[FOM] The Lucas-Penrose Thesis vs The Turing Thesis
A.P. Hazen
a.hazen at philosophy.unimelb.edu.au
Sat Oct 7 01:46:52 EDT 2006
This debate-- like many interesting but inconclusive philosophical
controversies-- suffers from being conducted in terms that have not
been defined precisely enough to make them suitable for use in
riforous proofs. Joe Shipman has, usefully I think, tried to make
some of them more precise.
Shipman:
>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: it's not obvious to me that this defines a
unique set. Suppose some version of the "many worlds" approach to
quantum mechanics is true and that the universe "splits" every time
a photon encounters a half-silvered mirror. Might not human
mathematicians come to accept different sets of set-theoretic
sentences in the histories arising out of the two sides of a split?
But since the definition HAS to refer to an idealized, rather than a
physically real, future development of mathematics (see below),
perhaps this isn't important.
Shipman:
>
>... 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.)
-------> Since there seems to be no apriori way of setting a bound
to HOW long it will take to discover and correct a "mistake"
[[[The relevant
mistakes will be things like accepting a new axiom, for
what seem like convincing intuitive reasons, which ultimately
turns out to be inconsistent.]]]
this seems to require that
the "historical" develoment of mathematics be projected arbitrarily
far into the future: "Human Mathematics" includes things that (e.g.)
the heat deat of the universe will keep actual humans from ever
discovering. I think we have to allow a similar idealization with
respect to space: surely there is a possibility of mistakes that
could only be discovered by examining deductions so large that, for
physical reasons, they can never be written down. (((Aside: There is
technical work on questions about inconsistent axiomatic systems
whose shortes proof of a contradiction is very, very long. Gödel's
"On the length of proofs," and Rohit Parikh's introduction to it in
v. I or G.'s "Collected Works," might be a starting point for the
pre-1980s part of a literature search. Mic Detlefsen's (1986) book
"Hilbert's Program" contains interesting, speculative, comments.
"Pavel Pudlak" would be a useful name to search under for later
literature.)))
Those who have read my earlier posts on this topic will not be
surprised to hear that I think Robert Jeroslow's notion of an
"experimental logic" (cf. his article in"J. of Philosophical Logic"
v. 4 (1975)) provides a useful mathematical model for "Human
mathematics" asdefined by Shipman. (Jeroslow's work is in the
framework of classical recursion theory: there is no need to question
Church's thesis here!)
Shipman:
>
>Proposition A: There exists a recursively enumerable and consistent set
>of sentences which contains "Human mathematics".
----> If forced to guess, I'd put my money on "yes". Precise
technical question: is the set of "permanent" sentences of an
experimental logic always a subset of some consistent r.e. set?
>
>Proposition B: There exists a recursively enumerable and consistent set
>of sentences which equals "Human mathematics".
------->Dubitable. The set of "permanent" sentences of an
exprimental logic is NOT always r.e. (Experimental logics are not
always equivalent to formalized axiomatic theories.)
Shipman (after two paragraphs of discussion I have not reproduced):
>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.
-------------> I ***LIKE*** this! I wouldn't just put it in terms
of "psychological impact," though. I would claim that a central
PHILOSOPHICAL error in Lucas and Penrose is that they havemistaken an
argument against B for a refutationof mechanism by ASSUMING that the
formalaxiomatic system is the ONLYrecursion theoretic model available
for Human mathematics.
--
Allen Hazen
Philosophy Department
University of Melbourne
(Hmmm. I have made three posts in about a week, ALL urging the
interested to look up Jeroslow's article. To avoid being a bore, I
suspect it is best if I not POST about the Lucas-Penrose stuff for a
while.)
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