Los Frupanishads

[Cain]

I'm going to add something on Mandlebrot tonight.  As you may be aware, I am not overly thrilled or enamoured with Chaos Theory in general, at least when compared to what most people call Chaos.

Stay tuned.

[LMNO]

Yes, i should reiterate that what mathemeticians (and many physicists) call "chaos" is much different than what most other people call "chaos", which is different than many Discordians at this site (and elsewhere) call "chaos".

To conflate all three to make some sort of point seems to be pointless.

[Triple Zero]

well yes, partly.

they are not the same.

but they all share unpredictability.

that was part of your point, and your problem with peregrineBF's statement. that mathematical chaos is not unpredictable. but it often is.
although the meanderings of the threadjack about mandelbrot showed that the mandelbrot set is a bad example of this (there is probably unpredictability in that set, but it's not very obvious or illustrative)

nurbldorff mentioned the logistic map, a much simpler fractal (doesn't use complex math) and demonstrating the prediction problems of mathematical chaos much clearer.

[LMNO]

Well, yeah.

I feel I could simplify it even more and say that as you begin to add more variables, it becomes harder to predict what the outcome will be beforehand, and eventually becomes nearly impossible until you actually perform the calculations 9substitute numbers fro variables.

For example, the best you can say about a trail of cigarette smoke before hand is that it will most likely travel upwards.  To know how it will actually behave would take an insane amount of computing power, plus a mountain of data about the environment, and the combustible material.



Gosh, I really hope i didn't just contradict myself there.

[Cain]

If you're in a pub, the smoke will automatically gravitate to the nearest off duty policeman, who will then issue you a £50 on the spot fine.

True story.

[LMNO]

The Man™:  the Overriding Variable.

[Triple Zero]

Well, yeah.

I feel I could simplify it even more and say that as you begin to add more variables, it becomes harder to predict what the outcome will be beforehand, and eventually becomes nearly impossible

yes. chaos and complexity are closely related.

until you actually perform the calculations 9substitute numbers fro variables.

until how what? sorry maybe i get confused by the typo but what are you trying to say here?

For example, the best you can say about a trail of cigarette smoke before hand is that it will most likely travel upwards.  To know how it will actually behave would take an insane amount of computing power, plus a mountain of data about the environment, and the combustible material.

Gosh, I really hope i didn't just contradict myself there.

i dunno if you did

but the point i've been trying to make for some time now, is it wouldn't only take an *insane* amount of computing power, but in order to predict it beyond a very finite and tangible timespan, it would require an *impossibly insane* amount of computing power. not "today impossible but maybe in 10 years" but more "impossible to get the initial state of up to enough precision due to the uncertainty principle" or "impossible to calculate in any reasonable timespan as compared to the current estimated age of the universe" or even "impossible to store the state of with enough precision, comparing the amount of memory chips needed to the size of our solar system".

i think i could probably say with certainty that -for example- we'll never be able to predict a reasonable weather forecast past the three weeks range. no matter what kind of computing power we'll throw at it.

(and as far as i understand quantum computing (which is hardly at all), it isn't going to help very much. maybe just to keep Moore's law going for another decennium or two -- which would be quite impressive nonetheless)


also, Cain

[LMNO]

gah.  fucking server.


LOL post ruint.

[Triple Zero]

that means i win?

so anyway, it's obvious that Benoit Mandelbrot invented the first breakbeats.

chaos = unpredictable = syncopated
QED

[LMNO]

Basically, i was saying instead of variable, plug in actual numbers.

The math equivalent of "doing the experiment".







Also, syncopated beats are predictable, but not uniform.

[nurbldoff]

The problem doesn't only lie in lack of computing power. The thing is, no matter how much computing power you've got, you still have the problem of accuracy of your initial conditions. A calculation that is iterating a formula over time will accumulate error. Usually this is OK, you can still make accurate predictions because your model is stable (the error may grow linearly or something like that), but in a "chaotic" or ("dynamical") system, this error will blow up very quickly and overtake the scale of the measurement itself. At which point the calculation becomes useless. This puts a cap on the predictions you can make in a real system, even with unlimited computing power.

In order to know where the cigarette smoke went, you'd have to get position and speed measurements of each particle of smoke when it leaves the cigarette, plus every molecule of air surrounding it, which in turn will be influenced by the movements of everyone in the room. Even if you could measure all of that simultaneously, you'd still only have finite accuracy - if nothing else, ultimately limited by Heisenberg's uncertainty principle.

I'm not sure about the actual numbers, but I saw some calculations that said something like, even if you had a set of really accurate sensors reading temperature, humidity, wind speed, etc every cubic meter in the entire atmosphere giving readings every second, you still couldn't even theoretically predict weather reliably (whatever that means) for longer than something like a week. The model is just too sentitive to errors. Within days you get to a point where the system can go either way, depending on differences in initial conditions smaller than the accuracy of your instruments - essentially the old "hurricane butterfly" thing.

Btw, I'm not even sure what the "everyday" definition of chaos is, if there is even such a thing... what do most people mean by "chaos", exactly?

[Bebek Sincap Ratatosk]

The problem doesn't only lie in lack of computing power. The thing is, no matter how much computing power you've got, you still have the problem of accuracy of your initial conditions. A calculation that is iterating a formula over time will accumulate error. Usually this is OK, you can still make accurate predictions because your model is stable (the error may grow linearly or something like that), but in a "chaotic" or ("dynamical") system, this error will blow up very quickly and overtake the scale of the measurement itself. At which point the calculation becomes useless. This puts a cap on the predictions you can make in a real system, even with unlimited computing power.

In order to know where the cigarette smoke went, you'd have to get position and speed measurements of each particle of smoke when it leaves the cigarette, plus every molecule of air surrounding it, which in turn will be influenced by the movements of everyone in the room. Even if you could measure all of that simultaneously, you'd still only have finite accuracy - if nothing else, ultimately limited by Heisenberg's uncertainty principle.

I'm not sure about the actual numbers, but I saw some calculations that said something like, even if you had a set of really accurate sensors reading temperature, humidity, wind speed, etc every cubic meter in the entire atmosphere giving readings every second, you still couldn't even theoretically predict weather reliably (whatever that means) for longer than something like a week. The model is just too sentitive to errors. Within days you get to a point where the system can go either way, depending on differences in initial conditions smaller than the accuracy of your instruments - essentially the old "hurricane butterfly" thing.

Btw, I'm not even sure what the "everyday" definition of chaos is, if there is even such a thing... what do most people mean by "chaos", exactly?

*ding!*

I think one interpretation of Chaos that I like fits well with what nurbldoff said.

Here we are with models and math (which is just a complex model). However, (let's all say it together ) the model is not the thing being modeled. Thus, the model doesn't have all of the data, thus there are unknowns. These unknowns, are denizens of the Void, children of Eris and that damned undefinable X.

[nurbldoff]

OK, I get what you're saying, but for correctness I have to clarify that the actual model can be chaotic. So, even if you had such a model that perfectly described a real system (OK, OK, a pure mind concept but bear with me) it would still behave chaotically. The fact that models are never perfect descriptions of reality is a separate problem, of course adding to the trouble with predictions.

But I guess "unpredictability" is a large part of what people generally mean when they talk of chaos.

[LMNO]

Doesn't the "unpredicatbility" stem from imperfect knowledge of the conditions?


If we had complete knowledge of the conditions, why wouldn't we be able to predict the motion?

[Bebek Sincap Ratatosk]

OK, I get what you're saying, but for correctness I have to clarify that the actual model can be chaotic. So, even if you had such a model that perfectly described a real system (OK, OK, a pure mind concept but bear with me) it would still behave chaotically. The fact that models are never perfect descriptions of reality is a separate problem, of course adding to the trouble with predictions.

But I guess "unpredictability" is a large part of what people generally mean when they talk of chaos.

Oh don't get me wrong, I agree with what you're saying. If we could create a Simulacrum of this Universe somewhere else in the Multiverse complete with monitors and measuring tools... we still wouldn't be able to predict the weather, and in that lies Chaos in one sense.

Chaos in another sense is that none of our models can actually get to the point where we could model all the data.

Further Chaos in that we can't even agree on which data should be there because of perception and interpretation. Yet more Chaos in the sense (as Old Uncle Al said) that in all of our models we swap X, Y and Z around in the values slots but we still don't actually get anywhere (There is no God, we came out of Primordial Soup is swapping X for Y and Panspermia is X for Z and the whole concept of abiogenesis was thought to be rubbish of the Middle Ages... until we wanted to stick it back in a model )

Doesn't the "unpredicatbility" stem from imperfect knowledge of the conditions?


If we had complete knowledge of the conditions, why wouldn't we be able to predict the motion?

Because Heisenberg was an ass and keeps sticking his fingers in from beyond the grave.