David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
Erkenntnis 65 (2):207 - 243 (2006)
In his Grundgesetze, Frege hints that prior to his theory that cardinal numbers are objects (courses-of-values) he had an “almost completed” manuscript on cardinals. Taking this early theory to have been an account of cardinals as second-level functions, this paper works out the significance of the fact that Frege’s cardinal numbers (as objects) is a theory of concept-correlates. Frege held that, where n>2, there is a one–one correlation between each n-level function and an n−1 level function, and a one–one correlation between each first-level function and an object (a course-of-values of the function). Applied to cardinals, the correlation offers new answers to some perplexing features of Frege’s philosophy. It is shown that within Frege’s concept-script, a generalized form of Hume’s Principle is equivalent to Russell’s Principle of Abstraction – a principle Russell employed to demonstrate the inadequacy of definition by abstraction. Accordingly, Frege’s rejection of definition of cardinal number by Hume’s Principle parallels Russell’s objection to definition by abstraction. Frege’s correlation thesis reveals that he has a way of meeting the structuralist challenge (later revived by Benacerraf, 1965) that it is arithmetic, and not a privileged progression of objects, that matters to the finite cardinals.
|Keywords||Philosophy Logic Ethics Ontology Epistemology Philosophy|
|Categories||categorize this paper)|
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
|Through your library|
References found in this work BETA
Paul Benacerraf (1965). What Numbers Could Not Be. Philosophical Review 74 (1):47-73.
Patricia A. Blanchette (1994). Frege's Reduction. History and Philosophy of Logic 15 (1):85-103.
George Boolos (1990). The Standard of Equality of Numbers. In Meaning and Method: Essays in Honor of Hilary Putnam. Cambridge University Press 261--77.
George Boolos (1993). Whence the Contradiction? Aristotelian Society Supplementary Volume 67:211--233.
Nino Cocchiarella (1985). Frege's Double Correlation Thesis and Quine's Set Theories NF and ML. Journal of Philosophical Logic 14 (1):1 - 39.
Citations of this work BETA
Kevin C. Klement (2012). Frege's Changing Conception of Number. Theoria 78 (2):146-167.
Similar books and articles
Arnold Cusmariu (1979). Russell's Paradox Re-Examined. Erkenntnis 14 (3):365-370.
Mark Textor (2010). Frege's Concept Paradox and the Mirroring Principle. Philosophical Quarterly 60 (238):126-148.
Howard Wettstein (1990). Frege‐Russell Semantics? Dialectica 44 (1‐2):113-135.
Michael Beaney (2007). Frege's Use of Function-Argument Analysis and His Introduction of Truth-Values as Objects. Grazer Philosophische Studien 75 (1):93-123.
Matthias Schirn (2003). Fregean Abstraction, Referential Indeterminacy and the Logical Foundations of Arithmetic. Erkenntnis 59 (2):203 - 232.
Marco Ruffino (2003). Why Frege Would Not Be a Neo-Fregean. Mind 112 (445):51-78.
Edward N. Zalta (1999). Natural Numbers and Natural Cardinals as Abstract Objects: A Partial Reconstruction of Frege"s Grundgesetze in Object Theory. [REVIEW] Journal of Philosophical Logic 28 (6):619-660.
Richard Heck (1993). The Development of Arithmetic in Frege's Grundgesetze der Arithmetik. Journal of Symbolic Logic 58 (2):579-601.
Boudewijn de Bruin (2008). Wittgenstein on Circularity in the Frege-Russell Definition of Cardinal Number. Philosophia Mathematica 16 (3):354-373.
Added to index2009-01-28
Total downloads28 ( #97,310 of 1,699,833 )
Recent downloads (6 months)4 ( #161,079 of 1,699,833 )
How can I increase my downloads?