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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)

Graduate Student / Many Worlder / Devil's Advocate

"I know what no one else knows" - 'Runaway Train', Soul Asylum

My URL: http://pages.nyu.edu/~jqm1584/

Received on Fri Oct 29 1999 - 17:03:12 PDT

Date: Fri, 29 Oct 1999 19:52:35 -0400 (EDT)

On Thu, 28 Oct 1999, Juergen Schmidhuber wrote:

No kidding. This shows how limited Turing machines are! There's

nothing wrong with reals, but Turing machines are not up to the task.

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)

Graduate Student / Many Worlder / Devil's Advocate

"I know what no one else knows" - 'Runaway Train', Soul Asylum

My URL: http://pages.nyu.edu/~jqm1584/

Received on Fri Oct 29 1999 - 17:03:12 PDT

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