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Topic: This Week's Finds in Mathematical Physics (Week 236)
Replies: 29   Last Post: Aug 24, 2006 9:00 AM

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Keith Ramsay

Posts: 1,745
Registered: 12/6/04
Re: This Week's Finds in Mathematical Physics (Week 236)
Posted: Aug 7, 2006 2:25 AM
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tchow@lsa.umich.edu wrote:
|It's still possible, of course, for someone---you mentioned Nik
Weaver---to
|come along and argue that our intuitive notion of predicativism, fuzzy
|though it is, can't possibly be identified with the level Gamma_0.
The
|reason you can't seem to decide immediately whether Weaver's position
is
|nonsensical or not is probably because the critical questions are not
|mathematical but philosophical, and of course it's usually harder to
arrive
|at definitive answers in philosophy than in mathematics.

My impression is that a lot of it has to do with Weaver's being
interested
in predicativism as a possible working philosophy, which is not quite
the
same as being interested in the concept of predicativity in the
abstract.

There's a basic problem with relating a philosophy to formal systems,
due to Goedel incompleteness. If we can exhibit a formal system S that
allegedly captures all the reasoning acceptable to a given philosophy
of mathematics, then it's a little hard to see how such a claim can be
supported. If someone holding that philosophy can see that this is so,
then how can they fail to see that S is consistent, which is unprovable
in S? And if the formal system S captures all the reasoning they accept
without their knowing it, how can this be established by anyone else?

Consider this for the prototypical "believer in ZFC". Proofs from ZFC
tend to be accepted as proofs without the author feeling the need to
include any of the axioms among the hypotheses, whereas proofs from
large cardinal axioms and the like tend to be stated with the
additional assumptions listed as explicit hypotheses. It's not very
credible to say, however, that provability in ZFC represents the exact
line between what can be accepted as proven and what can't, since,
in particular, if one manages to prove that a theorem follows from the
consistency of ZFC, this is as good a reason to believe it as its
provability from ZFC.

Weaver and Feferman regard this situation in different lights in the
case of predicativism. (And of course they differ on other issues.)

>From my point of view, we have to concede to the objection but only
so far. I don't think we can reasonably ask a predicativist to agree
that only the things that can be proven predicatively, in the sense of
the famous analysis of predicativity, can be validly proven. That would
be akin to asking the "ZFC believer" to agree not to believe that ZFC
is consistent. On the other hand, however, I'm unpersuaded by any
of the arguments that I've read that the predicativist is entitled to
go "very far beyond" the formal sense of predicativity, where I'm
just going to leave "very far beyond" as a vague idea here. I think
we can concede that some amount of reflection upon the concept
of predicative proof, taking the predicativist outside of the strict
formal limits on it, is compatible with the overall philosophy, without
having to concede that anything very "strongly impredicative" has
been justified in a way compatible with their philosophy.

I don't think it's so hard to see that the way one ordinarily proves
induction up to Gamma_0 is impredicative. It's not that it's impossible
to define it predicatively. Each computable ordinal can be defined
as an ordering on natural numbers, given by a primitive recursive
relation on them. The existence of this ordering isn't the problem.
The problem is proving induction up to it. The way that one ordinarily
does it makes reference to sets of ordinals. That's the gist of it. To
show that this is not a merely apparent obstacle to a predicative
proof is a longer story.

Keith Ramsay



Date Subject Author
7/26/06
Read This Week's Finds in Mathematical Physics (Week 236)
baez@math.removethis.ucr.andthis.edu
7/27/06
Read Re: This Week's Finds in Mathematical Physics (Week 236)
david.corfield@tuebingen.mpg.de
7/29/06
Read Re: This Week's Finds in Mathematical Physics (Week 236)
baez@math.removethis.ucr.andthis.edu
7/30/06
Read Re: This Week's Finds in Mathematical Physics (Week 236)
david.corfield@tuebingen.mpg.de
8/1/06
Read Re: This Week's Finds in Mathematical Physics (Week 236)
john baez
7/28/06
Read Re: This Week's Finds in Mathematical Physics (Week 236)
Jim Heckman
7/29/06
Read Re: This Week's Finds in Mathematical Physics (Week 236)
baez@math.removethis.ucr.andthis.edu
7/31/06
Read Re: This Week's Finds in Mathematical Physics (Week 236)
Jim Heckman
8/1/06
Read Re: This Week's Finds in Mathematical Physics (Week 236)
Tom Leinster
8/2/06
Read Re: This Week's Finds in Mathematical Physics (Week 236)
john baez
7/28/06
Read Re: This Week's Finds in Mathematical Physics (Week 236)
tchow@lsa.umich.edu
7/29/06
Read Re: This Week's Finds in Mathematical Physics (Week 236)
baez@math.removethis.ucr.andthis.edu
7/30/06
Read Re: This Week's Finds in Mathematical Physics (Week 236)
tchow@lsa.umich.edu
8/2/06
Read Re: This Week's Finds in Mathematical Physics (Week 236)
john baez
8/7/06
Read Re: This Week's Finds in Mathematical Physics (Week 236)
Keith Ramsay
8/8/06
Read Re: This Week's Finds in Mathematical Physics (Week 236)
john baez
8/8/06
Read Re: This Week's Finds in Mathematical Physics (Week 236)
tchow@lsa.umich.edu
8/24/06
Read Re: This Week's Finds in Mathematical Physics (Week 236)
Nik Weaver
7/28/06
Read Re: Order-preserving embeddings of ordinals in the real numbers
David Madore
7/28/06
Read Re: Order-preserving embeddings of ordinals in the real numbers
G. A. Edgar
7/29/06
Read Re: Order-preserving embeddings of ordinals in the real numbers
baez@math.removethis.ucr.andthis.edu
7/30/06
Read graphical representation of epsilon_0 (was: Re: Order-preserving embeddings of ordinals in the real numbers)
David Madore
8/1/06
Read Pictures of infinity
baez@math.removethis.ucr.andthis.edu
8/2/06
Read Re: Pictures of infinity
tony@mantis.co.uk
8/2/06
Read Re: Pictures of infinity
Alec Edgington
8/3/06
Read Re: Pictures of infinity
Dave L. Renfro
7/28/06
Read Re: Order-preserving embeddings of ordinals in the real numbers
Ian A. Mason
7/28/06
Read Re: This Week's Finds in Mathematical Physics (Week 236)
Kevin Buzzard
7/29/06
Read Re: This Week's Finds in Mathematical Physics (Week 236)
baez@math.removethis.ucr.andthis.edu

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