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

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

```Paul,

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
pins.

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.

-Ian.

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

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