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
JOINING post
My name is Lloyd David Raub. I'm a retired executive from Ohio State
University. I have a Ph.D. in Public Administration from Penn. State and my
interests now include TOE's, alternate universes, MWI, inflationary other
cosmologies {cyclic universes, quasi steady state, plasma,etc.} I am looking
forward to enjoying the discussions on this thread. thanks HELLO EVERYBODY.
Dave Raub
JOINING post
Hi all, I'm Ben Goertzel. This is my initial joining post I'm a math PhD originally, spent 8 years as an academic in math, CS and psych departments. Have been in the software industry for the last 5 years. My primary research is in Artificial General Intelligence (see www.realai.net) -- my friends and I are building a genuinely intelligent software program, a multi-year project that's been going on for some time. Am also working in bioinformatics, analyzing gene expression data Before building a thinking machine became an almost all-consuming obsession, I spent some time trying to create a unified physics theory. It was to be a discrete theory based on the discrete Clifford algebra the Cayley algebra. I'm also interested in the physics applications of the notion of mind creating reality while reality creates mind (John Wheeler and all that...). I studied quantum gravity, chromodynamics, string theory and lots of other fun stuff in the late 80's and early 90's, but haven't really kept current with technical physics, and all that stuff is pretty rusty for me now, but I still find it fascinating... Ben Goertzel www.goertzel.org/work.html
No Subject
My name is Lloyd David Raub. I'm a retired executive from Ohio State
University. I have a Ph.D. in Public Administration from Penn. State and my
interests now include TOE's, alternate universes, MWI, inflationary other
cosmologies {cyclic universes, quasi steady state, plasma,etc.} I am looking
forward to enjoying the discussions on this thread. thanks HELLO EVERYBODY.
Dave Raub
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.
