Against butterfly effect

It is known that, when you simulate a deterministic system described by nonlinear differential equations, small differences in the initial conditions can be exponentially amplified, resulting in huge differences in the final result.

To describe this phenomenon Edward Lorentz famously said that “a butterfly flapping its wings in Brazil can produce a tornado in Texas”. This quote, popularized by Gleick’s 1987 bestseller on chaos theory, came to mean that small events and small decisions can have huge and unpredictable consequences.

The problem with this conception is that it is extrapolating from only two data points a correlation that (almost surely) does not exist.

Let us suppose that some aliens run a simulation of our universe, starting in 1 january, with our present initial conditions x(0). This simulation could be deterministic or probabilistic, depending on your philosophical standpoint on how our universe works. The aliens go on simulating until 1 july, and observe that on 1 july there is no tornado in Texas. Then they run again the simulation, but this time they slightly modify the initial condition x(0) (a butterfly flips its wings). This time, on 1 july there is a tornado in Texas. Does this couple of observation mean, in any meaningful way, that the butterfly caused the tornado?

To answer this question, we must run many simulations, sampling all the possibile initial conditions. If our universe is not deterministic, it would make sense also to repeat many times the simulation for each initial condition. Then we could measure the correlation between the correlation between the event “butterfly flips wings on 1 january” and the event “tornado in Texas on 1 july”. But this correlation will be almost likely 0. The more the system is chaotic, the more correlations will decay exponentially with time.

There are systems (like human history) in which small decisions can have big consequences. For example, I guess that the aliens simulating our universe could detect some positive correlation between the events “Francis Ferdinand gets shot in 1914” and “New Zealand is at war in 1917”. I do not think that this correlation is very big, but it could be fairly greater than 0. But this is because Francis Ferdinand heir to the throne was a very special person, whose life had big and predictable correlations with the lives of millions of other people.

If the system is predictable, it is easy to think to cases of big correlations between small decisions and big events. Pressing a small button can start a factory. But you can not control weather by waving at the wind. The more the system is caothic, and the more it is exponentially unlikely to have big correlations.

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