[FOM] The Lucas-Penrose Thesis

[A.P. Hazen]

F.A. Muller writes:
>	"I have heard (or read), on another occasion, Penrose
>	making fun of the situation that everybody disagrees
	with him but the critics also disagree among themselves
	about what is wrong with the argument."

Part of the problem is that there are several  different versions of 
the  Lucas-Penrose argument (or: several different Lucas-Penrose 
arguments),and people responding to different versions can say very 
different things!

For example: two of the best  responses (responses which I think 
conclusively refute two versions  of  Lucas) are  by David Lewis 
("Lucas against mechanism," in "Philosophy"  v. 44 (1969), pp. 
231-233, and "Lucas against mechanism II," in "Canadian Journal of 
Philosophy" v. 9 (1979), pp. 373-376): they are very different, 
because Lewis addresses two very different interpretations of Lucas!

My "favorite" version is:
	(1) If the human mind is mechanical, human mathematics
	   is the product of a machine,
	(2) The product of a machine = a recursively enumerable
	   set = the set of theorems of a formalized axiomatic
	   theory,
	(3) No consistent theory incorporating basic number theory
	   has a theorem asserting its own consistency (Gödel),
	(4) But we human mathematicians can know that human mathematics
	   is (ultimately) consistent (because  we would revise our
	   axioms if we found a contradiction, so the contradiction
	   would -- ultimately -- be eliminated)
	(5) So the mind  is not a machine.
There's a fallacy here, but one best criticized by DEVELOPING a new 
theory. Step (2) depends for its plausibility on the fact that the 
most familiar way of conceiving of a set  as the "product"  of a 
machine is for the machine  to enumerate the set.  What we need,  in 
order to see through the argument, is an alternative conception of 
how a set of sentences might deserve to be called the "product" of a 
machine, one which makes (4) plausible but shows that it is not 
inconsistent with (3).  ...  So,though I don'tthink it actually 
MENTIONS Lucas, I think the best "refutation" of the Lucas thesis is 
Robert Jeroslow's "Experimental logics and Delta-0-2 theories," in 
"Journal of Philosophical Logic" v.  4 (1975), pp. 253-267. 
(Apologies to those who have seen me recommend this paper before!  I 
think it's good!)

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Allen Hazen
Philosophy Department
University of Melbourne