Mathematics? It's all just life.

[Carbonated_Brains]

All the kinetic energy of that ball is released by a tiny push.

Potential Energy, but let's not split hairs.

That was a damn good explanation, and had my 10 pages of rebuttal not been deleted when the stupid message board asked me for my password upon clicking "submit", I would have probably said something along those lines, with much more dilly-dallying and diagrams.

The computer thing is interesting though, the computer really wont repeat the same experiment and get the exact same answer if it's a truly chaotic model, because the tiny rounding errors made by computer processors are enough to propogate into the results over time.

The same chaotic "prediction" made by two computers will be hugely different due to minute changes in the way the processors are made, and handle math.

But you're right, bottom line is the butterfly does not CAUSE the situation, it merely represents a difference in initial conditions.

[Slartibartfast]

Originally posted by Carbonated_Brains


Potential Energy, but let's not split hairs.

Oops.

Originally posted by Carbonated_Brains


The computer thing is interesting though, the computer really wont repeat the same experiment and get the exact same answer if it's a truly chaotic model, because the tiny rounding errors made by computer processors are enough to propogate into the results over time.

Hmmmm, but if you feed the same computer the same variables on two seperate occasions, it is going to round them in exactly the same way each time it runs the simulation.


edit: added the word 'same' before the word 'computer'

[Carbonated_Brains]

Big, expensive problems yesterday.

Bad, bad day.

Found a screw in my tire when I got home from work, second time in 2 weeks. Also, somebody bent the shit out of my gas-nozzle cover on my car, and the door wont close properly.

After all that, I'm gonna take one huge crack at explaining how not only is chaos theory absolutely plausible, but it's probably the basic structure of our universe, so much so that "linear" systems are considered rare in comparison.

Oh, and I'm writing this shit in Notepad, and copying it over. I was ready to break this computer yesterday when everything got deleted. Here goes!

[Carbonated_Brains]

(oh, btw, the 'big expensive problems' were not gas-nozzle related, but $200,000-piece-of-equipment-stuck-10-metres-underground related)

[Catwoman]

Just caught up with this thread keep it going guys its fascinating! Don't be surprised if in a few years time you find yourselves quoted in a small-time (heh heh heh that's what you think) English novel.

[glatt]

I like the ball on the hill example.

You can also use the same idea to describe politics and international relations. Look at the 2000 elections. Those elections were very close. The country was split pretty much 50/50. Almost perfectly balanced. Some clerk in a little county in Florida designs a ballot one way, instead of another, and as a result, the entire world is sent on a drastically different course.

No-one knows what would have happend if the butterfly ballott was designed differently, and Gore was elected. Maybe 9/11 would have happened, maybe not. Iraq almost certainly wouldn't have happened. The oil prices might still be low, or something else could have happened to cause them to be even higher.

There's your "butterfly effect" or is it a "butterfly ballot effect?"

[Beestie]

Stay tuned: I think that CB is setting me up for the kill.

[Catwoman]

Knives at the ready... should be a good'un

[lumberjim]

Originally posted by Carbonated_Brains



Oh, and I'm writing this shit in Notepad, and copying it over. I was ready to break this computer yesterday when everything got deleted. Here goes!

you do know that the "back" button will bring back a dumped post if you fuck up like that, don;t you?

or right click, undo within the edit box if you accidentally delete text.

[Carbonated_Brains]

This is a work computer, and somehow the security is set high, and I guess I'm not accepting cookies, so if I do ANYTHING I have to re-enter my username and password
which is what happened when i clicked submit
so I clicked back, to see if the text would still be here, and again i was asked for the user/pass
typed it in, got this form, sans text.

[Beestie]

Originally posted by lumberjim
you do know that the "back" button will bring back a dumped post ...

That's what I would expect but sometimes, when I hit the back button, it reloads. I remember trying to go back to find a post that someone had deleted (I read it before they deleted it) and to my dismay, it reloaded the page instead of fetching the one in cache. That was some good blackmail stuff too, damnit.

[Carbonated_Brains]

In the spirit of a real debate, I'm going to try to address your individual concerns, Beestie, one by one.

"I'm not a big fan of chaos theory and, but for the beauty of the metaphor, the "butterfly effect" idea with its stretched-to-the-breaking-point logic..."

Logic is subjective, and if I can't bury you in the logic, I'll put you six feet under with cold, hard math ;-)

"To suggest that all of the 100 trillion (or whatever) variables were in perfect balance and the lone butterfly tilted the scale just enough is to remove one more shovel full of dirt from a (nearly) bottomless hole."

It just SEEMS implausible. After all, the universe is completely about balance! What could have created the miracle of human life, if not a trillion carefully balanced elements? If things were NOT hinged and balanced upon one another, entropy would rule, and the universe would be an evenly spaced bunch of atomic goo! For any cellestial "house of cards" to be built, a lot of crazy precarious things have to happen. But I digress...

If you want an identifyable, mathematical equivalent to the "butterfly effect", you need to look at what Edward Lorenz bumped into while he was studying weather patterns for MIT. He took a bunch of fluid dynamics formulae, and applied them to an atmospheric model. He boiled 'em down to the following three equations:

dx/dt = * (y - x)
dy/dt = r * x - y - x * z
dz/dt = x * y - b * z

is the Prandtl Number, which is a constant, Lorenz used the number 10. "r" represents the temperature gradient between the top and bottom of the volume of atmosphere Lorenz wanted to study. "b" is the width to height ratio of this volume "box", he used 8/3. X relates to the rate of rotation, Y is another temperature gradient, and Z is the deviation of the line from a graphed vertical temperature plot. The upshot is he graphed the sucker and ended up with this:

Bam!

Look familiar? It should!

Zoom!

The reason chaos theory is NEW, is because you couldn't really study it without a computer. It requires billions of complex calculations, something no mathematician would spend his life doing. Lorenz graphed that butterfly shape, known as the Lorenz Attractor, and found that the system he was graphing never precisely repeats itself...the trajectory of that line never traces over a previous trajectory, it loops forever and ever.

Lorenz, being a good scientist, initially said "No flippin way, I must be wrong." So he did a futher experiment, he made a waterwheel with 8 evenly spaced buckets, each on a swivel, and having a hole in the bottom. He opened a spout and began to fill the buckets, causing the waterwheel to spin at a somewhat constant rate. When he dropped the hammer on that spout, let the water flow really fast, the waterwheel started to shudder in one direction, jerk to a stop, move backward, shudder forward again, and dance around randomly. VERY randomly. Lorenz sat there for hours and recorded the waterwheel, and it never repeated its motion. If he graphed the waterwheel, he'd get a version of the Lorenz attractor; the waterwheel, and the atmospheric model, were both chaotic systems.

What's even weirder is that when you graph a chaotic system, you have to use dimensions which we never really thought existed. We're familiar with the first 3 dimensions, but graphed chaotic systems use non-integral dimensions. Which means you can be in dimension 2.8, or 7.1. What happens when you graph a chaotic mathematical model using non-integer dimensions? You get a fractal.

In fact, that's how Mandlebrot and Julia invented fractals. They are graphical representations of chaos theory.

This guy named Cantor did some funky stuff, too. He postulated that you can take a regular line segment and infinitely rip pieces out of it...draw a line, then erase the middle third. Then erase the middle thirds of the two line segments you got after the first erasing, etc, etc...until you have hundreds of tiny dots. These tiny dots, though infinite in number, have a combined length of zero. This is called "Cantor Dust".

Cantor dust is used in electrical engineering. Engineers looking at electronic transmission powerlines observed periods of error-free transmission, then bursts of errors, then periods of calm, etc...analysing the bursts, they found that these bursts contained small periods of error free transmissions, then sub-bursts, etc...in fact, it follows Cantor's model. Cantor Dust is essential in modeling intermittency.

There's even chaos washing machines. Goldstar, in 1993, invented one which used a tiny pulsator which rose and fall randomly using a chaotic algorhithm, on the premise that it would produce cleaner clothes with fewer tangles...and it sort of worked, to an extent.

Guess what else is a chaotic system...the stock market. Billions of variables, no predictability, yet patterns can rise and fall with time. The stock market is completely useless in the short term, yet serious profit can be made by analysing trends and participating in long term trading.

These are examples of nonlinear systems, which are also dynamic. Chaos theory is the study of these systems.

"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."

Chaos theory alleges something that physicists have been terrified of since Newton. Mathematicians and physicists believed that if you knew the initial conditions with great accuracy, you could get formulas to describe all events in the future, and explain the ones in the past. Like playing a movie backward and foreward, everything is explainable and calculatable based on those conditions.

Chaos theory threw that out the window. Now, scientists cannot predict complex celestial motion, because they cannot ever know the initial conditions. To say that has small ramifications is like saying Hurricane Mitch was just a spatter of raindrops.

I think someone earlier said it best...the butterfly flapping its wings, is the minute difference between two sets of initial conditions. Assuming everything else is the same, you still have two systems where the initial conditions are infinitesmally different. And this difference is enough to cause a mathematical spiral of the Lorenz attractor...in a year, you're in 2 completely different places if you make a prediction based on the 2 sets of initial conditions, no matter how similar they are.

Scientists now believe the universe is itself a chaotic system, and linearity is rare. This is making scientists sweat, and religious types pay attention.

Hawking said it best: "If we find the answer to that (the universe), it would be the ultimate triumph of human reason-for then we would know the mind of God."

My old physics teacher (smartest man I've ever met) always said, if you want to explain something, use science. But if you want to know WHY, and you want a philosophical answer, you need to look to faith.

Not making this argument theological, it's just interesting.

Now wiggle your rebut!

[Carbonated_Brains]

Betcha didn't expect a 1300 word reply =)

[russotto]

If you have a system which you know exhibits chaotic behavior, and where you don't know most of the initial conditions, nor the possible interactions among the parts of the system, how meaningful are your long-term predictions of the behavior of the system?

[Carbonated_Brainss]

...did I just get banned?

I can't post with the other account. I can't even email Undertoad.

What's going on?