🚀 go-pugleaf

In article <b3Yc02Cj8dl401@amdahl.uts.amdahl.com> kp@amdahl.uts.amdahl.com (Ken Presting) writes:

>...  I had focused on Penrose' arguments, which depend on
>the possible existence of uncomputable functions in the laws of physics.
>But I propose a simulation technique which depends on deduction rather
>than numerical solution of differential equations. The standard
>logical operation which represents the determinist metaphysical thesis

>is *deduction*, not computation.  Of course, computers are entirely
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>adequate to implement deductive systems, by the Church-Turing thesis.
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>If there is any way at all to predict a future state from a past state,
>computers can do it as well as any mere physicist.

I think you're suffering from a lack of imagination here. Penrose is
of course denying that computers are adequate to implement any
deductive system, if by "deductive system" you mean the process by
which real mathematicians establish new truths. If you're to understand
his argument, you have to try to imagine how this might be true.

Let's say that one day Penrose announces that he is able to solve, say,
the word problem for semi-groups - a well-known non-computable problem.
People give him instances of this problem. After a period of time
that goes up only reasonably with the size of the instance he announces
the answer: YES or NO. In those cases where the true answer is 
subsequently determined, he always turns out to be right. This holds
even for very difficult cases that require increasingly subtle 
arguments to establish that the answer is NO, as well as cases where
extremely complex reductions are needed to demonstrate that the answer
is YES.

I think it would be quite reasonable to conclude in the above situation
that Penrose can somehow perform non-computable operations, and hence
that the laws of physics must also be non-computable. You could construct
a similar scenario in which the word problem is solved by some physical
computer (obviously not Turing equivalent), rather than by a human being.

    Radford Neal