X-Message-Number: 23889
References: <>
From: Peter Merel <>
Subject: Constructivists and Formalists
Date: Thu, 15 Apr 2004 07:43:11 +1000

Thomas Donaldson writes,

> I will point out (for both Bob Ettinger and Peter Merel) that the
> underlying idea of constructive mathematics consists exactly of using
> a system in which proofs by contradiction aren't allowed

Thanks Thomas. Indeed constructivists reject the law of the excluded 
middle because it is necessarily ambiguous. They hold you cannot 
legitimately assert the truth of (P Or Not P) unless you can 
specifically affirm the truth of P, or you can specifically affirm the 
truth of Not P. Constructivists generally use a 3 valued logic (true, 
false, indeterminate) to include the possibility of ambiguity.

But this doesn't go far enough to beat Godel because it is possible to 
enumerate determinacy in a 3 valued logic through exactly the same 
procedure used to enumerate truth in a 2 valued one. One winds up with 
a statement that the system cannot consistently justify as determinate 
or indeterminate, and the nut is busted again. And you can keep adding 
values to your logic to account for the trail of busted nuts ... but 
Godel keeps busting 'em until you either get tired or run out.

There is an intuitionistic, or at least quasi-humanistic, path through 
Godel, but it doesn't come this way. Instead of dealing in predicates 
with values, and hence a machine with states blah blah blah, chuck that 
whole theoretical frame and pick one closer to empiricism. We don't 
find any "true" or "false" in life. We make distinctions concerning 
patterns of behaviors of our sensing processes when they interact with 
signals of other processes. Formalize the correspondence of process, 
behaviors, signals, and distinctions, and you can make a calculus in 
which there is no terminal encoding truth or falsity. Hence no nuts to 

Cryonet isn't the right forum for this, and anyway it's already covered 
reasonably well in Barnsley and in G. Spencer Brown. Naturally one 
hesitates to recommend the latter to a mathematician ...

Peter Merel.

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