That’s what you don’t understand - *every single one of them *has an effect on the outcome.
Well, in the case of double pendulum, yeah. It would actually. Weather, dunno.
Regardless of your doubts, the renowned mathematician and meteorologist Edward Lorenz established weather as the absolutely classic example of deterministic chaos. For his achievements that are now widely recognized and accepted, he was awarded a dozen major international honors including the Kyoto Prize for his pioneering work in chaos theory and its application to weather systems.
Fair enough. I won’t deny the quote. The double pendulum is a case where the grid is uniform. Any of the huge space the thing operates in, gravity is pulling down the same at all points.
So it logically makes sense that a tiny perturbation would affect the outcome because there is a huge amount of valid locations for the pendulum at any future moment in time.
With the weather, if the energy gradient says that Dallas is getting some cold air, it’s hard to see how sending the air to Cleveland is an equally reachable location.
I guess it comes down to what we *mean *by the weather being chaotic. Two locations within a mile of each other might or might not get rained on if there’s a 50% chance of rain in a given area. (on average the rain clouds will miss one of those two locations). I could see a tiny nudge affecting the later track of the rain clouds and making the other location get the rain.
But the butterfly is in Kuala Lumpur and the hurricane forms two years later just west of the Canaries. There are millions of intervening causal vectors. Blaming the primary cause on the butterfly is absurd.
Storms are ultimately driven by the earth’s net energy budget, but what you’re not appreciating is that the energy flows and their distribution can be influenced by trivially minor events. The failure of your “energy gradient” argument is that no one disputes that once those systems are formed, important aspects of their behavior are no longer subject to small perturbations. The accuracy of modern weather forecasts arises from the fact that they’re based on the observation of systems and events already in play. But the initial formation of weather systems is one of the most dramatic practical examples of chaotic systems and the validity of chaos theory. On longer timescales, the predictability of energy balances places constraints on the evolution of global climate, but even there, the development of regional climates can still be chaotic.
The butterfly is not on trial. Its legal culpability is not the same as its mathematical effect, which is that because of its actions, each of those millions of other events now successively operates on different initial conditions that diverge more and more from what they otherwise would have been.
The point (or at least a point) of the butterfly metaphor is that the butterfly flapping its wings is a **necessary **condition for the hurricane to form. No one is arguing that it is a **sufficient **condition for the hurricane to form.
Disturbingly, you actually hold that power.
But consider the bright side: we now all picture you as Jean Claude van Damme.
It’s more complicated than that. The conditions for a hurricane to form depend on the actions of every butterfly, and of course on every other perturbation of butterfly-size or larger. Just as it’s possible for a butterfly to cause a hurricane, it’s also possible for it to prevent one, or for some other butterfly to do either.
Ah!
I assume that the initial formation is similar to how a crystal gets started. You have a super-saturated solution and a crystal can in theory start at any point within the volume. But it takes a tiny bit of dirt or a seed of the same kind of crystal for the system to actually choose that tiny seed as the start location.
Similarly, what you are saying is that the energy conditions must be there for the weather to start. And then for a narrow window of time, over a large area in theory the center of the weather system could have started at any point over that area. But some small perturbation ends up starting it forming.
This was actually the same argument I made earlier. When snow or sand accumulates for an avalanche, only because the system was primed and ready for one allows a single grain of sand/single snowflake to actually start the avalanche.
Similarly, it sounds like weather systems almost all of the time “ignore” butterflies. Only at an exact time and place could such a butterfly flap have any real effect. The rest of the time it doesn’t matter.
That’s what I was trying to say.
That is indeed the same argument you made earlier. And as a characterization of chaos theory, that argument is fundamentally wrong. That misconception also led you to doubt that chaos theory was an important factor in weather, where it’s so fundamentally important that it’s where the theory originated in the first place.
You keep wanting to regard the butterfly metaphor as a trigger that precipitates an event for which all the conditions are in place. But what it really refers to is an early precursor that, over time, created those conditions in the first place. Weather is an excellent example of how Lorenz summed up the meaning of chaos: when the present determines the future, but the approximate present does not approximately determine the future (because you didn’t include that butterfly).
I agree that time travel is not practically possible for the reasons already enumerated here and which need no repetition. For this we can truly be thankful because I can’t even imagine the chaos that would ensue because of people going back in time, changing events and creating time paradoxes. I would go as far as to say that a time machine would end up being a true Dooms Day Device.
Recognize that, in any given time travel situation, it is not guaranteed that a paradox will happen, just that it might happen. One proposed solution is that all the “oops we have a paradox” outcomes simply can’t happen, which means that one of the “whew, everything is fine” outcomes must result, no matter how unlikely they seem. For example, you try to shoot your own grandfather but the gun jams.
If there is any chance at all that the time machine would malfunction in such a way as to destroy the universe, that would be a rather elegant solution for avoiding paradoxes. But a much simpler solution would be for the machine to malfunction in a way that merely destroys itself and/or its occupants. Every time anyone tries to switch on a time machine, it immediately blows up.
Ok, so in the avalanche/chemistry physical systems, you agree that in both examples small chaotic events did not set up the situation for an avalanche or a crystal to grow. It took some effect that systematically has been raising the salinity in a solution over time to the point that a crystal is possible. Or has been systematically adding and adding sand to a pile.
For a weather system like a hurricane, that would be the sun acting and acting over months to create the gradients that allow for a hurricane to form.
I’m not sure if this fits your definition of chaos or not. Because once the system is “primed” there can be a period of time where it teters in balance. Where a hurricane can form but has not yet.
Actually it takes more like a few months for such a tiny perturbation to significantly alter large scale state of the weather in chaotic models of the weather as the perturbation grows exponentially. The point is not about the butterfly being the “primary cause”, but certainly the butterfly is as big a factor on the state of system as anything else after a certain time.
It would be upsetting for weather prediction because the development up until the storm system formation would be a smooth steady change in temperatures. And the development after the storm system is fully built and moving would be a predictable “impact point” and maximum size depend on the track it follows. But until that mass of air is actually existing and has moved far enough to measure it’s velocity you wouldn’t know the target of a hurricane.
With that said I don’t know if weather uses the same rules that simpler systems like crystals and avalanches use. I freely admit this whole topic is out of my depth.
But that is still wrong though: it is a characteristic of chaotic system that small scale perturbations grow exponentially over time and it doesn’t require the system to be in a special “primed” state for small perturbations to grow in this way.
Or, to put it simply, weather does not behave like piles of sand.
Blaming the hurricane on the butterfly misses the point. A major component of chaos theory is “sensitivity to initial conditions”. Image parallel Earths E1 and E2 where the butterfly flaps its wings at a slightly different moment in time and the same chain of events takes place. By the time the hurricane makes landfall thousands of miles away, would the exact same trees be knocked down on E1 and E2? Probably not. There are bound to be hundreds of people whose houses either are or aren’t destroyed by a falling tree because the butterfly flapped its wings at 3:12:09 vs 3:12:17. And consider E3 where the buffalo never sneezes at all, but the hurricane still happens anyway. The exact location and wind speed would be slightly different and once again a completely different set of trees would be knocked down. Even if there’s only a single tree that falls on E1 but doesn’t fall on E2, that’s a big deal for whatever animals made nests in that particular tree.
The other factor, which the butterfly -> sneeze -> hurricane analogy doesn’t make clear, is that small effects can add up to large effects given enough time. If two asteroids collide in outer space and their angle of impact changes by a tiny millionth of a degree, sending the debris off in a slightly different direction which differs from the original path by just one millimeter per hour… just give that vector a million years to work and the trajectory is off by almost 9,000 km. That’s enough to make the difference between hitting a planet (causing an extinction event) or missing the planet entirely.
It’s something I also grapple with. When things get too deep in the weeds, I start to ask myself: should I stay or oregano? If I stay there will be trouble, if oregano there will be double.
A thymeless conflict.
That sounds a lot more like “wishful thinking” than a theory based on science.
I travel back in time because I want to see where my grandmother grew up. As I drive down the street, I am eagerly looking for the correct house number because I know I’m almost there! Unfortunately, that makes me very distracted. Suddenly, a little girl appears in front of my car and, before I can stop, I run her over and kill her. That little girl is (was? would have become?) my mother.
The paradox is obvious. If I’ve killed my mother, I would have never been born, which means I wouldn’t have ended up back in time killing my mother, which means I would have been born after all and … Well, you get the point.
To say some magical force would have prevented the accident seems very bogus to me. It’s a way of avoiding the problem, not a way of solving it.
Of the existing theories, the one I believe is that you actually create an alternative universe because, out of the infinite number of Quantum possibilities that exist at any given moment, a different “choice” has been made than the one that was made in your own universe. The only way to avoid the paradox is for BOTH occurrences to exist in two separate universes because the paradox precludes the possibility of both of them existing in the same universe.