Erkenntnis 59 (2):203 - 232 (2003)

Matthias Schirn
Ludwig Maximilians Universität, München
In Die Grundlagen der Arithmetik, Frege attempted to introduce cardinalnumbers as logical objects by means of a second-order abstraction principlewhich is now widely known as ``Hume's Principle'' (HP): The number of Fsis identical with the number of Gs if and only if F and G are equinumerous.The attempt miscarried, because in its role as a contextual definition HP fails tofix uniquely the reference of the cardinality operator ``the number of Fs''. Thisproblem of referential indeterminacy is usually called ``the Julius Caesar problem''.In this paper, Frege's treatment of the problem in Grundlagen is critically assessed. In particular, I try to shed new light on it by paying special attention to the framework of his logicism in which it appears embedded. I argue, among other things, that the Caesar problem, which is supposed to stem from Frege's tentative inductive definition of the natural numbers, is only spurious, not genuine; that the genuine Caesar problem deriving from HP is a purely semantic one and that the prospects of removing it by explicitly defining cardinal numbers as objects which are not classes are presumably poor for Frege. I conclude by rejecting two closely connected theses concerning Caesar put forward by Richard Heck: (i) that Frege could not abandon Axiom V because he could not solve the Julius Caesar problem without it; (ii) that (by his own lights) his logicist programme in Grundgesetze der Arithmetik failed because he could not overcome that problem.
Keywords Philosophy   Philosophy   Epistemology   Ethics   Logic   Ontology
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Reprint years 2004
DOI 10.1023/A:1024634404708
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References found in this work BETA

Frege.Michael Dummett - 1975 - Teorema: International Journal of Philosophy 5 (2):149-188.
The Limits of Abstraction.Kit Fine - 2002 - Oxford, England: Oxford University Press.
Frege's Conception of Numbers as Objects.Crispin Wright - 1986 - Studia Logica 45 (3):330-330.

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