[hist-analytic] Methods of Proof: Re: Clarity Is Not Enough
steve bayne
baynesrb at yahoo.com
Sun Feb 22 11:51:23 EST 2009
This, and Danny's earlier remarks, are interesting. Bishop
raises an interesting matter of the relation of the methods
of proof and confidence in the proof's validity.
Although I wouldn't qualify as a "C" level logicians, I have
always had an interest, particularly in the relation of set
theory an logic. For example, a theorem in some systems
can be derived in more ways that one. In another system
the number of available proofs varies. So, in a sense, one
can imagine the possibility of a metric for redundancy in
methods of proof. Optimally a method of proof would have
no redundancy. Another idea crossed my mind, which I
never pursued (probably for good reason).
If we consider propositions as points and proofs as
finite sets of points, then one wonders about the possibility
of using certain theorems in topology in proof theory. Here
is a "wild" case, one for which I make no claims except to
illustrate the general point. If we think of all the propositions
in propositional calculus as contained within a bounded
interval, and a proof as a sort of 'cover' in the sense of a
subclass of open intervals which contains the union of
the members of all propositions, then couldn't we use
something like the Heine-Borel Theorem to prove that a
proof covers all subclasses of propositions? A far more
general question might be: Is something like Godel numbering
essential for some of these metatheorems?
I just thought I'd throw this out. I'm nt working in logic, but
Roger raises an interesting point about the relation of
proof methods and epistemic adequacy. (I'll regret
bringing up this silly idea of Heine-Borel by monday).
Anyway, this is a good exchange.
Regards
Steve Bayne
--- On Sun, 2/22/09, Roger Bishop Jones <rbj at rbjones.com> wrote:
From: Roger Bishop Jones <rbj at rbjones.com>
Subject: Re: Clarity Is Not Enough
To: "Danny Frederick" <danny.frederick at tiscali.co.uk>
Cc: hist-analytic at simplelists.com
Date: Sunday, February 22, 2009, 10:42 AM
Danny,
Thanks for the clarification of your position, which is now
less puzzling to me.
I shall abstain from commenting about critical- and pan-critical-
rationalism for the present since this would take up more time
and energy than I can spare for it right now.
However, I am unable to resist responding to your observation
about the status of mathematics.
On Saturday 21 February 2009 16:02:01 Danny Frederick wrote:
> 4. there are plenty of disagreements in mathematics (fortunately),
> between formalists, intuitionists, etc., as well as over particular
> theorems, proofs or methods of proof, such as Gentzen's transfinite
> induction;
These disagreements are disagreements belonging to the philosophy
of mathematics or to the foundations of mathematics.
The practice of mathematics is remarkably unscathed by such
controversy.
In particular, "Gentzen's transfinite induction" is I believe
controversial only because he used it to prove the consistency
of arithmetic. The controversy is not about whether
transfinite induction is sound or about whether arithmetic
is consistent, but rather about whether the use of the
former to prove the latter adds anything to our confidence
in that proposition.
In mainstream mathematics, the full resources of ZFC are
deployed without controversy, and the induction principles
available in that context are very much stronger than
are needed to prove the consistency of arithmetic.
Progress in mathematics does not depend upon, or even make
use of, criticism of established results, and I would be
interested to know of any examples you have of disagreement
among mathematicians about the truth of supposedly proven
mathematical propositions.
This is of course connected to my as yet unsatisified request
for an example of a proposition of mathematics which has
been accepted as refuted by some empirical observation.
I do aknowledge that there is controversy about what methods
of proof are acceptable. This is not the same as controversy
about the truth of propositions of mathematics, and again,
perhaps surprisingly, does not arise from or lead to any
such controversy.
Roger Jones
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