From: andersw+@pitt.edu (Anders N Weinstein)
Date: 14 Nov 1996 00:33:45 GMT
Organization: University of Pittsburgh

In article , Jim Balter  wrote:
>In article <328A2BEC.70E1@ix.netcom.com>,
>be true, we cannot necessarily see the Godel sentence *as* true.  That is, if
>presented with the Godel sentence, there is no reason to think that we will be
>able to see that it is true, unless we can show that it *is* the Godel
>sentence for that system.  And, just as we can "see" that Godel's theorem is
>true (after all, it is a theorem), there are, contrary to popular belief,
>formal consistent systems in which Godel's theorem can be derived (after all,
>it is a theorem).

Be careful to distinguish Godel's *theorem* from the Godel sentence
used in the original proof. (Or *a* Godel sentence constructed for a
particular system).

It pays to remember that Godel's theorem states a *conditional*. It was 
originally demonstrated in application to a particular variant of 
Principia Mathematica, call it PM. It states that 

   IF 
	PM is consistent 
   THEN 
	G ("the Godel sentence") is neither provable nor refutable 
        within PM.

G is a statement of a branch of number theory contrived as it were 
with malice aforethought -- the arithmetized proof theory of PM syntax.
G was constructed so as to be a true statement of number theory just
in case G itself is not provable in PM. That means that if we 
accept that PM is consistent we may detach and go on to assert G as
a further truth of number theory.

Anyway, the point is that Godel's first incompleteness theorem -- the
conditional above -- can itself be formally represented ("coded") in PM
itself and can be shown to be a PM-theorem itself. That is, PM *itself*
can prove Godel's theorem for PM; it can as it were assert the
conditional Godel proved. What it can't do is detach, on pain of
inconsistency. So it cannot prove the Godel sentence proper, nor can it
prove its own consistency. This is the basis of Godel's second
incompleteness theorem.

Now one good question with regard to the Godelian argument against
mechanism is how we can be assumed to know the consistency of PM or
other sufficiently strong systems of arithmetic. For we need that
further premise in order to detach ("intuit the truth of") G after
reading Godel's paper.

One should also remember that a formal system is not an algorithm. The
rules of a formal system themselves are "non-deterministic" and do not
give a method for searching for proofs. Of course there are associated
proof-enumerating algorithms that can be considered. 

In general, I think it takes quite a bit of work to know how to connect
the abstractions Godel's theorem dealt with, such as consistent formal
systems, with flesh and blood human beings (or their brains). So it is
very hard to derive an exciting conclusion about "mechanism" from it.
For example, few people would committed to the claim that human beings
instantiate consistent proof-enumerating algorithms.

Of course, by the same token I think it is equally hard to derive any
substantive thesis from the slogan that "human beings are computers" or
instantiate algorithms.  So one should be just as suspicious of the
claim that we instantiate *inconsistent* proof-enumerating algorithms,
since the problem is what it means to say that we instantiate *any*
algorithms at all. (If so, we would have to be modelled as algorithms
that have formal inputs and outputs, another difficult problem to say
what those are).

I think the rules of interest here are not those of an algorithm, but
rather "non-deterministic" normative constraints of what we might call
after Chomsky a competence theory. But competence in this sense is
*already* quite far from any "mechanical" concept I think.

>>	It follows that no machine can be a complete or adequate
>>	model of the mind, that minds are essentially different
>>	from machines.
>
>Totally wrong.  There are consistent systems that can derive Godel's theorem.
>Nonetheless, they cannot derive their own Godel sentences.
>Lucas does not show that we cannot be such a system.

Although I think this is the wrong line of objection to Lucas/Godel (and
you evidently think there are stronger ones) I will raise a question anyway. 

I take it the Lucas response might be this: if you in some way
instantiate a consistent formal system (whatever that might mean) then
you could be presented with a Godelian proof using "your own" Godel
sentence (i.e.  one constructed via Godel's technique for the system
that is you).  You read along, repeat the reasoning yourself, and then
what? One would think, says our Lucas, you would come then to add G to
the stock of things you believe about numbers and so transcend,
angel-like, the limits of mechanism (Dennett).  Or do you expect to
rather sputter, emit smoke, and explode, like computers in some
movies?

The real answer is that you would have to assume your own consistency
as a premise in the reasoning process by which you come to add G to
your stock of truths about numbers in the course of your reading.  And
that leap of faith means engaging in some kind of reasoning not
embraced by any formal system.