Re: Metaphysics of Potential Infinity

examachine_at_gmail.com
Date: 01/25/05 

Date: 25 Jan 2005 05:52:51 -0800

Mitch Harris wrote:
> exanews@gmail.com wrote:
> >
> > My clarification of the distinction does not mean to undermine the
> > concept of actual infinity. Rather, it serves to establish the
strictly
> > metaphysical nature of actual infinity, while unbounded-ness has a
> > perfectly physical definition.
>
> OK. That seems a perfectly reasonable position. Not necessarily
> supportable but at least we know what

Yes, it's just a position among so many others.

> > You say my commentary is copious. Yes, it is copied from one great
> > thinker known as Aristotle if you mean copious in a lighter sense.
>
> I intended copious to mean "extensive" as in "a lot" as opposed to "a
> little". (I'm not sure what lighter sense of the word could be
> connected with copying)

My bad. I am no native speaker, sometimes my semantics analysis module
gets mixed up.

> > However, I give a finer account of the distinction based on a
> > contemporary talk of metaphysics.
> > I will reply to you point-by-point as soon as possible. Meanwhile,
I
> > beg you to bear in mind that this is the most trivial of all
> > distinctions that a metaphysician would concern himself with.
>
> There are mathematically technical ways of making the distinction and
> consistently using the two concepts distinctly. But I take it that
you
> have much more in mind?

A possible mistake in this kind of discussion is confusing the
philosophical and mathematical concepts. Once, a professor warned me
that I *cannot* replace every instance of "and" in a philosophical text
with the mathematical symbol "\land" (logical and). I am slowly
beginning to grasp why.

We are talking about the concept of infinity. This would ultimately
include the mathematical concept. However, we can consider, for
instance in philosophy of computation, or philosophy of mind, or
philosophy of language the concept of infinity *before* we get to the
philosophy of mathematics.

I eventually want to get at the mathematical theory. However, I must
first find out about what infinity means generally. Then, I can proceed
to make sense of mathematical theories, otherwise they are just
symbolic systems of some sort.

Here the philosophical theory does not "touch" the mathematical theory,
it just makes sense of it. Some people take foundational concepts for
granted (like Torkel), probably because they have been taught so. I
cannot take these things for granted. I have to argue for each and
every one of them.

In this case, the mathematical distinction(s) can be given differing
philosophical interpretations. Let's say I want to give these a
particularly reasonable interpretation.

For a starter, I am *not* referring to the distinction between the
cardinality of Z and cardinality of R. That is, if I were to try to
apply my metaphysics to mathematics, which I have *not* carried out in
full detail yet. But I will argue that the metaphysics is sound.

Instead, I am referring to the distinction between potential infinity
and actual infinity. That has absolutely nothing to do with set theory!
(I am talking about Aristotle, a great philosopher! I am not talking
about a deist who wanted to make his beliefs into a mathematical
theory!) This distinction does *not* correspond to the difference
between the extents of Z and R, which are *both* actually infinite in
_one_ sense! So let's just stop comparing apples and oranges.

I am saying that in a finite universe, something can be unbounded but
finite. That is something else than Z as you can imagine, of course. I
explain this by being unable to find a constant bound on the extent of
this something. There are better ways to explain it I am sure, but I'll
stick with this for a while. This could of course be used to *reify*
the concept of natural numbers, but that's something else I don't want
to get into yet because, you see, the talk of Z already assumes a
certain Platonist interpretation! (It's built in the language,
unfortunately. That is why, I first have to talk independently of Z or
R or anything in set theory whatsoever!)

If any of this was confusing, I would be willing to clarify.

> > That is
> > why, I feel utmost pain when I am having to talk about it. My
trouble
> > is that matters of greater importance cannot even be mentioned
unless
> > such trivial matters are settled. (As it happens, such matters are
> > fundamentally important for philosophy of computation as well, and
I
> > believe even such respectable luminaries of computationalism like
Aaron
> > Sloman were not able to address these issues as well as I would
have
> > liked)
>
> what issues? (so we can see where you really want to go with this, to

> see where these distinctions might make a difference)

About trivial issues which were of interest in this discussion: Some
people seem to think that Turing Machines can have no meaning, because
they are merely symbol manipulation. Some people say absurd things like
physical computers are NOT computationally equivalent to Turing
Machines, because TMs have an infinite tape! This is catastrophic for a
philosopher, because then I would have to ask, what on earth are
physical computers? Or better: what is the extension of a Turing
Machine then? You see, you might _have_ to accept a bad version of
substance dualism, known as epiphenomenalism, if you said that the
physical computers are not TMs, but still, TMs exist in some sense. You
have to say TMs exist in some sense, otherwise the theorems about TMs
could not possibly be true. You could also offer us the obsolete
Platonism! But all of this stuff clearly contradicts physicalism which
is a respected view in philosophy! Just to give a quick run-down of the
philosophical problems that would arise from simple misconceptions.
(So, no philosophy is isolated. You cannot easily say that TMs do _not_
exist, but still theorems of them are true. What a strange theory of
truth that would be!)

That is why I wrote the post "Misunderstanding computability", in
Stephen Harris's outrageous misconception that physical computers
cannot be TMs because what, "TMs can compute things that no physical
computer can"! Such a theory is wrong on several grounds, one of which
is getting confused about potential/actual infinity distinction, the
other is confusing the concept of X and X!!! Here are two trivial but
huge mistakes one could make in a single sentence!

There are non-trivial issues, too. An important matter is the reason
why Turing machine is an adequate model of mental processes. Sloman
explains convincingly the origin of _mathematical_ computation as
simulation of physical processes. However, he objects to TMs in
cognitive sciences for several reasons which I didn't find convincing.
Certain others reject that a theory of computation is possible at all,
which I think a computer scientist would find surprising! Much more
difficult issues can be brought up, for instance the use of algorithmic
complexity to explain subjective experience, etc.

Most of these issues would have absolutely no impact on the way we
carry on our daily work of mathematics, they have impact only on our
understanding of computation, mathematics and mind.
Regards, 

--
Eray Ozkural

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