##
Chaos Theory *January 29, 2009*

*Posted by pjd4891 in Mathematics, Metaphysics.*

trackback

trackback

In today’s class, we discussed how fluid dynamics is not researched thoroughly even though it is one of the most important fields. It is applicable for a wide variety of disciplines. Ocean flow could be mapped more accurately, which has a direct correlation to world temperature, air flow, and pressure. Turbulence has chaotic properties. Since turbulence is so chaotic, can we even accurately map ocean flow?

The Coastline Problem was formulated by Mandelbrot in the 1960’s. We discussed this in class. If you keep zooming into the coastline, the length will approach infinity. As you zoom in, the coastline appears more jagged, because you are focusing on smaller and smaller particles. What is the smallest particle we can focus on? Is there anything smaller than a quark? Chaos theory says you cannot have infinite accuracy, so the must be something smaller than a quark.

These fractal examples appear chaotic, but when you make a small change (zooming in), it shows massive changes.

Wikipedia defines a fractal as:

- It has a fine structure at arbitrarily small scales.
- It is too irregular to be easily described in traditional Euclidean geometric language.
- It is self-similar (at least approximately or stochastically).
- It has a Hausdorff dimension which is greater than its topological dimension (although this requirement is not met by space-filling curves such as the Hilbert curve).
- It has a simple and recursive definition.

The Julia set is an interesting example as well:

We discussed that chaos theory states that we cannot accurately understand the universe. Chaos theory questions science at the core. Since we cannot make accurate predictions, what is the goal of science? How can we achieve the goal of science?

All through school, we are taught simplified versions of natural events. I do not believe that mathematical functions exist for all natural events. I think we can get closely model the flow of the ocean, but not completely accurately. There will always be factors that we will have no way of predicting or testing. Life is not supposed to be predicted and science tries to do just this. It tries to rid human lives of emotion in order to give an objective view of how the world works. While science has shown progress throughout time it may be progressing towards something that may not be the correct answer. But then this also gets into the problem with the idea of what is correct and what isn’t. I think science has no self analytical skills in certain circumstances and that it may not see the error of its ways.

Says who?

And this is why it will fail to predict ocean flow?

I as well take issue with “Life is not supposed to be predicted” (where might I find a reference for codes/formatting allowed in comments?).

I think it’s clear that you place some sort of value on emotion, of the sort that places it higher than objectivity and science. I’m not quite sure why, either. Sure, science does try to remain detached.. but what’s wrong with that? That’s one of the reasons it works so well. Emotional feelings do not equate to, nor do they all that often lead to, truth and understanding. And, as Greg pointed out, ridding our research of emotion (actually, I’d say most scientists’ research is quite emotional, to them) does not in any way mean our research will fail to explain things. Seems a non sequitur, to me.

Well, anyway, I think we can at least agree that science is currently the best tool we have for understanding the world. No other method or process seems to be anywhere near as successful. This is not to say there is no better way (though I suspect there isn’t), but it certainly has a very good track record – much better than anything else in human history.

I think I may have gotten caught up on your closing paragraph. Back to Chaos theory.. I think it’s a very interesting idea, though I’m not entirely sure why there’s even a term for it. That is to say, it almost seems to state the obvious. But I suppose it makes more sense in the mathematical domain, where it belongs, and we’re just getting all “emotional” about it 😛

Awesome pictures, by the way. The strange attractors and turbulence pics are my faves.

So chaos theory questions the validity of our scientific predictions based on the idea that a small alteration in the accuracy of our data can vastly affect the outcomes of our calculations. An example was given that changing where you truncate a numerical value by one decimal place can radically change a long term outcome, even if the two outcomes converge early on.

However, I read today about the constant pi:

“…truncated to 39 decimal places is sufficient to compute the circumference of any circle that fits in the observable universe to a precision comparable to the size of a hydrogen atom.”

(Source: http://en.wikipedia.org/wiki/Pi#Numerical_value)

Similarly, elsewhere in that article, it says that we have calculated pi to a trillion decimal places. Assuming Moore’s Law (http://en.wikipedia.org/wiki/Moores_law) holds true, or at least true-ish, this will only ever improve.

So suppose that the more decimal places we can keep track of, the longer our predictions will hold true. Say hypothetically that we can presently calculate with enough accuracy to know the motion of the cosmos with perfect (plus or minus an atom) accuracy for the next ten seconds. As computing power increases and measurement accuracy follows suit, maybe that figure will eventually double, to say twenty minutes, then to forty, then to a day, a month, and so on. Once the span of our prediction’s accuracy surpasses the time it takes to make that prediction by a certain factor, who cares if our model breaks down a few months out.

If I had a computer that could locate the position of every molecule of water on earth, but not the corresponding atoms, and predict the next year’s worth of motion of all those molecules, and do so in five minutes, then why would it matter if my model diverged from reality after the first year? I could just run the calculation again when the new year approaches.

Does anything I said strike anyone as valid, or are these just the rantings of a sleep deprived hack?