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:


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