From: Christian Szegedy (szegedy@or.uni-bonn.de)
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
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.
> 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.
This archive was generated by hypermail 2.1.5 : Tue Feb 21 2006 - 04:22:43 MST