Chaotic Thoughts

Tick-Tock, goes the clock…

Last time, in the Determined Thoughts post, I talked about physical determinism, which is the idea that the universe is a machine — like a clock — that is ticking off the minutes of existence. The famous French mathematician, Pierre-Simon Laplace (the “French Newton”), was the first (in 1814) to articulate the idea of causal determinism.

We now know that quantum mechanics makes it impossible to know both the position and motion of particles, so Laplace’s Demon isn’t possible at the sub-atomic level. (It might be possible at the classical or macro level — that’s an open question.) Sometimes the issue of chaos theory is proposed as a counter-argument to determinism, so I thought I’d cover what chaos theory is and how it might apply.

If you want to skip to the punchline, the answer is it doesn’t apply at all.

What is Chaos Theory?

When last seen he’d turned over a new leaf and gone to work on a cruise liner.

Usually when people speak of “chaos” they mean “disorder.” For instance, I used to refer to my buddy’s kids as Agents of Chaos (not to be confused with Agents of Kaos, although there were certain similarities).

But chaos theory is a field of mathematics that studies systems that exhibit chaotic behavior. The Earth’s weather is the canonical example of a chaotic system. The term, “butterfly effect,” applies to how such systems work. The main idea is that tiny variations in the system evolve over time to very large differences.

The key aspect of chaotic systems is that the math used to describe them involves repeating the same calculation over and over. As the number of repetitions grows, tiny variations in the starting conditions cause the result to diverge more and more.

Thus, as the saying goes, a butterfly flapping its wings in Brazil ultimately causes a rainstorm in New York.

The beautiful Mandelbrot!

The famous Mandelbrot fractal is a (stunningly beautiful) visual example of chaos theory in action. I wrote about the Mandelbrot, and touched on chaos theory, some time back. I recommend you at least glance over it now.

History of Chaos Theory

Chaos theory has early origins in the late 1800s with regard to studying the orbits of planets. It’s fairly easy to use Newton’s laws of motion to calculate how two bodies orbit each other, but when you add a third body (or, worse, more) things get hugely more complicated. (In fact, complicated enough to have earned the well-known title, “The Three-Body Problem.”)

As computers came along, chaos theory became more formalized. A key milestone in chaos theory came when mathematician Edward Lorenz was using a computer to run weather calculations. The math for modeling weather involves repeated calculations that evolve current weather data into the future thus “predicting” whether weather involves rain or shine tomorrow.

The Lorenz “strange attractor” which looks much like a butterfly and is sometimes called “The Lorenz Butterfly.”

Lorenz wanted to test his weather model by repeating its calculations, so he re-entered his data, but to save time he rounded off the values from six digits of precision to three. (So a number like 0.123456 was re-entered as 0.123.)

Since the difference fairly small, he expected that the second result should turn out to be fairly similar to the first result. At least enough to test his model.

It didn’t. The new results were wildly different!

In seeking to fully explain this unexpected result, formal mathematical chaos theory was born (and Lorenz coined the term “butterfly effect”).

Lorenz also gave us the notion of a Lorenz System and “strange attractor,” which is another example of chaos math in action. They demonstrate how a system evolves chaotically given variations in initial conditions.

How Chaos Theory Applies

The key characteristic of chaotic systems is that minute differences in input values result in huge and surprising differences in output values. But there is nothing random or indeterminate about chaos math. Identical inputs always result in the same outputs.

Zooming in on the Mandelbrot fractal shows beauty in chaos. (See below for videos!)

If Lorenz had re-run his weather calculations using the same input data, he would have gotten the same output data (every time).

That means that, at least in principle, if we had a valid weather model, and if we could supply completely accurate input data, then we could predict the weather with 100% accuracy. And we could do so to any point in the future. We could know with certainty that it will rain at 3:00 PM exactly one year from now.

But here’s the rub. Actually there are several, but one is a show-stopper.

One problem is whether our weather model is valid. Over time we’ve created better and better models, and there’s no reason to think we wouldn’t someday have one that is entirely valid. There’s nothing stopping us from completely understanding how weather works.

You can blame chaos!

Another problem is gathering enough data. An accurate weather model might require knowing the temperature and pressure and wind for every cubic inch. Laplace posited the idea of knowing about every particle, and it might be that predicting the weather accurately might require knowing about every “particle” of air.

That’s a tough nut to crack, but maybe someday science could find some way to gather that much data. Or maybe it wouldn’t require that much; maybe every cubic inch turns out to be enough. Probably not, but it might be enough to get accurate results for weeks or even months.

The show-stopper is chaos. Even if we could gather enough data, in order to use that data we need to enter it into a computer. To even get the data, we need to record it. And therein lies the problem.

My thermostat tells me the temperature in my house (or, at least, at the thermostat). But it only tells me how many degrees. The temperature doesn’t actually jump from 68 to 69. It slides from 68 to a whole lot of between values until it reaches 69. (Fahrenheit)

It’s 68 o’clock!

For example, at some point it’s 68.1, then 68.2, then 68.3, and so on. And to get to that first number, at some point it’s 68.01, followed by 68.02, and so on. That continues down to a quantum level: 68.001, 68.0001, 68.00001, you get the idea.

If I’m going to take weather data, at some point my measuring device says, “Well, it’s currently 68.5849615438276451904 degrees.” (I have a very accurate temperature gauge in this case.)

But it’s not accurate enough, because the real temperature has a lot more digits. The real value may have nearly an infinite number. (The only reason it isn’t infinite is that, at some point, quantum mechanics does kick in, and at that point we can’t go any further.)

Only three digits doesn’t even begin to be accurate enough!

So our measuring device, like my thermostat, shows us the data at some resolution, some number of decimal digits the device is capable of.

Well, we just rounded off reality, and now we’re screwed. We’ve introduced small — perhaps very small — differences between our data and the reality. But that difference, no matter how small, severely limits our ability to calculate.

But let’s say our device could measure down to the lowest level, and let’s say our computing device could take inputs down to that level.

We’re still screwed, because calculations with that data result in new numbers, and those numbers are in the Real number domain. They aren’t restricted by quantum jumps — they can have any value. And no digital computer can handle that. To calculate, the numbers have to be fixed; they can’t go on forever.

I prefer cake.

Imagine trying to add 1 (one) to pi. The problem is that pi is a (transcendental) real number whose decimal digits go on forever. Our calculation starts: 4.141592653… and never ends. If we force it to end, we’ve rounded off the value, and in chaos math we just blew our foot off.

The bottom line is that our weather model may be accurate, but there’s no way to actually use it, except crudely. The key is that, the longer we calculate — the further out we try to predict — the worse it gets.

That’s why it is impossible practically to know the weather a year from now. It may be possible in principle — chaos mathematics is fully determined — but there’s just no way to actually pull it off.

Deep, deep, deep into the Mandelbrot (17 decimals).

It turns out many real life processes are chaotic: the weather, the stock market, the orbits of multiple bodies.

Chaos may even apply to our minds. It may be that the reason the future seems unknowable is that chaos makes it all but impossible (in any practical sense) to calculate.

It’s likely that the only “computer” that can simulate reality is reality itself.

But that doesn’t mean the Reality Machine is random or undetermined. Chaos mathematics is just as determined as 1+1=2. Chaos may make the future surprising, but it doesn’t make it uncertain, let alone random.

You can watch Dr. Holly Krieger from MIT explain the Mandelbrot:

Or you can just take a ride on “the deepest animation ever”: