[FOM] The Lucas-Penrose Thesis
Robbie Lindauer
robblin at thetip.org
Thu Oct 5 13:42:28 EDT 2006
On Oct 4, 2006, at 5:26 PM, John McCarthy wrote:
>>
>> On Sep 29, 2006, at 2:51 PM, John McCarthy wrote:
>>
>>> Program: I can use any system you like, although mostly I
>>> use a system based on a variant of ZF and descended from
>>> 1980s work of David McAllester.
>>
>>
>> It is almost certainly not true that the program can "use any system
>> you like". At least not any actual computer, and certainly not the
>> traditional turing machine. Different actual computers have different
>> limitations.
>
> My "any system you like" did not refer to imitating a specific person.
> The systems the program offered to "Penrose" in the dialog were PA and
> its extensions, ZF and its extensions, and other definite logical
> systems "Penrose" might propose.
>
> I'm not sure what Lindauer means by "Different actual computers have
> different limitations." If they are all universal, their limitations
> are only in amount of memory and speed.
I think we're teetering around this point now, also well played out in
Lucas:
There is a question of LOGICAL possibility here. It's LOGICALLY
possible for computer C to solve problem P, is the contention in
question.
The mechanist and non-mechanist agree that for any computer and program
combination C, there exists a problem P which it can not possibly solve
NO MATTER WHAT. If time and capacity are infinite, then C can not
solve P.
This is as far as I know an undisputed consequence (please see caveat
below before blowing up!) of Godel's incompleteness theorem along with
the additional assumption "Computers are implementations of the type of
system to which Godel's system applies". (undisputed within the
"classical logic" and "classical mathematics" although I think some
computer scientists are aware that their computers are rather obviously
NOT universal turing machines being absolutely limited in time and
resources. Ultrafinitists I think see the obvious shortcomings of such
generalizations noting that no computer COULD BE such a machine.)
The mechanist must also, therefore, assert that for any person C' that
there is a problem P' that they can not solve. Otherwise, obviously,
the computer and the person would be non-identical (there being a
difference in potentialities there).
More strongly, the mechanist must maintain that for every problem (P,
P', P'', P''', etc.) that their computer can not solve, neither can the
human to which it corresponds solve them.
The problem is that when the problems in question are actually
presented, the human can solve them at the very least, by fiat.
Personally, I think this is demonstrative of the non-formal nature of
actual human reasoning. In any case, as long as this is an open
LOGICAL possibility, then the FUNCTIONALIST mechanist remains unable to
press the point, I think. As long as there is, theoretically, a
sentence, the undecidable sentence for a given system S, then then
there also remains the possibility of some other system S' (logical or
non-logical) being able to decide that sentence. (This remains, again,
under the strongly infinitist metaphysics of mathematics which are
questionable to begin with.) Obviously, also, non-formal machines
(like brains, for instance) might be able to do these things where
formal machines can not. This may have been behind the
quantum-suggestion of Penrose - MAYBE QM makes our brains
unformalizable and those effects extend to human reasoning? Maybe....
My original point above was about REAL computers not LOGICAL computers.
As a long time programmer I know that there are REAL limitations to
ACTUAL computers and that they have very little to do with the
limitations associated with people.
Best Regards,
Robbie Lindauer
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