Le 07-nov.-06, à 20:10, Tom Caylor a écrit :

> > Brent Meeker wrote: >> Tom Caylor wrote: >>> Brent Meeker wrote: >>>> Tom Caylor wrote: >>>>> Bruno has tried to introduce us before to the concept of universes >>>>> or >>>>> worlds made from logic, bottom up (a la constructing elephants). >>>>> These >>>>> universes can be consistent or inconsistent. >>>>> >>>>> But approaching it from the empirical side (top down rather bottom >>>>> up), >>>>> here is an example of a consistent structure: I think you assume >>>>> that >>>>> you as a person are a structure, or that you can assume that >>>>> temporarily for the purpose of argument. You as a person can be >>>>> consistent in what you say, can you not? Given certain assumptions >>>>> (axioms) and inference rules you can be consistent or inconsistent >>>>> in >>>>> what you say. >>>> Depending on your definition of consistent and inconsistent, there >>>> need not be any axioms or inference rules at all. If I say "I'm >>>> married and I'm not married." then I've said something inconsistent >>>> - regardless of axioms or rules. But *I'm* not inconsistent - just >>>> what I've said is. >>>> >>>>> I'm not saying the what you say is all there is to who >>>>> you are. Actually this illustrates what I was saying before about >>>>> the >>>>> need for a "reference frame" to talk about consistency, e.g. "what >>>>> you >>>>> say, given your currently held axioms and rules". >>>> If you have axioms and rules and you can infer "X and not-X" then >>>> the axioms+rules are inconsistent - but so what? Nothing of import >>>> about the universe follows. >>>> >>> >>> Yes, but if you see that one set of axioms/rules is inconsistent with >>> another set of axioms/rules, then you can deduce something about the >>> possible configurations of the universe, but only if you assume that >>> the universe is consistent (which you apparently are calling a >>> category >>> error). A case in point is Euclid's fifth postulate in fact. By >>> observing that Euclidean geometry is inconsistent with non-Euclidean >>> geometry (the word "observe" here is not a pun or even a metaphor!), >>> you can conclude that the local geometry of the universe should >>> follow >>> one or the other of these geometries. >> >> No, you are mistaken. You can only conclude that, based on my >> methods of measurement, a non-Euclidean model of the universe is >> simpler and more convenient than an Euclidean one. >> >>> This is exactly the reasoning >>> they are using in analyzing the WIMP observations. >> >> The WIMP observations are consistent with a Euclidean >> model...provided you change a lot of other physics. >> >>> Time and again in >>> history, math has been the guide for what to look for in the >>> universe. >>> Not just provability (as Bruno pointed out) inside one set of >>> axioms/rules (paradigm), but the most powerful tool is generating >>> multiple consistent paradigms, and playing them against one another, >>> and against the observed structure of the universe. >> >> Right. As my mathematician friend Norm Levitt put it,"The duty of >> abstract mathematics, as I see it, is precisely to expand our >> capacity for hypothesizing possible ontologies." >> > > This quote is basically what I've been trying to get at. The possible > ontologies are the multiple self-consistent paradigms that I was > referring to. When we keep finding that "using abstract math to > hypothesize" actually works in guiding us correctly to what to look > for, then we have to start believing that there's got to be some kind > of truth to math that is greater than trivial self-consistent logical > inference. I think this is what Bruno is getting at with the border > between G (provable truth) and G* (provable and unprovable truth). > Math helps us find not just G, but we can also explore the border of G > and G*. Yes. Note that a lobian machine M1 can *deduce* the G and the G* corresponding to a simpler lobian machine M2, but can only "infer" or "hope" or "fear" ... about its own G*. Remark: I recall for others that G is the modal logic which axiomatizes completely the self-referential provable discourse of "sufficiently powerful classical proving machine", and G* formalize completely (at some level) the true discourse (the provable one and the inferable one). It corresponds to the third person point of view (the second hypostase of Plotinus). G is the discursive, G* is the "divine" one (true). The main axiom of G is B(Bp -> p) -> Bp" and its arithmetical interpretation is lob theorem. Exercise: deduce from Godel's theorem it (I have already answer it but ask if you don't find the answer). B represents here Godel's provability predicate: Godel's theorem = ~Bf -> ~B(~Bf) (If the false is not provable, then that fact itself is not provable). >> Agreement would be great. But the proof of scientific pudding is >> predicting something suprising that is subsequently confirmed. >> >> Brent Meeker Tom: > I would like to hear Bruno's thoughts on comp with respect to > prediction of global aspects such as geometry, as I brought up in the > above paragraph from a previous post. A sort of physical geometry should arise from the Bp & Dp (& p) povs. Mathematical geometry can occur through Minkowski geometry of numbers, or even just the traditional cartesian mapping between numbers and geometry. > > Also, a thought comes to mind that Bruno once said something about > reality (physics, sensations?) arising from our ignorance of the > absolute border between G and G*. Recall that the gap between G and G*, i.e. between proof and truth (by and about the machine) is lifted to "intelligible matter (Bp & Dp)" and "sensible matter (Bp & Dp & p)", that is the "probability 1" logic define in G. I will say more later. > This brings me back to the analogy > of the Mandelbrot set. We can never know the actual absolute border of > the Mandelbrot set. If we were asked to point out even one > (non-trivial) point on the complex plain that is exactly on the border, > we wouldn't be able to do it. However, if we take a finite number of > iterations of the recursive equation, we get a definite border, which > is an approximation. We get an actual instantiation/shape we can > interact with, something that kicks back, something akin to reality. > Again, as I originally proprosed, perhaps this is like the measurement > process in physics, or even the process of setting your coffee cup on > the table... In my view the table doesn't kick back any more than the > Mandelbrot set. I completely agree with you. Actually I have a precise argument that the Mandelbrot set is "turing universal", and Blub Shub and Smale give a proof of this in the frame of the theory of the partial recursive functions defined on a ring (like R or C). Bruno 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 everything-list@googlegroups.com 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 -~----------~----~----~----~------~----~------~--~---