# Re: Turing vs math

From: Jacques M. Mallah <jqm1584.domain.name.hidden>
Date: Fri, 29 Oct 1999 19:52:35 -0400 (EDT)

On Thu, 28 Oct 1999, Juergen Schmidhuber wrote:
> Jacques:
> > I don't really know what you mean, but it sounds like you're
> > saying, you don't think real-valued quantities can exist because real
> > #'s can't be described with a finite # of bits.
> > But as you admit, *sets* of real #'s can easily be described by us.
>
> Even the set of all real #'s is not formally describable. You cannot
> write a program that lists all reals. In infinite but countable time
> you can write down a particular computable real, but not all reals.

No kidding. This shows how limited Turing machines are! There's
nothing wrong with reals, but Turing machines are not up to the task.

> You can indeed write down rules for manipulating symbols and generating
> proofs. But a symbol doesn't care for whether you think it stands for,
> say, ``all reals''. The symbol string ``all reals'' is just another
> symbol string. Your mental representation of ``all reals'' is describable
> by another finite string, according to UTM theory. There is no compelling
> reason to believe in some sort of non-describable ``continuous reality''.

Using logic of your type, and I do so to illustrate its
circularity, one might equally say:

You can indeed write down rules for manipulating symbols and generating
proofs. But a symbol doesn't care for whether you think it stands for,
say, ``a Turing machine''. The symbol string is just another
symbol string. Your mental representation of ``Turing machine'' is
describable by a set of real-valued quantities, evolving according to the
laws of physics. There is no compelling reason to believe in some sort of
non-continuous ``Turing machine''.

The fact is *all* mathematical stuctures should be considered.
The existence of some should be rejected *only* if there is a good reason.
The only possible reason I can think of is that it may be necessary to
reject some in order to have a well defined measure distribution.

Let me give you another example since you seem so devoted to your
antirealist position. Another perfectly good mathematical structure is a
non-Turing computer, e.g. (a simple example) one that cycles through the
states (1,2,...,20) and has no additional states. Sure, a Turing machine
can emulate such a machine, but it can not *be* that machine. There is no
reason, though, that a structure should not exist which *is* that machine.

- - - - - - -
Jacques Mallah (jqm1584.domain.name.hidden)