From: "Phil Roberts, Jr." 
Subject: Re: Robot Evolution
Date: Tue, 26 Dec 2006 00:18:54 -0500 (EST)

Tim Tyler wrote:
> John Lucas's 'Godel' argument has been much-criticized - and
> Penrose's views in this area are essentially a variation on it.

I concede that there is a clear majority who
disagree with the Lucas/Penrose position.  On
the other side of the equation, however, we have:

a. Hofstadter, Dennett, Penrose and Chaitin,
   in various ways acknowledging that Godel at least
   SUGGESTS a disconnect between formalism and
   mathematical reasoning.

b. Little unanimity as to what exactly is wrong with
   the Godel argument, with dozens and dozens of different
   sorts of objections, many based on impenetrable

c. Papers still being published criticizing the Godel
   argument against mechanism almost 80 years after Godel
   first published his theorem.

d. The universal abandonment of Hilbert's program of
   formalizing mathematical reasoning by mathematicians
   all over the world subsequent to Godel's proof.

e. Intersubjectively reproducible empirical evidence
   (feelings of worthlessness) suggesting that not even
   Mother Nature herself seems to be able to constrain
   rationality within a formalism (the program for
   "trying to stay alive").

f. Evidence (e.g., Parfit, 'Reasons
   and Persons', p. 12) that any theory that attempts
   to constrain rationality within a formal structure
   (e.g., a fixed objective) can be shown to sanction
   rational irrationality (i.e., can be shown to be

> Brief version of what's wrong:
> ``A mathematician often makes judgments about what
>    mathematical statements are true. If he or she is not more
>    powerful than a computer, then in principle one could write
>    a (very complex) computer program that exactly duplicated
>    his or her behavior. 

Assumes what is being questioned.

>    But any program that infers

Programs don't infer, they model logical relations that
have been found to underly human inferences on most
occasions.  As to whether these relations are actually
being followed or simply EMBEDDED IN our inferences remains
to be seen.

>    mathematical statements can infer no more than can be proved
>    within an equivalent formal system of mathematical axioms
>    and rules of inference, 

True, but Lucas/Penrose assumes we can go beyond this, that
the intuiting of mathematical truth is not simply a matter
of logical proof:

    The immediate consequence is that truth cannot be
    defined in terms of provability.  In any serious
    intellectual endeavor we shall be able to formulate
    simple mathematical arguments, and thus shall be
    subject to Godel's incompleteness theorem.  However
    far we go in formalizing our canons of proof, we
    shall be able to devise propositions which are not,
    according to those canons, provable, but are none
    the less, true.  So it is one thing to be provable,
    and a different thing to be true.  Truth outruns
    provability.  (J.R. Lucas).

>    This argument won't fly if the set of axioms to which the
>    human mathematician is formally equivalent is too complex
>    for the human to understand. 

What is the basis for the assumption that the intuiting
of mathematical truth is based on a set of axioms, let
alone that they must be too complex to understand?

>    These are amazing claims, which Penrose hardly bothers to
>    defend. Reviewers knowledgeable about Godel's work, however,
>    have simply pointed out that an axiom system can infer that
>    if its axioms are self-consistent, 

An axiom system can infer?

>    then its Godel sentence
>    is true. An axiom system just can't determine its own self-
>    consistency. But then neither can human mathematicians know
>    whether the axioms they explicitly favor (much less the
>    axioms they are formally equivalent to) are self-consistent.
>    Cantor and Frege's proposed axioms of set theory turned out
>    to be inconsistent, and this sort of thing will undoubtedly
>    happen again.''

Agreed.  But we can nonetheless "know" them to be true in
the sense that we all agree we have good reason to believe.

> As to what this has to do with evolution - if humans can
> do things no machine can do - or will ever be able to do -
> that may impact the hypothesis that machine-based organisms
> may replace humans as the dominant life form on earth over
> the next century or so.

More importantly, it would mean that there is reason to
suspect that E. O. Wilson may have gotten it wrong in
asserting genetic determinism:

    Can the cultural evolution of higher ethical values
    gain a direction and momentum its own and completely
    replace genetic evolution?  I think not.  The genes
    hold culture on a leash.  The leash is very long, but
    inevitably values will be constrained in accordance
    with their effects on the human gene pool (E. O.

and that  Dawkins may have actually gotten it right in
asserting the converse:

    We, alone on earth, can rebel against the tyranny of
    the selfish replicators" (Dawkins, 1976, p. 215).

> However, this particular argment for the qualitative
> superiority of humans is simply wrong - and (IMO) rather
> obviously so for anyone who knows anything about Godel's
> work.

Why then is one of the papers you referenced
written in 2004?  Shouldn't this have all been over and
done with decades ago for a flaw that is so "obvious"?

[quote from Penrose]

   The many arguments that computationalists and
   other people have presented for wriggling around
   Godel's original argument have become known to me
   only comparatively recently; perhaps we act and
   perceive according to an unknowable algorithm,
   perhaps our mathematical understanding is
   intrinsically unsound, perhaps we could know the
   algorithms according to which we understand
   mathematics, but are incapable of knowing the
   actual roles that these algorithms play.  All
   right, these are logical possibilities.  But are
   they really plausible explanations?

   For those who are wedded to computationalism,
   explanations of this nature may indeed seem
   plausible.  But why should we be wedded to
   computationalism?  I do not know why so many
   people seem to be.  Yet, some apparently hold to
   such a view with almost religious fervour.
   (Indeed, they may often resort to unreasonable
   rudeness when they feel this position to be
   threatened!)  Perhaps computationalism can indeed
   explain the facts of human mentality -- but perhaps
   it cannot.  It is a matter for dispassionate
   discussion, and certainly not for abuse!

   I find it curious, also, that even those who argue
   passionately may take for granted that
   computationalism in some form -- at least for the
   objective physical universe -- HAS to be correct.
   Accordingly, any argument which seems to show
   otherwise MUST have a "flaw" in it.  Even Chalmers,
   in his carefully reasoned commentary, seeks out
   "the deepest flaw in the Godelian arguments".
   There seems to be the presumption that whatever
   form of the argument is presented, it just HAS
   to be flawed.  Very few people seem to take
   seriously the slightest possibility that the
   argument might perhaps, at root, be correct!
   This I certainly find puzzling;.

   Nevertheless, I know of many who (like myself) do
   find the simple "bare" form of the Godelian
   argument to be very persuasive.  To such people,
   the long and sometimes tortuous arguments that I
   have provided in 'Shadows of the Mind' may not
   add much to the case -- in fact, some have told
   me that they think that they effectively weaken
   it!  It might seem that if I need to go to
   lengths such as that, the case must surely be a
   flimsy one.  (Even Feferman, from his own
   particular non-computational standpoint, argues
   that my diligent efforts may be "largely
   wasted!)  Yet, I would claim that some progress
   has been made.  I am struck by the fact that none
   of the present commentators has chosen to dispute
   my conclusion G (in 'Shadows', p. 76) that "Human
   mathematicians are not using a knowably sound
   algorithm in order to ascertain mathematical
   truth".  (Roger Penrose, 'Psyche' Vol 2)


                             Rationology 101
How the Author of Genesis Got It Right (and the Golden Rule Got It Wrong)