Re: Which mathematical structure -is- the universe in Physics?
Colin Hales wrote: Hi Brian, I was wondering if you could connect (in the paper) the maths with our universe? As an example. What set operations or structures correspond to the standard particle model entities, what constitutes a chemical reaction or energy, what space is made of... that kind of thing. Maybe this is supposed to be obvious... if it is then sorry... but you've lost (as an audience) the entire world above mathematical physics...especially biofolks. I think this is what usual theoretical physics is trying to do. As an example, Torgny mentioned R^4 as being a relevant structure in General Relativity. In string theory, the relevant structure is, as far as I've read in the lay literature, some 11 (give or take) dimensional manifold. As such, connecting the ultimate context structure I am working towards to specific structures that represent things like particle interaction would constitute a complete theory of physics and, therefore, I myself am unable to see how this would be done. It perhaps can be done but I lack the knowledge to do so. Perhaps one thing to keep in mind is that this is a step towards a mathematical representation of the so called -level 4- multiverse, by which I'm referencing material here: http://space.mit.edu/home/tegmark/multiverse.html In Tegmark's ultimate ensemble paper, there is a diagram of physics and maths structures, part of which is here: http://space.mit.edu/home/tegmark/toe.gif All structures, including those in the top row, which are the ones I think you're asking about Colin, would have the property of being elementarily embeddable within the ultimate structure I'm investigating. (Keep in mind the deficency I mentioned in my previous post.) Roughly speaking, to quote a wiki article, In model theory http://en.wikipedia.org/wiki/Model_theory, an *elementary embedding* is a special case of an embedding http://en.wikipedia.org/wiki/Embedding#Model_theory that preserves all first-order formulas. In short, the sub-structures, so to speak, of the ultimate structure I am working towards that are relevant to Quantum Field Theory or General Relativity (such as R^4) are covered in other texts. This paper I am working on is to provide an answer to the question which is the subject of this thread, raised my Tegmark. I'm afraid I don't know enough about mathematical physics to be more explicit. I am a quintessentially visual/spatial thinker.. math does not speak very well to me unless I can 'see' the operations happening. in my mind. I don;t manipulate symbols. I manipulate 'stuff' and then retrofit symbols. I would also like to see how an observer with qualia might be constructed of it. In other words...how a universe thus constructed might create its own scientist describing it in the way you doHaving looked at the paper I hold some hope that it might contain a formalism I can use to construct the set theoretic basis of my own model... it might be within yoursmaybe... not sure. This is an excellent line of questioning and one I have high hopes to one day seeing answered. In Tegmark's first of two papers along the lines of a Mathematical Universe, he mentions what he calls Self Aware Structures (SAS's). I have spent a lot of time wondering what type of mathematical structures would have self-awareness. Two candidates that might be just fumbling in the dark are these: David Wolpert of NASA has written some interesting articles on what he calls devices. These devices are mathematical models of scientists plus investigative tools of scientists. In this mathematical device (not completely unlike a Turing machine), it starts with a question and ends with an answer; his papers form a theory of how devices operate. One of his papers is entitled the physical limits of inference. Anyway, he talks at some point about self aware devices, and my understanding is that these devices X are ones who correctly answer the question is X a device? That is at least some form of self awareness. For a pseudo-second example of mathematical self awareness, I was thinking of self-referential first order logical formulas that, in essence, say I have property P, but let P be the property X is a 1st order formula so these special self-referential 1st order formulas would essentially be equivalent to I am this 1st order formula. To simplify, that is like the sentence I am this sentence. The open question is what is the nature of SAS's that corresponds to human self-awareness. I think constructing an observer with qualia mathematically would be a most excellent step and a necessary one to solve that open problem raised in Tegmark's first MUH paper. --~--~-~--~~~---~--~~ 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
Re: Which mathematical structure -is- the universe in Physics?
I believe I have a working candidate for a plausibility case for a structure being literally the universe, assuming the MUH. It is the structure U(U), where the first U is script and the second is blackboard bold, on page 3 of the following document, listed under conjecture 4. http://www.universalsight.org/conference_abstract/00-02-00.pdf --~--~-~--~~~---~--~~ 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 was skimming though a book by Roberto Cignoli, Itala D'Ottaviano, and Daniele Mundici called Algebraic Foundations of Many-Valued Reasoning. Recall that I conjectured that the Physicist's universe has an MV-algebra structure. I probably should have said that the Physicist's universe is the category of all MV-algebras, or some such. In this book I'm studying, I have lifted some facts which might prove interesting when settling my conjecture (which obviously might be as insignificant as the conjecture 0+1=1). From book: Let A be the category of l-groups (lattice-ordered Abelean groups) with a strong distinguished unit. Let M be the category of MV-algebras. (I think a briefer way to say that would be let M be MV-algebra.) OK, now... Chapter 7 of the aforementioned book has as its goal proving the following statement: There is a natural equivalence between A and M, meaning that there is a functor, call it F, between A and M. In other words, between A and M, there is a full, faithful, and dense functor F. Thus another way to state my conjecture is this: The universe is an (or at least has the structure of an) l-group with a strong distinguished unit. Does this ring any bells with physicists? What, physically or observably, is this strong distinguished unit, if so? --~--~-~--~~~---~--~~ 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?
Dear Brian, have you had a look at Universal logic? http://en.wikipedia.org/wiki/Universal_logic Maybe there are points of interest in there for you (the wikipedia article is only a stub, but contains some names to google). Cheers, Günther Brian Tenneson wrote: I was skimming though a book by Roberto Cignoli, Itala D'Ottaviano, and Daniele Mundici called Algebraic Foundations of Many-Valued Reasoning. Recall that I conjectured that the Physicist's universe has an MV-algebra structure. I probably should have said that the Physicist's universe is the category of all MV-algebras, or some such. In this book I'm studying, I have lifted some facts which might prove interesting when settling my conjecture (which obviously might be as insignificant as the conjecture 0+1=1). From book: Let A be the category of l-groups (lattice-ordered Abelean groups) with a strong distinguished unit. Let M be the category of MV-algebras. (I think a briefer way to say that would be let M be MV-algebra.) OK, now... Chapter 7 of the aforementioned book has as its goal proving the following statement: There is a natural equivalence between A and M, meaning that there is a functor, call it F, between A and M. In other words, between A and M, there is a full, faithful, and dense functor F. Thus another way to state my conjecture is this: The universe is an (or at least has the structure of an) l-group with a strong distinguished unit. Does this ring any bells with physicists? What, physically or observably, is this strong distinguished unit, if so? -- Günther Greindl Department of Philosophy of Science University of Vienna [EMAIL PROTECTED] http://www.univie.ac.at/Wissenschaftstheorie/ Blog: http://dao.complexitystudies.org/ Site: http://www.complexitystudies.org --~--~-~--~~~---~--~~ 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?
Great reference, thanks! I'm investigating a problem I can phrase two ways, given that the category of MV-algebras is equivalent to the category of lattice-ordered Abelian groups with a distinguished strong unit. So one way to phrase my question, and I'm guess it has been answered before Take LOAGDSU to be the set of wffs that define lattice-ordered Abelian groups with a distinguished strong unit. Now consider the collection of all models of LOAGDSU. The question I have to anyone who knows is this: How many nonisomophic models can (the completion of) LOAGDSU have? I might be redundant there if the theory generated by the wffs in LOAGDSU is already complete (I haven't looked into that aspect today yet). Either way, for the ease of expression, let's say that my question is this: How many nonisomophic models can LOAGDSU have? Are any of them in an interesting sense non-standard, such as so called nonstandard models of arithmetic can give rise to a 'realm' that all natural numbers are in and other 'things' are in this realm that are in every other way elementarily equivalent to the usual model of (N, +, ., ), yet these unlimited numbers are larger than every standard natural number. Hyper-natural numbers these are called. My point is, do non-standard models of LOAGDSU exist and what is even standard about LOAGDSU that could be pushed into a non-standard line of thought. But the main question is how many non-isomorphic models can LOAGDSU have. In other news, I will try to apply to give a presentation on the promising connections between logic, algebra, and the muh in physics at this conference: http://www.mat.unisi.it/~latd2008/ I just need to concoct the best 2 page abstract I can and submit it. I am crossing my fingers. Back to the point at hand. Asking how many different models LOAGDSU has is in a natural way equivalent to asking how many models MV-algebra has. THat is because of the theorems in chapter 7 of the book referenced in my preceding post about their realization that there is a deep connection between MV-algebras and those certain l-groups. When I think of the 'things', denoted with variables, in an MV-algebra I think those are elements in the truth set. Ex/ in Classical Logic, the cardinality of the truth set is two. When I think of the 'things', also denoted by letters, in these l-groups, I think of groups (which are containers). However, due to the deep and categorical connection between those two systems, and combine that with my suspicion that the universe mentioned in Tegmark's paper about the MUH, I then see 'things' in MV-algebras and l-groups (with equipment) as -worldlines- of other 'things'. These structures, like MV-algebras, provide some of the laws of Physics as they would be under the MUH. So I guess my next peek will be into what a standard model of the theory of MV-algebras is (like) and see if it would be fruitful to investigate nonstandard models of the theory of MV-algebras. Günther Greindl wrote: Dear Brian, have you had a look at Universal logic? http://en.wikipedia.org/wiki/Universal_logic Maybe there are points of interest in there for you (the wikipedia article is only a stub, but contains some names to google). 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?
In an attempt to recruit the help of a friend from school, he writes this in an email in response: quote So, about your question, I've actually never heard of a lattice-ordered abelian group, so I don't think I can help you there. I can tell you about the connection of category theory to physics, though (although you may already know this): when you talk about open string theory (i.e. adding D-branes to the theory), depending on whether you consider the A or B twist, the D branes are supposed to form a derived Fukaya category for the A twist, or a category of derived coherent sheaves on the B twist. In categorical language, the objects are the D branes, and the morphisms are (open) strings stretching between D branes. If you wanted to then make some (tenuous at best) connection to the real universe, assuming that string theory is actually true, since all particles are supposed to be strings (strings are a subset of D branes), this means that theoretically the entire universe could be described by a category of D branes. The problem with this, though, is that D branes are not fully described by even the derived Fukaya/coherent sheaf setup, so before that kind of connection can be made, (1) string theory has to be proven true, (2) a complete mathematical description of D branes has to be worked out. /quote --~--~-~--~~~---~--~~ 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 26-avr.-08, à 06:55, nichomachus a écrit : 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 is vague. Do you think that natural numbers are human-specific concepts? You cannot axiomatize the natural numbers in a way such that it avoids other objects obeying your axioms. Even arithmetical truth (the set of first order true arithmetical propositions seen as a theory) has no standard models. Computability theory/ recursion theory is the best, imo, way to get a human independent, even a machine or formalism independent, mathematics (despite non standardness). ... doubly so with the explicit use of the (classical) Church's thesis. 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. If you say yes to the doctor, accepting a digital brain/body, you identify yourself (your 3-self) locally with a finite linguistic (et least finitely 3-person presentable) structure. 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. This is hard to interpret because I don't know your theoretical background. I say a few more words below. 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 (RE) axiom systems would exist in A. A is ambiguous. Strictly speaking Peano Arithmetic is an axiomatization, in first order predicate logic, of elementary number theory. It contains 3 axioms for the notion of succession, 4 axioms for addition and multiplication, and an infinite (but RE) set of axioms of induction. It is known that we cannot formalize
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 cardinality of A. (Scare qutoes since cardinality is a property of sets and these may not be sets if that would imply contradiction.) It is B that is
Re: Which mathematical structure -is- the universe in Physics?
quote 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 quote 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?
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: 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 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. 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. 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 The cardinality of B is not greater than the cardinality of A. (Scare qutoes since cardinality is a property of sets and these may not be sets if that would imply contradiction.) It is B that is interesting from the point of this discussion since it is believed (I don't know of any proof of this) that only systems in B could produce the type of rational and orderly physical existence capable of containing observers who can think about their existence as we do (SASs, or Self-Aware Substructures). The collection of all those systems capable of containing SASs is the most interesting from the point of view of the present discussion, and must have a cardinality not greater than that of B, since many axiom systems are too simple to contain SAS, and the ones with them are expected to predominate. The idea of this ensemble so propounded does not seem to entail an ad absurdum paradox such as you gave above. Further, didn't I see you say somewhere that you don't even believe in sets? I apologize if I am mistaken, but if that is true, I can't see how that statement would reconcile with sincere belief in the validity of
Re: Which mathematical structure -is- the universe in Physics?
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 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!! On Apr 24, 9:26 am, nichomachus [EMAIL PROTECTED] wrote: 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. 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
Re: Which mathematical structure -is- the universe in Physics?
On Thu, Apr 24, 2008 at 10:08:16AM -0700, Brian Tenneson 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. ... 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!! Isn't the sort of everything you have in mind a bit like omnipotence (which has problems such as creating the immovable object, then moving it). Perhaps such an everything really is logically impossible. The sorts of everything we've discussed here on the list are much more modest beasts - even Tegmark's all mathmatics tends to be viewed in terms of recursive enumerable structures (or finite axiomatic systems). Cheers -- A/Prof Russell Standish Phone 0425 253119 (mobile) Mathematics UNSW SYDNEY 2052 [EMAIL PROTECTED] Australiahttp://www.hpcoders.com.au --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
