Frege's cardinals as concept-correlates

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)
DOI 10.1007/s10670-005-4308-2
 Save to my reading list
Follow the author(s)
My bibliography
Export citation
Find it on Scholar
Edit this record
Mark as duplicate
Revision history
Request removal from index
Download options
Our Archive

Upload a copy of this paper     Check publisher's policy     Papers currently archived: 27,636
Through your library
References found in this work BETA
What Numbers Could Not Be.Paul Benacerraf - 1965 - Philosophical Review 74 (1):47-73.
Principles of Mathematics.Bertrand Russell - 1903 - Cambridge University Press.
The Basic Laws of Arithmetic.Gottlob Frege - 1964 - Berkeley: University of California Press.

View all 17 references / Add more references

Citations of this work BETA

Add more citations

Similar books and articles

Monthly downloads

Added to index


Total downloads

36 ( #142,885 of 2,169,144 )

Recent downloads (6 months)

1 ( #345,568 of 2,169,144 )

How can I increase my downloads?

My notes
Sign in to use this feature

There  are no threads in this forum
Nothing in this forum yet.

Other forums