Re: One-stop version of my postings

Re: One-stop version of my postings

[Vaughan Pratt]

Robert J. MacG. Dawson wrote:

> ...fatally flawed. (Why do the circles in Euc.I.1 intersect? None of his
> axioms assert that any pair of circles whatsoever do so.)

In axiomatic mathematics, everything that is not forbidden is permitted.
Circles can climb trees and drape themselves over branches in Dali's
axiomatization of geometry, not because he says they can but because he
is not the strict disciplinarian that Euclid is.  Euclid insists that
his circles shape up or else.  This creates many problems for the
circles but few problems for mathematicians as their managers.  Dali
runs a looser ship, which lets the circles lead a less structured life
while creating more problems for mathematicians.  This is a win-win
situation: the circles end up with fewer neuroses while the
mathematicians thrive, problems being their lifeblood.  Thank god for
Lobachevsky and the others we can't remember because Tom Lehrer didn't.

Vaughan Pratt

Re: One-stop version of my postings

[Robert J. MacG. Dawson]

Vaughan Pratt wrote:
> Robert J. MacG. Dawson wrote:
>
>> ...fatally flawed. (Why do the circles in Euc.I.1 intersect? None of his
>> axioms assert that any pair of circles whatsoever do so.)
>
>
> In axiomatic mathematics, everything that is not forbidden is permitted.

	Yes, in a sense... but theorems along the lines of "there exists a
model of X such that Y" were a long, long way in the future <grin>

	-Robert

One-stop version of my postings

[Robert J. MacG. Dawson]

Eduardo having preferred (wisely) not to try to edit down both sides of
a  discussion with a lot of quoting, I will follow his lead and
summarize one side of it with context quotes from the other side.

Eduardo Dubuc wrote (in part)

 > I am right about the fact that introducing Einstein and differential
 > geometry into our present discussion on the interaction of string theory
 > and category theory was an infantile attempt to attack Motl's views.
Worst
 > than that, it introduces a distraction to Marta's principal issues.

I responded:

     Sorry, Eduardo. I refuse to write out a hundred times "I will not
attempt to attack the views of Lubos Motl".

     As for introducing distractions, Marta suggested (Mar. 12) that we
should as a community not "quietly accept getting discredited by a
minority of us presumably applying category theory to string theory",
and that we should "react and point out that this is not what (all of)
category theory is about" and "give a thought about what we, as a
community, can urgently do to repair this damaging impression."  On Mar.
14, it was suggested that meeting organizers might want to take this
into account while choosing invited speakers.

     Now, I do accept that there are such things as bad mathematics and
bad physics. However, this call for collective action against an entire
field of research seems uncomfortably close to an organized boycott, an
extreme enough breach of tradition that only an emergency - if that -
could justify it. And, in such circumstances, the question of whether
the emergency really  exists is certainly relevant, and not (_pace_
George W. Bush) a presumptuous distraction.

     My response (Mar. 14) was twofold. I suggested that the general
credibility of category theory does not depend on the credibility of its
applications to physics.  But I also pointed out, which I think is even
more relevant, that physicists are quite used to tentative theories
being studied for many years before they are accepted or rejected, and
that there is no loss of credibility among physicists for those involved
in this process.  Neils Bohr is not remembered as "that crackpot who
thought the atom was like a solar system".

     Note that only credibility among physicists is at stake here.
Mathematicians are unlikely to base a negative judgement of _any_ branch
of mathematics on what physicists may be doing with it, and in
particular know perfectly well that this is not "what all of category
theory is about". The rest of the world will not have a clue what all of
category theory is about, and will never know the debate took place.

     We expect mathematics to be correct by the time it sees print;
internal consistency is a comparatively easy goal.  Physics, despite
using almost as many equations, has a real and sometimes uncooperative
universe to contend with, and cannot afford the luxury of progressing
only through unassailable, permanently-established truths.  Mathematics
was once in a similar position:  we recognize the genius of Euclid's
axiomatic system  even while realizing that (by our standards) it was
fatally flawed. (Why do the circles in Euc.I.1 intersect? None of his
axioms assert that any pair of circles whatsoever do so.)  Ramanujan is
perhaps alone among modern mathematicians in having been permitted, by
general consent, something like the latitude to err freely that the
physics community customarily grants itself - and his was a very special
case. Had he lived to see the greater part of his work in print, he too
would have proved, retracted, or labelled as conjecture many of the
speculative claims that ended up as his legacy.

     I mentioned Einstein in passing, assuming that readers would have
some idea of the initial skepticism with which relativity was viewed,
and the length of time it took before that skepticism ceased to be
justified. (Actual hostility was a separate, later, phenomenon, more
related to antisemitism and Stalinism than to physics.)  If I assumed
too much, and should have explained more, I apologize.

Eduardo then responded (in part)

 > Einstein using differential geometry to develop general relativity is
not at the same level that John Baez using category theory to develop
and/or understand string theory. I imagine John himself is probably the
first to laugh at such a comparison.

and I responded to that (in part)

     If category theory happens to be the approach that works, and
significant progress is made in fundamental physics as a result, it will
be very much on the same level as relativity.

     If it isn't, it will be, from most physicists' point of view, an
honorable attempt on the same level as many other theories such as  the
steady-state universe, tachyons (Remember tachyons? No? _They_ remember
_you_...), magnetic monopoles,  or the luminiferous ether, that were
seriously considered in the past by respectable physicists.

     Until the results are in, it is, I think, not unreasonable to make
comparisons with Einstein's early work on relativity - at a time when
Einstein could not get results consistent with observation and many of
his talented contemporaries had serious doubts about whether he was
using the right approach. (Not whether he was a fool or a charlatan,
just whether he was using the right approach.)  Or, equally, with
Einstein's later work on quantum mechanics, when history seems to have
shown that he backed the wrong horse - but it was not so clear at the time.

		-Robert Dawson