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Full Idea
In Cantor's abstractionist account there can only be two numbers, 0 and 1. For abs(Socrates) = abs(Plato), since their numbers are the same. So the number of {Socrates,Plato} is {abs(Soc),abs(Plato)}, which is the same number as {Socrates}!
Gist of Idea
After abstraction all numbers seem identical, so only 0 and 1 will exist!
Source
Kit Fine (Cantorian Abstraction: Recon. and Defence [1998], §1)
Book Ref
-: 'Journal of Philosophy' [-], p.4
A Reaction
Fine tries to answer this objection, which arises from §45 of Frege's Grundlagen. Fine summarises that "indistinguishability without identity appears to be impossible". Maybe we should drop talk of numbers in terms of sets.
| 9146 | After abstraction all numbers seem identical, so only 0 and 1 will exist! [Fine,K] |
| 9148 | I think of variables as objects rather than as signs [Fine,K] |
| 9149 | To obtain the number 2 by abstraction, we only want to abstract the distinctness of a pair of objects [Fine,K] |
| 9150 | We should define abstraction in general, with number abstraction taken as a special case [Fine,K] |
| 9152 | If green is abstracted from a thing, it is only seen as a type if it is common to many things [Fine,K] |