[FOM] The Lucas-Penrose Thesis vs The Turing Thesis
joeshipman at aol.com
Thu Oct 5 15:01:36 EDT 2006
It seems to me that there are several different questions being debated
here, which are not precisely formulated. Here is my attempt at
reframing things in simpler terms.
Define "Human mathematics" as the collection of formalized sentence in
the language of set theory which are logical consequences of statements
that will eventually come to be accepted by a consensus of human
mathematicians as "true".
(Remark: if ZFC is consistent, this collection includes ZFC, and it
includes many sentences which will never ACTUALLY be accepted by human
mathematicians as true simply because they are too complicated to be
written down in our universe, but which are still consequences of
sentences we accept as true. The word "eventually" means that any
mistakes leading to inconsistency will ultimately be found and
corrected, and the referenced consensus is one that persists forever.
Thus "Human mathematics" is consistent, because anything which leads to
an inconsistency will eventually be rejected, but we can't tell in a
finite time whether an arbitrary sentence is part of human mathematics
Proposition A: There exists a recursively enumerable and consistent set
of sentences which contains "Human mathematics".
Proposition B: There exists a recursively enumerable and consistent set
of sentences which equals "Human mathematics".
If human mathematics were entirely the product of our brains, I would
think A has a chance to be true, but it could also be false if Church's
thesis is incorrect. This depends on considerations external to our
brains. It is possible that our investigations of physics could lead us
to the belief that a mathematically definable but nonrecursive set can
be generated by physical processes, and this in turn would lead to an
inexhaustible source of new axioms for mathematicians to adopt,
falsifying A (but not "refuting" A, since our belief in the correctness
of the physics would not attain the same level of certainty as our
belief in mathematical axioms given to us by pure intuition).
It is not clear whether Lucas and Penrose claim to refute B or to
refute A. While A could be false, I don't think A can actually be
refuted: consider the theory T axiomatized by taking all true sentences
of length less than googlplex in a finitely axiomatizable extension of
ZFC such as VNBG. (I don't want to consider the theory axiomatized by
taking all true sentences of length less than googlplex in ZFC, because
that fails to include some axioms of ZFC which we "offically" accept
even though we can't write them down, but passing to VNBG seems to do
the trick.) It is very hard to see how humans could ever "know" a
sentence in the complement of T to be true. Beings whose brains were
big enough to comprehend and transcend T may exist in some realm, but
they are not "human" as the term is commonly understood.
So I don't think Lucas and Penrose can have refuted Proposition A, but
Proposition B is stronger and possibly easier to refute. However,
refuting B doesn't have the psychological impact refuting A would,
since there could still a be machine that is mathematically superior to
us in that it can prove everything we can and other (consistent) things
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