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>So theoretically speaking, they are deterministic, but practically they are unpredictable. The three body problem is REFERING to the ideal case where only a p

>So theoretically speaking, they are deterministic, but practically they are unpredictable. The three body problem is REFERING to the ideal case where only a perfect model is considered. Within this model we don't even know the math to calculate it. So, in short, we lack a "deterministic theory" about this problem at all. What's going on here is an assumption. We assume that an idealistic scenario will always produce the same result. But we don't actually know because we don't even have a proper model. From a certain angle what this paper is kind of saying is that this assumption is WRONG and that the underlying model of the ideal case of the three body problem IS probabilistic. I'm not a math/physics guy but the fact that this "theory" involves probability seems sort of like the same initial cop out that came with quantum theory. It's like we can't explain mathematically why a particle behaves this way but it seems to be obeying a probability so let's make probability the basis of the theory! Problem Solved! Men are by probability more likely to join engineering than women. Because of this probability should we make up a theory called "The fundamental theory of men and women joining engineering" that is described by a probability equation? OR is it better to find an underlying more "deterministic" explanation for why this occurs? It may sound like I'm denigrating the probabilistic path here, but really theories like this usually only come about when it's basically impossible to come up with a deterministic version. And technically speaking we can never actually know whether the foundations of the universe are probabilistic or deterministic.

konschubert 5y ago
No. We know there is a deterministic solution, even if we can't calculate it.
- deltasixeight 5y ago
  Prove it. I guarantee you 100% you don't know how and you can't ever prove it. You are completely and utterly wrong and you don't know what you're talking about. That is a fact. In fact I'm willing to put money on it. $1000 over venmo to you, a random stranger. You down? Give me a formal and correct proof that if a solution exists and that it is deterministic and I swear I'll venmo $1000 to whatever address you wish. Can you do the same? No, you can't. If not then I'm right. We don't KNOW whether it's deterministic because our BEST model is probabilistic. We really should raise the stakes and seriousness of HN comments to prevent random uneducated people who really don't know anything from posting random comments. Like come on... HN is better than reddit but some random set of "rules" against unsubstantiated comments doesn't stop people like this guy from posting unsubstantiated comments.
  - spekcular 5y ago
    You think Newtonian physics is non-deterministic? I haven't checked but I think existence and uniqueness for the classical three body problem likely follows from standard theorems on ODEs. If you can articulate why this is not the case, please do. edit: Yes, see section 2.1 of this term paper: https://sites.math.washington.edu/~morrow/336_13/papers/peter.pdf https://sites.math.washington.edu/~morrow/336_13/papers/pete.... In fact, more is known: the solution exists and is analytic. From Wikipedia: "However, in 1912 the Finnish mathematician Karl Fritiof Sundman proved that there exists an analytic solution to the three-body problem in the form of a power series in terms of powers of t1/3. This series converges for all real t, except for initial conditions corresponding to zero angular momentum. In practice, the latter restriction is insignificant since initial conditions with zero angular momentum are rare, having Lebesgue measure zero. "
    - deltasixeight 5y ago
      I didn't say that. For the three body problem we don't know the closed form solution under Newtonian physics. So if we don't know it, then a closed solution may not even exist. If we don't know whether a closed form solution exists then how can we even know if it's deterministic. You can prove determinism by finding me a solution that's closed and deterministic. My answer is the same as stated: we don't know, and that the claim that we do know, is completely wrong. >In fact, more is known: the solution exists and is analytic These are special cases, under special conditions. A general solution is not actually known. We don't know. We are only assuming that it's deterministic. Also the solution you provided is analytic not closed.
      - spekcular 5y ago
        "If we don't know whether a closed form solution exists then how can we even know if it's deterministic. You can prove determinism by finding me a solution that's closed and deterministic." Why would it be necessary to have a closed form solution to know that the solution is deterministic? By the first reference in my previous comment, two identical initial conditions lead to identical time evolutions. That is literally the definition of determinism.
- deltasixeight 5y ago
  https://sites.pitt.edu/~jdnorton/Goodies/Dome/ https://sites.pitt.edu/~jdnorton/Goodies/Dome/ we don't.
kergonath 5y ago
We can have these problems simply with classical mechanics’ equations of motion (Newton, Lagrange, Hamilton, whatever). These equations are deterministic, there is no doubt about this. We just don’t have an analytical form. We know that exactly the same initial conditions will lead to the exact same trajectory. What we also know is that the tiniest error will make the trajectories diverge exponentially. It is still deterministic. AFAICT, their random walk idea is not in the trajectories themselves, but in the sampling of the possible trajectories to assign them a statistical weight, a bit like we do commonly in statistical Physics.
- deltasixeight 5y ago
  >We know that exactly the same initial conditions will lead to the exact same trajectory. Do we have a proof of this? Or is it that we just assume this?
  - throwaway81523 5y ago
    Yes we have a proof (in the classical case). It is called the Picard–Lindelöf theorem. The "dome" in your example is called an unstable equilibrium: the ball starts rolling because of an unpredictable perturbation. Small motions are amplified according to the Lyapunov exponent, which in the unstable case is large. Hirsch and Smale and Devaney's book on differential equations and dynamical systems is a good place to learn about this. In classical mechanics we model a zero-diameter point at the top of the dome, but a real physical dome would be made of molecules vibrating randomly and that starts the ball rolling. At the end of the day, classical mechanics is nonphysical: it doesn't describe nature except as an approximation. Quantum mechanics on the other hand is non-deterministic. Is there a deeper, so-called superdeterministic reality underneath quantum mechanics? Some people think yes, and it has not been disproved, but it is pretty far out of the mainstream from what I can tell.
    - deltasixeight 5y ago
      No read the dome example again. You are wrong. Nortons dome is literally demonstrating there is proof for the non deterministic case. I've discussed this in other places but the theorem you reference only applies to functions that are Lipschitz continuous. Not all functions have this property globally and the dome and also the three body problem are two examples of things that are not Lipschitz continuous. Read it carefully. Nortons Dome is saying that the mathematical model described by newtons laws of classical mechanics is in itself non-deterministic. > Is there a deeper, so-called superdeterministic reality underneath quantum mechanics? Some people think yes, and it has not been disproved, but it is pretty far out of the mainstream from what I can tell. Never made this claim. Not even Einstein made this claim. Simply put, it felt wrong to Einstein simply because probability is a sort of bayesian outlook on the world. It's an admission that we lack knowledge about a system. Such is the case for much of science but not quantum mechanics? Hence the quote by Einstein: "God does not play dice with the universe." Either way not saying that quantum mechanics is crap and wrong but this is a possibility the mainstream definitely considers in a very speculative philosophical fashion. The quantum model works extraordinarily well but at the same time it's inconsistent with relativity and you gotta admit something is a bit off here when considered from the Bayesian angle.
dang 5y ago
We detached this subthread from https://news.ycombinator.com/item?id=28179014 https://news.ycombinator.com/item?id=28179014.

konschubert 5y ago

No. We know there is a deterministic solution, even if we can't calculate it.

deltasixeight 5y ago
Prove it. I guarantee you 100% you don't know how and you can't ever prove it. You are completely and utterly wrong and you don't know what you're talking about. That is a fact. In fact I'm willing to put money on it. $1000 over venmo to you, a random stranger. You down? Give me a formal and correct proof that if a solution exists and that it is deterministic and I swear I'll venmo $1000 to whatever address you wish. Can you do the same? No, you can't. If not then I'm right. We don't KNOW whether it's deterministic because our BEST model is probabilistic. We really should raise the stakes and seriousness of HN comments to prevent random uneducated people who really don't know anything from posting random comments. Like come on... HN is better than reddit but some random set of "rules" against unsubstantiated comments doesn't stop people like this guy from posting unsubstantiated comments.
- spekcular 5y ago
  You think Newtonian physics is non-deterministic? I haven't checked but I think existence and uniqueness for the classical three body problem likely follows from standard theorems on ODEs. If you can articulate why this is not the case, please do. edit: Yes, see section 2.1 of this term paper: https://sites.math.washington.edu/~morrow/336_13/papers/peter.pdf https://sites.math.washington.edu/~morrow/336_13/papers/pete.... In fact, more is known: the solution exists and is analytic. From Wikipedia: "However, in 1912 the Finnish mathematician Karl Fritiof Sundman proved that there exists an analytic solution to the three-body problem in the form of a power series in terms of powers of t1/3. This series converges for all real t, except for initial conditions corresponding to zero angular momentum. In practice, the latter restriction is insignificant since initial conditions with zero angular momentum are rare, having Lebesgue measure zero. "
  - deltasixeight 5y ago
    I didn't say that. For the three body problem we don't know the closed form solution under Newtonian physics. So if we don't know it, then a closed solution may not even exist. If we don't know whether a closed form solution exists then how can we even know if it's deterministic. You can prove determinism by finding me a solution that's closed and deterministic. My answer is the same as stated: we don't know, and that the claim that we do know, is completely wrong. >In fact, more is known: the solution exists and is analytic These are special cases, under special conditions. A general solution is not actually known. We don't know. We are only assuming that it's deterministic. Also the solution you provided is analytic not closed.
    - spekcular 5y ago
      "If we don't know whether a closed form solution exists then how can we even know if it's deterministic. You can prove determinism by finding me a solution that's closed and deterministic." Why would it be necessary to have a closed form solution to know that the solution is deterministic? By the first reference in my previous comment, two identical initial conditions lead to identical time evolutions. That is literally the definition of determinism.
deltasixeight 5y ago
https://sites.pitt.edu/~jdnorton/Goodies/Dome/ https://sites.pitt.edu/~jdnorton/Goodies/Dome/ we don't.

kergonath 5y ago

We can have these problems simply with classical mechanics’ equations of motion (Newton, Lagrange, Hamilton, whatever). These equations are deterministic, there is no doubt about this. We just don’t have an analytical form. We know that exactly the same initial conditions will lead to the exact same trajectory. What we also know is that the tiniest error will make the trajectories diverge exponentially. It is still deterministic. AFAICT, their random walk idea is not in the trajectories themselves, but in the sampling of the possible trajectories to assign them a statistical weight, a bit like we do commonly in statistical Physics.

deltasixeight 5y ago
>We know that exactly the same initial conditions will lead to the exact same trajectory. Do we have a proof of this? Or is it that we just assume this?
- throwaway81523 5y ago
  Yes we have a proof (in the classical case). It is called the Picard–Lindelöf theorem. The "dome" in your example is called an unstable equilibrium: the ball starts rolling because of an unpredictable perturbation. Small motions are amplified according to the Lyapunov exponent, which in the unstable case is large. Hirsch and Smale and Devaney's book on differential equations and dynamical systems is a good place to learn about this. In classical mechanics we model a zero-diameter point at the top of the dome, but a real physical dome would be made of molecules vibrating randomly and that starts the ball rolling. At the end of the day, classical mechanics is nonphysical: it doesn't describe nature except as an approximation. Quantum mechanics on the other hand is non-deterministic. Is there a deeper, so-called superdeterministic reality underneath quantum mechanics? Some people think yes, and it has not been disproved, but it is pretty far out of the mainstream from what I can tell.
  - deltasixeight 5y ago
    No read the dome example again. You are wrong. Nortons dome is literally demonstrating there is proof for the non deterministic case. I've discussed this in other places but the theorem you reference only applies to functions that are Lipschitz continuous. Not all functions have this property globally and the dome and also the three body problem are two examples of things that are not Lipschitz continuous. Read it carefully. Nortons Dome is saying that the mathematical model described by newtons laws of classical mechanics is in itself non-deterministic. > Is there a deeper, so-called superdeterministic reality underneath quantum mechanics? Some people think yes, and it has not been disproved, but it is pretty far out of the mainstream from what I can tell. Never made this claim. Not even Einstein made this claim. Simply put, it felt wrong to Einstein simply because probability is a sort of bayesian outlook on the world. It's an admission that we lack knowledge about a system. Such is the case for much of science but not quantum mechanics? Hence the quote by Einstein: "God does not play dice with the universe." Either way not saying that quantum mechanics is crap and wrong but this is a possibility the mainstream definitely considers in a very speculative philosophical fashion. The quantum model works extraordinarily well but at the same time it's inconsistent with relativity and you gotta admit something is a bit off here when considered from the Bayesian angle.

dang 5y ago

We detached this subthread from https://news.ycombinator.com/item?id=28179014 https://news.ycombinator.com/item?id=28179014.