Imagine a hundred trillion butterflies that each flap their wings in one synchronized movement, generating a massive gust of wind which is strong enough to topple buildings flatten mountains. If they were positioned correctly, they’d probably also be able to create a tornado that would not have occurred if the butterflies were not there flapping their wings, just by pushing air currents into place. Would that tornado be “caused” by the butterflies? I think most people would answer yes. If the swarm had not performed their mighty flap, the tornado would not have occurred.
Now, imagine that there’s an area where the butterfly-less conditions are almost sufficient to trigger a tornado at a specified location. Again, without any butterflies, no tornado occurs. A hundred trillion butterflies would do the job, but it also turns out that fifty trillion butterflies can also trigger a tornado using the same synchronized flap technique under these conditions. Then you find that a hundred butterflies would also trigger the tornado, and finally, it turns out that the system is so sensitive that a single butterfly’s wing-flap would be sufficient for the weather conditions to lead to a tornado. The boolean outcome of tornado vs no tornado, in this case, is the same for a hundred trillion flaps as it is for one. So if a hundred trillion flaps could be considered to cause a tornado, why can’t the one?
Of course, there are an uncountable number of things which are “causing” the tornado to occur. It would be ridiculous to say that the butterfly is solely responsible for the tornado, but the butterfly flap can be considered to be one of the initial conditions of the weather, and chaotic systems are by definition sensitive to initial conditions.
A deterministic system simulated by a perfectly accurate deterministic simulator would, given a set of inputs, produce the same outputs every time. If you change the value of a butterfly flap and the output is a tornado that otherwise would not occur, that does indeed mean that by some causal chain of events, the flap results in a tornado. A perfectly accurate deterministic simulator is, indeed, the only way a causal relationship between one event and another can be established with absolute certainty, because it is the only way to completely isolate a single variable to determine its effects on a system. Imagine the simulations as an experiment. The hypothesis is “This specific wing-flap of a butterfly in this specific environment causes a tornado in Texas in three months.” The simulator generates two simulations: one with the wing-flap and one with no wing-flap. The simulation with no wing-flap is the control simulation, and the simulation with the wing-flap is the experimental simulation. Because every single input variable other than wing-flap or no wing-flap is the same between the two simulations, and only the wing-flap simulation has the tornado, it must be that the wing-flap caused the tornado. This applies only to that specific wing-flap in that exact position and time. We cannot, for example, extrapolate that wing-flaps cause tornados in general.
If the universe is nondeterministic, then chaos theory doesn’t apply and neither does the butterfly effect.
Hi, I think I see what you mean. You can certainly say that the flap, as a part of the initial conditions, is part of the causes of the tornado. But this is true in the same sense in which all of the initial conditions are part of the cause of the tornado. The flap caused the tornado together with everything else. All the initial ocean temperatures, the position of the jet streams, the northern annular mode index, everything. If everything is the cause, then “being the cause of the tornado” is a property which carries exactly 0 bits of information, since everything is the cause.
I prefer to think that an event A “caused” another event B if the probability of B, conditioned on A happening, is at least greater than the prior probability of A.
The point is that in this scenario, the tornado does not occur unless the butterfly flaps its wings. That does not apply to “everything”, necessarily, it only applies to other things which must exist for the tornado to occur.
Probability is an abstraction in a deterministic universe (and, as I said above, the butterfly effect doesn’t apply to a nondeterministic universe.) The perfectly accurate deterministic simulator doesn’t use probability, because in a deterministic universe there is only one possible outcome given a set of initial conditions. The simulation is essentially demonstrating “there is a set of initial conditions such that when butterfly flap = 0 there is no Texas tornado, but when butterfly flap = 1 and no other initial conditions are changed, there is a Texas tornado.”
I see, but you are talking about an extremely idiosyncratic measure (only two points) on the space of initial conditions. One could as easily find another couple of initial conditions, in which the wing flip prevents the tornado.
If there were a prediction market on tornadoes, its estimations should not change in neither direction after observing the butterfly.
“there is a set of initial conditions such that when butterfly flap = 0 there is no Texas tornado, but when butterfly flap = 1 and no other initial conditions are changed, there is a Texas tornado.”
Phrased this way it is obviously true.
However, why are you saying that chaos requires determinism? I can think of some Markovian master equations with quite a chaotic behavior.
Imagine a hundred trillion butterflies that each flap their wings in one synchronized movement, generating a massive gust of wind which is strong enough to topple buildings flatten mountains. If they were positioned correctly, they’d probably also be able to create a tornado that would not have occurred if the butterflies were not there flapping their wings, just by pushing air currents into place. Would that tornado be “caused” by the butterflies? I think most people would answer yes. If the swarm had not performed their mighty flap, the tornado would not have occurred.
Now, imagine that there’s an area where the butterfly-less conditions are almost sufficient to trigger a tornado at a specified location. Again, without any butterflies, no tornado occurs. A hundred trillion butterflies would do the job, but it also turns out that fifty trillion butterflies can also trigger a tornado using the same synchronized flap technique under these conditions. Then you find that a hundred butterflies would also trigger the tornado, and finally, it turns out that the system is so sensitive that a single butterfly’s wing-flap would be sufficient for the weather conditions to lead to a tornado. The boolean outcome of tornado vs no tornado, in this case, is the same for a hundred trillion flaps as it is for one. So if a hundred trillion flaps could be considered to cause a tornado, why can’t the one?
Of course, there are an uncountable number of things which are “causing” the tornado to occur. It would be ridiculous to say that the butterfly is solely responsible for the tornado, but the butterfly flap can be considered to be one of the initial conditions of the weather, and chaotic systems are by definition sensitive to initial conditions.
A deterministic system simulated by a perfectly accurate deterministic simulator would, given a set of inputs, produce the same outputs every time. If you change the value of a butterfly flap and the output is a tornado that otherwise would not occur, that does indeed mean that by some causal chain of events, the flap results in a tornado. A perfectly accurate deterministic simulator is, indeed, the only way a causal relationship between one event and another can be established with absolute certainty, because it is the only way to completely isolate a single variable to determine its effects on a system. Imagine the simulations as an experiment. The hypothesis is “This specific wing-flap of a butterfly in this specific environment causes a tornado in Texas in three months.” The simulator generates two simulations: one with the wing-flap and one with no wing-flap. The simulation with no wing-flap is the control simulation, and the simulation with the wing-flap is the experimental simulation. Because every single input variable other than wing-flap or no wing-flap is the same between the two simulations, and only the wing-flap simulation has the tornado, it must be that the wing-flap caused the tornado. This applies only to that specific wing-flap in that exact position and time. We cannot, for example, extrapolate that wing-flaps cause tornados in general.
If the universe is nondeterministic, then chaos theory doesn’t apply and neither does the butterfly effect.
Hi, I think I see what you mean. You can certainly say that the flap, as a part of the initial conditions, is part of the causes of the tornado. But this is true in the same sense in which all of the initial conditions are part of the cause of the tornado. The flap caused the tornado together with everything else. All the initial ocean temperatures, the position of the jet streams, the northern annular mode index, everything. If everything is the cause, then “being the cause of the tornado” is a property which carries exactly 0 bits of information, since everything is the cause.
I prefer to think that an event A “caused” another event B if the probability of B, conditioned on A happening, is at least greater than the prior probability of A.
The point is that in this scenario, the tornado does not occur unless the butterfly flaps its wings. That does not apply to “everything”, necessarily, it only applies to other things which must exist for the tornado to occur.
Probability is an abstraction in a deterministic universe (and, as I said above, the butterfly effect doesn’t apply to a nondeterministic universe.) The perfectly accurate deterministic simulator doesn’t use probability, because in a deterministic universe there is only one possible outcome given a set of initial conditions. The simulation is essentially demonstrating “there is a set of initial conditions such that when butterfly flap = 0 there is no Texas tornado, but when butterfly flap = 1 and no other initial conditions are changed, there is a Texas tornado.”
I see, but you are talking about an extremely idiosyncratic measure (only two points) on the space of initial conditions. One could as easily find another couple of initial conditions, in which the wing flip prevents the tornado.
If there were a prediction market on tornadoes, its estimations should not change in neither direction after observing the butterfly.
Phrased this way it is obviously true.
However, why are you saying that chaos requires determinism? I can think of some Markovian master equations with quite a chaotic behavior.