Le 17-oct.-05, à 12:57, atillagurel (FOR list) a écrit :


--- In [EMAIL PROTECTED], "uv" <[EMAIL PROTECTED]> wrote:
[snip]

the paper (by Chris Small in fact)
http://www.stats.uwaterloo.ca/~cgsmall/conscious.html
on consciousness is quite long but, to me, makes some very good
points
about consciousness.

The author makes a distinction between mind and consciousness and
claims that consciousness is not someting algorithmic although it
may be an alghoritmic process how human mind operates. However he
doesn't give further details on the latter.

I want to explain why doubts on the latter as formulated below
are unjustified

"Since Godels proof shows that human mind can recognize the truth of
a statement although it cannot formally be proven to be true, we
must conclude that working of human mind must have an aspect, that a
computer programme never can have".


Goedel introduces a numbering system that assigns uniquely a natural
number to any possible mathematical symbol , a collection of symols
(a statement for example) or collection of statements (a proof for
example). Lets call it Godel's number.

The  proof is based on the construction of the following statement :

"The statement with Godel's number x is unprovable."

And it happens "accidentally" that the above statement has Goedel's
number x.

Thus the statement is self referencing.

That we recognize the truth of the statement x is an algorithmical
process namely:

1. Statement x is self referencing.
2. A self referencing statement is unprovable .
3. Statement x is unprovable.
4. Statement x is true.


Since self reference is a property that can be checked easily
algorithmiacally by a computer, I don't see why a computer should
not arrive the same conclusion above as the human mind if the
program checks each constructed statement on self reference before
proceeding with the next step.


Bravo. Your reasoning, despite some minor details, is quite accurate. It is due to the mechanizability of the Godel diagonal argument.

Note that there is no equivalent of Church thesis (CT, which bears on the notion of computability) for the notion of provability. So with CT computability is an absolute notion, but provability is always *relative* to a theory (or any mechanical theorem prover or checker). So instead of saying "the statement with Godel's number x is unprovable", it is better to say "the statement with Godel's number x is unprovable by T" (and T is the theory (machine) you want to prove incomplete).

An important point is that although the computer can generate (by the mechanical diagonalization) its unprovable truth, the computer can not take those derivation as proof (if not the statement would be false and the machine by proving them) would be inconsistent!

I think Emil Post is the first one to understand this clearly in its 1920 notes!

Best regards,

Bruno



http://iridia.ulb.ac.be/~marchal/


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