[FOM] The Lucas-Penrose Thesis vs The Turing Thesis

[joeshipman@aol.com]

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 
or not.)

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 
too.

-- JS


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