Hi Chris, Le 09-juin-07, à 13:03, chris peck a écrit :

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> > 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 universalPost said ³complete²normal (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 easyone double diagonalization step, see aboveconsequence 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 levelwhich 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 positiveopen 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/ --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---