[...] AFAIK , mathematical structures emerge from the self-ordering mathematical interactions of relational values even in a chaotic environment.
Chaos theory explains the emergence of self-ordering patterns from a dynamic and chaotic environment (fields).
Does that not suggests that the implicated future is as yet a dynamically disordered (probabilistic) environment and order might become expressed in a variety of patterns, based on the actual conditions present at the time of physical change in an otherwise probabilistic future reality?.
This is departing to something outside MUH, though.
Probability approaches might be considered a substitute for accurate or precise prediction when the latter is impossible for humans to achieve. Rather than those reflecting or espousing a metaphysical view that the future is ambiguous or unsettled. (Unless this was rubbing shoulders with some "multiverse" scenario or interpretation of each of us consciously branching off into different timelines and parallel worlds, or whatever. That would certainly be an "unsettled future" in terms of what was individually and personally encountered.)
As for chaos: Nonlinear systems are deterministic (not truly random). But it would require absolute knowledge of a current state's configuration to precisely or accurately predict the system's future long-term, due to a sensitivity to initial conditions.
A definition of "genuine randomness" would be that such concerns events that do not adhere to (or adhere wholly to) any pattern or principle whatsoever. Thus making them truly unpredictable even in theory. IOW, even a god would only know those future events via being able to literally access the future -- and not prediction by means of calculation.
IOW, there seems to be no justification for genuine randomness (if such were the case) being dependent upon the future not existing, or the future being unsettled. For instance, if experts could omnisciently declare in the present that an _X_ was truly random (not conforming to any pattern or rule), then _X_ will still carry that "random" assessment as it becomes part of the distant past. Even in the latter context, there would still be no pattern or principle subsuming it.
Tegmark explores the possibility of the universe being a simulation in MUH, but even in that context it is not a view of "computation" that is dependent on time "having a flow" (i.e., some mysterious substance circulating through the extra-dimensional structure), and only the past and specious present existing.
Max Tegmark: ... Lloyd has advanced the intermediate possibility that we live in an analog simulation performed by a quantum computer, albeit not a computer designed by anybody — rather, because the structure of quantum field theory is mathematically equivalent to that of a spatially distributed quantum computer. In a similar spirit, Schmidhuber, Wolfram and others have explored the idea that the laws of physics correspond to a classical computation. Below we will explore these issues in the context of the MUH.
[...] Suppose that our universe is indeed some form of computation. A common misconception in the universe simulation literature is that our physical notion of a one-dimensional time must then necessarily be equated with the step-by-step one-dimensional flow of the computation. I will argue below that if the MUH is correct, then computations do not need to evolve the universe, but merely describe it (defining all its relations).
The temptation to equate time steps with computational steps is understandable, given that both form a one-dimensional sequence where (at least for the non-quantum case) the next step is determined by the current state. However, this temptation stems from an outdated classical description of physics: there is generically no natural and well-defined global time variable in general relativity, and even less so in quantum gravity where time emerges as an approximate semiclassical property of certain “clock” subsystems.
Indeed, linking frog perspective time with computer time is unwarranted even within the context of classical physics. The rate of time flow perceived by an observer in the simulated universe is completely independent of the rate at which a computer runs the simulation.
Moreover, as emphasized by Einstein, it is arguably more natural to view our universe not from the frog perspective as a 3-dimensional space where things happen, but from the bird perspective as a 4-dimensional spacetime that merely is.
There should therefore be no need for the computer to compute anything at all — it could simply store all the 4-dimensional data, i.e., encode all properties of the mathematical structure that is our universe. Individual time slices could then be read out sequentially if desired, and the “simulated” world should still feel as real to its inhabitants as in the case where only 3-dimensional data is stored and evolved.
[...] In conclusion, the role of the simulating computer is not to compute the history of our universe, but to specify it. ... Each relation of the mathematical structure is thus defined by a computation. In other words, if our world is a well-defined mathematical structure in this sense, then it is indeed inexorably linked to computations, albeit computations of a different sort than those usually associated with the simulation hypothesis: these computations do not evolve the universe, but define it by evaluating its relations. --The Mathematical Universe (paper - either 2007 or 2019)
IMO, the confusing factor in a mathematically ordering universe is that the mathematics are influenced by the dynamical nature of spacetime, that prevents the perfect explication of the mathematics that guide each individual system.
Example: A perfect circle is an abstract Platonic object, but in a dynamic environment most circular objects and trajectories are expressed in elliptical torm, a distortion from purely circular. The earth itself may appear spherical but is a variable ellipsoid.
If we used guiding "railroad tracks" as a crude analogy, those do not follow ideal geometrical patterns or shapes, either (curving and winding all over the place sometimes). But the spaghetti "paths" through Tegmark's 4D structure would seemingly have to feature fewer imperfections than our flat highways and railroad tracks in order for us to abstract regulating laws and principles from them at all that were reliable.
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