Re: Which mathematical structure -is- the universe in Physics?
On Apr 25, 5:27 am, Bruno Marchal <[EMAIL PROTECTED]> wrote: > Le 24-avr.-08, à 18:26, nichomachus a écrit : > > > > > > > > > On Apr 22, 11:28 pm, "Brian Tenneson" <[EMAIL PROTECTED]> wrote: > >> Perhaps Hilbert was right and Physics ought to have been axiomatized > >> when he > >> suggested it. ;) Then again, there might not have been a motivation > >> to > >> until recently with Tegmark's MUH paper and related material (like by > >> David > >> Wolpert of NASA). > > > The logical positivists were motivated to axiomatize in the predicate > > calculus the laws of scientific theories in the early 20th century, > > first because they believed that it would guarantee the cognitive > > significance of theoretical terms in the theory (such as the > > unphysical ether of maxwell's electromagnetism), and then later > > because it had evolved into an attempt to specify the proper form of a > > scientific theory. In practice this had too many problems and was > > eventually abandoned. One of the consequences of this program was that > > axiomatizing the laws of a theory in first order predicate calculus > > with equality was that such a formulation of a theory always implied > > various unintended interpretations. The amount of effort needed to > > block these unintended interpretations was out of proportion with the > > benefit received by axiomatization. > > It is a bit weird because it is just logically impossible to block > those unintended interpretations. And This should not be a problem. > The reason why physical theories are not axiomatize is more related to > the fact that axiomatization does not per se solve or even address the > kind of conceptual problem raised by physics. Also to this point, that it is impossible to identify a theory with any particular linguistic formulation of it. Theories are not linguistic entities. And since we’re on the subject: according to Max Tegmark, given the apparent direction of inter-theoretic reduction, one may assume that the foundational physics of our universe should be able to be expressed in a completely “baggage-free” description that is without reference to any human-specific concepts. This presumed most basic law of the universe would be capable of being axiomatized without unintended implications since the mathematical structure expressing the most basic law would be isomorphic with the law itself to the degree that it may appropriately be identified with it. The mathematical laws which describe the phenomena of all of the emergent levels or organization diverge from this ideal more and more the further one proceeds from this unknown foundational theory. > > Also, I > > personally remain unconvinced that there is anything problematic about > > the exitence of the universe of universes, or the ensemble of all > > possible mathematical structures, thought it may not be well defined > > at present. I don't believe that this is simply the union of all > > axiomatic systems. If trying to define the Everything as a set implies > > a contradiction, then fine -- it isn't a set, it's an ensemble, which > > doesn't carry any of the connotations that are implied by the use of > > "set" in the mathematical sense. Therefore each entity in the ensemble > > is a unique collection of n axioms that has no necessary relationship > > to any other axiom collection. What happens in an axiom system stays > > in that axiom system, and can't bleed over to the next one on the > > list. Some of these may be equivalent to each other. > > > A = The collection of all finite axiom systems > > B = The collection of all consistent finite axiom systems > > I guess you mean "recursively enumerable" instead of finite. You would > loose first order Peano Arithmetic (my favorite lobian machine :). Really? It would seem that all recursively enumerable axiom systems would exist in A. > Note also that SAS occurs very quickly. SAS occur in theories which are > much weaker than the SAS themselves (ex: SAS occur in Robinson > Arithmetic, i.e. when you can define successor, addition and > multiplication. SAS themselves need induction. I don’t understand. Are you saying that Self Aware Substructures exist in the Robinson Arithmetic? --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: Which mathematical structure -is- the universe in Physics?
On Apr 24, 12:08 pm, "Brian Tenneson" <[EMAIL PROTECTED]> wrote: > I was attempting to -invalidate- that argument against the existence of the > universe, actually, by saying that in three truth values, which the > Physicists can't rule out as being the more accurate logic of their > universe, the argument "reductio ad absurdum" is not a tautology and, > therefore, can't necessarily be applied. > > However, in binary logic, the Physicist's universe (or whatever Everything > means) can't exist. I take your point about the reductio not working in three valued logic. I am not convinced that the problem you are attempting to solve is necessarily a problem since I haven’t been able to construe the proposed reductio ad absurdum argument in a way that seems coherent to my way of thinking. However, you may be on to something with the general program that you have embarked upon. Maybe there is a need for a mathematics to describe the everything ensemble. Something along those lines is likely the only way to define the everything with any sort of rigor. I think it is a good idea. Set theory does seem to be too rich for the job. Determining what type of formalism is apropriate is a task. I think that such a mathematical formalism may be precisely what is called for in order to define the everything, as well as analyze it any useful sort of way. I am still confused by what you mean by certain terms. What is meant by the Physicist’s universe? Even more to the point, what is meant by saying that it cannot exist in binary logic? The propositional calculus, for example, does not even satisfy the conditions the Godel theorems, i.e. there are no undecidable propositions possible in it. To think that the axioms of any two valued logic could be sufficient to produce a physical existence for self-aware substructures is distinctly overstepping what Max Tegmark suggests in his metaphysical theory. > > I doubt self-reference is inherently the problem in light of things like > Tarski's fixed point theorems which provide concrete examples of wffs that > are self-referencing, in terms of Godel numbers, if I recall. That proof I > was exposed to was not an existence proof of self-referencing wffs merely by > "logical flamboyancy" but by the providing an example of an actual -class- > of self-referencing wffs. Obviously, the above argument does not explicitly > involve wffs (it does, however, implicitly), and I am -only- making a case > for plausibility at this particular moment. > > I see no problems with the argument given that in binary logic, their > universe can't exist; this, to me, convinces me that the Physicist's > universe can't operate on binary logic by Occam's Razor as -none- of the > data in any experiment would fit the result that confirms their speculation > that their universe exists. > > Ergo, the Physicist's universe must operate on at least three truth values. > (Consequently, it exists.) This to me is a more elegant solution to the > argument than citing self-referencing issues as automatically damning. If > natural language can be used to prove the Heine-Borel theorem, without the > need for wffs, then why must a statement about Everything be formalized in > machine-level code with wffs? > > If there is further objection to my line of thinking, -please- point it out > to Everyone (which I hope is well-defined or else no one would know what I > mean, right?) ;) > > Thank you for your remarks; I find all input extremely productive!! I too appreciate the chance to talk about such interesting ideas. --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: Quantum Immortality = no second law
> The focus of my paper is on theories in principle fully describing universes > (or u-reality). The term 'logically possible' is intended to contrast with > 'physically possible' and refers to descriptions (theories) being internally > non-contradictory (more in note 4 in my paper). OK >Classical logic is usually > intended in these kinds of cases, and I can't actually see from what I know > of other logics how they might relevantly extend the range of possible > inhabitable universes beyond those describable by formal systems operating > according to classical logic. (There is also the issue of their additional Brent already mentioned paraconsistent logics, here a nice link: http://en.wikipedia.org/wiki/Impossible_world (the article links to an article by Zalta, the one who is responsible for this http://plato.stanford.edu/ wonderful resource. Cheers, Günther --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: Which mathematical structure -is- the universe in Physics?
I think we have no choice in the matter (once we assume the "unbelievable comp hyp."). The physical is not just a mathematical structure among others. The physical emerged from a sort of sum pertaining on the whole of the mathematical possible histories. If this does not give the empirical physics, then comp will be refuted. But preliminary results give already a sort of quantum topology. The one I have more or less extracted from the comp hyp, at the modest propositional level, has not yet been prove to be be equivalent to universal quantum topology, but they are clues indicating that comp could be the promising path. It is quasi obvious that comp entails many consistent histories, and the math gives reasons why such histories interferes statistically in a "quantum way", i.e. with a perpendicularity relation on the possible incompatible states/stories. Ah yes the "truly" parallel realities are perpendicular, but this is already the case with quantum mechanics and its "scalar product". What is hard, and on which I am stuck since years is to find the (arithmetical) needed tensor product, or how does a first person plural reality occur. Mathematically it is enough to assume at some place a linearity condition. But this is cheating; we have to justify that linearity from comp only, as comp justifies we have to do. Sorry if I am a bit short. bruno In the sense of David Wolerpt's (of NASA) omniscient devices and oracles, I think a theorem is this: Inconsistency in some sense (like answering a question as neither yes nor no, but something like MU in Eastern thought), is a -necessary- condition for omniscience. Or, phrased differently, omniscience implies inconsistency. In a -binary- logical universe, that is. What is "This"? --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: Which mathematical structure -is- the universe in Physics?
Le 24-avr.-08, à 18:26, nichomachus a écrit : > > On Apr 22, 11:28 pm, "Brian Tenneson" <[EMAIL PROTECTED]> wrote: >> Perhaps Hilbert was right and Physics ought to have been axiomatized >> when he >> suggested it. ;) Then again, there might not have been a motivation >> to >> until recently with Tegmark's MUH paper and related material (like by >> David >> Wolpert of NASA). > > The logical positivists were motivated to axiomatize in the predicate > calculus the laws of scientific theories in the early 20th century, > first because they believed that it would guarantee the cognitive > significance of theoretical terms in the theory (such as the > unphysical ether of maxwell's electromagnetism), and then later > because it had evolved into an attempt to specify the proper form of a > scientific theory. In practice this had too many problems and was > eventually abandoned. One of the consequences of this program was that > axiomatizing the laws of a theory in first order predicate calculus > with equality was that such a formulation of a theory always implied > various unintended interpretations. The amount of effort needed to > block these unintended interpretations was out of proportion with the > benefit received by axiomatization. It is a bit weird because it is just logically impossible to block those unintended interpretations. And This should not be a problem. The reason why physical theories are not axiomatize is more related to the fact that axiomatization does not per se solve or even address the kind of conceptual problem raised by physics. > > >> >> I was trying to answer Bruno's objections regarding set theory being >> too >> rich to be the 'ultimate math' the MUH needs to propose what the >> universe is >> and I quipped that that was because math is invented or discovered to >> further its own end by logicians, for the most part, and that >> metamathematicians such as Cantor had no apparent interest in physical >> things or furthering the pursuit of Physics. >> >> Another question of Bruno's was my motivation. I started this quest >> hoping >> that three truth values were sufficient to develop a set theory with a >> universal set that was in a classical logic sense consistent to ZFC >> set >> theory. Or, if not true, prove that and figure out why. Perhaps >> more truth >> values would solve that. My main motivation has definitely not been >> to >> "rescue" a major apparent shortcoming in the MUH as I started this >> on-and-off quest in 2003 with no internet connection or resources >> such as a >> deluge of journals (ie, a good library). How it started was that >> someone >> online in a place such as this used Russell-like arguments to -prove- >> that >> the Physic's universe -does not exist- for essentially the same >> reasons a >> universal set can't seem to be non-antimonious. >> >> Suppose Everything is well defined along with its partner, >> containment (such >> as the earth is contained in the solar system by the definitions of >> both). >> Then Everything does not exist. Proof: >> Consider the thing, call it "this something," that is the qualia of >> all >> things that do not contain themselves. >> Then this something contains itself if and only if this something >> does not >> contain itself. > > I am suspect of the claim that a logical argument such as the above, > which relies on a paradox of self-reference, could be used to > demonstrate the non-existence of the so-called Everything. Indeed. It will just prevent the "Everything" to be a thing (to belong to Everything). > Also, I > personally remain unconvinced that there is anything problematic about > the exitence of the universe of universes, or the ensemble of all > possible mathematical structures, thought it may not be well defined > at present. I don't believe that this is simply the union of all > axiomatic systems. If trying to define the Everything as a set implies > a contradiction, then fine -- it isn't a set, it's an ensemble, which > doesn't carry any of the connotations that are implied by the use of > "set" in the mathematical sense. Therefore each entity in the ensemble > is a unique collection of n axioms that has no necessary relationship > to any other axiom collection. What happens in an axiom system stays > in that axiom system, and can't bleed over to the next one on the > list. Some of these may be equivalent to each other. > > A = The collection of all finite axiom systems > B = The collection of all consistent finite axiom systems I guess you mean "recursively enumerable" instead of finite. You would loose first order Peano Arithmetic (my favorite lobian machine :). Note also that SAS occurs very quickly. SAS occur in theories which are much weaker than the SAS themselves (ex: SAS occur in Robinson Arithmetic, i.e. when you can define successor, addition and multiplication. SAS themselves need induction. > > The "cardinality" of B is not greater than the "cardin
