Re: Human mind not Turing computable according to Eliezer?

From: Christian Szegedy (
Date: Fri Oct 08 2004 - 03:39:30 MDT

Hi Bill,

Now I have read your writing thorougly and my opinion is
that neither your nor Penrose's arguments are conclusive.


First, why do I think that your refutation attempt is flawed:

Your wrote that the construction of Penrose can not
be performed since the human mind is not a Turing machine,
but a finite state machine.

However, every finite state-machine can be modelled by
a Turing machine, so I think that the the construction
of Penrose can be still performed.

More concretely, you wrote:

> Here is where the argument breaks down. With Turing machines,
> we said there must be some integer k such that the Turing machine
> TM_k will give the same answer to the question encoded by n that
> TM_b gives to question Q2. The integer k exists because we can
> construct a Turing machine TM_x that converts any positive integer
> n into the index of question Q2, and we can combine TM_x and
> TM_b to get TM_k. But there is no finite state machine that can
> convert an arbitrary integer n into the index of question Q2'.

In fact, for Penrose's argument to work, it is irrelevant whether TM_k
is a finite state-machine. The only important point is whether the
reader can be modelled by a Turing machine TM_b or not. Everything
else is irrelevant.

You don't solve anything by answering that his argument is flawed
because the human brain is an even more restricted type of Turing
machine. Then, he could ask: "How comes that an even more
restricted type of Turing machine can solve such a hard problem?"
Does not it really show that the human brain in fact is *not* a finite
state machine?

Still, I don't think that Penrose is right on this point.
But my opinion is that it does not have anything to do with
the question whether we are FSMs or not.


Now, let me come to my refutation of the argument of Penrose:

Assume that we have a Turing Machine AFRM (average formal
reasoning machine) which is capable of formal reasoning about
as well as an average theorem prover today
(it can even be a finite state machine, but it does not matter).

Furthermore, we formalize the situation described by Penrose
informally in his book. Then ask AFROM the same question
formally we were asked informally by Penrose.

I would bet that AFRM would find the correct answer quite
quickly. (This is an experiment that could be
performed in reality and would be a very impressive
demonstration why his argument fails)

How does it come? Simply, because the correct answer can be
easily formally deduced from the premises that are the
*real input*.

I think that the error in the argument of Penrose is that the
problem was posed to the reader in a different format than he
pretended to pose it.

This format is easily interpreted and solved. In its real format:
when k is explicitely written down (assuming of course that
we are Turing machines, since k depends on the index of the
TM modelling the reader) it can not be answered (and this is a
mathematical fact).

So, the reason why his argument fails is that he does not
actually ask the reader the question
"Will TM_k stop on input k?"
"Take an extremely complicated true statement S the answering
of which exceeds your mental capabilities. Is S true or not?"

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