From: Eliezer Yudkowsky (sentience@pobox.com)
Date: Mon Aug 16 2004 - 04:42:46 MDT
Christian Szegedy wrote:
> Eliezer Yudkowsky wrote:
>
>> Marc Geddes wrote:
>>>
>>> Given that the language of science used to describe the physical
>>> world is mathematical, and given the Turing arguments (showing the
>>> mapping between maths and algorithms), it follows that any of the
>>> equations being used to describe a finite portion of physical
>>> reality are back translatable into an algorithm.
>>
>> This does not follow automatically because physics is continuous,
>> while Turing machines and Church's lambda calculus are discrete.
>
> Your statement is true, but it uses the wrong argument: computability
> does not depend on discreteness/continuity. It is true, that
> computability can only be defined for discrete sets, but there can be
> continous models whose all discrete manifestations are Turing
> computable. There are mathematically definable discrete models that are
> not computable.
My "because" should be understood in the sense of exhibiting a conceptual
counterexample to the assertion that a mathematical language of physics is
alone a sufficient condition to prove Turing computability. For this it
suffices to point out that a mathematical language can be interpreted as
describing the interaction of fully arbitrary continuous functions; we
would then have the abstract fact that some possible models of this
language would not be exactly computable, albeit we could never think of them.
>> Albeit it was already a mathematical theorem that if our universe
>> ultimately consists of a finite or countable set of axioms (i.e.,
>> equations), and the axioms are satisfiable by any model, they must be
>> satisfiable by a countable model.
>>
> You refer to the theorem of Lowenheim and Skolem which is in my personal
> opinion (as a mathematician) interesting stuff, but does not have any
> practical relevance. Its philosophical relavance can be disputed too. A
> serious problem is that people tend to talk about it without a proper
> understanding of the exact notion of *model* in mathematical logic.
I'm still learning, but it looks to me like physics describes relations
that hold between points in quantum configuration space. Similarly, an
axiom set, if we map it to a model, describes relations that hold between
elements of the model. This is what suggests to me that the TOE of physics
might be analogous to an axiomatic system. For me the fascinating thing is
that although mathematicians can get along just fine using nothing but
axioms, we appear to actually *exist* in the model, not the axioms. I am
still trying to wrap my mind around the Lowenheim-Skolem theorem as it
applies to this - would there be any conceivable physical test that would
tell us we were really in a uncountable model, or would the
Lowenheim-Skolem theorem rule that out?
If that was gibberish, do let me know.
(I'm aware that mathematicians usually do not speak of sentient beings
"within the model" or "within the axioms", but complex enough axioms
apparently generate a model including complex subprocesses that attempt to
deduce the axioms by watching other model elements interact.)
-- Eliezer S. Yudkowsky http://intelligence.org/ Research Fellow, Singularity Institute for Artificial Intelligence
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