[hist-analytic] Clarity Is Not Enough
Roger Bishop Jones
rbj at rbjones.com
Sun Feb 22 10:42:16 EST 2009
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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