Thank you for the Pritchard reference, which I didn't know. I agree with
you about J. Klein's book.
It was not just the ancient Greeks who used ``arithmos'' for (something
like) finite sets. Many of Frege's criticisms of his predecessors and
contemporaries (including Euclid!) turn on their use of it that way and
his refusal to understand. E.g. in Hume's Treatise (Bk. I, Part III,
Sec. I) we have ``When two numbers are so combined as that the one has
always an unit answering to every unit of the other, we pronounce them
as equal''.
This is the source of the unfortunate `Hume's principle' to refer to
Cantor's definition of having the same cardinal number (for *arbitrary*
finite or infinite sets). But Frege finds difficulty even with this, the
source of which is the misunderstanding that I have already mentioned
and his reading of ``equal'' (he read it in German, I think: gleich) as
``identical''. (In Austin's English translation of Frege's Grundlagen,
the problems raised by the ambiguity of ``Gleichheit'' are increased by
his decision to always translate it as ``identity''.)
I discuss some of these things in a paper ``Frege versus Cantor and
Dedekind: On the concept of number''. It appears in a collection *Early
Analytic Philosophy: Frege, Russell, Wittgenstein: Essays in honor of
Leonard Linsky*, Open Court Press (1997), which I edited. (Its one way
to get your work published.) It appears also in a volume *Frege:
Importance and Legacy*, de Gruyter (1996), edited by M. Schirn. Given
his feelings about Frege, David Fowler might like to look at it: I think
of it as my trashing Frege paper---though the hostility is really more
directed at a certain style of English philosophy.