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Why are you assuming that S[n+1] takes S[n] as an input, rather than being completely unaffected by S[n]? Is n here a discretization of time or a "meta-time" th

Why are you assuming that S[n+1] takes S[n] as an input, rather than being completely unaffected by S[n]? Is n here a discretization of time or a "meta-time" that captures state progression? If the former, then you are treating time as a primitive, whereas the representation or even existence of time can be part of state and vary over state changes, so this does not work. If the latter, are you assuming S[n] is an input because a universe where that is not the case and state "transitions" are completely independent from each other is indistinguishable from a multiverse with inaccessible sub-universes and should be called as such? In the latter, you are also assuming linearity and discreteness of state changes, whereas it could be a DAG or more complex structure. Edit: On further reflection, I think you are proceeding with the standard post-Renaissance assumption that a universe must always be governed by fundamental rule(s) at the "lowest level," rather than a more chaotic or unstructured universe that behaves more like a disjoint set than an orderly structure. I think it makes sense to constrain to that assumption as departing from it makes reasoning difficult but it is worth stating explicitly as an axiom.

hctaw 5y ago
Not GP but I feel your analysis here doesn't follow. > rather than being completely unaffected by S[n]? Is n here a discretization of time or a "meta-time" that captures state progression? It would be remarkable if the "next state" of the universe did not depend or was so correlated to the "current state" that symbolically we represent it as a dependency. A ball only rolls downhill because it was once uphill, after all - one of the few things that unites various models of the universe was the existence of causality. > If the former, then you are treating time as a primitive, whereas the representation or even existence of time can be part of state and vary over state changes, so this does not work. This is interesting but I feel reductive. the "nth" state is not the state at time n, but the state that follows n-1 - it is dimensionless, but one might express it in spacetime if that is convenient. Nothing suggests it is uniformly sampled or even computable. > completely independent from each other is indistinguishable from a multiverse with inaccessible sub-universes and should be called as such No idea what this implies, but in a non-deterministic state machine there are multiple S(n + 1) for any given S(n). > you are also assuming linearity and discreteness of state changes, whereas it could be a DAG or more complex structure. GP does not suggest evolve(S) is linear or discrete (they suggest either continuous or discrete, actually). Neither of these imply anything about topology so I'm not sure what that has to do with a DAG. State transitions can encapsulate arbitrary topologies (and vice versa). They are duals. > rather than a more chaotic or unstructured universe that behaves more like a disjoint set than an orderly structure I feel GP covers this base by mentioning that state transitions may not be deterministic. Just because a system is chaotic or unpredictable does not imply we cannot formulate the nature of its chaos or unpredictability symbolically.
danbruc 5y ago
Why are you assuming that S[n+1] takes S[n] as an input, rather than being completely unaffected by S[n]? If different states would be independent, if there were not even statistical relations, it would just be a sequence of random states. It would not even really make any sense to say that S[n] evolved into S[n + 1]. Is n here a discretization of time or a "meta-time" that captures state progression? I did not want to imply any discretization, I just could not come up with a better notation. Proper mathematical treatment would probably use some index set with some kind of order defined on it. This would probably also overcome the linearity of state evolution implied by my notation. And I used n intentionally to not imply time so that time can be emergent. On the other hand the entire idea of the state of the universe evolving is very heavily influenced by our experience with things changing over time. So I am actually less worried about the details of the independent variable, whether it is time or true-time, how many dimensions it has, or what not. I think the real problem is that the entire idea of state evolution might be wrong, maybe there is no state or no evolution and our brains are just not capable of imagining the way the universe actually works On the other hand we have quite a few very general results like Turing-completeness, incompleteness theorems, uniqueness of the reals, and whatnot. And all this is built on top of rather simple axioms of logic without assuming much so it seems not that easy that we are totally missing some fundamental structure that we are not only unable to imagine but where we also lack the necessary logical or mathematical tools to even describe them. But it surly might be that there is a third option besides x is true and x is not true that humans are incapable of understanding and therefore our logic and everything build on top of it is missing.

hctaw 5y ago

Not GP but I feel your analysis here doesn't follow. > rather than being completely unaffected by S[n]? Is n here a discretization of time or a "meta-time" that captures state progression? It would be remarkable if the "next state" of the universe did not depend or was so correlated to the "current state" that symbolically we represent it as a dependency. A ball only rolls downhill because it was once uphill, after all - one of the few things that unites various models of the universe was the existence of causality. > If the former, then you are treating time as a primitive, whereas the representation or even existence of time can be part of state and vary over state changes, so this does not work. This is interesting but I feel reductive. the "nth" state is not the state at time n, but the state that follows n-1 - it is dimensionless, but one might express it in spacetime if that is convenient. Nothing suggests it is uniformly sampled or even computable. > completely independent from each other is indistinguishable from a multiverse with inaccessible sub-universes and should be called as such No idea what this implies, but in a non-deterministic state machine there are multiple S(n + 1) for any given S(n). > you are also assuming linearity and discreteness of state changes, whereas it could be a DAG or more complex structure. GP does not suggest evolve(S) is linear or discrete (they suggest either continuous or discrete, actually). Neither of these imply anything about topology so I'm not sure what that has to do with a DAG. State transitions can encapsulate arbitrary topologies (and vice versa). They are duals. > rather than a more chaotic or unstructured universe that behaves more like a disjoint set than an orderly structure I feel GP covers this base by mentioning that state transitions may not be deterministic. Just because a system is chaotic or unpredictable does not imply we cannot formulate the nature of its chaos or unpredictability symbolically.

danbruc 5y ago

Why are you assuming that S[n+1] takes S[n] as an input, rather than being completely unaffected by S[n]? If different states would be independent, if there were not even statistical relations, it would just be a sequence of random states. It would not even really make any sense to say that S[n] evolved into S[n + 1]. Is n here a discretization of time or a "meta-time" that captures state progression? I did not want to imply any discretization, I just could not come up with a better notation. Proper mathematical treatment would probably use some index set with some kind of order defined on it. This would probably also overcome the linearity of state evolution implied by my notation. And I used n intentionally to not imply time so that time can be emergent. On the other hand the entire idea of the state of the universe evolving is very heavily influenced by our experience with things changing over time. So I am actually less worried about the details of the independent variable, whether it is time or true-time, how many dimensions it has, or what not. I think the real problem is that the entire idea of state evolution might be wrong, maybe there is no state or no evolution and our brains are just not capable of imagining the way the universe actually works On the other hand we have quite a few very general results like Turing-completeness, incompleteness theorems, uniqueness of the reals, and whatnot. And all this is built on top of rather simple axioms of logic without assuming much so it seems not that easy that we are totally missing some fundamental structure that we are not only unable to imagine but where we also lack the necessary logical or mathematical tools to even describe them. But it surly might be that there is a third option besides x is true and x is not true that humans are incapable of understanding and therefore our logic and everything build on top of it is missing.