From: Christian Szegedy (email@example.com)
Date: Mon Aug 16 2004 - 04:56:18 MDT
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
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.
> 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.
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