eli5: Doesn’t chaos theory just prove we lack all the small details/data?


I don’t understand this concept of “chaos” in a universe governed by physics.

Just because something is nearly infinitely complicated, doesn’t mean predicting outcomes would be actually impossible. If the universe produces the outcome, doesn’t that mean it’s following a rule set?

Do I fundamentally not understand chaos theory?

In: 4

Chaos theory simply states “in certain systems small differences escalate to big differences” nothing more nothing less.

But the assessment “that means predicting the future is impossible” is also correct. To simulate physics forward in time we’d not only need all the details, we’d need them with infinite precision.

For example to predict the weather perfectly it isn’t enough to know the current temperature, you need it exact with infinite digits behind the decimal point. And then you need infinite measurements in space, knowing it’s 1 degree less a kilometer to the left isn’t enough, knowing it’s 0.0000001 degrees less a micrometer to the left also isn’t enough.

How chaotic a system is creates a finite horizon of predictability based on how precise you measure, because every error will grow exponentially. And every finite precision will lead to a finite time you can look into the future.

Fundamental to it is the idea that for a sufficiently complex system such complete knowledge and prediction is…perhaps the best word is infeasible.

Sure, if you were an omniscient god and could know everything, ignore uncertainty, etc… Maybe you could predict it but we’re not and likely never will be. At the very least we run into the uncertainty principle but there are hordes of things we are nowhere NEAR needing to worry about that on outside of ideal lab conditions.

It would be fair to say that we can grasp and predict the outcomes of MORE complex systems as time goes on, but A) chaos theory is an arguably valid model right up until we meet that point and B) there is a theoretical cap on what we can know.

Chaos theory lives at the margins of what we can sort of predict, but escapes our total comprehension for now. The better we get, the farther out it is pushed.

>Just because something is nearly infinitely complicated, doesn’t mean predicting outcomes would be actually impossible

Correct. It’s not a matter of complexity.

>If the universe produces the outcome, doesn’t that mean it’s following a rule set?

There’s a rule-set for sure. But the [Heisenberg’s Uncertainty Principle](https://en.wikipedia.org/wiki/Uncertainty_principle) shows us that it’s impossible to measure a thing without affecting the thing, changing it’s state. That includes interacting with it in any way. We will fundamentally never know the exact details of a thing to perfectly predict it’s future.

Without being able to know the starting state, we can only ever have a best guess at what it’s going to do in the future.

And with the butterfly effect and chaotic systems, lacking the small details leads to very large differences. Yes, chaos theory really does hinge on lacking the small details. We fundamentally cannot know the smallest of small details without scrambling the details. It’s a pair of dice that get shaken up every time we open the box.

Chaos theory, or the butterfly effect is an actual mathematical phenomena that is creates a butterfly like pattern called [Lorenz System](https://en.wikipedia.org/wiki/Lorenz_system) where a very, very minute unknown creates a huge effect and its result can be unpredictable. The system, in nature needs absolute perfect knowledge to be able to predict accurately. And considering that we can’t know everything — for instance the minute flapping of a butterfly’s wings not factoring in the calculation, there will always be a degree of uncertainty in the start of the equation, creating the chaos pattern.

Whenever you predict something you first have to measure the initial state. Then you use the model to predict the outcome. However every measurement has only finite precision, which means there is always a margin of error for the initial state and therefore there is a margin of error on the predicted outcome. If small margins of error on the initial state lead to small margins of error on the outcome the system is well behaived and predicting the outcome is possible. Otherwise the system is chaotic and it is impossible to predict the outcome, no matter how precisely measured the initial state was.

In both cases you can actually perfectly predict the outcome if the initial state was perfectly known but perfectly knowing the initial state is simply not possible in practice.