Re: Turing vs math

From: Juergen Schmidhuber <juergen.domain.name.hidden>
Date: Wed, 27 Oct 1999 09:35:24 +0200

>> Why assume non-computable stuff without compelling reason?
>> Shaved by Occam's razor.

Jacques:

>On the contrary. Why assume the lack of *any* given type of
>mathematical stucture? A true everything-hypothesis surely would not.
>Occam's razor says: don't add extra distinctions such as a restriction
>like that.

>Note also that, as I said, computability isn't the real issue. A
>Turing machine can not be a continuous (but computable) structure. Of
>course the non-computable stuctures should exist too in an everything -
>hypothesis.

The "non-computable structures" are just an ill-defined fidget of our
imagination. They do not exist in the sense that we cannot formally
describe them with a finite number of bits. Textbooks and theorems about
real numbers are computable (finite symbol strings, finite proofs), most
real numbers are not.

Occam's razor really says: do not add any bits beyond those necessary
to explain the data. Observed data does not require more than a finite
number of bits, and never will.

Non-computability is not a restriction. It is an unnecessary extension
that greatly complicates things, so much that we cannot even talk about
it in a formal way.

Juergen Schmidhuber www.idsia.ch
Received on Wed Oct 27 1999 - 00:37:16 PDT