One reason is the fact that it seems impossible to define _the_ set of
all Naturals in an unique way, when starting with a "best" finite set,
and subsequently invoking a limit process, via the potential infinite.
Apart from the difficulties already involved with taking these limits,
there is _no_ such best finite set, as has been pointed out by several
debaters in the thread. So the process of idealization from something
finite to something infinite is rather ambiguous at best. That is true
at least for the natural numbers. I'm not going to try this again.

But the insight is dawning that this situation is not really very much
different for the real numbers as well. Constructive mathematics _has_
done something in this area too. But I've always been quite reluctant,
because, as a physicist, I know very well that the results obtained by
constructivists are way to meager, in order to be of specific interest
to people who are involved with real world applications. Hence my pink
intuitionism, as the intuitionism without too much trouble. I've been
well aware that this stand would become untenable too in the long run.

Anyway, to have to consider sqrt(2) as a number that is "not finished"
and i.e. not being able to employ the law of excluded middle, those are
the kind of self-imposed restrictions that can hardly be defended to be
useful, when considered from the stand of an engineering mathematician.
Moreover, if these difficulties have not been overcome in the past 50
years, then it can hardly be expected that they will be overcome in the
50 years forthcoming. Indeed, I don't expect it to happen anymore.

So I'm giving up on all this. My excuse for having pursued it all _so_
long - and so intensively - is that I'm kind of a stubborn character in
the first place. On the other hand, I think it's not a bad thing to try
to go to the _extreme_. And explore everything that's in there. I may
have experienced more heights and depths than many of you conventional
types in 'sci.math'. If and only if you've done everything, right _and_
wrong, then maybe it's time, finally, to become (old and) wise. And no
longer fight where you cannot possibly win. But one must be _convinced_
before deciding so.

If you can't beat them, join them ? I don't think so.  Nobody can blame
_me_ for _not_ having gathered all possible evidence against mainstream
mathematics. It hasn't been an easy verdict. And I'm not satisfied with
it either. Because the question remains: in what direction to proceed?

Until now, we as "anti-Cantorians" only have been looking at the input
side of mathematics. We have always been thinking that something goes
wrong with the process of idealization, which transforms things in the
real world into mathematical entities:

                   idealization
        real world ============> mathematics

Now I don't say that this is an impossible stand: constructivism is an
example of how to accomplish things in a "neater" way than eventually
would be possible with mainstream mathematics. But a definite drawback
of some non-standard approaches has been that the mathematical objects
themselves i.e. the reals, become more difficult to handle in practice.
One wouldn't like to ponder about a intuitionistic course on calculus
for engineers, huh?

But, as I've said, we will not be considering anymore the forward path:
how to lay down the foundations of an alternative "scientific" sort of
mathematics, by defining kind of a "proper" idealization process, what
ever that should mean.

Rather we have decided to work the other way around: the _output_ side
of mathematics will be considered instead. The process that transforms
mathematical objects into real world objects is called materialization.
Post-processing instead of pre-processing:

                    materialization
        mathematics ===============> real world

I'm very well aware of the fact that "materialization" may sound like
a new phrase, but that doesn't mean it's a new concept altogether. As
long as mathematics has been applied - and that has been for a _long_
time - there have been ways to translate the outcome of a mathematical
theory back to the real world. Without such a mechanism, any progress
in the exact sciences, of course, would have been impossible already.

Our goal is to establish, though, that "materialization" may be _more_
than just a hollow phrase and it's covering more than some well known
habits of our scientists and engineers.

To be continued.
Any comments so far?