cheers Bruno. :)

>From: Bruno Marchal <[EMAIL PROTECTED]>
>Subject: Re: Penrose and algorithms
>Date: Sat, 9 Jun 2007 18:40:50 +0200
>Hi Chris,
>Le 09-juin-07, à 13:03, chris peck a écrit :
> >
> > Hello
> >
> > The time has come again when I need to seek advice from the
> > everything-list
> > and its contributors.
> >
> > Penrose I believe has argued that the inability to algorithmically
> > solve the
> > halting problem but the ability of humans, or at least Kurt Godel, to
> > understand that formal systems are incomplete together demonstrate that
> > human reason is not algorithmic in nature - and therefore that the AI
> > project is fundamentally flawed.
> >
> > What is the general consensus here on that score. I know that there
> > are many
> > perspectives here including those who agree with Penrose. Are there any
> > decent threads I could look at that deal with this issue?
> >
> > All the best
> >
> > Chris.
>This is a fundamental issue, even though things are clear for the
>logicians since 1921 ...
>But apparently it is still very cloudy for the physicists (except
>I have no time to explain, but let me quote the first paragraph of my
>Siena papers (your question is at the heart of the interview of the
>lobian machine and the arithmetical interpretation of Plotinus).
>But you can find many more explanation in my web pages (in french and
>in english). In a nutshell, Penrose, though quite courageous and more
>lucid on the mind body problem than the average physicist, is deadly
>mistaken on Godel. Godel's theorem are very lucky event for mechanism:
>eventually it leads to their theologies ...
>The book by Franzen on the misuse of Godel is quite good. An deep book
>is also the one by Judson Webb, ref in my thesis). We will have the
>opportunity to come back on this deep issue, which illustrate a gap
>between logicians and physicists.
>------ (excerp of "A Purely Arithmetical, yet Empirically Falsifiable,
>Interpretation of Plotinus¹ Theory of Matter" Cie 2007 )
>1) Incompleteness and Mechanism
>There is a vast literature where G odel¹s first and second
>incompleteness theorems are used to argue that human beings are
>different of, if not superior to, any machine. The most famous attempts
>have been given by J. Lucas in the early sixties and by R. Penrose in
>two famous books [53, 54]. Such type of argument are not well
>supported. See for example the recent book by T. Franzen [21]. There is
>also a less well known tradition where G odel¹s theorems is used in
>favor of the mechanist thesis. Emil Post, in a remarkable anticipation
>written about ten years before G odel published his incompleteness
>theorems, already discovered both the main ³G odelian motivation²
>against mechanism, and the main pitfall of such argumentations [17,
>55]. Post is the first discoverer 1 of Church Thesis, or Church Turing
>Thesis, and Post is the first one to prove the first incompleteness
>theorem from a statement equivalent to Church thesis, i.e. the
>existence of a universal‹Post said ³complete²‹normal (production)
>system 2. In his anticipation, Post concluded at first that the
>mathematician¹s mind or that the logical process is essentially
>creative. He adds : ³It makes of the mathematician much more than a
>clever being who can do quickly what a machine could do ultimately. We
>see that a machine would never give a complete logic ; for once the
>machine is made we could prove a theorem it does not prove²(Post
>emphasis). But Post quickly realized that a machine could do the same
>deduction for its own mental acts, and admits that : ³The conclusion
>that man is not a machine is invalid. All we can say is that man cannot
>construct a machine which can do all the thinking he can. To illustrate
>this point we may note that a kind of machine-man could be constructed
>who would prove a similar theorem for his mental acts.²
>This has probably constituted his motivation for lifting the term
>creative to his set theoretical formulation of mechanical universality
>[56]. To be sure, an application of Kleene¹s second recursion theorem,
>see [30], can make any machine self-replicating, and Post should have
>said only that man cannot both construct a machine doing his thinking
>and proving that such machine do so. This is what remains from a
>reconstruction of Lucas-Penrose argument : if we are machine we cannot
>constructively specify which machine we are, nor, a fortiori, which
>computation support us. Such analysis begins perhaps with Benacerraf
>[4], (see [41] for more details). In his book on the subject, Judson
>Webb argues that Church Thesis is a main ingredient of the Mechanist
>Thesis. Then, he argues that, given that incompleteness is an easy‹one
>double diagonalization step, see above‹consequence of Church Thesis,
>G odel¹s 1931 theorem, which proves incompleteness without appeal to
>Church Thesis, can be taken as a confirmation of it. Judson Webb
>concludes that G odel¹s incompleteness theorem is a very lucky event
>for the mechanist philosopher [70, 71]. Torkel Franzen, who
>concentrates mainly on the negative (antimechanist in general) abuses
>of G odel¹s theorems, notes, after describing some impressive
>self-analysis of a formal system like Peano Arithmetic (PA) that :
>³Inspired by this impressive ability of PA to understand itself, we
>conclude, in the spirit of the metaphorical ³applications² of the
>incompleteness theorem, that if the human mind has anything like the
>powers of profound self-analysis of PA or ZF, we can expect to be able
>to understand ourselves perfectly². Now, there is nothing metaphorical
>in this conclusion if we make clear some assumption of classical
>(platonist) mechanism, for example under the (necessarily non
>constructive) assumption that there is a substitution level where we
>are turing-emulable. We would not personally notice any digital
>functional substitution made at that level or below [38, 39, 41]. The
>second incompleteness theorem can then be conceived as an ³exact law of
>psychology² : no consistent machine can prove its own consistency from
>a description of herself made at some (relatively) correct substitution
>level‹which exists by assumption (see also [50]). What is remarkable of
>course is that all machine having enough provability abilities, can
>prove such psychological laws, and as T. Franzen singles out, there is
>a case for being rather impressed by the profound self-analysis of
>machines like PA and ZF or any of their consistent recursively
>enumerable extensions 3. This leads us to the positive‹open minded
>toward the mechanist hypothesis‹ use of incompleteness. Actually, the
>whole of recursion theory, mainly intensional recursion theory [59],
>can be seen in that way, and this is still more evident when we look at
>the numerous application of recursion theory in theoretical artificial
>intelligence or in computational learning theory. I refer the reader to
>the introductory paper by Case and Smith, or to the book by Osherson
>and Martin [14] [46]. In this short paper we will have to consider
>machines having both provability abilities and inference inductive
>abilities, but actually we will need only trivial such inference
>inductive abilities. I call such machine ³L obian² for the proheminant
>r-ole of L ob¹s theorem, or formula, in our setting, see below. Now,
>probably due to the abundant abuses of G odel¹s theorems in philosophy,
>physics and theology, negative feelings about any possible applications
>of incompleteness in those fields could have developed. Here, on the
>contrary, it is our purpose to illustrate that the incompleteness
>theorems and some of their generalisations, provide a rather natural
>purely arithmetical interpretation of Plotinus¹ Platonist, non
>Aristotelian, ³theology² including his ³Matter Theory². As a theory
>bearing on matter, such a theory is obviously empirically falsifiable :
>it is enough to compare empirical physics with the arithmetical
>interpretation of Plotinus¹ theory of Matter. A divergence here would
>not refute Plotinus, of course, but only the present arithmetical
>interpretation. This will illustrate the internal consistency and the
>external falsifiability of some theology.

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