘On the Philosophical Significance of Frege’s Theorem’, in Heck 1997a, pp. 201–244
C. Wright (1999) ArticleTitle‘Is Hume’s Principle Analytic?’ Notre Dame Journal of Formal Logic 40 6–30
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Schirn, M. Hume’s Principle and Axiom V Reconsidered: Critical Reflections on Frege and His Interpreters. Synthese 148, 171–227 (2006). https://doi.org/10.1007/s11229-004-2829-x
Issue Date:
DOI: https://doi.org/10.1007/s11229-004-2829-x