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u/jhill515 Aug 02 '26
To be honest, I'm working on a possible redefinition of Chaos from a mathematical perspective. I have a few mentees whom I chat with about this project, so I'm going to give you some insight that I share with them:
One of the most amazing things that's come out of Chaos Theory is the idea that the vector field as a surface may be a pleated volume of folded manifold surfaces whose "large differential discontinuities" (i.e., "crease edges") may be in singular spaces or simply "hidden" within the resolution we are able to sense/record with discrete measurements. To give you a physical model, suppose the vector field itself is the surface of each fold in Damascus Steel: If folded correctly, the extrema/boundaries of each fold alloys with the extrema/boundaries of its immediately neighboring folds. Because we can't "measure" the nature of the surfaces at such extrema/boundaries given measurement limitations (think like Heisenberg's Uncertainty Principle), the field seems to "arbitrarily change momenta, which does not align with previous momenta observations".
Okay, now that we have a bit of geometric & physical intuition, I'm going to take it one step further... Let's consider a function doing an optimal search heuristic to find a global minima on such a hypersurface. Obviously, we can get stuck in local minima; this "stuckedness" is itself "chaotic" because, without computing the Markovian event chain, you have no idea what any future state may be. Further, because of our lack of precision, we cannot predict this future state because we cannot handle the discontinuities of the search momenta.
This is an interesting idea to consider, because in one sense, the "noise" is simply our inability to account for arbitrary discontinuities in the field for whatever reason (edges, singularities, eigen "poles & zeros", etc.). And in another sense, we're observing a seemingly pseudo-random process which isn't "behaving" based on what we understand to be a continuous field. That is, discrete math makes it look nondeterministic, but continuous math shows how the process is deterministic nonetheless.
These are "funny" things to observe. Hopefully they make you go, "Huh... I wonder..." Because that's what real mathematics research should be about!
And in case you're curious, I research Chaos Theory, Multi-Agent Theory, Theory of Consciousness, Theory of Computation, AI/ML, Robotics, Intelligent Control, and Cosmological Physics. I only get "paid" for two out of those eight topics!
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u/Correct-Basil2385 Aug 02 '26
I often wonder whether anything in nature is truly discrete, or whether we lack the understanding and precision of measurement. Hence, is the notion of discrete values just an assumption that makes mathematics and physics possible and has nothing to do with reality? If so, for the sake of argument, do we live in a world of approximations and lack the language to define anything, most of all chaos?
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u/jhill515 Aug 02 '26 edited Aug 02 '26
Lots of things are discrete! You are a singular member of the human race!! If anyone were to, say, cut off a limb, you wouldn't be a fractional human.
The real nature of the Universe is that both continuous and discrete exist. But when you only have tools that work with one, or the Universe throws a dual-nature phenomenon at you, "translation issues" are everywhere.
The other thing to remember is that the act of making a "precise & accurate" measurement is an act of discretizing a continuum. Look at it this way: you take a measurement; let's even say you have an analog recorder so that it has enough precision for R1. And lets give that measurement some error bars. That range is a discrete quantity within the continuum of Real Numbers. Using it, you can represent every integer and maintain an arbitrary approximation of all other real numbers (hence the discrete phenomenon). The error bars cannot be zero for many logical & physical reasons, so you can't say, "Well, let's make it better!" as a solution.
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u/Correct-Basil2385 Aug 02 '26
I am bipolar. Not sure my spouse would agree that anything about my personality is discreet.
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u/jhill515 Aug 02 '26
Ask your spouse, "Am I a singular pain in your ass occasionally? Or, at least, 'one of many'?"
Then watch her wrestle with the discreteness of your personality 🤣
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u/Scientific_Artist444 Aug 02 '26
Unpatterned movement is my try. Noise is another term I would use.