Corcoran recommends Hambourger on the Frege-Russell number definition


Authors
John Corcoran
State University of New York, Buffalo
Abstract
It is widely agreed by philosophers that the so-called “Frege-Russell definition of natural number” is actually an assertion concerning the nature of the numbers and that it cannot be regarded as a definition in the ordinary mathematical sense. On the basis of the reasoning in this paper it is clear that the Frege-Russell definition contradicts the following three principles (taken together): (1) each number is the same entity in each possible world, (2) each number exists in each possible world, (3) some entities existing in the actual world do not exist in every possible world. Since these principles seem to be true, the paper is a refutation of the Frege-Russell definition. The paper does more. It shows that the contradictory of the Frege-Russell definition follows even when principles 2 and 3 are replaced by one considerably weaker principle. The ideas contained in the paper are related to two earlier objections to the definition. The first, sometimes attributed to the mathematician, C. S. Keyser, is that existence of the numbers as defined implies the existence of infinitely many particulars in each possible world. The second is, in effect, an idea which is said to have led Whitehead to reject the definition of number to which he had subscribed in Principia Mathematica. Whitehead is supposed to have said that he could not believe that the number two changes every “time twins are born”. The mathematician H. Jeffreys expressed similar ideas [Philos. of Sci. 5 (1938), 434–451]. One of the merits of the author’s work is that it refutes the Frege-Russell definition without the need to take sides on controversial points presupposed by the Keyser and Whitehead objections. The objections made by the author are therefore not to be identified with the Keyser and Whitehead objections. Even if the author’s work is to be regarded as a refinement and integration of previous ideas, it is nevertheless a contribution—not only because the basic points are well worth repeating but also because the refinements are logically significant improvements and because the author has stated them clearly and concisely in the idiom of contemporary philosophy.
Keywords FREGE  RUSSELL  WHITEHEAD  NUMBER  DEFINITION  LOGIC  ANALYTIC PHILOSOPHY  MATHEMATICS  POSSIBLE WORLD  SEMANTICS
Categories (categorize this paper)
Options
Edit this record
Mark as duplicate
Export citation
Find it on Scholar
Request removal from index
Revision history

Download options

Our Archive
External links

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

No references found.

Add more references

Citations of this work BETA

No citations found.

Add more citations

Similar books and articles

Frege's Definition of Number.Steven Wagner - 1983 - Notre Dame Journal of Formal Logic 24 (1):1-21.
Zamyšlení nad Fregovou definicí čísla.Marta Vlasáková - 2010 - Organon F: Medzinárodný Časopis Pre Analytickú Filozofiu 17 (3):339-353.
Frege‐Russell Semantics?Howard Wettstein - 1990 - Dialectica 44 (1‐2):113-135.
Number: From the Nyāya to Frege-Russell.J. L. Shaw - 1982 - Studia Logica 41 (2-3):283 - 291.
The Contemporary Interest of an Old Doctrine.William Demopoulos - 1994 - PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1994:209 - 216.
Is Frege's Definition of the Ancestral Adequate.Richard G. Heck - 2016 - Philosophia Mathematica 24 (1):91-116.
A Set Theory with Frege-Russell Cardinal Numbers.Alan McMichael - 1982 - Philosophical Studies 42 (2):141 - 149.
Russell And Frege On The Logic of Functions.Bernard Linsky - 2008 - The Baltic International Yearbook of Cognition, Logic and Communication 4:1-17.

Analytics

Added to PP index
2016-01-14

Total views
101 ( #76,663 of 2,253,738 )

Recent downloads (6 months)
9 ( #156,515 of 2,253,738 )

How can I increase my downloads?

Downloads

My notes

Sign in to use this feature