For any Turing machine there is an equivalent axiomatic system;
whether we could construct it or not, is of no significance here.
Reading your link I was impressed by Russell Standish's sentence:
'I cannot prove this statement'
and how he said he could not prove it true and then proved it true.
Isn't it more likely that the sentence is paradoxical and therefore
non propositional. This is what could make a difference between humans
and computers: the correspinding sentence for a computer (when 'I' is
replaced with the description of a computer) could not be non
propositional: it would be a gödelian sentence.
On Jun 28, 10:05 pm, "Jesse Mazer" <[EMAIL PROTECTED]> wrote:
> LauLuna wrote:
> >This is not fair to Penrose. He has convincingly argued in 'Shadows of
> >the Mind' that human mathematical intelligence cannot be a knowably
> >sound algorithm.
> >Assume X is an algorithm representing the human mathematical
> >intelligence. The point is not that man cannot recognize X as
> >representing his own intellingence, it is rather that human
> >intellingence cannot know X to be sound (independently of whether X is
> >recognized as what it is). And this is strange because humans could
> >exhaustively inspect X and they should find it correct since it
> >contains the same principles of reasoning human intelligence employs!
> But why do you think human mathematical intelligence should be based on
> nothing more than logical deductions from certain "principles of reasoning",
> like an axiomatic system? It seems to me this is the basic flaw in the
> argument--for an axiomatic system we can look at each axiom individually,
> and if we think they're all true statements about mathematics, we can feel
> confident that any theorems derived logically from these axioms should be
> true as well. But if someone gives you a detailed simulation of the brain of
> a human mathematician, there's nothing analogous you can do to feel 100%
> certain that the simulated brain will never give you a false statement. It
> helps if you actually imagine such a simulation being performed, and then
> think about what Godel's theorem would tell you about this simulation, as I
> did in this post:
> Make every IM count. Download Messenger and join the i'm Initiative now.
> It's free.http://im.live.com/messenger/im/home/?source=TAGHM_June07
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