indeed. there are a) misperceptions b) perceptions c) lack of perceptions d) impossibility of perception e) pseudo-perceptions.
It is interesting to check out what Penrose is talking about when he talks about Fashion, Faith, and Fantasy in theoretical physics. Fashion: String Theory Faith: Quantum Mechanics at all levels Fantasy: Inflationary Cosmology and other wild cosmological schemes On Jun 7, 8:01 am, "Stephen Paul King" <[email protected]> wrote: > Dear Bruno, > > From: Bruno Marchal > Sent: Monday, June 06, 2011 9:00 AM > To: [email protected] > Subject: Re: Mathematical closure of consciousness and computation > Hi Stephen, > > On 06 Jun 2011, at 05:27, Stephen Paul King wrote: > > Hi Bruno, Rex and Friends, > > My .002$... > > [BM] > No theories nor machine can reach all arithmetical truth, but few > people doubt that closed arithmetical propositions are either true or > false. We do share a common intuition on the nature of arithmetical > truth. > I have doubt on any notion of global mathematical truth. Sets, real > numbers, complex numbers, etc. are simplifications of the natural > numbers. They are convenient fictions, I think. If we are machine, it > is undecidable if ontology is more than N. > > [SPK] > > I think that there is some differences in opinion about this but it > seems to me that we need to look at some details. For example, there should > exist a theory that could reach all arithmetic truth given an eternity of > time or an unnamable number of recursions or steps. > [BM] > ? > > No this cannot exist. It is precluded by the incompleteness theorem. Eternity > can't help. Unless you take a non axiomatisable theory, or some God-like > entities. > > [SPK] > > Yes, you are correct. I miswrote. I had even developed an informal proof > of this in my critique of Leibniz’ Monadology. But this still presents a > challenge.. Umm, maybe this is where Cantor et al considered this idea in > terms of unnamable cardinals... > ** > > This by definition would put them forever beyond human (finite entity) > comprehension. Whether or not there is closure or a closed form of some > theory does not make it realistic or not. AFAIK, closed arithmetic > propositions are tautologies, no? > [BM] > They are not tautologies, unless you mean by this "propositions true in *all* > models of Peano Arithmetic. But then "tautology" means "theorem", and that > would be an awkward terminology. Ax(0 ≠ s(x)) is not a tautology (it is > already false in (Z,+), nor is Fermat last theorem. > > [SPK] > Yes, I did mean it that way, as in “propositions that are true in *all* > models” but not just of Peano Arithmetic. I was considering all Arithmetics, > especially Robinson’s. Usually one thinks of tautologies as A = A. What I am > trying to weaken is the way that the so called law of identity is usually > defined. I am working toward a notion of equivalence that allows for not just > strict equality but a more general notion of “bisimilarity”. In this way > theorems would be tautologies in this weaker form of Identity. > ** > > That we share a common intuition of truth may follow from a common local > measure of truth within each of us. (Here the "inside" implied by the word > "within" is the logical/Arithmetic/abstract aspect of the duality that I > propose.) > Additionally, we should be careful not to conflate a plurality of > fungible individuals with a multiplicity of non-fungible entities. We can set > up a mental hall of mirrors and generate an infinite number of self-images in > it, but this cannot *exactly* map to all of the selves that could exist > without additional methods to break the symmetries. > > I have been waiting a long time for you to state this belief of yours, > Bruno! That "Sets, real numbers, complex numbers, etc." are simplifications > of (mappings on/in?) the Natural Numbers. This seems to be the Pythagorean > doctrine that I suspected that you believed. > [BM] > Would you take the time to study the papers, you would have understood that > this is a result of comp. Comp transforms the very banal arithmetical realism > in an authentic Pythagorean neoplatonist theology, i.e. with some use of > OCCAM razor. > > [SPK] > I am studying the papers, but I need to clarify some ideas by asking > questions to the Professor. ;-) I do not think the way you do and must > translate your mental language into my own to understand them. > ** > > It has a long history and a lot of apostles that have quite spectacular > histories. I think that there is a deep truth in this belief, but I think > that it needs to be more closely examined. > [BM] > It can be derived from Church thesis and the assumption that "we" are Turing > emulable. > > [SPK] > OK, but would you allow me to say that it seems that you are considering > a form of Turing emulation that is vastly more sophisticated and subtle than > the purely mechanical one that Turing, for example, considered with his A > machines? The fact that you are considering infinities of computations as > “running” each instance of us, is pushing the idea of a recursive algorithm > into places it is never been before. > ** > > > > > Perhaps there is just human belief. > > [BM] > Jason said it. If you follow that slope you may as well say that there > is only belief by Rex. You can also decide that there is nothing to > explain, no theories to find, and go walking in the woods. Science, by > definition, assumes something beyond (human) belief. > > [SPK] > > I admit that I laughed out loud at this! Good point, Bruno! The > reduction of all truth to that which can be defined within a single human's > belief trivializes and renders it meaningless. That is one of the absurd > consequences that we lambast solipsism for, but I think that Rex should not > be to swiftly dismissed form maybe trying to make a deeper observation; he > has brought up a very good topic for discussion. > While it is absurd to reduce all truth to what a single finite entity > can "compute" - which is that we are actually saying if we follow the > Kleene-Turing-Church-Post road - > > [BM] > Careful. It can be said that all ontological truth will be generated, but the > epistemological truth will never be generated, but they will emerge in a not > completely computable way. Remember that arithmetic, seen from inside, is > *much* bigger than arithmetic see from outside. > > [SPK] > OK, but that poses a difficult problem because it is epistemological > truth that we consider as reality! What we “know” to be true, even by the > Bp&p definition, is by definition what is “real” of us individually and via > consensus, no? I am not understanding what you mean by “arithmetic seen from > the outside”. Are you saying that there is more to Existence than numbers? My > apologies, I am confused. > ** > > we are actually positing that "all truths can be defined in terms of N -> N > mappings". > > [BM] > That would contradict Gödel's incompleteness. Unless you mean *all* N -> N > mappings, which is far to bigger and trivialize the theory (making it non > testable, and unable to derive anything in physics, cognitive science, etc.). > > [SPK] > Not provable truths, just the ones that we can bet on. Yes, to extend to > *all* N->N mappings would be like what we see in superstring theory – the > landscape that has almost completely reduced SUSY to a Scholastic type of > theory.... > ** > > Many such mappings to be sure, but N to N mappings nonetheless. We are back > to that strange belief that Bruno explicitly, albeit inadvertently, stated. > But this is not really a "strange" belief, partly because it seems to > be almost universally the default postulate within the basket of beliefs that > people operate with in our every day world. I would like to pose the question > of whether or not we are inadvertently painting ourselves into a corner with > this belief. IT seems to me, and this is just a personal prejudice of mine, > that there exists truths that cannot be named or represented exactly in terms > of N->N maps. > [BM] > In this context, you should clearly stated if you take all N->N mappings, or > the total computable one, or the partial computable one. The non triviality > of comp entirely resides in such nuances. > > [SPK] > I do not know yet how to do that parse. I am still learning the > vocabulary. My apologies. > ** > > The source of this suspicion comes from what I have studied of G. Cantor's > work on transfinites and the histrionics of practitioners of mathematical > logic that have been examining the nature of cardinalities. > [BM] > With comp, the diagonalization of Kleene gives the information. Cantor's one > are far too crude. > > [SPK] > What text might you suggest that I study to understand Kleene’s > diagonalization? I have only found this paper on the > topic:www.columbia.edu/~hg17/naming-diag.pdf > ** > > Additionally there is my belief that the Totality of Existence must be, at > least, Complete (not in the Gödel sense of just 1st order logics), Bicomplete > (in the Category theory sense) and Closed (in the topological sense). This > implies the existence of unnamable truths, or at least Truths that cannot be > exactly represented in terms of recursive functions on the Integers. > > [BM] > That the totality of existence is complete seems to me to be a tautology, or > a truth by definition. > That truth is beyond machine's means, is a theorem. > That truth is beyond us is consequence of such theorem when we assume that we > are machines. > > [SPK] > Umm, not quite the same idea. I am following the reasoning in these > papers:http://arxiv.org/abs/math/0307090,http://arxiv.org/abs/physics/0212092and > my own ideas formed from studying Category theory. The difficulty that I see > in your definitions is that it makes the notion of a machine into something > altogether unknowable. People seem to interpret your word machine in the same > way that, for example, Descartes considered the idea of automata. I made that > mistake myself until I saw that you where not considering the notion of a box > full of levers, springs, gears and widgets. > ** > > The question becomes one of the implications of this on our > metaphysical assumptions about the ontologies that we are using in our > thinking about the issue of mathematical closure of computation and > consciousness. As I see it, and this very well could be just an eccentric > thought, is that we need to be very careful that we do not tacitly assume > that all of the minds of entities are replicating the same ideas as one’s > own. The fact that we are continuously > ... > > read more » -- 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.
