if reality was known, it wouldn't have to be stated... unless there was a mis-perception that needed to be corrected.... hence our theorem tautologies are evidence that reality is not known.... otherwise it would not need to be doubly and secondly stated for assurance and clarification.... that is a cover, we convice ourselves that our tautologies are true, but the only reason they are called for is because things are obviously unknown.

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"We accept the real so readily only because we sense that reality does not exist" Jorge Luis Borges "Last night I had a dream about reality. It was such a relief to wake up." Stanislaw J. Lec On Jun 7, 8:01 am, "Stephen Paul King" <stephe...@charter.net> wrote: > Dear Bruno, > > From: Bruno Marchal > Sent: Monday, June 06, 2011 9:00 AM > To: everything-list@googlegroups.com > 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 everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.