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A purely mathematical universe immediately brings to mind a question: Since chaotic systems evolve in a way provably intractable to calculation, how could any

A purely mathematical universe immediately brings to mind a question: Since chaotic systems evolve in a way provably intractable to calculation, how could any system based solely upon calculation address, well, almost everything that exists?

klank 9y ago
That's an odd definition of chaotic systems you are using. It's not so much that chaotic systems can not be calculated but instead that enough precision of starting conditions can't be obtained to ensure an accurate result (for example, highly non-linear systems). The double pendulum system is a perfect example where computation is simple yet the system is still chaotic.
- otakucode 9y ago
  I think it's an odd definition you are using. Chaos depends upon the sensitivity to those initial conditions, not what those initial conditions ARE. You do not need actual physical objects to see this. Conjecture that a double-pendulum system starts in a perfectly real-number-valued mythical reality where spacetime is perfectly continuous (a fiction). Now predict what state the system will be in at a bunch of 'close together' states. Then try to do some 'closer together' states. See if you can ever get them to line up. Do it on pencil and paper, of course, since you want to capture the full real-number-infinity effect. Of course, you can't write down the real-number result with infinite precision, can you? You'll have to chop it off at some point. That's OK. That's what error bars are for. Just keep going and keep track of your error. Oops, you'll find that your error grows larger than the prediction itself with remarkable speed! THAT is the chaos. It doesn't matter if you measured the starting conditions with literal infinite perfect precision. The error bars on your predictions are going to explode like a busy beaver function. You're not going to catch that. Nothing can. (I imagine the actual physical limit would be that your predictions can be accurate within a distance equivalent to the radius of a sphere expanding at the speed of light from the initial position but I haven't actually read anything suggesting that, it just seems sensible.) Then you've got stupid simple cellular automata like rule 31 1D systems. Totally discrete. Totally basic. No precision even needed anywhere. And yet, you can't predict it. Not without accumulating an error greater than the state in the system itself if you skip even the slightest act of computation necessary to evolve the system as a whole from one step to the next.
  - naasking 9y ago
    > You'll have to chop it off at some point. That's OK. That's what error bars are for. Just keep going and keep track of your error. Oops, you'll find that your error grows larger than the prediction itself with remarkable speed! THAT is the chaos. If this were true, it should be trivial to prove by observation that we're not in a mathematical universe, or that physics is not computable, even in principle. As far as I understand, physics as we know it is widely acknowledged to be computable, albeit with exponential slowdown on classical computers. By the Bekenstein bound and other thermodynamic constraints, a finite region of space necessarily contains a finite amount of information, or it would collapse into a black hole, ergo physics is at worst a finite state machine. So where's the disconnect?
tzahola 9y ago
Chaotic systems are only intractable if you measure the initial conditions with finite precision.
- otakucode 9y ago
  No, that is not true. Their intractability has absolutely nothing to do with the precision of measurement. Assume the initial condition with infinite precision and the further evolution of the system is still intractable. Ignoring the Heisenberg Uncertainty Principle for a moment and getting to the heart of chaos theory the problem is the production of information at a rate faster than any existant (and I believe, though I don't understand it well enough to give exact details) or fundamentally possible future system of mathematics can compensate for. Nonlinear systems (ie, essentially everything in existence) display chaos (extreme sensitivity to initial conditions, period-doubling, whatever you wish to call it) on every scale. Were the universe mathematics and mathematics alone, all things which are provably outside of the grasp of mathematics would also be non-existent. Such things, however, do exist. While we may not be able to predict the location of a single molecule of air in a mere 10,000 years with a rough accuracy of a cubic femtometer, the universe does it. And our inability to predict it is not because we need a bigger computer. Our inability to predict it is because in order to predict it, we would have to simulate the entire universe and every single particle within it, and every interaction between all of them, to infinite (ehhh... maybe) precision. Which would have an information load exactly as big as the universe itself and increase entropy (which implies it takes as much time) as much as actual evolution of the entire universe over that course of time. To predict one molecule of air. And the universe does it for all the molecules of air. Along with all the other particles.
  - Koshkin 9y ago
    > the universe does [predict] But it does not.
    - otakucode 9y ago
      That's a very strange quote you did... did you honestly think that I meant the pronoun 'it' to mean 'predict'? I absolutely did not. The universe does it. It executes. It evolves along the pathway and 'calculates' (if you wish) the end result. A thing which no mathematics can, or will ever be able to, do.

klank 9y ago

That's an odd definition of chaotic systems you are using. It's not so much that chaotic systems can not be calculated but instead that enough precision of starting conditions can't be obtained to ensure an accurate result (for example, highly non-linear systems). The double pendulum system is a perfect example where computation is simple yet the system is still chaotic.

otakucode 9y ago
I think it's an odd definition you are using. Chaos depends upon the sensitivity to those initial conditions, not what those initial conditions ARE. You do not need actual physical objects to see this. Conjecture that a double-pendulum system starts in a perfectly real-number-valued mythical reality where spacetime is perfectly continuous (a fiction). Now predict what state the system will be in at a bunch of 'close together' states. Then try to do some 'closer together' states. See if you can ever get them to line up. Do it on pencil and paper, of course, since you want to capture the full real-number-infinity effect. Of course, you can't write down the real-number result with infinite precision, can you? You'll have to chop it off at some point. That's OK. That's what error bars are for. Just keep going and keep track of your error. Oops, you'll find that your error grows larger than the prediction itself with remarkable speed! THAT is the chaos. It doesn't matter if you measured the starting conditions with literal infinite perfect precision. The error bars on your predictions are going to explode like a busy beaver function. You're not going to catch that. Nothing can. (I imagine the actual physical limit would be that your predictions can be accurate within a distance equivalent to the radius of a sphere expanding at the speed of light from the initial position but I haven't actually read anything suggesting that, it just seems sensible.) Then you've got stupid simple cellular automata like rule 31 1D systems. Totally discrete. Totally basic. No precision even needed anywhere. And yet, you can't predict it. Not without accumulating an error greater than the state in the system itself if you skip even the slightest act of computation necessary to evolve the system as a whole from one step to the next.
- naasking 9y ago
  > You'll have to chop it off at some point. That's OK. That's what error bars are for. Just keep going and keep track of your error. Oops, you'll find that your error grows larger than the prediction itself with remarkable speed! THAT is the chaos. If this were true, it should be trivial to prove by observation that we're not in a mathematical universe, or that physics is not computable, even in principle. As far as I understand, physics as we know it is widely acknowledged to be computable, albeit with exponential slowdown on classical computers. By the Bekenstein bound and other thermodynamic constraints, a finite region of space necessarily contains a finite amount of information, or it would collapse into a black hole, ergo physics is at worst a finite state machine. So where's the disconnect?

tzahola 9y ago

Chaotic systems are only intractable if you measure the initial conditions with finite precision.

otakucode 9y ago
No, that is not true. Their intractability has absolutely nothing to do with the precision of measurement. Assume the initial condition with infinite precision and the further evolution of the system is still intractable. Ignoring the Heisenberg Uncertainty Principle for a moment and getting to the heart of chaos theory the problem is the production of information at a rate faster than any existant (and I believe, though I don't understand it well enough to give exact details) or fundamentally possible future system of mathematics can compensate for. Nonlinear systems (ie, essentially everything in existence) display chaos (extreme sensitivity to initial conditions, period-doubling, whatever you wish to call it) on every scale. Were the universe mathematics and mathematics alone, all things which are provably outside of the grasp of mathematics would also be non-existent. Such things, however, do exist. While we may not be able to predict the location of a single molecule of air in a mere 10,000 years with a rough accuracy of a cubic femtometer, the universe does it. And our inability to predict it is not because we need a bigger computer. Our inability to predict it is because in order to predict it, we would have to simulate the entire universe and every single particle within it, and every interaction between all of them, to infinite (ehhh... maybe) precision. Which would have an information load exactly as big as the universe itself and increase entropy (which implies it takes as much time) as much as actual evolution of the entire universe over that course of time. To predict one molecule of air. And the universe does it for all the molecules of air. Along with all the other particles.
- Koshkin 9y ago
  > the universe does [predict] But it does not.
  - otakucode 9y ago
    That's a very strange quote you did... did you honestly think that I meant the pronoun 'it' to mean 'predict'? I absolutely did not. The universe does it. It executes. It evolves along the pathway and 'calculates' (if you wish) the end result. A thing which no mathematics can, or will ever be able to, do.