From: Marc Geddes (marc_geddes@yahoo.co.nz)
Date: Sat Oct 09 2004 - 01:16:58 MDT
--- Christian Szegedy <szegedy@or.uni-bonn.de> wrote:
> Bill Hibbard wrote:
>
> >But his argument depends on a property of Turing
> machines, namely
> >the ability to do arithmetic with arbitrary
> integers, not shared
> >by finite state machines. So his argument fails by
> assuming a
> >capability that humans do not have.
> >
> No, he does not assume any capability. He
> *exhibits* a capability:
> that you can decide something you are not supposed
> to be able to.
>
> His argument is based on a proof by contradiction.
> It goes by:
>
>
> 1) Assume that your brain can be modeled by a Turing
> machine
> (This is true even if it is a finite state
> machine)
>
> 2) Let us consider the following problem that this
> TM cannot
> solve.
>
> 3) But, you can clearly solve it => Contradiction!!!
>
> 4) Since we had a contradiction, your brain cannot
> be modelled by a TM.
>
>
> You cannot refute this line of argumentation by
> saying that our brain is
> an FSM
> changing his proof and say that the modified proof
> is flawed.
>
> You must refute the *original proof*.
>
>
One of the most irritating things about Penrose and
his arguments is that whilst most people agree that
he's wrong, the reasons that his critics as to *why*
Penrose is wrong are all different to one another ;)
I agree that Finite State Machines are not the issue.
What *is* the issue is the empirical fact that as far
as we know, there is no such thing as an uncomputable
finite physical process. That is, we think that all
finite physical processes are computable.
But empirical issues aside, what is logically wrong
with the Penrose argument summarized above?
My answer:
There is an ambiguity in the word 'solve'. Look at
(2) again:
> 2) Let us consider the following problem that this
> TM cannot
> solve.
All the Godel Theorem shows is that there is no
algorithmic procedure for *solving* certain classes of
problem WITH 100% ACCURACY
So in other words, Penrose is using a very limited
definition of the word 'solve'. By 'solve' he means
reach a conclusion that we can be 100% of.
If we drop the requirement that we have to limit
oursleves to axiomatic reasoning yielding certain
conclusions, then the Godel Theorem no longer limits
us.
You see that Penrose's argument now falls apart.
There is no reason why a computer cannot make guesses,
approximations and draw probablistic conclusions about
any of the so-called 'uncomputable' problems. Take
any 'uncomputable' function. There is no reason why a
computer could not compute this to say 99% accuracy.
All 'uncomputable' means is that no finite algorithm
can be 100% accurate.
You see that Penrose's argument now falls apart.
Penrose is trying to con us into thinking that humans
can draw infallible conclusions about maths truths.
But there is no reason for thinking this is the case
at all. Humans *can't* reach infallible answers about
uncomputable functions, any more than computers can.
Humans simply GUESS at the answers to 'uncomputable'
problems, using statistical (Bayesian) reasoning to
draw probabilistic conclusions. The point is that we
are NOT in fact certain that arithmetic is consistent.
We humans simply GUESS that it is (or at least we
assign some probability which is less than 100%).
So there is nothing special about human reasoning.
There is nothing to Penrose's arguments at all. He is
just plain flat wrong.
=====
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