🚀 go-pugleaf

In article <90Mar3.152728est.6160@neat.cs.toronto.edu> radford@ai.toronto.edu (Radford Neal) writes:
 >In article <b3Yc02Cj8dl401@amdahl.uts.amdahl.com> kp@amdahl.uts.amdahl.com (Ken Presting) writes:
 >>Of course, computers are entirely
 > ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
 >>adequate to implement deductive systems, by the Church-Turing thesis.
 > ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

 >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.

And if pigs had wings they could fly.  So what?
--
Richard Kennaway          SYS, University of East Anglia, Norwich, U.K.
Internet:  jrk@sys.uea.ac.uk		uucp:  ...mcvax!ukc!uea-sys!jrk