[FOM] The Lucas-Penrose Thesis (long)

[A.P. Hazen]

The current string on "the" Lucas-Penrose argument has generated more 
responses than I expected!  Several people have  posted  the 
"orthodox" solution (going back to Gödel), that if there is a machine 
that simulates  our, human, mathematical ability, we will never  know 
of this machine THAT it simulates our ability.  (And so we will never 
be in a position to spring the anti-mechanist trap by recognizing the 
consistency of its output.)  A defender of the Lucas-Penrose argument 
has objected that this appeal to inevitable ignorance is 
unscientific,  that it puts the  defender of mechanism in the 
position of the scientist in the Harris cartoon whose black-board 
full of equations has "Here  a miracle occurs" in the middle!  So 
perhaps it would  help to give some  more detail on this point.  I 
believe that this "inevitable ignorance of our own identity as 
machines" (A) is plausible in itself,  (B) is  well-nigh demonstrable 
from an a priori plausible bunch of mechanist assumptions, and (C) 
when further analyzed helps show the relevance of Jeroslow's notion 
of  an "experimental logic" to the Lucas-Penrose  argument.  I 
apologize in advance for the length of this post: (A) contains little 
logic, and the perhaps more interesting (B) and (C) are largely 
independent of it.

	(A) INTRINSIC PLAUSIBILITY.  Why should we EXPECT that we 
should be able to learn "what machine we are"?   Perhaps the most 
attractive ground is optimism about neuroscience: can't we anticipate 
that the full "wiring diagram" of the human brain will one day be 
mapped out?  (Gödel himself didn't share this optimism: he told Wang 
he thought it likely that empirical research would show that the 
brain wasn't complex enough to be a basis for the mind!  But his view 
seems to have become a  minority one.)
					But this seems inadequate. 
Never mind the uniqueness presuppositions of the definite articles in 
"the  full wiring diagram of the human brain."  (Though if we allow 
the brain to evolve we perhaps approach the idea of Gödel's late 
objection to Turing's Thesis, "Collected Works" vol. II, p. 306....) 
It's still inadequate.  For some purposes the conception of the 
mathematical subject pursuing its thoughts solipsistically is not too 
bad a distortion, but real human mathematics is a cooperative 
enterprise, so at the very least we'd need, not just the wiring 
diagram of a brain,  but a theory of the interactions of 
(arbitrarily?) large numbers of brains.  And, leaving aside 
contentious questions of the status of "computer proofs" (e.g. that 
of the Four Colour Theorem), human mathematicians have long used 
pencil and paper as aids  in the calculational side of their work: so 
we'd need a  theory of groups of brains with "external memory" 
devices.  And real mathematics has often taken inspiration from 
problems in physics, so maybe identifying the "machine" which 
simulates human mathematics would  require a knowledge of all the 
physical phenomena that might INSPIRE groups of brains to come  up 
with  new approaches.  ...   In short, it seems to me that it may be 
far from easy to attain even EMPIRICAL confidence that any particular 
machine simulates ALL human mathematics!  But the Lucas-Penrose 
argument doesn't apply to, e.g., the mathematical thought of X as of 
date D: there is no reason to think that such a limited mathematical 
theory is consistent (remember Martin  Davis's Honor Roll of 
logicians who have proposed inconsistent foundational theories!). 
The Lucas-Penrose argument has to be about the idealized limit, human 
mathematics as it would develope if pursued forever.

	(B) NECESSITY.  Suppose-- as the Lucas-Penrose argument 
supposes the mechanist must suppose-- that there  is a formal, 
axiomatic, theory, HM, whose theorems express all and only the 
mathematical **truths** that have been or ever will be discovered by 
humans, or ever would be if human mathematics continued indefinitely 
into the future.
				The restriction to **truths**  may 
seem extrinsic to the problem, but we don't WANT a theory that 
captures the mistakes mathematicians make: such a theory wouldn't 
even be consistent!  Perhaps the requirement of truth can be replaced 
by a weaker one, but the Lucas-Penrose  argument, turning as it does 
on the claim that humans can know THEIR mathematics to be consistent, 
needs SOMETHING like it.
							What would HM 
be like?  Perhaps it  would amount to an extension of ZFC by some 
(recursive) set of strong axioms of infinity: axioms even Harvey 
Friedman hasn't considered yet!  HM would BE consistent, but would we 
KNOW that it was consistent?  By Gödel's Theorem, we wouldn't have a 
mathematical proof that HM was consistent: all our mathematical 
proofs are, by definition, proofs of theorems of HM.  Our confidence 
that HM was consistent would be at best empirical, and based on the 
consensus of expert opinion.  (And, since it would include the most 
extreme and speculative true axioms ever proposed, I doubt the 
consensus of the expert would be clear or unanimous!)
			Would we know of it that it was the 
systematization of human mathematics?  Again, our assurance would be 
at best empirical and conjectural.  The key move in the Lucas-Penrose 
argument is that we can "recognize" the consistency of human 
mathematics, but the ground for this recognition is the principle 
that we are smart enough that if we ever encountered an inconsistency 
in our  mathematics we would re-think and abandon some  axiom(s), so 
our **real** mathematics would be a consistent subsystem of the 
inconsistent one we  had  previously TAKEN  to be our mathematics. 
So any doubt about the consistency of HM -- and  I have just argued 
that there would be SOME doubt of this -- would  yield  a doubt that 
HM was **really** equivalent to HUMAN MATHEMATICS.

	(C) FURTHER ANALYSIS.  What could neuroscience discover? 
(Ignoring the complications under (A).)  The assumption of the 
Lucas-Penrose argument is that IF "mechanism" is true, then science 
can discover what Turing Machine simulates Human  Mathematics. 
Suppose this is so, that there is a Turing  machine, THM,  that 
adequately models Human  Mathematics.  How does it do this?  Maybe by 
printing  out, in correct chronological order, everything that will 
ever be published in a Human mathematics journal.  Suppose 
neuroscience  discovers that this is so (as -- to paraphrase Post-- 
an  empirical law about human mathematical thinking).
			(((Note that the agreement is supposed to be 
lawlike: the abstract machine, HTM, doesn't stop at the point when 
bird flu or global warming or a new dark age or someone selling 
nuclear technology to a TRULY mad dictator stops Human  mathematical 
development in its tracks.  It prints -- or would  print if it were 
ever physically instantiated -- all that WOULD be published by human 
mathematicians if they went on forever.)))
								But 
note  that mathematics journals don't JUST contain theorems: they 
contain RETRACTIONS as well!
	Leaving my final question: what would be the relation of the 
Turing machine, THM, to the axiomatic theory HM described above in 
(B)?  Would it be (or would it be convertible into) a machine that 
enumerates the theorems of HM?  Conceivably, YES.  Maybe we are lucky 
and on the right track and a time will come (or would  come if 
mathematics endured long enough) after which no further retractions 
will  ever be necessary!  But also conceivably, NO.  Maybe there 
will be ever new additions  to Martin Davis's honor roll: logicians 
who advance mathematics by proposing new and stronger axioms, some of 
which will  later -- perhaps MUCH later -- be discovered to be 
inconsistent.  THM, it seems to me, would amount to an EXPERIMENTAL 
LOGIC, in the sense of Jeroslow's 1976 "Journal  of Philosophical 
Logic" paper: thre is no particular reason, it seems to me, to assume 
that it would  be (or would ever be known to be) equivalent to an 
axiomatic formal system.  And this, it seems to me, is the most 
fundamental problem with the Lucas-Penrose argument.  It assumes that 
"Mechanism" is committed to a certain relationship  between the human 
machine and the human mathematical theory: that the machine 
enumerates the theorems of the theory.  But it isn't.  THM might 
exist and not enumerate the theorems of HM.  THM might exist and have 
a Delta-0-2 but not Sigma-0-1 set of "permanent" theorems, in which 
case the theory HM would not exist.

---

Allen Hazen
Philosophy Department
University of Melbourne