It has been recorded (though rarely) of a species going extinct and then a relative species occupying the vacant niche where they then begin to selectively evolve back into the exact same species with the same qualities behaviours and anatomy. — Benj96

What species would that be?

Butterflies moving their wings as a root cause of hurricanes are a figure of speech. If someone hadn't uttered that phrase, once upon a time, we'd all be infinitesimally better off.

All the misnamed "butterfly effect" means is that in a discrete deterministic iterative system, very small changes in the inputs can lead to huge changes in the outputs. It's mathematically true and easily reproduced. The Mandelbrot set provides a striking example. Starting points extremely close together may have strikingly different evolutions under repeated applications of the transformation rule.

The idea that you go back in time and kill a butterfly and thereby bring about the election of Donald Trump is a popularization and not to be taken too literally. For one thing, there's no evidence that reality is a discrete deterministic iterative system; whereas computer simulations of weather or calculations of the state of the solar system under Newtonian gravity, are.

The evolution of the solar system under Newtonian gravity is a good example of how even though a system is deterministic, its future can't be predicted. The idea that "if we knew the position and velocity of every particle of the universe, we could predict the future," turns out to be flat out false. Even if we know the initial conditions with perfect accuracy, accumulated rounding errors in the computation make it impossible to even know whether the solar system is stable.

https://en.wikipedia.org/wiki/Stability_of_the_Solar_System

Considering speculative theories of history, the butterfly effect is one extreme. Another is Stanislaw Lem's ergodic theory of history: There are social "attracting fixed points" so powerful that minor or even major alterations of history have no substantial effect on a particular outcome. For instance, had Hitler been assassinated early on, the outcome - a disastrous WWII - might very well still have happened, more or less in the way it happened.

Ergodic theory of history

The Grandfather Effect is fun to contemplate. An outlandish form of multiple universe theory is that the universe is constantly splitting into uncountable alternative states, so that murdering your grandfather when he is young would simply lead to an alternative world history. :cool:

All the misnamed "butterfly effect" means is that in a discrete deterministic iterative system, very small changes in the inputs can lead to huge changes in the outputs. It's mathematically true and easily reproduced. The Mandelbrot set provides a striking example. Starting points extremely close together may have strikingly different evolutions under repeated applications of the transformation rule. — fishfry

I tend to agree. The Butterfly Effect is observed in isolated abstract systems but in the real world there are a myriad of butterfly effects cancelling each other out. eg in a galaxy of stars there are numerous Lagrange Points where gravity is cancelled. One Butterfly Effect can be cancelled by another before it gets going.

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There's no evidence nature is a specific kind of (mathematical) dynamic system.

Depends on the butterfly. I don't think all butterflies change the course of history. Pick the right one, though...

I'd appreciate it if you could unpack this a bit. The more I read it, the more I can't quite figure it out. — tim wood

As I thought about responding, I soon got into thickets I couldn't figure out either. Here are some idle thoughts for what they're worth.

I'll offer a guess, on the off-chance I get it. To wit: Nature is unpredictable, either because it is essentially unpredictable, or predicative/calculative power will never be up to the task of being able to make exact predictions. — tim wood

Exactly. This is the point I got to in my thinking. Is chaos an ontological or an epistemological phenomenon? Is nature itself actually chaotic? Or does chaos only arise from the finite computer approximations we make to the differential equations that describe the world?

If chaos is only an artifact of the physical limitations of computers; then in fact a butterfly flapping its wings may not make much difference to the future. I actually don't know what the experts think about this; and as I Googled around, I didn't find anything on point. Perhaps nobody's noticed the question.

To sum up and emphasize this point: There is nature itself; and then there are computer approximations to the differential equations that model nature. The later may be chaotic with or without the former being chaotic as well. I believe that's the question we're both asking.

Or both. — tim wood

Yes. It might be that the computer approximations are chaotic AND the "real" universe is too.

* An example that comes to mind is the Mandelbrot set. This is a purely mathematical phenomenon and NOT a physical one so it's useful as an idealized case.

Mandelbrot is purely mathematical; and it is inherently chaotic. Points that are arbitrarily close to each other can have dramatically different evolutions under the transformation rule, following the math itself. It's a fact about the Cartesian plane; and not about the computer models. In other words if we used "God's computer" and could manipulate real numbers exactly, the Mandelbrot set still exhibits chaos. The strange and beautiful patterns are inherent in the equations and not in the physical computation of them.

The mathematical chaos of Mandelbrot is contrasted with the physical modeling aspect.

According to Wiki, chaos theory applies to dynamical systems. These are systems that evolve in deterministic ways and whose behavior can be modeled as a set of differential equations.

The classic example is the three body problem in Newtonian gravity. We can solve the equations for two bodies but not three. Let alone all the stuff bouncing around the solar system.

Since we can't solve the equations we have to approximate the solutions using iterative methods implemented on physical hardware. That's where the chaos comes from.

* Aha! It occurs to me, it might be that just like the Mandelbrot set, it might be that the equations themselves are chaotic; and not just the computer approximations. In other words Newtonian gravity might be "inherently chaotic" by virtue of the differential equations themselves, and not because of rounding errors in the computer.

This is a distinction that had not previously been clear in my mind but I think it's important.

And if we ask, Is Newtonian gravity chaotic in nature? And then of course we realize that the question itself is a mistake. Nature doesn't do Newtonian gravity. The issue is moot.

* This is pretty much how I see all this without being a specialist in the least.

But I think we can conclude that the butterfly story may be about computations than about the world. Perhaps in nature the butterfly effect is false. Nearby points have nearby fates.

Or perhaps the butterfly effect is inherent in the differential equations Lorenz used for his weather predictions. But that doesn't really apply to butterflies. The popularization is really kind of muddled.

In the end nobody knows whether the butterfly effect is part of nature OR just a feature of the differential equations OR just a feature of the computer approximations to the equations.

Notwithstanding that gross and imprecise predictions are made all the time and that in the aggregate more-or-less dependable predictions help to get the world's work done. — tim wood

The stability and seeming coherence of the world are deep philosophical mysteries. I could just be a Boltzmann brain, a momentary coherence in an otherwise random and formless world.

After all bridges seem to stay up most of the time, even though quantum theory tells us that with nonzero probability, half the bridge could move a mile away in an instant. Modern physics doesn't give us much to stand on.

If you want to work in a reference to the strange attracters of chaos theory, I'd be glad to read that too. — tim wood

Can't help there.

Ok tl;dr on all this:

Three levels:

* Chaos in nature itself;

* Chaos inherent in the differential equations that we use to model nature;

* Chaos arising from approximating the solutions to those equations on physical computers.

It's good to think about which level is being talked about in any given situation. And there's no reason to believe that the butterfly effect is true in the real world. It was true of Lorenz's computer simulation of the weather. That's a specialized domain. The public is misinformed on the butterfly effect. This is my opinion.

There's no evidence nature is a specific kind of (mathematical) dynamic system. — jgill

You summed up my thoughts a lot more efficiently than I did!! Thanks!