From: Marc Geddes (marc_geddes@yahoo.co.nz)
Date: Mon May 31 2004 - 02:09:33 MDT
--- Ben Goertzel <ben@goertzel.org> wrote: >
>
> I know what you mean about randomness, Eli. I tried
> a long time to show
> that the concept of randomness is mathematically
> inconsistent, but just
> wound up with various variants of Godel's Theorem
> instead!
>
> The real mystery here, to me, is that introducing
> such whacky concepts
> as uncomputable numbers appears to drastically
> improve the elegance of
> our theories about the physical world. So, the
> physical world
> apparently is a finite computational-type system --
> yet to model some of
> the more basic aspects of it, it's apparently most
> convenient to assume
> these uncomputable entities (like "truly random
> numbers"). Very weird.
> Yet inarguable: for instance, ordinary
> differentio-integral calculus,
> resting on the implicit assumption of uncomputable
> numbers, is a lot
> more elegant for making physics calculations than
> "computable calculus"
> that does the same thing using only computable
> numbers.
>
> Making consistent but Godelishly-weird mathematical
> models of the
> unmeasurable and nonexistent appears to be a great
> strategy for
> understanding the measurable and existent.
>
> And this is a different point from the weirdness of
> quantum
> probabilities.
>
> Weird universe, huh?
>
> Oh wait a minute -- maybe the problem is with our
> brains? ;-p
>
> -- Ben G
>
Well, I doubt that it is correct to talk about
uncomputable 'entities' as such. The way I interpret
it is simply that there is an infinity of entirely
computable entities. Any apparent 'uncomputability'
is simply an artefact or 'illusion' which arises when
one wrongly attempts to treat an infinity as an
entity.
If you have some broad class of well defined
mathematical problem people might call it
'uncomputable', but any particular finite sub-set of
that class will in fact be entirely computable. So
there is really no constructive way of defining an
'uncomputable' entity. All mathematical entities with
objective existence are computable. It is simply the
case that there is an infinite number of such entities.
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