cheers Bruno. :)
>From: Bruno Marchal <[EMAIL PROTECTED]> >Reply-To: [EMAIL PROTECTED] >To: [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 >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/ > > >> _________________________________________________________________ Win tickets to the sold out Live Earth concert! http://liveearth.uk.msn.com --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. 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