Re: [OPE-L] is algebra dialectical and vice versa?

From: Ian Wright (iwright@GMAIL.COM)
Date: Mon Sep 19 2005 - 12:46:02 EDT


Great paper! Very nice examples. I learnt things. 

The practical realisability of an effective computation is a really 
important point to stress. The onus is on those who claim that the 
mathematical objects labelled "super-Turing" are *real*. That demonstration 
requires that they *build* such machines. In other words, it requires 
experimentation. Otherwise, it is so many angels dancing on the heads of 

I agree that Aristotle's notion of the difference between potential and 
actual infinite is crucial, but perhaps this points to more fundamental 
philosophical issues and questions.

For example, an area you don't go into, but is relevant, is the role of 
diagonal proofs, which I believe assume the existence of (actual) infinite 
sets. Cantor used a diagonal proof to argue there is a hierarchy of 
infinities, Godel used a diagonal proof to derive his incompleteness 
theorems, and Turing used it in the halting problem. Without getting into 
details, the diagonal proof method is a proof by contradiction. It is 
philosophically controversial that a proof by contradiction can demonstrate 
the existence of a mathematical object. Showing that *if* a statement "S " 
is true a contradiction ensues, does not imply that "not S" is true. This is 
the debate over the validity of the law of excluded middle, which assumes 
that either "S" or "not S" is true, and in a way denies the possibility of 
dynamic contradiction (your oscillator). Intuitionistic logic and variants 
deny its validity, and it is no accident that these philosophy of math 
approaches are close to computational approaches. I think it would be 
interesting to peer a little deeper into the respective uses of the diagonal 
proof by Cantor, Godel and Turing. To what extent are the conclusions 
regarding the existence of: (i) the existence of a hierarchy of inifinites; 
(ii) the non-existence of a complete axiomatization of the natural numbers; 
and (iii) the non-existence of super-Turing machines that can solve the 
halting problem, reliant on the law of excluded middle applied to actual 
infinite sets? My question is whether some of these conclusions are in a 
sense "undecidable" in theory, as the proofs of existence and non-existence 
are subject to the critique of the law of excluded middle. 


P.S. Typo of "inherentaly" on P4.

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