[FOM] The Lucas-Penrose Thesis
A.P. Hazen
a.hazen at philosophy.unimelb.edu.au
Thu Sep 28 04:59:01 EDT 2006
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!)
---
Allen Hazen
Philosophy Department
University of Melbourne
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