X-Message-Number: 21718
From: 
Date: Tue, 6 May 2003 20:29:30 EDT
Subject: A little more about Godel

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Although probably few on this list are interested, the Godel mob and the 
anti-cryonics mob have some things in common. Here I'll just reiterate in 
extreme brevity the deflation of Godel and add something I think I omitted 
previously.

Here's what's wrong--grotesquely wrong--with the famous incompleteness 
theorem(s), or rather with their interpretation.

Although there are moderately interesting questions about the limitations of 
formal systems of the type he studied, no one--not Godel, not anyone, as far 
as I know--has ever claimed that there exists a single clear-cut proposition 
of (say) arithmetic, or anything else coherent or meaningful, that is 
unverifiable and unfalsifiable in principle. 

In the formal system, a statement equivalent to "This statement is 
unprovable" is indeed unprovable, so it follows that such statements are true 
(in our heads, in the metasystem) although unprovable (in the formal system). 
This has been misconstrued as showing that there will always be truths about 
the real world that mind can never fathom, among other wild pseudo-metaphors.

But this particular limitation of a class of formal systems is only one of 
several limitations, and it says nothing about the real world--or even about 
all formal systems--and it certainly says nothing about any limitations of 
mind.

Almost all scholars, as far as I recall, somehow fall into the trap of 
assuming that a formal system of sufficient power must be similar in most 
respects to those we know, typified perhaps by  the Principia Mathematica of 
Whitehead and Russell. But the fact is that there is a huge universe of 
possible formal systems, with different rules and limitations, and it is 
entirely possible--I consider it nearly certain--that some of them will prove 
better than the ones in use, because they will be constructed with the intent 
of avoiding language traps.

(I think it is fair to say that all "philosophical" problems are language 
problems, or problems of the linguistic aspects of science.)

Incidentally, some think that, because the unsolvability of the Halting 
Problem in computers parallels the unprovability of Godelian sentences, and 
because we use computers (embodying formal systems) to make inferences about 
the world, it follows that our potential knowledge of the world is likewise 
limited. 

But it does not follow. There is never one-to-one correspondence between a 
formal system, or a computer system, and the physical world. If you want a 
kindergarten example of a misleading alleged limitation, consider arithmetic 
using decimal notation. You can't even write the number 1/3 in a finite 
number of decimal places! But the human mind can understand the number 1/3, 
and a computer using (say) the duodecimal system can also handle it exactly. 
If it were worth the bother, we could also use programs that switch from one 
base to another when convenient.

Robert Ettinger 


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