Alien Mathematics(troynikov.io) |
Alien Mathematics(troynikov.io) |
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 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.
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.
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.
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.
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).
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.
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.
Or can we prove that there is no simple exact 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).
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.
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.
- 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!
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.
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?
“ 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 rejects this idea as shallow and fallacious.
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?
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!
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.
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.
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.