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“Prediction is difficult, especially the future.” — Niels Bohr
“Prediction is difficult, especially the future.” — Niels Bohr
Okay, in my post last week, I revealed that I was a deterministic Newtonian, and my reasoning was about two hundred years old. I posited, “If I could identify all the forces and weights and measures of a leaf blowing in the wind, what’s to prevent me from hooking into the local quantum computer to predict exactly how that leaf will behave as it is falling to the ground?” It seemed to me—and to Sir Isaac Newton— that this should be possible. Newton believed that every interaction of bodies and force should be able to be predicted and determined using mathematics.
Newton’s Cradle: We all know what happens next
This is what calculus is all about, right? I thought that made intuitive sense. Scientists and mathematicians agreed with me—up until the end of the 1800s when they uncovered some difficult non-linear differential equations to solve; in particular, the horribly difficult and outstanding problem of predicting the behavior of three mutually gravitationally attracted bodies—the so-called “three-body problem”. Turns out, this can’t be solved (or, at least, not yet).
The Three-Body Problem: We can’t predict with 100% accuracy what will happen next
And here’s the rub: we don’t live in a Newtonian world. We live in a quantum one. You simply can’t model how any leaf behaves in the wind because they are made up of molecules, and you can’t model every molecule accurately because you don’t know the position and velocity of every particle because of …irony, irony, (because I was talking about the power of quantum computing)… quantum mechanics. It is fundamentally not possible to model the world to arbitrary accuracy.
And that makes intuitive sense, too. Of course it’s true (and thank you, Paul, for first pointing this out to me in a way that I could understand). No matter how much quantum computing power we have access to, quantum physics itself prevents us from making mathematical models to predict anything with 100% accuracy. We can make models that approximate the behavior of complex systems, but before too much time passes after we begin the model, the accuracy of that model will degrade.
This is chaos theory—the theory that within the apparent randomness of chaotic complex systems, there are underlying patterns, constant feedback loops, repetition, self-similarity, fractals, self-organization, and reliance on programming at the initial point known as sensitive dependence on initial conditions (thank you, Wikipedia). This is why weather models’ predictions degrade after a few days; the tiniest variable (such as the flap of a butterfly wing in China) at the beginning of the model may affect the grandest of systems (say, a hurricane in Texas), to the point that the model becomes useless.
Chaos is indeterminism at its core, embodying three important principles:
This indeterminism forms the basis of the fact that there will never be any universal equation, and we will never be able to predict the future with any kind of real accuracy.
You know what a fractal is—a fractal is a geometric object that is like itself on all scales. If you zoom in on a fractal object it will look exactly like the original shape. This property is called self-similarity.
Examples of Fractals. Left: Constructing a Sierpinski triangle; Center, a Koch Snowflake; Right, a Barnsley Fern
(A neat thing about a Koch Snowflake and a Barnsley Fern is that they enclose a finite area with an infinite perimeter! Now you know the answer when the question comes up on your next trivia night.)
You see examples of fractals all over the place, from the growth of a tree to the way a river behaves to swirls in a seashell to how a mountain forms. Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Driven by recursion, fractals are images of dynamic systems—the pictures of chaos.
I love time travel stories, although chaos theory would never permit it. (Suspension of disbelief is a powerful force.)
In Back to the Future, can you imagine the fallout of the smallest event of Marty McFly going back to 1955? Think of the very first thing that happened when he went back in time:
He hits a scarecrow and then crashes in the barn of a local farmer. But think of the birds that wouldn’t have been scared off by the scarecrow the following day, which led the birds to eat grain in another field, which led to a lower grain yield of the other field, which led to something else… Or imagine the time that the farmer had to spend cleaning up the barn was time spent that he didn’t spend doing something else that didn’t get done, which led to some other kind of consequences… Or think about the effects of the disturbed sleep of the family, which could have resulted in lower test scores of the kids the next day, which led to lower grades, which led to goodness knows what… And don’t even get me started about thinking of the effects of the family seeing what they interpret to be a space alien. These micro-events were just thrown away in the movie. (But how could they be treated with any seriousness? It was a comedy, after all.)
The best treatment of time travel I have read illustrates another kind of butterfly effect: Ray Bradbury’s A Sound of Thunder, a science fiction short story first published in 1952. Long story (well, short story) shorter, the main character travels back in time to the Late Cretaceous period, and accidentally steps on a butterfly. When he returns to the present, words are spelled and spoken strangely, and a local election went a different way than before he left. The death of the butterfly had apparently set a vast series of subtle changes that affected the nature of the present. See? Chaos theory in practice.
The Butterfly Effect
You see it everywhere. Have you ever been stuck in traffic without any discernable cause? That’s chaos theory. About a month ago I got a flat tire in a very inconvenient and dangerous place on the highway, and police officers closed off traffic behind me, so I could limp to a safe place to put on my spare. My flat tire caused the traffic to back up, which likely caused someone to be late to a meeting, which then caused goodness-knows-what. I feel guilty about causing that traffic tie-up, but should I? It wasn’t my fault, it was the fault of whoever lost that nail that embedded itself into my tire.
Fault is a ridiculous concept when you’re thinking about chaos theory.
A famous proverb
In the last three weeks, I have been written about what quantum computing is, some possible effects of quantum computing on our world, and the limitations of unlimited computing power due to the certainty of uncertainty. I think that it’s safe to say that considering the remarkable variables that are in play in this complicated world, there will always be unintended consequences in whatever we do, whether we’re simulating an IC design or driving to work one morning. The best we can do is roll with whatever is thrown in our direction, and be aware that for every action, there is an equal and opposite reaction.
Or not. Your actions can snowball. You can count on it.
Some websites that helped me to write about this stuff: