Measure of Chaos: When Uncertainty Runs Amok

-April 09, 2013

Perfect measurements are limited to counting. The number of people in your office can be measured exactly. Your weight cannot. There is always another decimal point of accuracy.

You might argue, “What if the scale says my weight is exactly 150 pounds and it really is?” Yes, I’d counter, it could happen, but there’s no way for you to know if it did happen because there are no perfect scales.

Measurement precision is the root of chaos, the reason that predictions of nonlinear systems like weather, economies, the result of climbing sea levels, the final position of a gas molecule after escaping a popped balloon, even consciousness itself, are one step beyond predictable.

“Nonlinear” is all but synonymous with chaos, but it’s just jargon. The important thing about complex nonlinear systems is that, in order to predict how they will behave, you have to know the initial conditions precisely. By "initial conditions" I mean the entire state at one time. Think of a soccer player kicking a ball. Newton taught us how to calculate the ultimate destination of the ball if we know the initial velocity and angle with which the ball is kicked (include wind and air pressure, if you like). Those are the initial or boundary conditions. For a linear system like a soccer ball, if the initial conditions aren't very precise, our prediction will reflect that lack of precision and the ball will land within the uncertainty of our prediction.

For a nonlinear system, like an air molecule in a tornado, a specific smoke molecule, or your response to a given situation, the prediction of the ultimate fate of the air molecule, the price of a share of stock, or your destiny, can change radically with small variations in the initial conditions. It's also called the "butterfly effect:" a tiny variation in the initial conditions leads to a radically different final state. For example, if the soccer ball were a complex nonlinear system and I mis-measured the initial velocity of the soccer ball by 1%, instead of flying 40 +/- 2 meters across the field, it could be whisked off by the wind and ultimately end up in orbit.

The point is that to determine the fate of a complex nonlinear system, the initial conditions must be made with incredible precision. The more complex the system, the greater the precision.

Could a butterfly flapping its wings in Japan cause a tornado in Oklahoma?

The butterfly’s wings alter the initial conditions of the atmosphere a tiny, tiny amount, so, yes, they could alter the state in such a way that a tornado happens – but there’s no way to tell. And not just “no way to tell because we can’t measure it accurately enough and climate science isn’t perfect” but no way to tell, ever, no matter what!

The chaotic nature of nonlinear systems means that even when a system’s conditions can be measured with arbitrary accuracy, there will always be uncertainty in the outcome. Mind you, this has nothing to do with quantum mechanics or Heisenberg's Uncertainty Principle, it is simply the reality of measurements.

The advent of quantum physics in the early 1900s introduced the idea that science can only predict probable outcomes, but it had always been true. Even if we lived in a purely classical universe where Newton’s laws of motion and Maxwell’s laws of electromagnetism held sway, we would still be capable of only probabilistic predictions of complex systems.

(By the way, my weight hasn’t been within 7 pounds of 150 in quite some time and the reason has nothing to do with my scale. I do, however, blame chaos.)

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