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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.
> 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
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'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.
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.
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.
> 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.
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.
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.
> 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).
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?
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).
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 double pendulum equation”, and be done with it.
Which, by the way, is how one defines the exponential, trigonometric, even n-th roots! Not to say the erf, Bessel, hypergeometric functions etc.
The fact that there is not a “closed solution involving only elementary functions” is irrelevant as long as the equations ones is solving have a unique solution.
Edit: toned down.
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.
That does not matter: either the solutions are unique or not, and ODEs (out of singular points) have a unique aolution for any set of initial conditions.
The sensitivity to initial conditions has nothing to do with regular ODEs and uniqueness.
> ODEs (out of singular points) have a unique solution for any set of initial conditions
I am not sure if that statement is to weak. In general you can only guarantee the solution of an explicit ODE over some interval if the right hand side is Lipschitz continuous.
Well, yes. But the author is assuming it and his equations are C-infinity as a matter of fact.
The author has confused several mathematical concepts, unfortunately.
- 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!
Nice piece meant for one to pause and think a little bit about limitation of mathematics and our perception of reality. There, however, I would submit that I find the mixture of several limitations too confusing to even allow (in my view) the question the article outlines to be a valid one. First, our perception limitation (that is a multi-level problem, going from personal all the way to entire species). Im not sure at which level the author wants to start and stop. Second, limitations of mathematics explaining physical phenomena as we perceive them and can measure them. Third, mathematical models are not the reality and never have been considered to represent reality as is. They are necessary abstractions for us to have our linear and local time predictability.
I would further submit for a thought that as of right now there is nothing we understand from first principles using mathematical models. This is not a conjecture or speculation, but a fact. See eg. Wheeler's "More is different", ("More is really different" by another author), or R. Laughlin's book Different Universe which with simple logic shows physical laws cannot be build from first principles, because, well, "more is different" (emergent phenomena acquire characteristics not in the original constituents). Lets think about weather, clima, planet formation, galaxy evolution, economics, etc ... It would be foolish to even begin such enterprise...(imho). But, in any case this is issue is not to be closed for discussion and interpretations and learning...that I agree with the author.
The article incorrectly describes the equations of kinematics of the double pendulum as describing its dynamics. They do not; describing its dynamics requires taking time into account and considering how its angles and positions change over time.
It also considers the existence of a general theory of nonlinear mathematics. There's a famous quote to the effect that this makes as much sense as the study of non-elephant mammals. And I think the argument that our everyday experience is linear is very weak; our bodies are made of non-Newtonian fluids, fluid friction is nonlinear, everyday physical phenomena like solid bodies coming into contact with one another are wildly nonlinear.
So in a sense I would criticize this article in the same way it criticizes Deutsch's book: it's flawed and weak, but still thought-provoking. Even if we can't predict the double pendulum's position several Lyapunov times in the future with any degree of certainty, are there things we can say about it that haven't yet been said? Is there a way we can look at things (or that some physical system could look at things) that would enable useful cognition?
From the article -
“ The double pendulum has the special property that very small changes in initial conditions result in very large changes in eventual outcome. And that means small approximation errors compound much faster than we can deal with them - the system diverges**.”
- isn’t this just a problem of lack of computing power rather than weakness of the equations themselves. Imagine that there one unit change in the source results in a 1000 new combination , but having computing power that can scale horizontally can solve it. I am aware that current computing power (even in the cloud) is limited , but in future we may have quantum computing or something similar that can accommodate modeling these kind of divergence problems.
Big Brain is told it might not be a universal understander.
Big Brain rejects this idea as shallow and fallacious.
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 true source of the “Stuff” ? Could the trees be made of logic?
This will take me a while to read :) but thanks for sharing
a bit vague and hand wavey. also I'm baffled how the author didn't use the term "chaos theory" in the text.
Is disappointing because there is a potential for good deep thought experiments here, but when the author focuses on linear/nonlinear he sounds like he has no idea what he’s talking about and would have done better to stay at the level of metaphysics.
Part of this feels like nonsense, but it does chime with what I heard Stephen Wolfram talking about the other day: If we take the world to operate as he describes, which I do, then there are many ways to interpret the world, many different basis that could be constructed that paint a coherent picture. In this sense I think that the fire in the sky aspect of this article makes some sense. I just think that looking for these radically different interpretations of reality by postulating different kinds of mathematics, it doesn't work. imho.
Perhaps a better explanation for glowing orbs in the sky is a natural process, self sustaining plasma balls perhaps, or maybe electromagnetic solitons? Maybe even plasma life forms, self sustaining balls of interacting electromagnetic energy feeding off stuff. Crazy thoughts, but even these crazy thoughts are less crazy than things from other interpretations of reality, those things are weirder, think "cosmic horror".
Let's look into weird weather phenomena, then look into alien technology/plasma life forms. Then when all those things are done we might consider beings from another interpretation of reality!