The interesting bit is that if you could actually use Solomonoff induction as prior, I’m pretty sure you’d be an irreparable aether believer and ‘relativity sceptic’, with confidence that has so many nines nothing would ever convince you. That’s because the absolute-speed irreversible aether universes like 3d versions of Conway’s game of life are so much more computationally compact than highly symmetric universe with many space-time symmetries (to the point of time and space intermixing in the equations) and the fully reversible fundamental laws of physics. Saying this to you as applied mathematician with experience actually implementing laws of physics in software.
The priors are only a free parameter if all you care for is not getting Dutch-booked on your bets about physics but do not give a damn to have any useful theories.
That’s because the absolute-speed irreversible aether universes like 3d versions of Conway’s game of life are so much more computationally compact than highly symmetric universe with many space-time symmetries (to the point of time and space intermixing in the equations) and the fully reversible fundamental laws of physics.
Can you explain why this is the case? Also, when you say “aether” universes are more computationally compact than “relativity” universes, is this before or after taking into account our observations (i.e., are you restricting your attention to universes that fit our observations, or not)?
Saying this to you as applied mathematician with experience actually implementing laws of physics in software.
Is it possible that what you said is true only if we want the laws of physics to run fast on current computers? I’m afraid that most software people, including myself, probably have bad intuitions about Solomonoff Induction because we only have experience with a very small subset of possible computations, namely those that can be run economically on modern computers. Perhaps the laws of physics can be implemented much more compactly if we ignored efficiency?
As intuition pump consider the reversible vs irreversible cellular automata. If you pick at random, vast majority will not be reversible. Ditto for the symmetries. (Keep in mind that in Solomonoff probability we feed infinite random tape to the machine. It is no Occam’s razor. Elegant simplest deterministic things can be vastly outnumbered by inelegant, even if they are most probable. edit: that is to say you can be more likely to pick something asymmetric even if any particular asymmetric is less likely than symmetric)
Also, when you say “aether” universes are more computationally compact than “relativity” universes, is this before or after taking into account our observations (i.e., are you restricting your attention to universes that fit our observations, or not)?
There can always be a vast conspiracy explaining the observations… ideally if you could simulate whole universe (or multiverse) from big bang to today and pick out the data matching observations or the conspired lying, then maybe it’d work, but the whole exercise of doing physics is that you are embedded within universe you are studying. edit: and that trick won’t work if the code eats a lot of random tape.
Is it possible that what you said is true only if we want the laws of physics to run fast on current computers?
I don’t think relaxing the fast requirement really helps that much. Consider programming Conway’s game of life in Turing machine. Or vice versa. Or the interpreter for general TM on the minimal TM. It gets way worse if you want full rotational symmetry on discrete system.
Of course, maybe one of the small busy beavers is a superintelligence that likes to play with various rules like that. Then I’d be wrong. Can not rule even this possibility out. Kolmogorov/Solomonoff name drop is awesome spice for cooking proven-untestable propositions.
One could argue that second order logic could work better, but this is getting way deep into land of untestable propositions that are even proven untestable, and the appropriate response would be high expectations asian father picture with “why not third order logic?”.
edit: also you hit nail on the head on an issue here: i can not be sure that there is no very short way to encode something. You can ask me if I am sure that busy beaver 6 is not anything, and I am not sure! I am not sure it is not the god almighty. The proposition that there is a simple way is a statement of faith that can not be disproved any more than existence of god. Also, I feel that there has to be scaling for the computational efficiency in the prior. The more efficient structures can run more minds inside. Or conversely, the less efficient structures take more coding to locate minds inside of them.
Discrete universes introduce an unobserved parameter (a fundamental length) as well as a preferred frame. Abandoning complex numbers for a countable group in QM would also be very difficult. There is a complexity theory for real-valued objects and if Solomonoff induction could be adapted to that context, continuum theories ought to be favored.
Yep. Well, in any case, the point is that the Solomonoff probability is not in any way ‘universal prior’, you have to pick the machine, then you can’t actually compute anything useful because its uncomputable and you’ll never get anything non-trivial all the way down to minimum length.
If you go for changing the machine, you could use laws of physics as you know them as the machine, too (then you’ll be irreparable sceptic of any new physics). In the context of laws of physics its just the emulation of one universal computation (laws of physics) with another (what ever stuff you pick). We are only discussing this because some highly self-important people heard of Solomonoff induction and Kolmogorov complexity, modelled those with something ill defined and fuzzy (lacking not only sufficient understanding of those things but the understanding of the concepts in which to understand those, and so on several layers deep, as is usually the case when you choose something really complicated without having spent a lot of time studying towards it) and then used it as universal smart sounding word to say (e.g. here or here or here ). Name-dropping and nothing more.
Hence, we have a way of getting hold of the concept of Euclidean
angle, starting from a purely combinatorial scheme
...
The central idea is that the system
defines the geometry. If you like, you can use the conventional description
to fit the thing into the ‘ordinary space-time’ to begin with, but then the
geometry you get out is not necessarily the one you put into it
It seems that euclidean space, at least, can be derived as a limiting case from simple combinatorial principles. It is not at all clear that general relativity does not have kolmogorov complexity comparable to the cellular automata of your “aether universes”.
Kolmogorov complexity of GR itself (text of GR or something) is irrelevant. Kolmogorov complexity of universe that has the symmetries of GR and rest of physics, is. Combinatorial principles are nice but it boils down to representing state of the universe with cells on tape of linear turing machine.
The interesting bit is that if you could actually use Solomonoff induction as prior, I’m pretty sure you’d be an irreparable aether believer and ‘relativity sceptic’, with confidence that has so many nines nothing would ever convince you. That’s because the absolute-speed irreversible aether universes like 3d versions of Conway’s game of life are so much more computationally compact than highly symmetric universe with many space-time symmetries (to the point of time and space intermixing in the equations) and the fully reversible fundamental laws of physics. Saying this to you as applied mathematician with experience actually implementing laws of physics in software.
The priors are only a free parameter if all you care for is not getting Dutch-booked on your bets about physics but do not give a damn to have any useful theories.
Can you explain why this is the case? Also, when you say “aether” universes are more computationally compact than “relativity” universes, is this before or after taking into account our observations (i.e., are you restricting your attention to universes that fit our observations, or not)?
Is it possible that what you said is true only if we want the laws of physics to run fast on current computers? I’m afraid that most software people, including myself, probably have bad intuitions about Solomonoff Induction because we only have experience with a very small subset of possible computations, namely those that can be run economically on modern computers. Perhaps the laws of physics can be implemented much more compactly if we ignored efficiency?
As intuition pump consider the reversible vs irreversible cellular automata. If you pick at random, vast majority will not be reversible. Ditto for the symmetries. (Keep in mind that in Solomonoff probability we feed infinite random tape to the machine. It is no Occam’s razor. Elegant simplest deterministic things can be vastly outnumbered by inelegant, even if they are most probable. edit: that is to say you can be more likely to pick something asymmetric even if any particular asymmetric is less likely than symmetric)
There can always be a vast conspiracy explaining the observations… ideally if you could simulate whole universe (or multiverse) from big bang to today and pick out the data matching observations or the conspired lying, then maybe it’d work, but the whole exercise of doing physics is that you are embedded within universe you are studying. edit: and that trick won’t work if the code eats a lot of random tape.
I don’t think relaxing the fast requirement really helps that much. Consider programming Conway’s game of life in Turing machine. Or vice versa. Or the interpreter for general TM on the minimal TM. It gets way worse if you want full rotational symmetry on discrete system.
Of course, maybe one of the small busy beavers is a superintelligence that likes to play with various rules like that. Then I’d be wrong. Can not rule even this possibility out. Kolmogorov/Solomonoff name drop is awesome spice for cooking proven-untestable propositions.
One could argue that second order logic could work better, but this is getting way deep into land of untestable propositions that are even proven untestable, and the appropriate response would be high expectations asian father picture with “why not third order logic?”.
edit: also you hit nail on the head on an issue here: i can not be sure that there is no very short way to encode something. You can ask me if I am sure that busy beaver 6 is not anything, and I am not sure! I am not sure it is not the god almighty. The proposition that there is a simple way is a statement of faith that can not be disproved any more than existence of god. Also, I feel that there has to be scaling for the computational efficiency in the prior. The more efficient structures can run more minds inside. Or conversely, the less efficient structures take more coding to locate minds inside of them.
Discrete universes introduce an unobserved parameter (a fundamental length) as well as a preferred frame. Abandoning complex numbers for a countable group in QM would also be very difficult. There is a complexity theory for real-valued objects and if Solomonoff induction could be adapted to that context, continuum theories ought to be favored.
Yep. Well, in any case, the point is that the Solomonoff probability is not in any way ‘universal prior’, you have to pick the machine, then you can’t actually compute anything useful because its uncomputable and you’ll never get anything non-trivial all the way down to minimum length.
If you go for changing the machine, you could use laws of physics as you know them as the machine, too (then you’ll be irreparable sceptic of any new physics). In the context of laws of physics its just the emulation of one universal computation (laws of physics) with another (what ever stuff you pick). We are only discussing this because some highly self-important people heard of Solomonoff induction and Kolmogorov complexity, modelled those with something ill defined and fuzzy (lacking not only sufficient understanding of those things but the understanding of the concepts in which to understand those, and so on several layers deep, as is usually the case when you choose something really complicated without having spent a lot of time studying towards it) and then used it as universal smart sounding word to say (e.g. here or here or here ). Name-dropping and nothing more.
Consider Penrose’s “Angular momentum: An approach to combinatorial space-time” (math.ucr.edu/home/baez/penrose/Penrose-AngularMomentum.pdf)
It seems that euclidean space, at least, can be derived as a limiting case from simple combinatorial principles. It is not at all clear that general relativity does not have kolmogorov complexity comparable to the cellular automata of your “aether universes”.
Kolmogorov complexity of GR itself (text of GR or something) is irrelevant. Kolmogorov complexity of universe that has the symmetries of GR and rest of physics, is. Combinatorial principles are nice but it boils down to representing state of the universe with cells on tape of linear turing machine.