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Tegmark's Mathematical Universe Hypothesis (MUH) is actually pretty much a computational theory of the universe. It basically makes the assumption that there ex

by eref 9y ago

Tegmark's Mathematical Universe Hypothesis (MUH) is actually pretty much a computational theory of the universe. It basically makes the assumption that there exists somewhere the simplest process imaginable: One that counts through and runs all mathematical formulas which includes all programs, which, in turn, includes all (computable) universes. This assumption explains why our universe is oddly complex: If all universes exist, then a weird one such as ours merely exists because all universes do. This line of reasoning is also called the "anthropic principle". I think, Tegmark does not talk about computations per se because the hypothesis is agnostic about what kind of computer our universe is. It could, for example, be a geometric computer in which the position of objects can be determined with infinite precision (i.e. using the real numbers). Such a computer would be strictly more powerful than a Turing machine or equivalent (i.e. all computational models that can be described and run inside a Turing machine). Enumerating the space of all formalisms (i.e. the domain of mathematics) is as agnostic as you can be.

gfodor 9y ago
I didn't take Tegmark's argument to be a computational theory. I took it to be an inductive one based upon the trend of the physics over the last decades towards the more and more generified multiverse theories. Ultimately, you land on a model where all consistent mathematical representations of a reality exist. At that level of abstraction, there is no where to go: it is the limit of the induction. There's a rather satisfying sense of self-evidence about it, but that doesn't mean it's falsifiable or true. IIRC there are no arguments about computation or what underlying medium(s) are involved, it seems like a separate question.
- eref 9y ago
  I am unsure now whether he describes the set of mathematical objects by enumeration and evaluation. He definitely describes the set of universes this way, as being “run”, so it is a computation. But you are right that he never discussed the substrate that they are run on. He assumes the existence of that enumeration to be axiomatic, like an uncaused cause, somewhere in the space of all mathematical objects that simply exist. I guess, if we had a proof that all non-computable formal systems are inconsistent then we could also skip one layer and assume the enumeration of all programs as axiomatic in order to be maximally agnostic.
- naasking 9y ago
  > Ultimately, you land on a model where all consistent mathematical representations of a reality exist. [...] IIRC there are no arguments about computation or what underlying medium(s) are involved, it seems like a separate question. These are actually related by Goedel's incompleteness theorems and the Curry-Howard isomorphism. Computation and mathematics are inextricably linked. The Computable Universe is a necessary restriction on the mathematical universe hypothesis to avoid accepting inconsistent universes.
  - eref 9y ago
    It links a certain axiom set with computations, not all of mathematics (i.e. the set of all axiom sets), doesn’t it?
    - naasking 9y ago
      If you accept that every mathematical structure of interest has an intuitionistic construction (as is currently believed to be the case), then every mathematical structure has an expression as a computer program.
otakucode 9y ago
>This assumption explains why our universe is oddly complex That is, to me, a weird mixing of scales. The universe only seems complex to us because of the scale at which we experience it in terms of time and physical extent as well as the convoluted way our own systems of understanding developed and then were later mutated and stretched into different forms to be better suited, etc. By definition the universe itself can't really be more or less complex than anything. At least not anything we can experience. The idea of a geometric computer is surprisingly close to things I've been considering recently, but 'position of objects can be determined with infinite precision' seems strange. It may be just that I don't understand what is fully meant when you say 'geometric computer' but I would expect 'position' would be mostly meaningless in that space except as an emergent/illusory 'property'... and I don't think arbitrary precision would necessarily be implied either.
- eref 9y ago
  I think the universe could easily be less complex if you would simply remove everything below Newtonian physics and basic chemistry. We would have evolved in pretty much the same way. But our universe is also complex in that the currently most fundamental laws of physics depend on a lot of seemingly arbitrary constants that are finely tuned (i.e. they become incompatible with our observations if they would be set slightly differently). Of course, it could be that we are missing a much simpler unifying theory, but currently it seems to have a quite high entropy such that we cannot explain it by saying it is in some sense canonical, i.e. that it is the only way a universe can be. That is also supported by the fact that we can build tiny universes in our computers (e.g. video games and Convey’s Game of Life).
  - otakucode 9y ago
    I understand that perspective, but to me it seems a failure of history. If we had simply thrown out all of our previous knowledge when we discovered 'lower level' understandings and reformulated them in those terms, we would see no 'special values'. Those values would be 1. And all else would be derived from them, and the things at our scale would look complex and maddening. Temperature measured in Kelvin, velocity measured in ratios off the speed of light, distance based on the planck length, things like that (although I know you're talking more about the fine structure constant and things of that nature, but I think it could be done with them as well). The question, in my mind at least, is how to show that the quantum understanding leads, on the macro scale, to the emergent phenomena we observe and once thought unitary. That's a great degree of precision and division between entities to arise out of the nonlinear interactions of bunches of 'particles'. And given the scale on which such things emerged (the universal one), clearly a proper system of understanding would make the emergence of things like atoms, molecules, chemistry, astronomy, geologies, planets, solar systems, galaxies, stars, and all those things self-evident. But... save for a few exceptions (more all the time though which is encouraging!) we rarely even look at things en-masse. It's all well and good to know how 2 bodies interact under gravitation. But when your system falls apart with 3, and you really need to be able to toss in a trillion with dozens of other nonlinear relationships... It really ought to be more obvious, I would think, that we're missing something.

by eref 9y ago

gfodor 9y ago

I didn't take Tegmark's argument to be a computational theory. I took it to be an inductive one based upon the trend of the physics over the last decades towards the more and more generified multiverse theories. Ultimately, you land on a model where all consistent mathematical representations of a reality exist. At that level of abstraction, there is no where to go: it is the limit of the induction. There's a rather satisfying sense of self-evidence about it, but that doesn't mean it's falsifiable or true. IIRC there are no arguments about computation or what underlying medium(s) are involved, it seems like a separate question.

eref 9y ago
I am unsure now whether he describes the set of mathematical objects by enumeration and evaluation. He definitely describes the set of universes this way, as being “run”, so it is a computation. But you are right that he never discussed the substrate that they are run on. He assumes the existence of that enumeration to be axiomatic, like an uncaused cause, somewhere in the space of all mathematical objects that simply exist. I guess, if we had a proof that all non-computable formal systems are inconsistent then we could also skip one layer and assume the enumeration of all programs as axiomatic in order to be maximally agnostic.
naasking 9y ago
> Ultimately, you land on a model where all consistent mathematical representations of a reality exist. [...] IIRC there are no arguments about computation or what underlying medium(s) are involved, it seems like a separate question. These are actually related by Goedel's incompleteness theorems and the Curry-Howard isomorphism. Computation and mathematics are inextricably linked. The Computable Universe is a necessary restriction on the mathematical universe hypothesis to avoid accepting inconsistent universes.
- eref 9y ago
  It links a certain axiom set with computations, not all of mathematics (i.e. the set of all axiom sets), doesn’t it?
  - naasking 9y ago
    If you accept that every mathematical structure of interest has an intuitionistic construction (as is currently believed to be the case), then every mathematical structure has an expression as a computer program.

otakucode 9y ago

>This assumption explains why our universe is oddly complex That is, to me, a weird mixing of scales. The universe only seems complex to us because of the scale at which we experience it in terms of time and physical extent as well as the convoluted way our own systems of understanding developed and then were later mutated and stretched into different forms to be better suited, etc. By definition the universe itself can't really be more or less complex than anything. At least not anything we can experience. The idea of a geometric computer is surprisingly close to things I've been considering recently, but 'position of objects can be determined with infinite precision' seems strange. It may be just that I don't understand what is fully meant when you say 'geometric computer' but I would expect 'position' would be mostly meaningless in that space except as an emergent/illusory 'property'... and I don't think arbitrary precision would necessarily be implied either.

eref 9y ago
I think the universe could easily be less complex if you would simply remove everything below Newtonian physics and basic chemistry. We would have evolved in pretty much the same way. But our universe is also complex in that the currently most fundamental laws of physics depend on a lot of seemingly arbitrary constants that are finely tuned (i.e. they become incompatible with our observations if they would be set slightly differently). Of course, it could be that we are missing a much simpler unifying theory, but currently it seems to have a quite high entropy such that we cannot explain it by saying it is in some sense canonical, i.e. that it is the only way a universe can be. That is also supported by the fact that we can build tiny universes in our computers (e.g. video games and Convey’s Game of Life).
- otakucode 9y ago
  I understand that perspective, but to me it seems a failure of history. If we had simply thrown out all of our previous knowledge when we discovered 'lower level' understandings and reformulated them in those terms, we would see no 'special values'. Those values would be 1. And all else would be derived from them, and the things at our scale would look complex and maddening. Temperature measured in Kelvin, velocity measured in ratios off the speed of light, distance based on the planck length, things like that (although I know you're talking more about the fine structure constant and things of that nature, but I think it could be done with them as well). The question, in my mind at least, is how to show that the quantum understanding leads, on the macro scale, to the emergent phenomena we observe and once thought unitary. That's a great degree of precision and division between entities to arise out of the nonlinear interactions of bunches of 'particles'. And given the scale on which such things emerged (the universal one), clearly a proper system of understanding would make the emergence of things like atoms, molecules, chemistry, astronomy, geologies, planets, solar systems, galaxies, stars, and all those things self-evident. But... save for a few exceptions (more all the time though which is encouraging!) we rarely even look at things en-masse. It's all well and good to know how 2 bodies interact under gravitation. But when your system falls apart with 3, and you really need to be able to toss in a trillion with dozens of other nonlinear relationships... It really ought to be more obvious, I would think, that we're missing something.