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 
Hofstadter!).

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.

Best,

Bruno


------ (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.

....

------





http://iridia.ulb.ac.be/~marchal/


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