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Article 2363 of comp.ai.philosophy:
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>From: zeleny@zariski.harvard.edu (Mikhail Zeleny)
Newsgroups: sci.philosophy.tech,sci.logic,sci.math,comp.ai.philosophy
Subject: Re: Penrose on Man vs. Machine
Summary: penrose hasn't been refuted
Keywords: analytic arguments, reflection principle, standard model
Message-ID: <1991Dec22.131401.6869@husc3.harvard.edu>
Date: 22 Dec 91 18:13:58 GMT
References: <1991Dec19.041945.27038@oracorp.com>
Organization: Dept. of Math, Harvard Univ.
Lines: 211
Nntp-Posting-Host: zariski.harvard.edu

In article <1991Dec19.041945.27038@oracorp.com> 
daryl@oracorp.com writes:

DMC:
>Mikhail Zeleny has asked that I reprint some private email I sent
>to him so that it can be debated in front of the whole Net. Rather
>than posting the letters verbatim, I'll try to summarize my side of
>the argument, and Mikhail can do the same for his side. Does that
>sound fair, Mikhail?

Very well, Daryl, I'll do likewise; however recall that it was your
original contention that philosophical arguments ought to be stated in a
precisely formulated fashion in order to make clear their logical
structure.  You claimed to have done so in your email arguments; I claimed
that some of your assumptions were unwarranted.  If you wanted to reopen
this issue, perhaps you should have done so *more geometrico*, in the
original, more rigorous format.  Most importantly, in order to give this
issue a fair hearing, you shouldn't have limited yourself to the inbred
audience of AI cheerleaders that constitutes comp.ai.philosophy.  I've
taken the liberty to correct the last oversight by cross-posting.

DMC:
>Penrose is a brilliant man, and since I am a physicist by training, I
>am properly awed by his contributions to the field. However, having
>had the pleasure of hearing Penrose speak on his book, The Emperor's
>New Mind, I have come to the conclusion that what he has to offer to
>the field of mathematical logic is not much. His speculations about
>the relationship between quantum mechanics, gravity, and the mind are
>fun reading, but the part of his book that I really think is
>fundamentally mistaken is his arguments that the human mind must be
>more than a computer. Unlike Searle's argument, which is too murky
>either to be refuted or to be convincing, Penrose' arguments were
>models of clarity, although mistaken.

Although I disagree with you on Searle, I would like to emphasize that my
own semantical argument bears only a remote resemblance to his own.
Briefly, I appreciate his realism about propositional attitudes, as well as
his rejection of functionalism; however I don't accept his apparent
materialism, seeing it as incompatible with the former.  My own position is
somewhere between the Popper-Eccles dualism and the O'Shaughnessy dual
aspect theory, generally favoring the latter.

DMC:
>I will take two examples:
>
>I. "How to outdo an algorithm", on pages 64-66 of my edition contains
>the following paragraph:
>
>     Suppose we have some algorithm which is sometimes effective for
>     telling us when a Turing machine will not stop. Turing's procedure,
>     as outlined above, will explicitly exhibit a Turing machine
>     calculation for which that particular algorithm is not able to 
>     decide whether or not the calculation stops. However, in doing so,
>     it actually enables us to see the answer in this case! The particular
>     Turing machine calculation that we exhibit will indeed not stop.
>
>A little background: The halting problem is the question of deciding,
>for arbitrary numbers n and m, whether Turing machine #n will ever
>halt when given input m. An algorithm H(n;m) is said to solve the
>halting problem if H(n;m) = 1 if n halts on m, and H(n;m) = 0 if n
>never halts on m. A well-known proof of Turing's shows that there is
>no algorithm that can solve the halting problem; every algorithm H(n;m)
>must either give the wrong answer on some inputs n and m, or else will
>fail to give any answer (that is, H(n;m) may never halt). Penrose calls
>an algorithm H(n;m) "sometimes effective" for solving the halting problem
>if it always gives the right answer whenever it halts (although it may
>never halt for some inputs). This is the same notion as the notion of
>a "partially correct" program in computer science, so I will use that
>terminology.
>
>Penrose is saying in the paragraph above that for any partially correct
>algorithm H(n;m) for solving the halting problem, we can
>find numbers n and m such that
>1. We can see that Turing machine n never halts on m.
>2. H cannot (less anthropomorphically, H never returns an answer).
>Therefore, we can beat algorithm H (by answering at least one question
>that H cannot). Since we can do this for any H, then we can beat any
>algorithm.
>
>It may appear at first that Penrose has shown that no algorithm can do
>as well as the human mind at solving the halting problem, and
>therefore the human mind cannot be an algorithm. Penrose himself seems
>to believe that he has demonstrated this. However, he has shown no
>such thing. Penrose' method for finding the numbers n and m relies on
>the *assumption* that algorithm H is partially correct.  If we are
>given H but are *not* told that it is partially correct, the Penrose'
>recipe doesn't give us any insight into an example of something that
>we can do but H cannot do.
>
>The non-algorithmic insight, if there is any, is in knowing which H's
>are partially correct and which are not, and Penrose tells us nothing
>that would suggest that humans are any better at this than Turing
>machines.
>
>The *strongest* conclusion that we can legitimately draw from Penrose'
>argument is that we can beat any algorithm that we know to be partially
>correct, so if we are algorithms then there must be some algorithms that
>we don't know whether they are partially correct or not.

As you undoubtedly are aware, all analytic arguments contain their
conclusion in their premisses.  Penrose argues that we are capable of
outdoing any algorithm; in order to do so he has to assume an equally
powerful, but intuitively more plausible premiss.  It appears to me that he
has succeeded in doing so, since the premiss that we can *potentially*
determine the partial correctness of an arbitrary program by semantic
reflection appears to me as intuitively unexceptionable as A.A.Markov's
abstraction of potential realizability, which stipulates e.g. that the
successor operation can be applied to an arbitrarily large integer.

In other words, by rejecting the assumption made by Penrose, you are
implicitly committing yourself to ultra-intuitionism and its concept of
feasible numbers.  I have long maintained that not only the Soviet
democratic movement, but also the US AI community owes an intellectual debt
to the rediscoverer of glasnost and ultra-intuitionist extraordinnaire
Alexander Yessenin-Volpin; perhaps you can honor this obligation by
underwriting his research program.  All grateful parties are hereby invited
to contact the dissident logician through Gabriel Stolzenberg,
gabe@zariski.harvard.edu.

DMC:
>Does Penrose give any reasons to believe that humans are better than
>algorithms in sorting out the partially correct programs from the
>incorrect programs? Yes, he does, in a different section of The
>Emperor's New Mind. Penrose makes the claim that the insight given by
>reflection is somehow nonformal, and gives us abilities beyond
>those of mere algorithms: on page 110, he says:
>
>     Reflection principles provide the very antithesis of formalist
>     reasoning.
>
>However, the only examples of reflection that Penrose cites are
>*perfectly formal*! He says:
>
>     The insight whereby we concluded that the Godel proposition
>     P_k(k) is actually a true statement in arithmetic is an example
>     of a general type of procedure known to logicians as a reflection
>     principle...
>
>(The sentence P_k(k) is the famous Godel sentence for first order
>arithmetic, a sentence that is true but unprovable.)
>
>As in the case with "How to outdo an algorithm" above, Penrose glosses
>over a crucial step: on the bottom of page 107 and top of 108, he mentions
>in passing
>
>     Our formal system should not be so badly constructed that it actually
>     allows false propositions to be proved!
>
>The reasoning leading to the conclusion that P_k(k) is true depends in
>an essential way on this assumption that the original system is sound
>(doesn't allow false propositions to be proved). The crucial,
>non-algorithmic step in the reasoning is in this very assumption (it
>is non-algorithmic to decide which formal systems are sound and which
>are not). The reflection principle, allowing us to deduce that P_k(k)
>is true, given that our original system is sound, is not the
>antithesis of formalist reasoning; it is completely formal! It is even
>provable in ZFC.

Yes, the reflexion principle for first-order arithmetic is provable in ZFC,
particularly after you avail yourself of Gentzen's consistency proof for
the former within the latter.  Likewise, the semantic notions of
first-order arithmetic, in particular the crucial notion of finitude, can
be expressed in the first-order ZFC; however due to the existence of
non-standard, countable models of the latter, guaranteed by the
L\"owenheim-Skolem theorem, these notions won't be determined on the purely
syntactic, proof-theoretic level.  It is by reflecting on the semantic
aspects of our theory, i.e. by contemplating its standard models, that we
find proper justification for applying reflection principles.  In
particular, the insight whereby we conclude that the Godel proposition
P_k(k) is actually a true statement of elementary arithmetic, is the
antithesis of formal reasoning, precisely because by adhering to the
formalism of the first-order PA, we can't tell whether the P_k(k) is true
or false, being that neither itself nor its negation is provable in this
formalism.  It is only by reflecting on the standard models of arithmetic,
all of which satisfy P_k(k), that we can conclude its truth.

DMC:
>Therefore, in these cases, Penrose' arguments amount to the following:
>
>    (1) Assuming that we can tell which Turing machines H are partially
>        correct for solving the halting problem, then our reasoning
>        is nonalgorithmic.
>
>    (2) Assuming that we can tell which theories are sound, then our
>        reasoning is nonalgorithmic.
>
>In other words, assuming that we can do things that no machine can do,
>then we can do things that no machine can do.

Quite so.  As I stated earlier, all analytic arguments can be construed as
a form of begging the question by an opponent who doesn't accept the
intuitive validity of their premisses.  In the end, we may have to agree to
disagree; however I hope that you would understand that your claim about
the purely formal nature of reflexion principles as illustrated by Penrose
was made in error.

>Daryl McCullough
>ORA Corp.
>Ithaca, NY.

`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'
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: Mikhail Zeleny                                                     :
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