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Herman Jurjus says...
>
>Daryl McCullough wrote:
>> Set-theoretic statements are finite bits of syntax. Set-theoretic
>> proofs are finite sequences of statements. You don't have to have
>> access to anything infinite in order to work with set theory.
>
>But then you reduce sentences about sets to sentences of the form
>'ZFC proves ...'. And such sentences have a very concrete meaning.
>For example, no ontological commitment to the existence of, say, P(P(N))
>is needed to make sense of such statements.
I think that's true. There is no ontological commitment needed to
work with set theory.
>In a way, your argument even confirms Petry's opinion: in the end, the
>only thing that's really 'exact' is what can be reduced to concrete,
>computable formalism. The rest is vague, fantasy.
I don't see how it confirms Petry's opinion. He is asserting
1. We should limit mathematics to what can be "explained to a
computer".
2. If we agree to 1. above, then we must throw out Cantorian
set theory.
I'm saying, in contrast, that 2 does not follow from 1. Cantorian
set theory can perfectly well be explained to a computer (inasmuch
as anything can be).
>You can use it alright, for heuristic purposes. But don't claim that it
>is well-defined beyond suspicion or has meaning in itself, without first
>coding or translating it into more basic notions.
I didn't say that it should be coded or translated into more
basic notions.
>What bothers me more with David Petry is: where's the mathematics?
>The way it proceeds, it appears to remain just some philosophical point.
It seems to me that David Petry's notion of mathematics is a proper
subset of the standard notion of mathematics. It doesn't produce
anything new. The advantage to adopting David's position is that
if all the most talented mathematicians stopped doing stuff with
infinite sets and noncomputable functions, then that would free
them up to work on more important things.
David isn't really proposing any new research program, he's proposing
the elimination of *existing* research programs.
--
Daryl McCullough
Ithaca, NY
--
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