FOM: human well-being; constructivism; anti-foundation
amah8857 at brain.math.fau.edu
Tue Jan 2 08:23:12 EST 2001
Some responses to Matthew Frank's recent post:
>Some recursive function theorists (Nerode, for instance) think that
>Bishop-style math is about computability, and that recursive mathematics
>is a more powerful, more precise, and less confusing way of discussing it.
I think I agree with the view that recursive mathematics is more precise
than Bishop's mathematics, I also think that the inherent beauty or power
of Bishop's mathematics is it being less precise which allows it to have
more interpretation as was pointed out more than once in this list. But
still I don't see the objection of it less powerful, Abelian group theory
is more powerful than group theory can this be a objection to group theory?
More laws naturally makes a more powerful system. Regarding confusing, I
can't comment I assume it to be very subjective.
>Many formalists believe in the principle of the excluded middle and
>therefore find constructive math pointless.
This is not especially a objection by any formalist, but all the people who
believes in the law of excluded middle, like classical mathematicians, I
can guess that some of them also have the same objection. Even if you
believe in LEM there can be reasons to do constructive mathematics, similar
to the reason many people would try to eliminate axiom of choice from proof
of a theorem which uses one.
>Hilbert responded to Brouwer
>(according to Reid's biography, p. 184) with a slight variation of this:
>"with your methods most of the results of modern mathematics would have to
>be abandoned, and to me the important thing is not to get fewer results
>but to get more results."
If this is an objection to constructive mathematics, then I quote from
Bishop, page 13 Bishop-Bridges:
"It is no exaggeration to say that a straightforward realistic approach to
mathematics has yet to be tried. It is time to make an attempt".
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