David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jack Alan Reynolds
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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.
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References found in this work BETA
Paul Benacerraf (1965). What Numbers Could Not Be. Philosophical Review 74 (1):47-73.
Bertrand Russell (1903). Principles of Mathematics. Cambridge University Press.
Gottlob Frege (1964). The Basic Laws of Arithmetic. Berkeley, University of California Press.
Citations of this work BETA
Kevin C. Klement (2012). Frege's Changing Conception of Number. Theoria 78 (2):146-167.
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