Mathematics? It's all just life.

[Beestie]

What I tried to say is that I have a hard time establishing any cause-and-effect relationship between an independent (air movement from a butterfly flapping its wings) and a dependent variable (weather in Japan) in a chaotic system.

Let me put it another way: is anyone saying that had the butterfly not flapped its wings that it would not have rained in Japan? It sounds to me like that is what the example is suggesting and I'm saying that the idea is ridiculous.

I was saying that a scenario whereby the butterfly is the straw that broke the camel's back is (to put it mildly) not plausible because of the number and frequency of interactions that occur between the butterfly and the rainstorm. To adopt the butterfly theory is to ascribe an equal liklihood to each of the variables which is to render each of them essentially negligible.

To assume that one variable out of a trillion can propogate through the entire system - reinforcing everything in its path or its successor's path so as to rival the magnitude of a weather pattern defies any application of common sense.

The 3-body problem, imho is a different problem - the variables are discreet in number and, therefore, traceable. In a chaotic system, the variables are not traceable therefore, the notion of cause and effect has no application. While the butterfly may have flapped its wings and it may have rained in Japan, wether the two events are dependent or independent is unknowable.

In Zen, we can speak about chaos but never of chaos.

[Carbonated_Brains]

What I tried to say is that I have a hard time establishing any cause-and-effect relationship between an independent (air movement from a butterfly flapping its wings) and a dependent variable (weather in Japan) in a chaotic system.

But how is this different than you pushing a line of dominos, resulting in the last domino falling? It seems like you're arbitrarily "deciding" that a butterfly cannot possibly have an affect a year later around the world. Chaos theory highlights the dependency of massive outcomes on tiny events, that's the entire basis behind it.

Let me put it another way: is anyone saying that had the butterfly not flapped its wings that it would not have rained in Japan? It sounds to me like that is what the example is suggesting and I'm saying that the idea is ridiculous.

That's exactly what the theory states. Allegorically of course. Always keep in mind this is an allegory, the butterfly just REPRESENTS an infinitismally small alteration of the initial conditions.

One initial condition, A, is the butterfly does flap its wings. It rains in Japan a year later.

Initial condition B, is the butterfly does NOT flap its wings. Japan goes through drought.

The initial conditions, tiny and almost identical, have enormous impact on the final outcome: That little deviation, between flapping and not flapping, exponentially propogates into a massive change.

I was saying that a scenario whereby the butterfly is the straw that broke the camel's back is (to put it mildly) not plausible because of the number and frequency of interactions that occur between the butterfly and the rainstorm.

That may be true, but it's not the point. Whether it's the butterfly that actually causes the storm is completely irrelevant. The allegory is demonstrating that something (whatever it is) as small as a butterfly flapping its wings, can determine whether a huge event does or does not happen.

A tiny event results in a massive difference in outcome.


To adopt the butterfly theory is to ascribe an equal liklihood to each of the variables which is to render each of them essentially negligible.

Again, there is no butterfly theory. Chaos theory does not ascribe equal likelyhood to all variables, but states that all variables are capable of enormous impact on the end result, even tiny ones.

To assume that one variable out of a trillion can propogate through the entire system - reinforcing everything in its path or its successor's path so as to rival the magnitude of a weather pattern defies any application of common sense.

Why does it defy common sense? Just because it seems unlikely? I think common sense would say that if you knock over the first domino, the last one eventually falls. There's nothing that should suggest a major event can't start from a tiny, tiny key event.

The 3-body problem, imho is a different problem - the variables are discreet in number and, therefore, traceable.

They are no more traceable than the butterfly scenario. The variables in a 3 body problem could be as numerous as the atoms in each celestial body.

In a chaotic system, the variables are not traceable therefore, the notion of cause and effect has no application. While the butterfly may have flapped its wings and it may have rained in Japan, wether the two events are dependent or independent is unknowable.

It's only unknowable due to the fact that you can't be infinitely accurate. If you had a set of infinitely accurate measuring tools, you could definitely spend an infinite number of years doing calculations that eventually trace back to a butterfly flapping its wings.


In Zen, we can speak about chaos but never of chaos.


Chaos is the supreme ideal of Taoism. Chaos is wholeness, oneness, and Nature. Chaos represents the natural state of the world. Digging holes in the head of Chaos means destroying the natural state of the cosmos.

[russotto]

Originally posted by Carbonated_Brains

They are no more traceable than the butterfly scenario. The variables in a 3 body problem could be as numerous as the atoms in each celestial body.

Nyet. There are eighteen variables in the three body problem (position and velocity, in three dimensions, of the three bodies, with respect to the center of mass of the three). There are also three constants (mass of the three bodies). If you start worrying about the atoms in the bodies, it's not a three-body problem anymore.

[Carbonated_Brains]

I worded that wrong, methinks.

What I mean to say is, the main variables you mentioned are a direct result of things like atom configurations and on which side of the fridge you store the marmalade, etc.

Each major variable is an amalagamation of the exact positions and states of trillions of tiny components, an error in measurement of any one of those components can propogate and exponentially increase.

So if some football player in Kansas jumps into the air right when you're trying to do your calculation, the infinitesimally small amount of force he exerts to jump, and the resultant gravitational pull which brings the two bodies (him and the earth) together again, resulting in a tiny change in acceleration of the Earth, resulting in a temporary change of initial conditions...that could be enough to throw off the calculation.

I doubt I'm explaining this any better than before.

[Beestie]

I am admitedly hard-headed but I still do not think that your examples demonstrate chaos theory or explain why my position is flawed.

Neither the domino example nor the 3-body problem are analogous to the butterfly example.

The domino example falls short for several reasons: a) there is only one element (the dominos), there is no intereference from other elements (reinforcing or dampening) and there is no provision for amplifying the magnitude of the effect as it ripples through the system.

The three body problem is also different than the butterfly example. The 3-body problem is one of sensitivity to initial assumptions. For example, how the universe formed (galaxies with space between them) is entirely dependent on (among other things) the mass of electrons and anti-electrons. Change either of those masses in a nearly infinitely small amount and the post- Big Bang universe is very different. Why? Because there are so many electrons and anti-electrons. Change the mass of a single electron (a single butterfly) and nothing happens.

It seems to me that what you are argueing as a chaotic system is more like the following example: a germ microbe infects a flea that bites a mouse that then goes nuts and chews through some wires in a nuclear submarine causing a meltdown that brings down the sub. So, in this example, a butterfly flapping its wings which blew the microbe left instead of right thereby infecting the flea did, in fact, lead directly and traceably to a nuclear explosion. This example, however is not an example of chaos theory in action.

Ultimately, a theory has to aid in our understanding of an event. My problem with the butterfly example is and remains that it really doesn't do or say anything nor does it prove or even allege anything that we didn't already know. When we see it raining in Japan, it is not illuminating to postulate that the rain might have been caused by a minute event that happened over a year ago. Obviously something caused the rain - but chaos theory as explained in the butterfly example brings us not one inch closer to understanding what or why. Nor does it eliminate any false notions of what caused the rain. Basically, its useless -we are no closer to the truth nor are we any further away from a lack of understanding than we were before the "theory" was introduced.

That is why I have such an academic contempt for this example - it sounds really smart but its as empty as outer space.

[Carbonated_Brains]

If you're game, I wouldn't mind taking another crack at this argument, but I feel some might think it's getting old or redundant.

Wanna keep it up, or should I respectfully drop the topic and we'll agree to disagree?

[Beestie]

I'm interested.

If your email link is active (mine is), we can take it off line so as not to bore the rest of the Cellar to sleep

[wolf]

Just because we aren't all commenting doesn't mean we aren't entertained.

[Beestie]

Great. Let's keep it here, CB.

I thought of two other problems with your examples that I touched on but did not address directly. In the butterfly example, something small (a butterfly sneeze or whatever) is deemed to have caused something large (a hurricane). In the domino example, the first domino falling causes the last domino to fall but the events are of equal magnitude. Connecting the first and last domino is much more realistic than connecting the sneeze with the hurricane. No one has addressed my concern that the initial small event continues to increase in magnitude (conveniently without interruption or interference) throught the system from beginning to end so as to terminate in something as big as a hurricane. I throw a rock in a lake and the way of the world is that the waves get smaller as it propogates. Why? Because the elements it encounters require energy to change their state - the energy of the rock hitting the water is slowly dissapated by creating the waves until enough water is encountered to have used up all the energy the rock had and the waves disappear completely. The sneeze causing a hurricane argues nearly the opposite. The sneeze actually has to pick up steam as it ripples through the world. Now, waves can be reinforced as they travel through a medium by other activity also in the medium. But, that is really stretching it - hence my invocation of common sense to put an end to the endless chain of increasingly preposterous suppositions necessary to connect a sneeze to a hurricane.

Additionally, in the three body problem, another way to demonstrate my unwillingness to accept it as an example of chaos can be illustrated by the difference between temperature and heat. Raise the termperature of one atom by one degree and the increase in heat is negligible as are the effects of the increase in heat. However, raise the temperature of every atom in the universe by one degree and the heat gain is nearly immeasurable. So, the "small change produces a big event" idea seems to hold but what is being called a small change is really a big change. A butterfly sneeze is a small change - one atom's temperature increase is a small change. In addition, an error in an initial setting is a factual error - the reality of the situation did not vary - only our understanding of it. The effect of the error has a direct and instantaneous effect on the outcome. The sneeze however, is but one event out of a trillion that help determine the weather.

Furthermore, I do not accept the premise that the absence of the sneeze prevents the hurricane because it presupposes something that was never established - that the sneeze caused it in the first place. And the weakness of chaos theory, at least in this example, is that it is powerless to prove me wrong or even suggest why I might be. Not to be overly dramatic about it but chaos theory - at least what I have heard/read so far is little more than a repackaging of "cause and effect" but with enough hot air to float a blimp. Its predictive value is zero as far as I can tell and that is, after all, what theories are supposed to be for.

[Carbonated_Brains]

"It's predictive value is zero"

Damn right, that's the entire point behind the theory; chaotic systems cannot be predicted.

But let me form a better, more detailed response, I'm currently at work and the engineers around here are in a huff about something.

[Carbonated_Brains]

btw, how mathy can I get?

[wolf]

as mathy as you wanna be ...

(come on, you were ALL thinking it. )

[Beestie]

Originally posted by Carbonated_Brains
...chaotic systems cannot be predicted.

So the butterfly really didn't cause the hurricane then. Because the same analytic tools that confirmed it could have predicted it.

And a model that cannot predict anything is a fiction - it exists outside the realm of the verifiable.

[xoxoxoBruce]

No one has addressed my concern that the initial small event continues to increase in magnitude (conveniently without interruption or interference) throught the system from beginning to end so as to terminate in something as big as a hurricane.

I really don't know shit about this but it struck me that the sneeze could tip the balance between two larger, virtually equal forces. The result of which tips the balance between two slightly larger forces. In that way the result of the sneeze gathers force. Just a thought.

[Slartibartfast]

Originally posted by Beestie


Furthermore, I do not accept the premise that the absence of the sneeze prevents the hurricane because it presupposes something that was never established - that the sneeze caused it in the first place. And the weakness of chaos theory, at least in this example, is that it is powerless to prove me wrong or even suggest why I might be. Not to be overly dramatic about it but chaos theory - at least what I have heard/read so far is little more than a repackaging of "cause and effect" but with enough hot air to float a blimp. Its predictive value is zero as far as I can tell and that is, after all, what theories are supposed to be for.

Yes, it is impossible to say a butterfly caused a hurricane, but it is a thought experiment that demonstrates how chaotic systems work. Taking two systems with almost identical initial conditions, you will get vastly diverging results because over time differences magnify rather than dissipate. Comparing two 'systems', one in which a butterfly flaps its wings, and another in which it doesn't, is a cool way of saying these two systems are identical- with the exception of the tiniest infinitesimal difference.

And chaos theory has been shown to happen in weather systems, using less variables than the near infinite of the real weather system of the earth. Scientists can run identical simulations over and over with perfect accuracy using computers. It is real easy to show that changing one variable a tiny amount will result in a vastly different end result.

Weather is unstable. Bruce just explained it really well. Imagine a ball perfectly balanced on the top of a smooth hill. The slightest wind in any direction will force the ball to fall in any number of directions. All the kinetic energy of that ball is released by a tiny push.