What is Chaos Theory?

[Leigha]

Chaos theory assumes that there is order behind seemingly random events, and that very small occurrences within a system can induce significant changes. Chaos explores the transitions between order and disorder, which often occur in surprising ways.

Are there practical uses for chaos theory? Is it more of a mathematical theory, or a philosophy? Are our lives demonstrations of chaos theory in action? (Minor occurrences/decisions add up to create significant changes)

Not much talk about Chaos Theory, but I find it interesting. 

[Ostronomos]

A truly random system transitions from pure chaos to pure order. Chaos as defined by me is the absence of order. It is where logic meets contradiction and incoherence.

A beautiful and elegant system such as our universe is free from contradiction, and so it may be defined as orderly. Its primordial origin may be defined as chaotic.

[Magical Realist]

https://www.youtube.com/watch?v=3orIIcKD8p4

James Gleick on Chaos Theory.

[Yazata]

Chaos theory assumes that there is order behind seemingly random events, and that very small occurrences within a system can induce significant changes.

I tend to think of this in terms of non-linear dynamics.

Imagine a mathematical function. This is a one-to-one mapping of mathematical objects (typically real numbers) to other mathematical objects (real numbers). It can be thought of as a black box, where if you input a number, another number pops out. Functions can be illustrated in graphs, there the x-axis is the input number and the y-axis the output number.

All of those mathematical hieroglyphs that theoretical physicists love so much are defining functions that they believe or hypothesize define the relationship between different  physical variables.

Typically, these graphs are lines. Meaning that if we change the x input an infinitesimal amount, the y output only changes an infinitestimal amount. The smaller the change in x, the smaller the change in y.

But not always. Sometimes the graphs of functions aren't lines at all, but rather dusts. A small difference in x might produce a huge difference in y, such that the points of the graph are scattered all over seemingly randomly.

It isn't entirely undetermined since if there's still a function defining where each y falls given a particular x, there would still be a deterministic function. But the non-linearity makes extrapolation and interpolation impossible.

And if we only have measurements of y, a distribution that appears random, the assumption that the y values are indeed all determined by x's through precise one-to-one functions is basically an expression of metaphysical faith, of how we a-priori imagine the universe to operate. I'm not sure how one could demonstrate it.

The bottom line is that with some functions, even an infinitesimal difference in x can be associated with an arbitrarily large difference in y.

Chaos explores the transitions between order and disorder, which often occur in surprising ways.

The question then is how many of what seem to be totally random (the word used is stochastic) processes in nature are in fact determined by nonlinear functions? (And how to recognize the action of a non-linear function experimentally when we see it.)

Are there practical uses for chaos theory?

Back in the 1990's this kind of stuff created tremendous excitement and many universities created research units to study it. But I don't believe that many useful results ever emerged and much of the excitement has dissipated. They did succeed in producing some interesting (though perhaps not definitive) mathematical tests for chaos though.

Is it more of a mathematical theory, or a philosophy?

I don't really distinguish between philosophy, mathematics and physics at the level of fundamentals. It's all three and of interest to all three.

Are our lives demonstrations of chaos theory in action? (Minor occurrences/decisions add up to create significant changes)

I can't prove it, but I'm inclined to think (very strongly) yes.

Quantum mechanics might conceivably have big implications for this. If some of the functions describing the evolution of physical systems and the unfolding of physical reality itself are indeed non-linear, such that even infinitesimal differences in the value of input variable x can be associated with huge differences in how the universe unfolds (the value of subsequent variable y)... and if some physical values on the microscale are undefined, don't always have precise values and can sometimes be probabilistic, then we could perhaps argue that there aren't any mathematical functions (in the 'if a particular x, then a particular y' sense) governing these events. They might just be stochastic by their fundamental nature.

If quantum mechanics make the idea of precise x's problematic, and if nonlinear dynamics can amplify even infinitesimal differences in x into big differences in y (in how the universe unfolds), then the 19th century deterministic picture of physical reality might start to collapse.

That's one reason why I'm not inclined to imagine time as a single one-dimensional line extending from a determined past (what happened happened) to a determined future (what is going to happen must happen precisely that way).

I'm more inclined to imagine the future as an infinitely branching tree, where countless states of affairs are consistent with the (imprecise on the microscale) state of the universe now.

[Ostronomos]

Yazata,

You raise an interesting point about non-linear dynamics. Suppose that there is a dimension that contains chaos. I am talking about an actual container for chaos. What are your thoughts about this?