# Interpretation of the butterfly effect

It is said that in certain circumstances, a tiny change, like the flap of a butterfly's wings, can lead to enormous changes, like a tornado somewhere. However, it should be clarified what "change" means.

If the constantly changing state of the universe is determined by certain mathematical equations, then the flap has been part of the plan. The previous states of the the universe has dictated the flap.

Updated: If the butterfly didn't flap its wings, then it's not our universe. One might say I'm being too pedantic. So let's assume that the flap event is the only parameter changed in the state of the universe at the moment t. This assumption leads to one minor and one major problem.

1. The minor problem: Physical laws only allow consistent initial states. Changing only one parameter could result in an inconsistent state, so it will probably be necessary to change other parameters too, which might not be a tiny change anymore.

2. The major problem: If the mathematical equations predict that a tiny change in the state of the universe at the moment t will make the future course of the universe to increasingly diverge, what can be said if the same mathematical equations are used to compute the past of the universe? If they result in an increasingly diverging past course for the universe, then it's definitely not our universe.

Note: This question has been majorly edited under the influence of the great answers given.

• Where do we need to think about violations or even laws? The principle is that small causes can have very large effects. Had the butterfly flapped its wings at a slightly different time, things might turn out quite differently. That basic idea does not involve addressing the laws of physics as a whole or considering whether they get violated.
– user9166
Apr 2 '18 at 23:20
• You should read Wikipedia's Butterfly Effect. It means the same as sensitivity to initial conditions, and there is no need for violations, tiny or otherwise. There is a difference between laws and initial conditions, even if the latter are also "part of the plan". Apr 3 '18 at 0:18
• Mathematical equations do not determine the exact state of the universe because, according to our current understanding (quantum mechanics, etc.), the universe is not deterministic. Rather, those mathematical equations determine the boundaries of what is possible. But even more so, our knowledge of the universe's current state is (necessarily) incomplete, and therefore suppositionally, we can speculate on whatever events we want (even in a deterministic universe) so long as they do not contradict our incomplete knowledge. Apr 3 '18 at 14:21
• "The previous states of the the universe has dictated the flap" That was classically true but quantum effects blow that out of the water. The universe is governed by probabilities. This is why (annoyingly) predeterminism is a largely vacant philosophical concept. Apr 3 '18 at 16:02
• Ultimately to say 'the butterfly' flapping it's wings caused the tornado is to pick an arbitrary event in the whole sequence of causality. You could Just as well say 'The butterfly being born caused a tornado, or 'the big bang caused a tornado' Apr 5 '18 at 9:46

I think you might be confusing determinism with what’s called ‘chaos.’ Chaotic systems are deterministic, nonlinear systems, which are characterized as ‘chaotic’ because of their extreme sensitivity to initial conditions. ‘Nonlinear,’ just means that they can be described by nonlinear differential equations. The so-called ‘butterfly effect’ is a colorful way of illustrating this.

Nonlinear systems can be so sensitive to initial conditions that differences in how two computers handle floating point rounding errors can result in wildly different computational outcomes. This leads to a general inability to accurately predict nonlinear behavior, which can lead to confusion between chaotic and random systems . In practice, there may be no meaningful distinction, but in theory, chaotic systems are completely deterministic, whereas random systems (such as a quantum mechanical measurement) are taken to be fundamentally nondeterministic—meaning that the outcome is fundamentally not determined by the initial conditions.

• "In theory, there is no difference between theory and practice. In practice, there is."
– user32096
Apr 3 '18 at 12:11

You have clarified that you understand some descriptions of physics are about hypothetical well-isolated systems with different initial states.

However, when it comes to the Butterfly Effect you reject the concept of taking a well-isolated system with initial states, and demand that it explain the entire universe.

The Butterfly Effect describes a thought experiment involving a large apparatus (the Earth's entire atmosphere, at least) and posits that if it were possible to change the initial conditions (of the experiment - not of the entire universe), even a tiny change might lead to a vastly different conclusion.

It isn't intended to be an experiment that is feasible to conduct. It is intended to be a vivid model to explain some of the concepts of chaotic systems, that have real implications to modelling of real systems.

• Thanks. Let's discuss the thought experiment. An advanced civilization is able to do this experiment on the Earth. Is it possible to have only "one" parameter changed in the initial conditions? Apr 3 '18 at 15:32
• @Asmani: You're taking it way too literally. The point of a thought experiment is that it is not a real experiment. It's often not even possible, neither to conduct nor to validate the results, but ignores the impossibility in order to simply ask "what if". But yes, if you control the initial conditions, then by definition you can indeed change one parameter. It just might not be the one you're so fixated on.
– cHao
Apr 3 '18 at 16:25
• This is a question of physics: Is it possible for a physical system to have two consistent states only differing in one parameter? Consider 2 electrons. If you change the position of one of them, then the electrical force that the second electron feels changes. It means two changes, not one, right? Apr 4 '18 at 8:45
• @Asmani That's about definitions, not reality. You can consider the electrical force between two electrons to be separate from the position o the two electrons, or not. In reality ("to the best of our knowledge" etc.), each electron is a disturbance in the underlying electron quantum field, and the force between them is transferred via a different disturbance in the electro-magnetic quantum field. The behavior of both is governed by one equation. You can't change the two independently, so we don't consider them independent unless there's some reason to (e.g. for a thought experiment). Apr 4 '18 at 11:17
• @Asmani So if our guess is indeed right and all that's around you is part of one equation that determines everything (and we figure out how to fill in the blanks we know about), you can't really change anything independently of anything else. But that doesn't mean the concept of change is worthless - it just means it isn't literally true, which is kind of true of most (all?) physics and science. We'd just say "change the position of one electron", and all the other "dependent changes" are implied in that. We use tools that help us, not tools that hinder our progress. Apr 4 '18 at 11:20

The butterfly effect is formally captured mathematically. Consider a chaotic system (such as a mathematical equation of the weather) and an initial state. If we use those equations, we can calculate what the state will be at some future time (say, 1 month in the future). For this deterministic system, that is the only possible state we can be at in 1 month.

Now perturb the initial state slightly. Pick a "nearby" state, such as a state where the butterfly is about to flap its wings instead of staying still. Now use the same equations to calculate the state 1 month in the future. For this deterministic system, there will be one and only one possible state that we can be at in 1 month. However, because the system is chaotic, this state may be drastically different from the start in the first scenario.

Now this mathematical result points out two interesting physical outcomes. The first is to note that the state of the weather in 1 month is incredibly hard to predict. In theory, it's future state depends on how polarized or depolarized a few neurons within a butterfly's brain are -- the neurons which control wing flapping. The idea that the future state of a system as large as the globe is highly sensitive to such a small detail is I, in my opinion, a pretty significant philosophical statement.

The other major result of this has us noting that we never measure anything perfectly. There's always some error in our measurements. This means we cannot predict the future state of such chaotic systems accurately. It is a practical limitation to the capabilities of science.

It also implies that if we got the laws of the universe slightly wrong, we could measure everything perfectly and still not be able to predict the future state of chaotic things like the weather. Or, as you suggest, if the consistency of the world's laws were suspended for a brief moment in a tiny place, the effects could still change very large systems.

As a final thought, the butterfly theory does indicate that if you were able to suspend the rules of physics in a deterministic chaotic universe and change one thing to a state that it wasn't, you would indeed cause widespread changes in the overall state of the universe. God "changing a beetle" may indeed result in a tornado. One might say "the beetle caused the tornado," or maybe one might just say "God caused the tornado, and used a beetle to do it."

As for cause and effect, I think mathematicians generally agree that it is an ineffective concept in chaotic systems. It's not impossible to use, as you noted with the beetle causing a tornado, but it becomes less meaningful. I find the more meaningful use of cause and effect is in situations which are stable in the presence of perturbations, where identifying a "cause" permits you to mostly ignore large numbers of other details which won't have a large change on the effect. In chaotic systems, you can't ignore the other effects (because they could cause huge changes as well), so the benefit of identifying a "cause" is limited.

• Pi is a transcendental number.
– user32096
Apr 3 '18 at 12:13
• I believe this question centers around how one would go about "perturbing" the initial state without modifying the states that led to that initial state (and so on, to infinity), without violating some laws of physics. They are arguing that the initial state is predetermined due to physics of everything that preceded it. - I think the answer comes down to the Butterfly Effect being a theory of models, not the universe. Apr 3 '18 at 12:27
• @BradleyUffner, thank you, that's exactly what I was asking. Maybe write your comment as an "Answer" and I will check it as the answer. Apr 3 '18 at 14:41
• @Asmani OddThinking's answer seems to pretty well align with what I was trying to get at. Unfortunately, I don't have the time right now to type up an answer in way that would be up to standards. My feelings won't be hurt if you accept that one. Apr 3 '18 at 14:44
• I'd say its similar to how we might call the order of a well shuffled deck "random." It's not random. There is exactly one order you are going to draw the cards in, which is the order they appear in that shuffled deck. However, the shuffling process is designed to be chaotic, making it so that you can't effectively model the order of the cards in the shuffled deck any better than random. Apr 3 '18 at 16:14

The "butterfly effect" appears to be a modern variant of the ancient philosophical axiom "Parvus error in principiis, magnus in conclusionibus" or "Parvus error in principio, magnus est in fine":

A small error in the beginning (or in principles) leads to a big error in the end (or in conclusions).

See St. Thomas Aquinas De Ente et Essentia, proemium, which references Aristotle De Cœlo bk. 1, specifically 271b:

…the least initial deviation from the truth is multiplied later a thousandfold. Admit, for instance, the existence of a minimum magnitude, and you will find that the minimum which you have introduced, small as it is, causes the greatest truths of mathematics to totter. The reason is that a principle is great rather in power than in extent; hence that which was small at the start turns out a giant at the end.

Upon which St. Thomas commentates (In De caelo lib. 1 l. 9 n. 4 [97.]):

…one who makes a slight departure from the truth in his principles gets 10,000 [i.e., many] times farther from the truth as he goes on. This is so because all things that follow depend on their principles. This is especially clear in an error at the crossroads: for one who at the beginning is only a slight distance from the right road gets very far away from it later on.* And he gives, as an example of what he is talking about, the case of those who posited a smallest magnitude, as Democritus posited indivisible bodies. By thus introducing a least quantity, he overthrew the most important propositions of mathematics — for example, that any given line can be cut into two halves. The reason for this effect is that a principle, though small in stature, is nevertheless great in power, just as from a small seed a mammoth tree is produced. Hence it is that what is small in the beginning becomes multiplied in the end, because it reaches unto all that to which the power of the principle extends, whether this be true or false.

*St. Thomas's example here is exactly that of Chaos: A Mathematical Adventure, ch. 2 "Vector Fields", 9:10ff.; see also ibid. ch. 7 "Strange Attractors & the Butterfly Effect".

Even if you assume determinism without quantum complications, I think a better interpretation (more consistent with Lorenz's original math) is this: Given a complete description of the state of the Universe at one point in time, calculating all the causes and effects forward may come to a point with, say, a hurricane in Florida. Suppose that we rewound back to that same initial state, and then made a small change--like removing the flap of a butterfly's wing in China. It's quite likely, then, that performing those exact same calculations on this new state for the same amount of time will result in a world without that hurricane.

• But sooner or later there would be a hurricane somewhere. The atmosphere is unstable. Unpredictable. We can totally rely on that.
– user32096
Apr 6 '18 at 13:43

In addition to the excellent answers given so far, you view this question in the light of something you call a "plan" or "laws", which you just pulled out of thin air.

Question your presupposition. The butterfly effect is also an excellent philosophical tool to illustrate that there is no plan and cannot be. (the exact proof here is lengthy, but trivial)

Now what you might mean by these words is determinism, but if we have learnt anything in the past century, it is that determinism only exists as a law of averages on the macro scale. The main philosophical effect of the butterfly effect is to remind us of this, and of the fact that under certain circumstances, a tiny change in the underlying causes can cause a ripple effect that causes dramatic changes in the result.

The butterfly effect is only the most extreme example of that. Another great example from everyday life is what Gottmann calls "sliding doors". Imagine catching the train, and you are almost late. The fraction of a second that decides if you just make it or the doors close just in front of you can make you late for one hour (or more, depending on how often the train goes). That's a factor of 10,000 or so between cause and effect.

Maybe that more down-to-earth example helps to understand the butterfly effect also in a less abstract way.

"How one would go about 'perturbing' the initial state without modifying the states"

The difference in initial conditions is generally an imaginary one. Because what the effect really points to, is that any uncertainty in measurement of the current universe limits our ability to make predictions.

Determinism relies on the idea that current complexity came out of earlier simplicity that could be computationally manageable, and in principle predict now. But complexity of iterating from the simple inital condition would fundamentally that - not enough bits in the world to represent the world.

So determinism was always just a fantasy anyway, complete knowledge of the past or future was always impossible from inside the universe. It is really of theological concern only, whether God 'plays dice' from outside.

• Even if determining the future is impossible in practice, the question of whether the universe is fundamentally deterministic or not is important - and I don't mean for theological/philosophical reasons. Even if determinism isn't 100% real, it's still a thing worth studying. It's not a black-and-white proposition, there's all sorts of shades of grey. You may not be able to predict accurately the path a baseball will take, but you still aim when throwing, don't you? You take into account reasonable risks, and do your best. Apr 4 '18 at 11:30
• As for determinism being a fantasy, you don't need the complete knowledge of the past. You just need a snapshot, and a set of rules to follow. If you needed the whole past and future, what would be the difference between a deterministic and a non-deterministic universe? Both would give you the exact same answers regardless. The same way, you don't necessarily need to take into account the whole universe, if you're not looking for 100% accuracy - the butterfly effect is interesting in that it assumes a deterministic universe, and still shows that it doesn't necessarily mean predictability. Apr 4 '18 at 11:33
• @Luaan You can't get a snapshot, that was my point - except possibly of infinitesimal microseconds after the big bang, which can't be iterated from - not least because the tiniest error from simplification needed to do calculations within the universe, could ripple into huge errors. Apr 4 '18 at 14:46
• Because the fundamental constants are transcendental. See? It solves the whole question.
– user32096
Apr 6 '18 at 13:37

The point of the thought experiment is that, there are things that are "either or," and some very small change may push some parameter of a system from one macro-state to another macro-state.

For a more vivid example, a specific human being's head may be 1mm away from the trajectory of a bullet, and the human survives. Later, that human may invent a cure for the common cold. A very small change to the initial state, if the head had been one inch off to the "wrong" side, would have had a very different outcome, and the common cold would not have been cured.

Other examples include a nail on the road in front of a car, being at enough of an angle to cause a flat tire, versus being slightly less than that angle and not causing a flat tire. Or a cat missing a jump up onto a ledge just barely and falling backwards/off, or just barely making it with enough space to claw its way up. In either of these cases, there is SOME "thin line" that determines the difference between success or failure. In fact, these kinds of systems are the vast majority of systems in our daily life, and thus we can't really "measure everything" and "run the computation" to simulate/predict the future.

These thought experiments are there to illustrate the vast outcome multiplier of seemingly small changes in input state in systems that are "chaotic" or "divergent." The experiment is not "flap or not flap a wing," or "move or not move a head," but instead, illustrations of the principle.

There exist chaotic systems we can actually run the A/B experiment on. It's most easily done in computer simulations, because there you can know it's 100% the same initial conditions, except for the one thing you change. But, as others have said, this is a tought experiment, intended to illustrate the concept of "large changes being predicated on very small differences in chaotic/divergent systems."

Finally:

In order to be accurate, we can only say that the current state of the universe is caused by its previous states.

This is one place this train of thought can end up, and the final conclusion from that thought is not just that there are no gods, but also that there is no free will. I personally find this argument perfectly acceptable, and there's no testable hypothesis or experiment that has been able to demonstrate anything to the contrary.

• The original butterfly concept came from computer simulations. A "cue ball" world with only dry atmosphere and sun shining on it = predictable, boring. Add some water vapor... Unpredictable. Like the 3 body problem.
– user32096
Apr 6 '18 at 13:40

It is said that in certain circumstances, a tiny change, like the flap of a butterfly's wings, can lead to enormous changes.

It's an image, not to be taken literally. All the billions of flaps of Butterfly wings cannot lead to big changes, because the displacement of air they cause is immediately lost in the white noise of all other air circulation anyway.

Same as in : "The Straw That Broke the Camel's Back", never in history has any Camels back been broken by adding a single straw.

Imagine you have (fair) dice. You throw it twice in exactly the same way. Do you think it will show the same number twice? No matter how hard you try to throw it exactly the same way, it will end up showing a different number, because even the slightest difference (butterfly's wing flap) will influence the throw.

Now imagine you throw a baseball, trying to hit a glove. You throw it twice exactly the same. Do you think you (or a competent thrower) can hit the glove twice? Sure they can, because it takes significant difference to make the ball miss the glove. So no matter how many butterflies are present, a good pitcher can make the ball hit a target.

So the number shown by a throw of dice is very sensitive to small changes, making it hard to repeat the same result or to predict, whereas hitting a glove with a baseball is a rather stable system not too dependent on small changes.

Chaos theory produced this concept of the butterfly effect.

I think this is conceptually an illusion. In lots of systems there are triggers and tendencies that grow and build to a massive extent, or fluctuate continually without any change.

So a system can exist, which is just about to set off this chain of events which is released by a small change. It is not the small change that matters, but the whole build up of the system that was ready to be triggered by a small change.

In human systems take the liberation of Iraq, bringing democracy to a society that did not know what to do with it. The trigger was there, but not the system that could take advantage of it.

Or Oliver Cromwell over throwing the King of England, yet could not trigger democracy and freedoms, to see the King return, and things progress slowly on.

Or the settlers in the USA, which led to the foundation of the country and the principles of self determination. A small change in expectation but with a system that could implement it.