Re: No regrets (Was: something about respect)

From: Eckard Blumschein (blumschein_at_et.uni-magdeburg.de)
Date: 02/03/05 

Date: Thu, 03 Feb 2005 08:30:55 +0100

On 2/3/2005 3:47 AM, examachine@gmail.com wrote:

> There is no reason why I am not allowed to talk about 2^20000. I can
> even factorize the number, right here. The number exists right _here_
> in your mind, as a theoretical entity, and on the screen as program
> text. The better computationalist stance would be something like "that
> which can be fully conceived of". Obviously "2^20000" is a really short
> program to describe a large number 2^20000 that cannot be represented
> in "unary" notation of TMs.

I feel supported after feirce attacks by mathematician. Let me tell some
of my concerns:
To me as an engineer as well as to physicists, real numbers largely
belong to physical quantities, about in a manner as the scale at a
yardstick. Resolution is always limited in practice but the original
notion of continuity suggests open-ended resolution theoretically
corresponding to a twice (towards the small as well as towards the
large: aleph_1) unlimited amount of elements which could be imagined
like points or cuts.
Weierstrass's limit demands delta>0. To my understanding, it mutilates
the original continuum into a sampled one, no matter how tiny the
distance between two "pixels" may get.
On the other hand, it seems to be a tenet of mathematics to declare any
number exact without any doubt. Only this tenet justifies to clearly
distinguish between two numbers.
Mathematics also claims that a number must not be confused with its
representation. To my understanding, numbers cannot exists without
representation. Even if there are frequently many possible
representations of the same number, I am only interested in IR+ scale
whre 2^20000 cannot be represented. So, to my understanding, it does not
make sense to distinguish between a quantity >= 2^20000 and just
>2^20000. (So far I just made a correspondig caveat for very small
differences.)

> On the other hand, actual infinity cannot be "fully conceived of",
> including actual infinite division and joining.

If the so called "actual" infinity is defined by means of an elastic
limit, then I consider it actually not actual but finite and thus
conceivable while the genuine potential infinity evades imagination.

> That is, there is absolutely no way out of Aristotle's metaphysics if
> you want to talk about this. ;)

Agreed.

>
> See "Metaphysics of potential infinity" and "Theories of general
> machines" threads on comp.theory for a more rigorous presentation of
> some of these ideas.

Thank you for this hint.

>
> Another point: a better way to think of mathematics is "formal
> metaphysics". In my mind, the more sensible question is: Can we even
> imagine a world with entirely different physical laws, e.g. permitting
> such silly constructs as continuum? Or, whenever we try to do that, we
> end up in falsehood? (Which is something I suspect of)

While I do not deny the necessity of such ideas, I feel not in position
to contribute to them. On the other hand, I found a lot of imperfections
that I consider worth to be revealed and corrected in traditional theories.

Regards,
Eckard Blumschein

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