[FOM] The Lucas-Penrose Thesis
laureano luna
laureanoluna at yahoo.es
Thu Sep 28 13:58:41 EDT 2006
I would focus on the simple argument in 'Shadows of
the Mind' propounding that no knowably sound Turing
machine M can mirror human mathematical ability,
because for any such M we would know something that M
could not know, namely, that M wouldn't stop at the
code of M.
To accept this one has to accept that 'human
mathematical ability' is well defined. I think some
have argued against this point, like perhaps the late
Torkel Franzen in 'Inexhaustibility'.
But suppose we grant it is well defined. To go on to
conclude that human mathematical ability cannot be
mirrored by any Turing machine, we need to show that a
Turing machine mirroring human mathematical ability
would be knowably sound.
Once we have accepted that 'human mathematical
ability' has a well defined meaning, it is hard not to
admit that that ability has to be sound: any argument
to the contrary could easily turn out self-defeating;
nevertheless, I think I remember someone has tried
it(Putnam?).
But what about 'knowably sound'. Faced to the
description of the (hypothetical) Turing machine M
that mirrors human mathematical ability, would human
mathematical ability be able to recognize it as sound?
Any Turing machine has a finite description. The
description of M would include nothing else than our
mathematical and logical intuitions. Thus, why should
we be in principle incapable of knowing M sound?
Of course, the description of M could be too long or
intricate for any human to be able of coming to a firm
conviction about M's soundness. Let's assume it is so.
My question is now the following: could that
impossibility be deemed the same as the impossibility
of M to know of its own soundness? Can technical or
physical impossibility be equated to logical
impossibility?
(Further: would answering 'no' beg the question at
issue? would answering 'yes' provoke modal collapse or
result self-defeating?)
Anyhow, if mechanism is right and human mathematical
ability is (ultimately) sound, then there must be a
logical impossibility for humans to come to know the
soundness of the Turing machine that mirrors their
mathematical ability.
Mechanists would put it definitely clear for all if
they proved the existence of that impossibility. Has
anyone tried?
Best wishes,
Laureano Luna cabañero
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