FOM: Hersh's unfruitful attack on logicism and formalism
Edwin Mares
Edwin.Mares at vuw.ac.nz
Tue Sep 29 23:20:02 EDT 1998
At 22:11 29/09/98 -0400, Simpson wrote:
>Reuben Hersh writes:
> > I see that you can restate the goals of formalism and logicism
> > .... why not say that your are updating the goals?
>
>I'm *not* updating the goals, at least not intentionally. My
>statement of the goals was intended to be reasonably accurate.
>
>Let's take Frege. You say that Frege's goal was to reduce mathematics
>to logic. I say that Frege's goal was to investigate *the extent to
>which* mathematics is reducible to logic. From the scientific point
>of view, these are two ways of saying the same thing. But my way is
>better, because it's more fruitful. You must dismiss Frege's work as
>a failure, while I can build on Frege's genuine and remarkable
>achievements.
>
I'm not sure I agree with either of these interpretations of Frege. Frege
explicitly
wanted to reduce arithmetic to logic to prove that arithmetical statements are
analytically true. But he didn't think that this was true of geometry, say,
on which
he seems to have taken a rather Kantian line. Thus, it would seem that he
neither wanted
to reduce all of maths to logic nor even tried to find out which parts of
maths outside
arithmetic were so reducibe.
Ed Mares
Department of Philosophy
Victoria University of Wellington
P.O. Box 600
Wellington, New Zealand
Ph: 64-4-471-5368
____________________________________________
Theorem 1. Every horse has an infinite number of legs.
(Proof by intimidation.)
Proof. Horses have an even number of legs. Behind they
have two legs and in front they have fore legs. This makes
six legs, which is certainly an odd number for a horse.
But the only number that is both odd and even is infinity.
Therefore horses have an infinite number of legs.
Joel E. Cohen, "On the Nature of Mathematical Proofs"
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