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Alien Mathematics

troynikov.io

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Re: Alien Mathematics

#21

Post by adonovan »

i_cannot_hack wrote: post #3

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…

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?

Re: Alien Mathematics

#22

Post by nobodyandproud »

i_cannot_hack wrote: post #3

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 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.

Re: Alien Mathematics

#23

Post by freemint »

adonovan wrote: post #21
i_cannot_hack wrote: post #3

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…

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).

Re: Alien Mathematics

#24

Post by freemint »

pfortuny wrote: post #16
JackFr wrote: post #8

Earlier quoted context omitted.

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.

Re: Alien Mathematics

#25

Post by i_cannot_hack »

nobodyandproud wrote: post #22
i_cannot_hack wrote: post #3

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 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…

> 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.

Re: Alien Mathematics

#26

Post by la6471 »

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.

Re: Alien Mathematics

#27

Post by nobodyandproud »

i_cannot_hack wrote: post #25
nobodyandproud wrote: post #22

Earlier quoted context omitted.

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…

> 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.

Re: Alien Mathematics

#28

Post by i_cannot_hack »

nobodyandproud wrote: post #27
i_cannot_hack wrote: post #25

Earlier quoted context omitted.

> 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.

Re: Alien Mathematics

#29

Post by pfortuny »

freemint wrote: post #24
pfortuny wrote: post #16

Earlier quoted context omitted.

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.
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