5 ms·

> However let's reiterate what came from your initial post: "You think Newtonian physics is non-deterministic?" > I have proven this statement to not be someth

> However let's reiterate what came from your initial post: "You think Newtonian physics is non-deterministic?" > I have proven this statement to not be something that I think, but something that is completely true and thus I have validated what you "thought" was my initial point. That should provide some partial conclusions to your initial points and you have admitted that you agree. I'm been aware of the dome example for some time. I didn't think such pathologies were worth discussing because they are irrelevant to the current problem, where multiple solutions for the same initial condition do not exist. >> What there does not apply to all initial conditions with distinct mass positions? > The theory does not apply at the singularities. You stated this yourself, no solution exists. You are repeating what I said. Initial conditions with two or more masses at the same position don't have well-defined coefficients, so one can't meaningfully speak about there being a solution. I see no problems with determinism here. > Can you prove to me that past the point of collision the solution is deterministic? It doesn't make sense to talk about a solution past the point of collision. One doesn't exist. > After the collision there are multiple outcomes that can occur depending on factors you assume and make up in your regularization scheme. All of these possible paths are governed by classical mechanical laws as they are outside of the singularity but we have several possibilities here. Or maybe not. Who knows. My claim is nobody knows. But we do know the system is still ruled by physical laws so the space of possible solutions is bounded. Basically I'm asking you to prove to me that within this space there is one deterministic unique solution. I think this is where the root of your confusion lies. You write "after the collision there are multiple outcomes that can occur depending on factors you assume and and make up in your regularization scheme." The classical three-body problem (discussed in the article and the references I gave) is a problem about a system of differential equations. The formulation is entirely mathematical: I give you some differential equations and the initial conditions, and I want a solution. The physics only enters in the selection of those equations; once I write them down, it's a math problem. Further, it is a theorem that solutions do not exist past the point of collision. In summary, we know rigorously there is no non-determinism in the three-body problem, because it is a theorem that the same initial condition can never give rise to two distinct solutions.

deltasixeight 5y ago
>I'm been aware of the dome example for some time. I didn't think such pathologies were worth discussing because they are irrelevant to the current problem, where multiple solutions for the same initial condition do not exist. If you're aware that means you knew your own statement was false and you were attempting to argue a point you didn't believe. The pathology was expanded upon by you. I talked about the n body problem, your statement expanded it to all of newtons laws. >You are repeating what I said. Initial conditions with two or more masses at the same position don't have well-defined coefficients, so one can't meaningfully speak about there being a solution. I see no problems with determinism here. Yes I am. Because you asked me a question where the answer involves repeating it. I do see a problem with determinism, because the theorem that is invoked by the paper is about uniqueness. The theorem is called "the existence and uniqueness theorem" and it gives a set of conditions under which an initial value problem has a unique solution. One of these conditions is Lipschitz continuation which fails at the singularity. Thus the existence and uniqueness of a solution is not guaranteed and thus the theorem doesn't apply. No guarantee of existence means no guarantee of completeness. No guarantee of uniqueness means no guarantee of determinism. >It doesn't make sense to talk about a solution past the point of collision. One doesn't exist. This is not true. By definition if the particle is past the point of collision it is not in a singularity and therefore it CAN be modeled by Newtons laws of motion. We just don't know from Newtons laws when it emerges from the singularity or what the exact initial state is outside of the singularity. That being said the state of the particle is still bounded by newtons laws and thus there are certain things we can still say about the particle. For example while this is isn't technically newtons law we know that Energy/mass is conserved before, through and after the singularity. There is no confusion here. A singularity exists at the top of Nortons Dome and Norton is exactly describing the behavior of the particle AFTER it exits the singularity. The exact same phenomena is occurring in the n-body problem as nortons dome.
- spekcular 5y ago
  I have already explained why you are wrong. There is however a new error here: > There is no confusion here. A singularity exists at the top of Nortons Dome and Norton is exactly describing the behavior of the particle AFTER it exits the singularity. The exact same phenomena is occurring in the n-body problem as nortons dome. There's no singularity in Norton's dome. The equation d^2r/dt^2 = r^{1/2} is well-defined for all non-negative times t. This is not the case in the three-body problem, where a coefficient blows up in finite time when two bodies collide. The dome example is not relevant here and you are confusing yourself by conflating the two phenomena. >> >It doesn't make sense to talk about a solution past the point of collision. One doesn't exist. > This is not true. And just to repeat myself: it is true, it's a mathematical theorem, you can't continue the solution past the point of the collision. Please check the references. If you want a heuristic reason for this, note that conservation of energy implies infinite velocities at the point of collision. The situation is inherently unphysical due to it being a mathematical idealization. edit: If you're serious about the $1k thing, I propose the following. We agree on a precise mathematical formulation of the question "Is the three-body problem deterministic?," an expert, and a charity. We email the expert for comment. If I am right, you donate to the charity. If I am wrong, I will admit to it here, and perhaps donate some money myself. We can put the agreement on our personal websites and post it to HN to publicly commit.

deltasixeight 5y ago

>I'm been aware of the dome example for some time. I didn't think such pathologies were worth discussing because they are irrelevant to the current problem, where multiple solutions for the same initial condition do not exist. If you're aware that means you knew your own statement was false and you were attempting to argue a point you didn't believe. The pathology was expanded upon by you. I talked about the n body problem, your statement expanded it to all of newtons laws. >You are repeating what I said. Initial conditions with two or more masses at the same position don't have well-defined coefficients, so one can't meaningfully speak about there being a solution. I see no problems with determinism here. Yes I am. Because you asked me a question where the answer involves repeating it. I do see a problem with determinism, because the theorem that is invoked by the paper is about uniqueness. The theorem is called "the existence and uniqueness theorem" and it gives a set of conditions under which an initial value problem has a unique solution. One of these conditions is Lipschitz continuation which fails at the singularity. Thus the existence and uniqueness of a solution is not guaranteed and thus the theorem doesn't apply. No guarantee of existence means no guarantee of completeness. No guarantee of uniqueness means no guarantee of determinism. >It doesn't make sense to talk about a solution past the point of collision. One doesn't exist. This is not true. By definition if the particle is past the point of collision it is not in a singularity and therefore it CAN be modeled by Newtons laws of motion. We just don't know from Newtons laws when it emerges from the singularity or what the exact initial state is outside of the singularity. That being said the state of the particle is still bounded by newtons laws and thus there are certain things we can still say about the particle. For example while this is isn't technically newtons law we know that Energy/mass is conserved before, through and after the singularity. There is no confusion here. A singularity exists at the top of Nortons Dome and Norton is exactly describing the behavior of the particle AFTER it exits the singularity. The exact same phenomena is occurring in the n-body problem as nortons dome.

spekcular 5y ago
I have already explained why you are wrong. There is however a new error here: > There is no confusion here. A singularity exists at the top of Nortons Dome and Norton is exactly describing the behavior of the particle AFTER it exits the singularity. The exact same phenomena is occurring in the n-body problem as nortons dome. There's no singularity in Norton's dome. The equation d^2r/dt^2 = r^{1/2} is well-defined for all non-negative times t. This is not the case in the three-body problem, where a coefficient blows up in finite time when two bodies collide. The dome example is not relevant here and you are confusing yourself by conflating the two phenomena. >> >It doesn't make sense to talk about a solution past the point of collision. One doesn't exist. > This is not true. And just to repeat myself: it is true, it's a mathematical theorem, you can't continue the solution past the point of the collision. Please check the references. If you want a heuristic reason for this, note that conservation of energy implies infinite velocities at the point of collision. The situation is inherently unphysical due to it being a mathematical idealization. edit: If you're serious about the $1k thing, I propose the following. We agree on a precise mathematical formulation of the question "Is the three-body problem deterministic?," an expert, and a charity. We email the expert for comment. If I am right, you donate to the charity. If I am wrong, I will admit to it here, and perhaps donate some money myself. We can put the agreement on our personal websites and post it to HN to publicly commit.