[i_cannot_hack]

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 as defined in the article.

In the same way the ratio of a circle's circumference to its diameter is easily explained and understood, even if expressing it in the base-ten numeral system would require infinite digits.

The suggested equivalence between linear/nonlinear phenomena and inside/outside human perception was also tenuous and poorly justified.

[bmc7505]

> 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.

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.

https://youtube.com/watch?v=gvck7ssg9dE&t=11m30s

[matjet]

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 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_cannot_hack]

I'm not sure I follow you.

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.

[TheRealNGenius]

I agree, I just skimmed it but the examples they used and their arguments were weak. It reads as if they haven't actually dug deep into the material they are presenting.

When they mentioned not being able to calculate double pendulum, my mind immediately jumped to concepts of computability. For instance, we know that the Halting Problem cannot be solved by a Turing machine. We (or aliens) can introduce an oracle, but that would have it's own equivalent halting problem. These are truths that have been proven.

This follows for any and all theorems. You start with a set of axioms, and then successively arrive at your proof through logical steps. Aliens might come up with new questions, new answers, etc. But that has no bearing on the validity of our mathematics.

[denton-scratch]

> the base-ten numeral system would require infinite digits

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).

[nobodyandproud]

The point of the article was that we don’t have a way of predicting the position at a point in time, without calculating all of the interim steps.

An analogy would be the ancient mathematicians aversion to infinites (Calculus); also, being unable to imagine non-Euclidean planes.

There may be better tools out there that we haven’t considered, because it’s so far removed from our intuition.

Edit: Was it a Vernor Vinge book, where humans had a small (but great) advantage over more established, space-faring aliens because of their ability to handle infinites (calulus)? Whereas all aliens relied on numerical/computional approximations.

[i_cannot_hack]

> The point of the article was that we don’t have a way of predicting the position at a point in time, without calculating all of the interim steps.

Sure. But that is true for most linear systems of PDEs as well. It has very little to do with nonlinearity. And my point is that it does not contradict universality as defined in the article, as was claimed by the author.

[nobodyandproud]

Aren’t chaotic systems non-linear?

It’s the non-linear PDEs that get interesting, and one is reduced to iterative approximation or computational methods, way back when I went to school for this stuff.

[i_cannot_hack]

You are more often than not reduced to iterative numerical approximations for very simple linear PDEs as well.

Consider a simple linear PDE such as the heat equation du/dt = Δu + f(t). On a square or a circle you can solve this analytically in the frequency domain using separation of variables. But as soon as you consider an arbitrary domain (say, shaped like an elephant) you can no longer solve it in the frequency domain and need to use iterative numerical approximations.

[adonovan]

Is it possible that we might discover some new branch of (say) calculus that produces an exact formula for position as a function of time? (To my amateur eyes it looks like a tricky integral.) If so, this does appear to be merely a weakness of iterative simulation.

Or can we prove that there is no simple exact formula?

[freemint]

This is a good question and the answer depends on what you mean by formula.

One could probably show that there exists no formula using a finite number of +,*,/,^. However one might be able to define functions which together with a finite number of operations from above allow to express solutions. However it is likely that calculating those helper functions even when they are well studied and known is basically solving a slightly different differential equation (or as you said integration problem).