11–20 of 29 posts
a bit vague and hand wavey. also I'm baffled how the author didn't use the term "chaos theory" in the text.
I think this article was rather weak. The double pendulum is not "unexplainable" or "inexplicable behavior", in fact is is explained very well in this very article. It just requires an infinite degree of precision if you desire to simulate it numerically with infinitesimal error. It's (an easily explained) limitation of the numerical methods used, not a lack of explanatory power. It does not contradict universality a…
I think this article was rather weak. The double pendulum is not "unexplainable" or "inexplicable behavior", in fact is is explained very well in this very article. It just requires an infinite degree of precision if you desire to simulate it numerically with infinitesimal error. It's (an easily explained) limitation of the numerical methods used, not a lack of explanatory power. It does not contradict universality a…
Two simulations of chaotic systems (starting identically) with different step sizes will always diverge (The difference in eventual positions does not stabilise as steps are made smaller). For this reason, I am not even sure if infinitesimal steps would avoid divergence from (ideal) reality. Plus y'know, the whole issue of a simulation with infinitesimal steps never making ANY progress, regardless of how fast it runs.
Therefore, I conclude that infinite degrees of precision is not the issue or solution for numerical explanation of chaotic behavior.
I think this article was rather weak. The double pendulum is not "unexplainable" or "inexplicable behavior", in fact is is explained very well in this very article. It just requires an infinite degree of precision if you desire to simulate it numerically with infinitesimal error. It's (an easily explained) limitation of the numerical methods used, not a lack of explanatory power. It does not contradict universality a…
Numerical simulations that I know of, all use steps or iterations of some type. Even if we start with theoretical infinite/perfect precision of the double pendulum, the numerical simulation will diverge from (ideal frictionless!) reality. Two simulations of chaotic systems (starting identically) with different step sizes will always diverge (The difference in eventual positions does not stabilise as steps are made sm…
When you discretize a continuous equation for numerical analysis, you always make sure to use a consistent discretization. The point of a consistent discretization is that it can be proven its solution will converge to the exact solution of the continuous equation as the step size approaches 0.
Consistent discretizations are possible even for nonlinear equations.
Either way we are simply discussing limitations of a chosen numerical method, which doesn't really support the arguments in the article.
I think this article was rather weak. The double pendulum is not "unexplainable" or "inexplicable behavior", in fact is is explained very well in this very article. It just requires an infinite degree of precision if you desire to simulate it numerically with infinitesimal error. It's (an easily explained) limitation of the numerical methods used, not a lack of explanatory power. It does not contradict universality a…
Pi cannot be expressed precisely using any integer-based numeral system (I suppose one could contrive a number-base that incorporated pi, one in which one of the numerals represented pi).
> there is no equation (that we know of) that would allow us to write down where the pendulum would be at some point in the future, given its current position. This is a misunderstanding. The author uses the sine and cosine functions as if they were “functions which give us values” but if you are allowed to assume that (what is sin(1), by the way?), then one might as well define “Pend2(t)” as “the solution to the dou…
You misunderstand sensitive dependence on initial conditions, though admittedly it’s poorly elucidated in the article. It’s not about closed form solutions. It’s about how neighborhoods on the line are mapped.
The sensitivity to initial conditions has nothing to do with regular ODEs and uniqueness.
I think this article was rather weak. The double pendulum is not "unexplainable" or "inexplicable behavior", in fact is is explained very well in this very article. It just requires an infinite degree of precision if you desire to simulate it numerically with infinitesimal error. It's (an easily explained) limitation of the numerical methods used, not a lack of explanatory power. It does not contradict universality a…
The interesting thing is, while the double pendulum does exhibit randomness, it would not appear to be ergodic. If you look at a plot of the double pendulum in phase space, it has some interesting structure. There exists stable regions of phase space which seem to imply non-trivial solutions.
Unstoppable “Stuff”: A Fractal Synthesis of Light and Darkness
https://docs.google.com/document/d/1aLV89MuNTdk8hPNBXEEFSfAk...
TLDR: Do we live in Math World, or does Math World live in Philosophy Town?
No idea if it’s right, but it’s the best I’ve got with available information now.
What do you think is the true source of the “Stuff” ? Could the trees be made of logic?
Hey, I’ve fallen down the rabbit hole on this and stayed up all night writing a simpler v3 theory about it: Unstoppable “Stuff”: A Fractal Synthesis of Light and Darkness https://docs.google.com/document/d/1aLV89MuNTdk8hPNBXEEFSfAk... TLDR: Do we live in Math World, or does Math World live in Philosophy Town? No idea if it’s right, but it’s the best I’ve got with available information now. What do you think is the tr…
- The double pendulum is a chaotic system, which means that starting states which are close together can quickly diverge.
- This has nothing to do with nonlinearity as such; it is also true of many linear systems.
- The concept of well-posedness in differential equations addresses this question, which is (from one point of view) about whether it's even worth trying to numerically solve an equation, or whether cutoff errors will quickly destroy your solution. The time-reversed heat equation is the best example of an ill-posed linear system.
- None of this touches at all on the universality of mathematics!