Re: tortoise algorithms

["John S. Denker" <jsd@AV8N.COM>]

At 7:21 PM -0400 8/30/03, I wrote:
>
>>Zeno got this wrong.
>>
>>None of this has anything to do with Banach-Tarski.

On 08/30/2003 09:12 PM, Chuck Britton wrote:
>         I guess that this is where we disagree.
>
> Cantor was not the first to be seriously concerned with the
> fundamentals of the infinite/infinitesimal. The axiom of choice is a
> major link between Banach-Tarski and the axiomatic set-theoretic
> calculus of the continuum.

Does this mean to imply that Zeno _was_ "seriously
concerned with the fundamentals of the infinte /
infinitesimal"?

Also note that people can seem "serious" and still
be quite wrong.

As I see it, we can somewhat charitably summarize
Zeno's position as a syllogism:

Version 1:
  Major premise:  If we can't sum the series, Achilles
    loses the race.
  Minor premise:  We can't sum the series.
  Conclusion:  Achilles loses the race.

The first problem is that Zeno's major premise is
wrong and his conclusion is wrong.

The second problem is that even if the major premise
were correct, such a syllogism would not be proof of
the minor premise.  Something along these lines might
perhaps have served as an interesting *illustration*
of constructivist assumptions.  (But since it is wrong,
it doesn't even rise to that level.)

Note that I am using "constructivist" as shorthand
for "not using the axiom of choice".  I consider
constructivist mathematics to be respectable and
interesting, although it doesn't bestow direct
benefits to physics quite the way some other bits
of mathematics do.  (And I remain open-minded about
that;  the expendability of the parallel postulate
was known as mathematics for years before non-Euclidean
geometry became part of physics.  The expendability
of the axiom of choice may follow a similar path.)

It may be useful to consider a slightly different
syllogism:

Version 2:
  Major premise:  If there is no constructivist
    solution to the race problem, Achilles loses
    the race.
  Minor premise:  There is no constructivist solution.
  Conclusion:  Achilles loses the race.

In this version the *minor* premise is wrong also!  It
is trivial to construct a solution.  It's one linear
equation in one unknown.

Constructivist mathematics concerns itself with the
existence of constructivist solutions.
  1) The question of whether or not there exist
non-constructivist solutions to the same problem
is quite irrelevant to constructivist mathematics.
  2) Zeno exhibited a non-constructivist method that
leads to a *non-solution*.  This is doubly irrelevant
to constructivist mathematics.

================

To say the same thing yet again in different words:

Constructivist mathematics says:  "Suppose we can't
sum the series.  Then ......"  and goes on to say
some true things.  This is a step forward for the
theory.

Zeno says:  "We can't sum the series.  Therefore
......." and goes on to say false things.  I am
really not seeing how this is in any way a step
forward.

I would think that anybody who is "seriously concerned
with the fundamentals of the infinite / infinitesimal"
would consider Zeno the enemy, for bringing disrepute
upon an otherwise-respectable field.