Re: Tegmark's TOE Cantor's Absolute Infinity
Hal Finney wrote: I have gone back to Tegmark's paper, which is discussed informally at http://www.hep.upenn.edu/~max/toe.html and linked from http://arXiv.org/abs/gr-qc/9704009. I see that Russell is right, and that Tegmark does identify mathematical structures with formal systems. His chart at the first link above shows Formal Systems as the foundation for all mathematical structures. And the discussion in his paper is entirely in terms of formal systems and their properties. He does not seem to consider the implications if any of Godel's theorem. I still think it is an interesting question whether this is the only possible perspective, or whether one could meaningfully think of an ensemble theory built on mathematical structures considered in a more intuitionist and Platonic model, where they have existence that is more fundamental than what we capture in our axioms. Even if this is not what Tegmark had in mind, it is an alternative ensemble theory that is worth considering. Hal Finney Of course, and I express this point as a footnote to my Occam's razor paper (something to the effect of remaining agnostic about whether recursively enumerable axiomatic systems are all that there is). Somewhere I speculated that these other systems require observers with infinitely powerful computational models relative to Turing machines and that these observers are of measure zero with respect to observers with the same computational power as Turing machines... Cheers A/Prof Russell Standish Director High Performance Computing Support Unit, Phone 9385 6967, 8308 3119 (mobile) UNSW SYDNEY 2052 Fax 9385 6965, 0425 253119 () Australia[EMAIL PROTECTED] Room 2075, Red Centrehttp://parallel.hpc.unsw.edu.au/rks International prefix +612, Interstate prefix 02
Re: Tegmark's TOE Cantor's Absolute Infinity
I'm not so sure that I do perceive positive integers directly. But regardless of that, I remain convinced that all properties of them that I can perceive can be written as a piece of ASCII text. The description doesn't need to be axiomatic, mind you. As I have mentioned, the Schmidhuber ensemble of descriptions is larger than the Tegmark ensemble of axiomatic systems. Cheers Hal Finney wrote: But as an example, how about the positive integers? That's a pretty simple description. Just start with 0 and keep adding 1. From what we understand of Godel's theorem, no axiom system can capture all the properties of this mathematical structure. Yet we have an intuitive understanding of the integers, which is where we came up with the axioms in the first place. Hence our understanding precedes and is more fundamental than the axioms. The axioms are the map; the integers are the territory. We shouldn't confuse them. We have a direct perception of this mathematical structure, which is why I am able to point to it for you without giving you an axiomatic description. Hal Finney A/Prof Russell Standish Director High Performance Computing Support Unit, Phone 9385 6967, 8308 3119 (mobile) UNSW SYDNEY 2052 Fax 9385 6965, 0425 253119 () Australia[EMAIL PROTECTED] Room 2075, Red Centrehttp://parallel.hpc.unsw.edu.au/rks International prefix +612, Interstate prefix 02
Re: Tegmark's TOE Cantor's Absolute Infinity
At 19:08 -0400 29/09/2002, Wei Dai wrote: On Thu, Sep 26, 2002 at 12:46:29PM +0200, Bruno Marchal wrote: I would say the difference between animals and humans is that humans make drawings on the walls ..., and generally doesn't take their body as a limitation of their memory. It's possible that we will never be able to access more than a bounded volume of space. It depends on the cosmology of our universe. The UDA is intended to show that with comp it is the cosmology of our universe which is the result of an average on our unbounded computationnal stories, which are living, statically, in UD* (comp platonia). It is also the difference between finite automata, and universal computers: those ask always for more memory; making clear, imo, the contingent and local character of their space and time bounds. My point is that our inability to compute non-recursive functions is also a contingent bound. It's contingent on us not discovering a non-recursive law of physics. I agree. I think comp implies the existence (from our first plural point of view) of non recursive law of physics. The amazing fact, which would follow---empirically---from freedman NP paper or Calude beyond turing barrier paper, is that such non recursive phenomenon can be exploited. It is of course highly non trivial to show this from comp, although with UDA we know that we must extract this or the negation of this from comp (making comp completely empirically testable). I have read and appreciate a lot of papers by Shapiro. He has edited also the north-holland book Intensionnal Mathematics which I find much interesting than its case for Second-order Logic. It is not very important because, as you can seen in Boolos 93, basically the logic G and G* works also for the second order logic. Only the restriction to Sigma_1 sentences should be substituted by a substitution to PI^1_1 sentences. This can be use latter for showing the main argument in AUDA can still work with considerable weakening of comp, but I think this is pedagogically premature. I guess I'll have to take your word for it. Well, just look Boolos 93 chapter 14 ... (read the definitions, until you understand the enunciation of the theorems and take Boolos word for the proof ...). BTW, you never answered my earlier question of why Arithmetical Realism rather than Set Theoretic Realism. Is is that you don't need more than Arithmetical Realism for your conclusions? What do you personally believe? (I thought I did answer that question once (?)). With comp, arithmetical realism is enough for the basic ontological (different from substancial) basic level. Set theoretic realism can be used, except that I have no idea of what it could mean. That is, I believe that each sentence with the form ExAyEzArEtAuP(x,y,z,r,t,u ...) is true or false when the variable x, y, z, r .. are (positive) integers. And this independently of my ability to know the truth value. I have just no similar belief if the variable are allowed to represent arbitrary sets. Set theory is like group theory, I can be platonist on the groups (= models of group theory) and I can be platonist about the universes-of-sets (models of set theory). Now, if you ask me Is a * b = b * a in group theory, I will answer you by it depends on the group you are talking about. Similarly, if you ask me Is the Cantor Continuum Hypothesis true or false about sets, I will answer that it depends on which set-universe you talk about. If you answer me: Come on, I am talking about the standard model of ZF theory, I am just not sure I can know what you mean by standard. You can only define the word standard in a more doubtful theory. By a sort of miracle---akin to Church thesis---, I have a clear (but admittedly uncommunicable) understanding of the standard model of natural number theories. If you ask me if the prime twin conjecture is false or true, I will just answer that I currently do not know, but that I do find the question meaningful. I have no doubt the twin prime conjecture is true or is false. By the same token, I have no doubt a machine will stop, or not stop, independently of my ability to solve any stopping machine problem. Bruno
Re: Tegmark's TOE Cantor's Absolute Infinity
On Tue, Sep 24, 2002 at 12:18:36PM +0200, Bruno Marchal wrote: You are right. But this is a reason for not considering classical *second* order logic as logic. Higher order logic remains logic when some constructive assumption are made, like working in intuitionist logic. A second order classical logic captures a mathematical structure in a very weak sense. My opinion is that the second order *classical* logics are misleading when seen as logical system. Why not taking at once as axioms the set of all true sentences in the standard model of Zermelo Fraenkel (ZF) set theory, and throw away all rules of inference. This captures, even categorically, the set universe. But it is only in a highly technical sense that such a set can be seen as a theory. If we can take the set of all deductive consequences of some axioms and call it a theory, then why can't we also take the set of their semantic consequences and call it a theory? In what sense is the latter more technical than the former? It's true that the latter may require more computational resources to enumerate/decide (specificly it may require the ability to compute non-recursive functions), but the computability of the former is also theoretical, since currently we only have access to bounded space and time. Logically you are right, and what you said to Brent is correct. I just point here that the use of second order classical logic can be misleading especially for those who doesn't have a good idea of what is a *first order* theory. Some would argue that it's first-order theory that's misleading. See Stewart Shapiro's _Foundations without Foundationalism - A Case for Second-Order Logic_ for such an argument.
Re: Tegmark's TOE Cantor's Absolute Infinity
At 21:36 -0400 21/09/2002, [EMAIL PROTECTED] wrote: For those of you who are familiar with Max Tegmark's TOE, could someone tell me whether Georg Cantor's Absolute Infinity, Absolute Maximum or Absolute Infinite Collections represent mathematical structures and, therefore have physical existence. Hi Dave, Cantor was aware that his absolute infinity was strictly speaking inconsistent. I also deduce from letters Cantor wrote to bishops that his absolute infinity was some sort of un-nameable god. The class of all sets (or of all mathematical structures) can play that role in axiomatic set theory, but keep in mind that in those context the class of all set is not a set, nor is the class of all mathematical structure a mathematical structure. Formalization of this impossibility has lead to the reflection principle, the fact that if you find a nameable property of such universal class, then you get a set (a mathematical structure) having that property, and thus approximating the universal class in your universe (= model of set theory). Please read Rudy Rucker infinity and the mind which is the best and quasi unique popular explanation of the reflection principle. Now physical existence is another matter. With the comp hyp in the cognitive science, physical existence is mathematical existence seen from inside arithmetics. I agree with Tim and Hal Finney that mathematical existence is more, and different, from the existence of formal description of mathematical object. For example, arithmetical truth cannot be unified in a sound and complete theory, and if comp is true, arithmetical truth escape all possible consistent set theories even with very large cardinal axioms. The seen from inside, that is the 1-person/3-person distinction is the key ingredient missed by Schmidhuber and Tegmark (although Tegmark is apparantly aware of the distinction in his interpretation of QM). See also Rossler's papers or Svozil's one, for works by physicist who are aware of that distinction (under the labels exo/endo-physics). Bruno
Re: Tegmark's TOE Cantor's Absolute Infinity
At 22:26 -0700 21/09/2002, Brent Meeker wrote: I don't see how this follows. If you have a set of axioms, and rules of inference, then (per Godel) there are undecidable propositions. One of these may be added as an axiom and the system will still be consistent. This will allow you to prove more things about the mathematical structures. But you could also add the negation of the proposition as an axiom and then you prove different things. So until the axiom set is augmented, the mathematical structures they imply don't exist. Why? The *tree* of possible extensions of theories can exist (in platonia let us say). The tree of possible models of theories can also exists in Platonia. Actually both those trees are posets (or even categories). And those theories/models posets are related by categorical (in the category sense, not in the logical sense) so-called adjunction, relating theories and models of theories in a mathematically rich sense. What is true for Everett's worlds is a fortiori true for mathematical models or models' sequences. The splitting and relative 1-indeterminacy cannot be used against their ontological 3-atemporal existence, it seems to me. Bruno
Re: Tegmark's TOE Cantor's Absolute Infinity
At 2:19 -0400 22/09/2002, Wei Dai wrote: This needs to be qualified a bit. Mathematical objects are more than the formal (i.e., deductive) consequences of their axioms. However, an axiom system can capture a mathematical structure, if it's second-order, and you consider the semantic consequences of the axioms instead of just the deductive consequences. You are right. But this is a reason for not considering classical *second* order logic as logic. Higher order logic remains logic when some constructive assumption are made, like working in intuitionist logic. A second order classical logic captures a mathematical structure in a very weak sense. My opinion is that the second order *classical* logics are misleading when seen as logical system. Why not taking at once as axioms the set of all true sentences in the standard model of Zermelo Fraenkel (ZF) set theory, and throw away all rules of inference. This captures, even categorically, the set universe. But it is only in a highly technical sense that such a set can be seen as a theory. Logically you are right, and what you said to Brent is correct. I just point here that the use of second order classical logic can be misleading especially for those who doesn't have a good idea of what is a *first order* theory. Bruno
Re: Tegmark's TOE Cantor's Absolute Infinity
At 11:34 -0700 23/09/2002, Hal Finney wrote: I have gone back to Tegmark's paper, which is discussed informally at http://www.hep.upenn.edu/~max/toe.html and linked from http://arXiv.org/abs/gr-qc/9704009. I see that Russell is right, and that Tegmark does identify mathematical structures with formal systems. His chart at the first link above shows Formal Systems as the foundation for all mathematical structures. And the discussion in his paper is entirely in terms of formal systems and their properties. He does not seem to consider the implications if any of Godel's theorem. I still think it is an interesting question whether this is the only possible perspective, or whether one could meaningfully think of an ensemble theory built on mathematical structures considered in a more intuitionist and Platonic model, where they have existence that is more fundamental than what we capture in our axioms. Even if this is not what Tegmark had in mind, it is an alternative ensemble theory that is worth considering. ... and comp leads naturally toward such an alternative ensemble theory. You can look again at Tegmark's Chart, substitute formal system by machines, all the rest are machine dreams. But comp constraints forces us not only to put a measure on those dreams, but to extract the (1)-measure from Godel-Lob theorems, actually from the whole logic of self-reference. (This is what I have partially done, not exactly in those terms, because it was a long time before Tegmark wrote is paper). Bruno
Re: Tegmark's TOE Cantor's Absolute Infinity
Russell Standish writes: [Hal Finney writes;] So I disagree with Russell on this point; I'd say that Tegmark's mathematical structures are more than axiom systems and therefore Tegmark's TOE is different from Schmidhuber's. If you are so sure of this, then please provide a description of these bigger objects that cannot be encoded in the ASCII character set and sent via email. You are welcome to use any communication channel you wish - doesn't have to be email. And if you can't describe what you're talking about, why should I take them seriously? Well, first, I am not so sure of any of these matters. But as an example, how about the positive integers? That's a pretty simple description. Just start with 0 and keep adding 1. From what we understand of Godel's theorem, no axiom system can capture all the properties of this mathematical structure. Yet we have an intuitive understanding of the integers, which is where we came up with the axioms in the first place. Hence our understanding precedes and is more fundamental than the axioms. The axioms are the map; the integers are the territory. We shouldn't confuse them. We have a direct perception of this mathematical structure, which is why I am able to point to it for you without giving you an axiomatic description. Hal Finney
Re: Tegmark's TOE Cantor's Absolute Infinity
I have gone back to Tegmark's paper, which is discussed informally at http://www.hep.upenn.edu/~max/toe.html and linked from http://arXiv.org/abs/gr-qc/9704009. I see that Russell is right, and that Tegmark does identify mathematical structures with formal systems. His chart at the first link above shows Formal Systems as the foundation for all mathematical structures. And the discussion in his paper is entirely in terms of formal systems and their properties. He does not seem to consider the implications if any of Godel's theorem. I still think it is an interesting question whether this is the only possible perspective, or whether one could meaningfully think of an ensemble theory built on mathematical structures considered in a more intuitionist and Platonic model, where they have existence that is more fundamental than what we capture in our axioms. Even if this is not what Tegmark had in mind, it is an alternative ensemble theory that is worth considering. Hal Finney
Re: Tegmark's TOE Cantor's Absolute Infinity
On Monday, September 23, 2002, at 11:34 AM, Hal Finney wrote: I have gone back to Tegmark's paper, which is discussed informally at http://www.hep.upenn.edu/~max/toe.html and linked from http://arXiv.org/abs/gr-qc/9704009. I see that Russell is right, and that Tegmark does identify mathematical structures with formal systems. His chart at the first link above shows Formal Systems as the foundation for all mathematical structures. And the discussion in his paper is entirely in terms of formal systems and their properties. He does not seem to consider the implications if any of Godel's theorem. I still think it is an interesting question whether this is the only possible perspective, or whether one could meaningfully think of an ensemble theory built on mathematical structures considered in a more intuitionist and Platonic model, where they have existence that is more fundamental than what we capture in our axioms. Even if this is not what Tegmark had in mind, it is an alternative ensemble theory that is worth considering. I think this is exactly so, that Reality nearly certainly has more that what we have captured (or perhaps can _ever_ capture) in our axioms. Godel's results can be recast in algorithmic information theory terms, as Greg Chaitin has done, and has Rudy Rucker has admirably explained in Mind Tools. For example, a few excerpts (out of a full chapter, so my excerpts cannot do it justice): It turns out that there's a real sense in which our logic cannot reach out to anything more complicated than what it starts with. Logic can't tell us anything interesting about objects that are much more complex than the axioms we start with. [p. 286] Now we may reasonably suppose that the world around us really does contain phenomena that code up bit strings of complexity greater than three billion [Tim note: Rucker had earlier estimated that the complexity of all of modern math and science is reasonably explained and axiomatized in a thousand or so books, or about 3 billion bits, give or take]. Chaitin's theorem tells us that that our scientific theories have very little to say about these phenomena. On the one hand, our science cannot find a manageably short explanation for a three-billion-bit complex phenomenon. On the other hand, our science cannot definitively prove that such a phenomenon _doesn't_ appear to have a short, magical explanation. [p. 289] Discussion: It seems plausible that we ourselves will eventually have a knowledge base of more than 3 billion bits, perhaps hundreds of billions of bits (I expect diminishing returns, in terms of basic theories, hence an asymptotic approach to some number...just my hunch). Some Jupiter-sized brain may have a much richer understanding of the cosmos and may be able to understand and prove theorems about much more complicated aspects of reality. It seems likely that the current limits on our ability to axiomatize mathematics are not actual limits on the actual universe! (Unless one adopts a weird Distress-like model that future hyperintelligent beings will bring more and more of the mathematical structure of the universe into existence merely through their increased ability to axiomatize.) Personally, for what's it's worth, I vacillate/oscillate between a Platonist point of view that Reality and the Multiverse/Universe/Cosmos actually has some existence in the sense that there appears to be an objective reality which we explore and discover things about and a Constructivist/Intuitionist point of view that only things we can actually construct with atoms and programs have meaningful existence. (I believe in the continuum in the axiomatic sense, in the sense of the reals as Dedekind cuts, in the ideas of Cauchy sequences, limits, and open sets, but I don't necessarily always believe that in any existential sense there are infinities. The real universe does not appear to have an infinite number of anything, except via abstraction.) The two views--Platonism vs. Constructivism--are not necessarily irreconcilable, though. Paul Taylor's book, The Foundations of Mathematics, discussed the reconciliation. Lastly, the Schmidhuber approach, as I understand it, is closer to the Chaitin/Rucker point above than the Tegmark approach is. By considering all outputs of UTMs as string complexity increases, one is including ever-richer axiom systems. (Chaitin talks about Omega, which Rucker also discusses.) I don't want to diss Tegmark, but as I said when I first started posting to this list, Tegmark seems to have a fairly simple view of mathematics. His famous chart showing the branches of mathematics and then his hypothesis that perhaps the multiverse has variants of all of of the axioms of these branches, isn't terribly useful except as a stimulating idea (hence this list, of course). Naturally Tegmark is not claiming his idea is _the_ theory, so stimulation is presumably one of his
Re: Tegmark's TOE Cantor's Absolute Infinity
From: Osher Doctorow [EMAIL PROTECTED], Sat. Sept. 21, 2002 11:38PM Hal, Well said. I really have to have more patience for questioners, but mathematics and logic are such wonderful fields in my opinion that we need to treasure them rather than throw them out like some of the Gung-Ho computer people do who only recognize the finite and discrete and mechanical (although they're rather embarrassed by quantum entanglement - but not enough not to try to deal with it in their old plodding finite-discrete way). Mathematics and Physics are Allies, more or less equal. I prefer not to call the concepts of one inferior directly or to indirectly indicate something of the sort, unless they really are contradictory or something very, very, very close to that more or less. As for a computer, maybe someday it will be *all it can be*, but right now I have to quote a retired Assistant Professor of Computers Emeritus at UCLA (believe it or not, bureaucracy can create such a position - probably the same bureaucratic mentality that created witchhunts and putting accused thieves' heads into wooden blocks so that they could be flogged by passers-by in olden times), who said: *Computers are basically stupid machines.*We knew what he meant. They're very vast stupid machines, and sometimes we need speed, like me getting away from the internet or I'll never get to sleep. Osher Le Doctorow (*Old*) - Original Message - From: Hal Finney [EMAIL PROTECTED] To: [EMAIL PROTECTED]; [EMAIL PROTECTED] Sent: Saturday, September 21, 2002 7:18 PM Subject: Re: Tegmark's TOE Cantor's Absolute Infinity Dave Raub asks: For those of you who are familiar with Max Tegmark's TOE, could someone tell me whether Georg Cantor's Absolute Infinity, Absolute Maximum or Absolute Infinite Collections represent mathematical structures and, therefore have physical existence. I don't know the answer to this, but let me try to answer an easier question which might shed some light. That question is, is a Tegmarkian mathematical structure *defined* by an axiomatic formal system? I got the ideas for this explanation from a recent discussion with Wei Dai. Russell Standish on this list has said that he does interpret Tegmark in this way. A mathematical structure has an associated axiomatic system which essentially defines it. For example, the Euclidean plane is defined by Euclid's axioms. The integers are defined by the Peano axioms, and so on. If we use this interpretation, that suggests that the Tegmark TOE is about the same as that of Schmidhuber, who uses an ensemble of all possible computer programs. For each Tegmark mathematical structure there is an axiom system, and for each axiom system there is a computer program which finds its theorems. And there is a similar mapping in the opposite direction, from Schmidhuber to Tegmark. So this perspective gives us a unification of these two models. However we know that, by Godel's theorem, any axiomatization of a mathematical structure of at least moderate complexity is in some sense incomplete. There are true theorems of that mathematical structure which cannot be proven by those axioms. This is true of the integers, although not of plane geometry as that is too simple. This suggests that the axiom system is not a true definition of the mathematical structure. There is more to the mathematical object than is captured by the axiom system. So if we stick to an interpretation of Tegmark's TOE as being based on mathematical objects, we have to say that formal axiom systems are not the same. Mathematical objects are more than their axioms. That doesn't mean that mathematical structures don't exist; axioms are just a tool to try to explore (part of) the mathematical object. The objects exist in their full complexity even though any given axiom system is incomplete. So I disagree with Russell on this point; I'd say that Tegmark's mathematical structures are more than axiom systems and therefore Tegmark's TOE is different from Schmidhuber's. I also think that this discussion suggests that the infinite sets and classes you are talking about do deserve to be considered mathematical structures in the Tegmark TOE. But I don't know whether he would agree. Hal Finney
Re: Tegmark's TOE Cantor's Absolute Infinity
On 21-Sep-02, Wei Dai wrote: On Sat, Sep 21, 2002 at 10:26:45PM -0700, Brent Meeker wrote: I don't see how this follows. If you have a set of axioms, and rules of inference, then (per Godel) there are undecidable propositions. One of these may be added as an axiom and the system will still be consistent. This will allow you to prove more things about the mathematical structures. But you could also add the negation of the proposition as an axiom and then you prove different things. Are you aware of the distinction between first-order logic and second-order logic? Unlike first-order theories, second-order theories can be categorical, which means all models of the theory are isomorphic. In a categorical theory, there can be undecidable propositions, but there are no semantically independent propositions. That is, all propositions are either true or false, even if for some of them you can't know which is the case if you can compute only recursive functions. If you add a false proposition as an axiom to such a theory, then the theory no longer has a model (it's no longer *about* anything), but you might not be able to tell when that's the case. I was not aware that 2nd-order logic precluded independent propositions. Is this true whatever the axioms and rules of inference? Brent Meeker If a cluttered desk is the sign of a cluttered mind, what's an empty desk a sign of? --- Kenneth Arrow
Re: Tegmark's TOE Cantor's Absolute Infinity
On Sat, Sep 21, 2002 at 11:50:20PM -0700, Brent Meeker wrote: I was not aware that 2nd-order logic precluded independent propositions. Is this true whatever the axioms and rules of inference? It depends on the axioms, and the semantic rules (not rules of inference which is a deductive concept). Here's a good page for clarification between semantic concepts and deductive concepts: http://www.joh.cam.ac.uk/societies/moral/mathlogic.htm. It can be confusing because in first-order logic they happen to coincide, but that's not the case in second-order logic. And again, second order theories *can* be categorical, whereas first-order theories can not be.
Re: Tegmark's TOE Cantor's Absolute Infinity
Osher Doctorow wrote: From: Osher Doctorow [EMAIL PROTECTED], Sat. Sept. 21, 2002 11:38PM Hal, Well said. I really have to have more patience for questioners, but mathematics and logic are such wonderful fields in my opinion that we need to treasure them rather than throw them out like some of the Gung-Ho computer people do who only recognize the finite and discrete and mechanical (although they're rather embarrassed by quantum entanglement - but not enough not to try to deal with it in their old plodding finite-discrete way). Mathematics and Physics are Allies, more or less equal. I prefer not to call the concepts of one inferior directly or to indirectly indicate something of the sort, unless they really are contradictory or something very, very, very close to that more or less. As for a computer, maybe someday it will be *all it can be*, but right now I have to quote a retired Assistant Professor of Computers Emeritus at UCLA (believe it or not, bureaucracy can create such a position - probably the same bureaucratic mentality that created witchhunts and putting accused thieves' heads into wooden blocks so that they could be flogged by passers-by in olden times), who said: *Computers are basically stupid machines.*We knew what he meant. They're very vast stupid machines, and sometimes we need speed, like me getting away from the internet or I'll never get to sleep. Osher Le Doctorow (*Old*) ... So I disagree with Russell on this point; I'd say that Tegmark's mathematical structures are more than axiom systems and therefore Tegmark's TOE is different from Schmidhuber's. I also think that this discussion suggests that the infinite sets and classes you are talking about do deserve to be considered mathematical structures in the Tegmark TOE. But I don't know whether he would agree. Hal Finney If you are so sure of this, then please provide a description of these bigger objects that cannot be encoded in the ASCII character set and sent via email. You are welcome to use any communication channel you wish - doesn't have to be email. And if you can't describe what you're talking about, why should I take them seriously? Now from my point of view, the continuum exists, of course, but it exists as a collection of descriptions which make use of primitive concepts like limit. Each of these descriptions can be encoded in ASCII (or any other encoding system). I am open to the proposition that there is no enumeration of the set of all descriptions of the continuum - and indeed the enumeration of the set of all descriptions takes c steps to execute :) Anyone who is familiar with my postings would never categorise me as being a discrete bigot. Cheers A/Prof Russell Standish Director High Performance Computing Support Unit, Phone 9385 6967, 8308 3119 (mobile) UNSW SYDNEY 2052 Fax 9385 6965, 0425 253119 () Australia[EMAIL PROTECTED] Room 2075, Red Centrehttp://parallel.hpc.unsw.edu.au/rks International prefix +612, Interstate prefix 02
Tegmark's TOE Cantor's Absolute Infinity
For those of you who are familiar with Max Tegmark's TOE, could someone tell me whether Georg Cantor's Absolute Infinity, Absolute Maximum or Absolute Infinite Collections represent mathematical structures and, therefore have physical existence. Thanks again for the help!! Dave Raub
Tegmark's TOE Cantor's Absolute Infinity
For those of you who are familiar with Max Tegmark's TOE, could someone tell me whether Georg Cantor's Absolute Infinity, Absolute Maximum or Absolute Infinite Collections represent mathematical structures and, therefore have physical existence. Thanks again for the help!! Dave Raub
Tegmark's TOE Cantor's Absolute Infinity
For those of you who are familiar with Max Tegmark's TOE, could someone tell me whether Georg Cantor's Absolute Infinity, Absolute Maximum or Absolute Infinite Collections represent mathematical structures and, therefore have physical existence. Thanks again for the help!! Dave Raub
Re: Tegmark's TOE Cantor's Absolute Infinity
Dave Raub asks: For those of you who are familiar with Max Tegmark's TOE, could someone tell me whether Georg Cantor's Absolute Infinity, Absolute Maximum or Absolute Infinite Collections represent mathematical structures and, therefore have physical existence. I don't know the answer to this, but let me try to answer an easier question which might shed some light. That question is, is a Tegmarkian mathematical structure *defined* by an axiomatic formal system? I got the ideas for this explanation from a recent discussion with Wei Dai. Russell Standish on this list has said that he does interpret Tegmark in this way. A mathematical structure has an associated axiomatic system which essentially defines it. For example, the Euclidean plane is defined by Euclid's axioms. The integers are defined by the Peano axioms, and so on. If we use this interpretation, that suggests that the Tegmark TOE is about the same as that of Schmidhuber, who uses an ensemble of all possible computer programs. For each Tegmark mathematical structure there is an axiom system, and for each axiom system there is a computer program which finds its theorems. And there is a similar mapping in the opposite direction, from Schmidhuber to Tegmark. So this perspective gives us a unification of these two models. However we know that, by Godel's theorem, any axiomatization of a mathematical structure of at least moderate complexity is in some sense incomplete. There are true theorems of that mathematical structure which cannot be proven by those axioms. This is true of the integers, although not of plane geometry as that is too simple. This suggests that the axiom system is not a true definition of the mathematical structure. There is more to the mathematical object than is captured by the axiom system. So if we stick to an interpretation of Tegmark's TOE as being based on mathematical objects, we have to say that formal axiom systems are not the same. Mathematical objects are more than their axioms. That doesn't mean that mathematical structures don't exist; axioms are just a tool to try to explore (part of) the mathematical object. The objects exist in their full complexity even though any given axiom system is incomplete. So I disagree with Russell on this point; I'd say that Tegmark's mathematical structures are more than axiom systems and therefore Tegmark's TOE is different from Schmidhuber's. I also think that this discussion suggests that the infinite sets and classes you are talking about do deserve to be considered mathematical structures in the Tegmark TOE. But I don't know whether he would agree. Hal Finney
Re: Tegmark's TOE Cantor's Absolute Infinity
On Sat, Sep 21, 2002 at 09:20:26PM -0400, [EMAIL PROTECTED] wrote: For those of you who are familiar with Max Tegmark's TOE, could someone tell me whether Georg Cantor's Absolute Infinity, Absolute Maximum or Absolute Infinite Collections represent mathematical structures and, therefore have physical existence. I'm not sure what Absolute Maximum and Absolute Infinite Collections refer to (a search for Absolute Infinite Collections on Google gave no hits), but I'll take the question to mean whether proper classes (i.e. collections that are bigger than any set, for example the class of all sets) have physical existence. I think the answer is yes, or at least I don't see a reason to rule it out. To make the statement meaningful, we need (at least) two things, (1) a way to assign probabilities to proper classes so you can say I am observer-moment X with probability p where X is a proper class, and (2) a theory of consciousness of proper classes, so you can know what it feels like to be X when X is a proper class. (2) seems pretty hopeless right now. We don't even have a good theory of consciousness for finite structures yet. Once we have that, we would still have to go on to a theory of consciousness for countably infinite sets, and then to uncountable sets, before we could think about what it feels like to be proper classes. But still, it may not be impossible to work it out eventually. As to (1), Tegmark doesn't tell us how to assign probabilities to observer moments. (He says to use a uniform distribution, but gives no proposal for how to define one over all mathematical structures.) However, it does not seem difficult to come up with a reasonable one that applies to proper classes as well as sets. Here's my proposal. Consider a sentence in set theory that has one unbound variable. This sentence defines a class, namely the class of sets that make the sentence true when substituted for the unbound variable. It may be a proper class, or just a set. Call the classes that can be defined by finite sentences of set theory describable classes. Any probability distribution P over the sentences of set theory, translates to a probability distribution Q over describable classes as follows: Q(X) = Sum of P(s), over all s that define X Take P to be the universal a priori probability distribution (see Li and Vitanyi's book) over the sentences of set theory, and use the resulting Q as the distribution over observer moments. Of course this distribution is highly uncomputable, so in practice one would have to use computable approximations to it. However, computability is relative to one's resources. We have access to certain computational resources now, but in the future we may have more. We may even discover laws of physics that allow us to compute some non-recursive functions, which in turn would allow us to better approximate this Q. The point is that by using Q, instead of a more computable but less dominant distribution (such as ones suggested by Schmidhuber), in our theory of everything, we would not have to revise the theory, but only our approximations, if we discover more computational resources.
Re: Tegmark's TOE Cantor's Absolute Infinity
On 21-Sep-02, Hal Finney wrote: ... However we know that, by Godel's theorem, any axiomatization of a mathematical structure of at least moderate complexity is in some sense incomplete. There are true theorems of that mathematical structure which cannot be proven by those axioms. This is true of the integers, although not of plane geometry as that is too simple. This suggests that the axiom system is not a true definition of the mathematical structure. There is more to the mathematical object than is captured by the axiom system. So if we stick to an interpretation of Tegmark's TOE as being based on mathematical objects, we have to say that formal axiom systems are not the same. Mathematical objects are more than their axioms. I don't see how this follows. If you have a set of axioms, and rules of inference, then (per Godel) there are undecidable propositions. One of these may be added as an axiom and the system will still be consistent. This will allow you to prove more things about the mathematical structures. But you could also add the negation of the proposition as an axiom and then you prove different things. So until the axiom set is augmented, the mathematical structures they imply don't exist. Brent Meeker One way (of designing software) is to make it so simple that there are obviously no deficiencies and the other way is to make it so complicated that there are no obvious deficiencies. --- Tony Hoare
Re: Tegmark's TOE Cantor's Absolute Infinity
From: Osher Doctorow [EMAIL PROTECTED], Sat. Sept. 21, 2002 10:39PM I've glanced over one of Tegmark's papers and it didn't impress me much, but maybe you've seen something that I didn't. As for your question (have you ever been accused of being over-specific?), the best thing for a person not familiar with Georg Cantor's work in my opinion would be to read Garrett Birkhoff and Saunders MacLane's A Survey of Modern Algebra or any comparable modern textbook in what's called Abstract Algebra, Modern Algebra, Advanced Algebra, etc., or look under transfinite numbers, Georg Cantor, the cardinality/ordinality of the continuum, etc., etc. on the internet or in your mathematics-engineering-physics research library catalog or internet catalog. To answer even more directly, here it is. *Absolute infinity* if translated into mathematics means the *size* of the real line or a finite segment or half-infinite segment of the real line and things like that, and it is UNCOUNTABLE, whereas the number of discrete integers, e.g., -1, 0, 1, 2, 3, ..., is called COUNTABLE. If you accept a real line or a finite line segment or a finite planar geometric figure like a circle or a 3-dimensional geometric figure like a sphere as being *physical*, then *absolute infinity* would be physical. If you don't accept these as being physical, then you can't throw them out either - if you did, you'd throw physics out. So there are *things* in mathematics that are related to physical things by *approximation*, in the sense that a mathematical straight line approximates the motion of a Euclidean particle in an uncurved universe or a region far enough from other objects as to make little difference to the problem. There are also many things in mathematics, including the words PATH and CURVE and SURFACE, that also approximate physical dynamics. Do you see what the difficulty is with over-simplifying or slightly misstating the question? Osher Doctorow - Original Message - From: [EMAIL PROTECTED] To: [EMAIL PROTECTED] Cc: [EMAIL PROTECTED] Sent: Saturday, September 21, 2002 6:59 PM Subject: Tegmark's TOE Cantor's Absolute Infinity For those of you who are familiar with Max Tegmark's TOE, could someone tell me whether Georg Cantor's Absolute Infinity, Absolute Maximum or Absolute Infinite Collections represent mathematical structures and, therefore have physical existence. Thanks again for the help!! Dave Raub
Re: Tegmark's TOE Cantor's Absolute Infinity
On Sat, Sep 21, 2002 at 10:26:45PM -0700, Brent Meeker wrote: I don't see how this follows. If you have a set of axioms, and rules of inference, then (per Godel) there are undecidable propositions. One of these may be added as an axiom and the system will still be consistent. This will allow you to prove more things about the mathematical structures. But you could also add the negation of the proposition as an axiom and then you prove different things. Are you aware of the distinction between first-order logic and second-order logic? Unlike first-order theories, second-order theories can be categorical, which means all models of the theory are isomorphic. In a categorical theory, there can be undecidable propositions, but there are no semantically independent propositions. That is, all propositions are either true or false, even if for some of them you can't know which is the case if you can compute only recursive functions. If you add a false proposition as an axiom to such a theory, then the theory no longer has a model (it's no longer *about* anything), but you might not be able to tell when that's the case. Back to what Hal wrote: This suggests that the axiom system is not a true definition of the mathematical structure. There is more to the mathematical object than is captured by the axiom system. So if we stick to an interpretation of Tegmark's TOE as being based on mathematical objects, we have to say that formal axiom systems are not the same. Mathematical objects are more than their axioms. This needs to be qualified a bit. Mathematical objects are more than the formal (i.e., deductive) consequences of their axioms. However, an axiom system can capture a mathematical structure, if it's second-order, and you consider the semantic consequences of the axioms instead of just the deductive consequences.
