[Tom Bishop]

Well, it's still the same fallacy. You take a video of a guy who says it DOES work with computer simulations even though we don't have what he calls "a neat set of equations" (ie, a formal closed-form solution), he literally says "we can get around this". From that video, you infer that he admits it does not work. If you did that on Wikipedia, you would eventually get banned for backing your statements with citations that say exactly the opposite.

"It's an approximation" still isn't an equivalent of "it doesn't work". Perfect solution fallacy, once again.

"We can get around this" means that he needs to employ work arounds and cheats. Two body approximations for a three body system is hardly a solution to the three body problem, for example. By treating it as two problems, Earth and Moon and Sun and Earth, it totally ignores the problem of three bodies.

Appealing to a word "approximation" alone is not reasonable argument.

You need to show us what they are doing and demonstrate that the three body problem actually does work, to some degree, to explain the motions of the celestial systems. Rather, you are appealing to the word "approximations" without knowledge of what is actually being "approximated".

The number "one" is approximately "zero," but this is also rather dumb to state as a rule and invalidates a proof which shows that 1 + 1 = 2.

You are arguing that we can cheat our way through. Lets see the nature of these cheats you find acceptable.

But you guys cannot actually show us a working system, or a statement from a physicist which states what you are arguing, and must argue on basis of personal "inferences" of the statements that you read. Rather insufficient, TBH.

[GreatATuin]

I really don't get your point. An approximated solution is an approximated solution. It's not an absence of solution. Just like 3.14159265 is an approximated value of pi. It will be accurate enough for most purposes. If for some special purpose I need a more accurate value, I'll take the time and effort needed to find one.

We have approximated solutions to the n-body problem and they are accurate enough for the systems we want to simulate. It looks like you really don't want that to be true, but facts are stubborn things.

[Tom Bishop]

From the askamathematician page we saw that it was acknowledged that the three body problem doesn't work except for some goofy scenerios, and then the author went off on tangents talking about cheating ways of doing it:

https://www.askamathematician.com/2011/10/q-what-is-the-three-body-problem/

the problem with the 3-body problem is that it can’t be done, except in a very small set of frankly goofy scenarios (like identical planets following identical orbits).

...

So, if you want to calculate the orbits of all the planets, a “2-body approximation” will get you more than 99% of the way to the right answer.

...

But even with just mechanical pencil and paper there are cheats.  For example, although there are more than three bodies in the solar system (the Sun, eight planets, dozens of moons, and millions of asteroids and comets), almost everything behaves, roughly, as though it were in a two body system.

Two body approximations, cheats, etc.

How is a two body approximation a solution to this? This ignores the problem of three bodies and is a total cheat.

Again, lets see the nature of these "approximation" cheats that you claim exist, and are putting forward.

How about some real sources for what you are claiming for once, rather than citing yourself as a source?

[GreatATuin]

From the askamathematician page we saw that it was acknowledged that the three body problem doesn't work except for some goofy scenerios, and then the author went off on tangents talking about cheating ways of doing it:

https://www.askamathematician.com/2011/10/q-what-is-the-three-body-problem/

the problem with the 3-body problem is that it can’t be done, except in a very small set of frankly goofy scenarios (like identical planets following identical orbits).

...

So, if you want to calculate the orbits of all the planets, a “2-body approximation” will get you more than 99% of the way to the right answer.

...

But even with just mechanical pencil and paper there are cheats.  For example, although there are more than three bodies in the solar system (the Sun, eight planets, dozens of moons, and millions of asteroids and comets), almost everything behaves, roughly, as though it were in a two body system.

Two body approximations, cheats, etc.

How is a two body approximation a solution to this? This ignores the problem of three bodies and is a total cheat.

Again, lets see the nature of these "approximation" cheats that you claim exist, and are putting forward.

How about some real sources for what you are claiming for once, rather than citing yourself as a source?

OK, I'll begin with the same askamathematician page you quoted:

Despite that, we do alright, and happily, reality doesn’t concern itself with doing math, it just kinda “does”.  For example, quantum field theory, despite being the most accurate theory that ever there was, never involves exactly solving anything.  Once a physicist gets a hold of all the appropriate equations and a big computer, they can start approximating things.  With enough computing power and time, these approximations can be made amazingly good.  Computer simulation and approximation is a whole science unto itself.

Point is, this effect only shows up in systems with three or more bodies, it’s chaotic (in the chaos theory sense), and there is no way to predict it exactly.  That being said, we can still get computers to come pretty close (up to a point, because chaos is a punk), and there are even some mathematical tricks to get reasonable solutions that, while not perfect, are still pretty good (and can even get us well into that last “1% of weirdness”).

[Tom Bishop]

It's saying that the approximations can get pretty good, and only gives the method of the two body problem approximation for the orbital problems in the article.

This invalidates the inferences that you had to argue from.

Again, you guys just read what you want to read. Those quotes are in the section talking about approximations. The approximations can be good, but ultimately invalid and irrelevant for the three body problem.

[GreatATuin]

The source you chose literally says that the lack of a perfect formal solution isn't a big problem because we can get computer simulations to be very accurate. If you decide to read and understand anything else from this text, I'm afraid there isn't anything more I can say.

Edited for your edit: why would an approximation be irrelevant for the n-body problem, or any other problem? You never get a perfect solution from a computer simulation. The perfect solution is the real system.

[Tom Bishop]

The source you chose literally says that the lack of a perfect formal solution isn't a big problem because we can get computer simulations to be very accurate. If you decide to read and understand anything else from this text, I'm afraid there isn't anything more I can say.

Edited for your edit: why would an approximation be irrelevant for the n-body problem, or any other problem? You never get a perfect solution from a computer simulation. The perfect solution is the real system.

The only orbital approximation method discussed in the article is talking about two body problem approximations. They can approximate it by treating it as Sun-Earth and Earth-Moon.

The approximation is basically "The number two is close to the number three, so oh well, approximated"

It is bewildering how you can defend this as "not a perfect solution" but that it's almost there, and quote a section which calls those approximations "pretty good" and think that this is any way sufficient. The approximation method totally ignored the concept of three bodies.

[GreatATuin]

The only orbital approximation method discussed in the article is talking about two body problem approximations.

This is absolutely not what the article says. Read again:

Point is, this effect only shows up in systems with three or more bodies, it’s chaotic (in the chaos theory sense), and there is no way to predict it exactly.  That being said, we can still get computers to come pretty close (up to a point, because chaos is a punk), and there are even some mathematical tricks to get reasonable solutions that, while not perfect, are still pretty good (and can even get us well into that last “1% of weirdness”).

This "last 1% of weirdness" being what you can't explain with a 2 body simulation.

I rest my case, perfect solution fallacy. Anyway, a simulation is pretty much always an approximation. The map is not the territory. Ceci n'est pas une pipe.

[Tom Bishop]

No, it's already talking about approximations by that point in the article.

The previous paragraph even abandons the idea of any bodies:

"When you have even more bodies you can almost abandon the idea that there are any bodies at all, and move over to fluid dynamics.  Although, again, that’s just an approximation."

And before that " Once a physicist gets a hold of all the appropriate equations and a big computer, they can start approximating things.  With enough computing power and time, these approximations can be made amazingly good.  Computer simulation and approximation is a whole science unto itself."

By the time of your quote it's talking about how we can approximate our way out of it

The only approximation mentioned in the article is two body approximations. And how computer simulations can be approximated.

So, since your quote comes from a section about approximations, how does this help you?

[GreatATuin]

No, it's already talking about approximations by that point in the article.

The whole point of this article is that approximations are good enough for our use cases. Including for the n-body problem or other problems in physics that may not have a "perfect" mathematical solution either. You distort it to keep on with the same perfect solution fallacy and claim that since there is no perfect solution, there is no solution at all. This continues to be wrong, no matter how many times you repeat it.

[Pete Svarrior]

[You] claim that since there is no perfect solution, there is no solution at all.

Ah, how refreshing to hear that from an RE'er. Now, if only you could convince the rest of your side of this...

That said, pattern-based approximations are largely model-agnostic. If this is the best you have, then you no longer have an argument for RE's predictive powers being superior in this case.

This continues to be wrong, no matter how many times you repeat it.

Hey! That's my line!

[existoid]

[You] claim that since there is no perfect solution, there is no solution at all.

Ah, how refreshing to hear that from an RE'er. Now, if only you could convince the rest of your side of this...

That said, pattern-based approximations are largely model-agnostic. If this is the best you have, then you no longer have an argument for RE's predictive powers being superior in this case.

Depends on how approximate the predictions can get, though, right?  We can regularly predict with exceptional accuracy everything that happens in our solar system among the celestial bodies we know of - eclipses, comets, etc. - but more spectacularly the arrival of dozens of man-made objects sent off to various places such as a single moon of a distant planet.  That would be quite a trick if NASA and others weren't able to actually predict these things, but were just amazing guessers.

[Pete Svarrior]

That would be quite a trick if NASA and others weren't able to actually predict these things, but were just amazing guessers.

You've missed the point. The question isn't whether those predictions are accurate (even if imprecise), but what the source of them is. RE'ers like to claim that it's RET, but in this thread we have the smoking gun - it's an observation of patterns and computer modelling based on those patterns. Accuracy aside, as soon as you make this admission, it ceases to be evidence pointing towards RET.

[Tim Alphabeaver]

That would be quite a trick if NASA and others weren't able to actually predict these things, but were just amazing guessers.

You've missed the point. The question isn't whether those predictions are accurate (even if imprecise), but what the source of them is. RE'ers like to claim that it's RET, but in this thread we have the smoking gun - it's an observation of patterns and computer modelling based on those patterns. Accuracy aside, as soon as you make this admission, it ceases to be evidence pointing towards RET.

But NASA models not using patterns but 'RET' models (Newtonian Gravity, Relativity etc.). They take the equations of motion from these theories and put them into some code - am I missing something?

[existoid]

That would be quite a trick if NASA and others weren't able to actually predict these things, but were just amazing guessers.

You've missed the point. The question isn't whether those predictions are accurate (even if imprecise), but what the source of them is. RE'ers like to claim that it's RET, but in this thread we have the smoking gun - it's an observation of patterns and computer modelling based on those patterns. Accuracy aside, as soon as you make this admission, it ceases to be evidence pointing towards RET.

But NASA models not using patterns but 'RET' models (Newtonian Gravity, Relativity etc.). They take the equations of motion from these theories and put them into some code - am I missing something?

That is my understanding as well.  The "computer modelling" is not simple pattern recognition like some fancy AI, but simply highly complex and/or iterative math equations or something.

[Tim Alphabeaver]

I think the actual interesting question in this discussion is are there any real, physical systems that are analytically solvable? I think the answer is no. Can we discount all of physics, now?

[AATW]

Acceptance != proven. There were points in history where people accepted the existence of witches too.

So firstly, no scientific theory is ever proven in the strictest sense. The best we can say is that Einstein's theories work better than Newton's in certain situations. And by "work better" I mean they make predictions which more accurately match observations. For most normal situations there is no measurable difference.

You get hung up on the 3 body problem a lot and I don't understand what your issue is.
You're right in that there is no solution. There are no equations which have a variable "t" for time, you plug in the initial positions of the bodies, you put in a value for t and the equations give you the results in terms of where all the bodies will be 't' seconds into the future. I think if anything this is a failure of mathematics, or maybe it's an inherent problem with chaotic systems (chaotic in the mathematical sense). The weather is chaotic too - or our models of it are. That's why weather forecasts are often wrong and there is no such things as a truly long range forecast.

The best we can do is split the n-body problem into a series of 2 body problems. Solve those for a small time increment and then iterate. So yes, it's a bit of a workaround but the key here is it does work. It's got us to the moon, it's put craft on Mars. I don't think any of these models are simply using cycles, they're using Newton's equations to work out from initial positions and velocities what the next positions and velocities will be in small time increments and iterating. But because of this iteration if the initial values are incorrect - in the sense that they are not perfect, which they can't be - then over time those errors will build up.

So yes, this is a weakness of our models, BUT...a model does not have to be perfect to be useful. Someone else said that the value of pi cannot be used perfectly because it's an irrational number. But we know the value well enough to use it to make accurate calculations. If we used the value 3 then that wouldn't be good enough for most purposes, 3.14159265 (which is as much as I know off the top of my head) is good enough for pretty much any purpose.

The headline is while yes, we don't have "a solution" to the n body problem I don't see why you think that is a problem. We have ways of calculating it well enough to send craft into space and land them on other celestial bodies. I'd say that's a triumph of our models, not a failure.

[existoid]

I think the actual interesting question in this discussion is are there any real, physical systems that are analytically solvable? I think the answer is no. Can we discount all of physics, now?

I am not sure I fully understand what it means for something to be "analytically solvable."   But if what you are writing here is true, would your statement then equally apply to proposed FE physical systems?

[Pete Svarrior]

am I missing something?

You are. Namely: the source of these "approximate" equations.

[Tim Alphabeaver]

am I missing something?

You are. Namely: the source of these "approximate" equations.

The equations aren't approximations - just their solutions.