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

[A.P. Hazen]

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.)