FOM: North Pole Argument (Frege #1)

[JRG Williams]

In the passage Dean Buckner refers to, Frege is objecting to the notion that
`Abstract number is the empty form of difference'  (quotation from Jevons at GL
44). Frege asks: what are we to understand by `the empty form of difference' .
Frege suggests: perhaps the abstract number 2 is to be identified with (the
schematic proposition):

    not-(a=b)                (*)

Then he gives some objections, eg that there is no obvious way to relate `the
Earth has two poles' to (*). Eg `the north pole is different from the south
pole' won't do. Dean agrees with Frege thus far, I think.

Dean then says:
>The real equivalence is between "the Earth has two poles" (1) and
>"the Earth has a pole x and the Earth has a pole y and x is a
>different thing from y" (2).

OK. But that's not what Frege's argument at GL 44  was aimed against! You could
perhaps take (2) to be what Jevons meant by `the empty form of difference', and
so try to convict Frege of a misreading of his opponent's position (not an
unusual occurence.) But as it is Dean's point doesn't engage with Frege's
argument, which concerned (*) and not (2).

On the general question of analyzing uses of number in terms of quantification,
identity etc, I haven't yet heard what analyses Dean would give for the
following statements:

(a) there is exactly one even prime.
(b) there are infinitely many primes.

Until we get detailed suggestions for these, we can't really judge whether
Dean's got a coherent alternative to the literalist reading of these statements.
And only once we have the full alternative can we begin to assess whether or not
it's a better theory than Frege's.

best
Robbie

NB: Frege considers the analysis of numerically definite quantification in terms
of existential quantification, identity and negation  at GL 55-56. That's where
to look for his opinions on the subject. I don't think he disputes the truth of
the equivalence Dean notes (who would?): rather, he questions whether it's
adequate as a definition of number.

J. Robert G. Williams
Wolfson College, Oxford
B.Phil Candidate (master's in philosophy).



Dean Buckner wrote:

> This is the first in a series of short objections to Frege's arguments about
> number.
>
> #1 The North Pole Argument
>
> Frege argues, in discussing Jevons' and Schroder's theory of number (GL
> ~~40-44), that a proposition like "a is different from b" cannot "give" the
> number 2.  He objects that if so, "the Earth has two poles" must be true iff
> "the North Pole is different from the South Pole" is true.
> Whereas, he says, either can be true without the other.
>
> Of course, but this is quite spurious.  "The North Pole" (like "Dartmouth")
> is just a proper name in disguise.  The real equivalence is between "the
> Earth has two poles" and "the Earth has a pole x and the Earth has a pole y
> and x is a different thing from y".
>
> Thus his argument does not disprove that difference between objects (not
> concepts) can yield the concept of number.
>
> Counter-objections warmly invited.
>
> Dean Buckner
> 4 Spencer Walk
> London, SW15 1PL
> ENGLAND
>
> Work 020 7676 1750
> Home 020 8788 4273