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>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, whe

>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 have already explained why you are wrong. There is however a new error here: And I have countered your argument. I get it. You're trying to say that we're talking past each other here and that you're just repeating yourself. No. I addressed your point, you don't need to repeat yourself you only need to counter the new point. That being said, there is no new error. See this paper here which talks about the singularities at the top of the dome: http://philsci-archive.pitt.edu/3195/1/NortonDome.pdf http://philsci-archive.pitt.edu/3195/1/NortonDome.pdf Norton is 100% dealing with a singularity and therefore he is describing the exact same phenomenon of what happens to the particle after it exits the singularity. >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. I never said the mathematical theorem is not true. I said it doesn't apply. I did check the references. The theorem only applies for certain conditions and those conditions aren't met, so the theorem has nothing to say about the system at that point. Let's be clear. The theorem says one thing and one thing only that a unique solution exists if the conditions are met. When the conditions are not met (at the singularity) the theorem says absolutely nothing and therefore it does not apply. The theorem does not say that the solution does not exist, it just doesn't apply. >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. Doesn't matter. Infinite velocities is something you can't consider because the model is undefined at the singularity but it is defined both before and after the singularity. To give you a heursitic reason as well: imagine the successor function for peano arithmetic in number theory. We have a number X where X = 0 and subsequent numbers are defined as S(X) or S(S(X))) and so on. If X = 0 exists and A is any number, then A/S(X) still exists even though A/X doesn't exist (division by zero). This is what's going on here. Energy is conserved at the singularity (X = 0) but what that implies is undefined (1/X). But outside of the singularity the implication 1/S(X) is defined. There's no rule saying that 1/S(X) can't be defined just because 1/X is not defined. We are still in the purely mathematical realm here, we are not talking about physics in reality. Where the mechanics becomes nondeterministic is that (to continue with the heuristic analogy from above) 1/S(X) where X=0 is nondeterministic. Just like in number theory The literal math states that the system is undefined at the singularity. In other words the math DEFINES the solution to be UNDEFINED at the singularity. However the math itself makes zero statement about the system AFTER the particle exits the singularity. Hence we can say certain things about the particle if/when it exits the singularity even though many things are unknown. This is the reason why Norton is able to arrive at his conclusion of bounded nondeterminism. Certain things can be derived but newtons laws about the particle at 1/S(X) but not enough to arrive at a unique nondeterministic solution. The exact same thing applies to the n-body problem as it does to Nortons Dome. Also take a look at the paper I linked above and read the "Space Invaders" section. It addresses another possibility of in-determinism in a Newtonian n-body system based on how we arbitrarily choose to define it. This topic alone will initiate another angle in which to approach the question of determinism of an N-body system. >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. I am serious. The problem with experts on this topic is they all have different opinions on the topic. What if I pick the expert and author of the paper I linked above? There is no consensus among experts. I'll think about it but I don't want to be in a situation where the expert picks a solution and I remain unconvinced only to later find another expert who has the opposite opinion. I will acknowledge that there's an obvious $1000 bias on my side here and that a 3rd party levels the playing ground but unfortunately there is heavy bias on the "expert" side as well. I mean aren't you an "expert"? Hence your confidence on this topic. I think for now, unfortunately, we're going to have to settle with you convincing me and you'll have to take my word that if you can convince me I'll concede and try to look past my biases. Either way I'll think about your proposal. Also you should note my initial bet post was flagged so this entire thread is basically dead except for a couple people still responding to me.

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 have already explained why you are wrong. There is however a new error here: And I have countered your argument. I get it. You're trying to say that we're talking past each other here and that you're just repeating yourself. No. I addressed your point, you don't need to repeat yourself you only need to counter the new point. That being said, there is no new error. See this paper here which talks about the singularities at the top of the dome: http://philsci-archive.pitt.edu/3195/1/NortonDome.pdf http://philsci-archive.pitt.edu/3195/1/NortonDome.pdf Norton is 100% dealing with a singularity and therefore he is describing the exact same phenomenon of what happens to the particle after it exits the singularity. >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. I never said the mathematical theorem is not true. I said it doesn't apply. I did check the references. The theorem only applies for certain conditions and those conditions aren't met, so the theorem has nothing to say about the system at that point. Let's be clear. The theorem says one thing and one thing only that a unique solution exists if the conditions are met. When the conditions are not met (at the singularity) the theorem says absolutely nothing and therefore it does not apply. The theorem does not say that the solution does not exist, it just doesn't apply. >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. Doesn't matter. Infinite velocities is something you can't consider because the model is undefined at the singularity but it is defined both before and after the singularity. To give you a heursitic reason as well: imagine the successor function for peano arithmetic in number theory. We have a number X where X = 0 and subsequent numbers are defined as S(X) or S(S(X))) and so on. If X = 0 exists and A is any number, then A/S(X) still exists even though A/X doesn't exist (division by zero). This is what's going on here. Energy is conserved at the singularity (X = 0) but what that implies is undefined (1/X). But outside of the singularity the implication 1/S(X) is defined. There's no rule saying that 1/S(X) can't be defined just because 1/X is not defined. We are still in the purely mathematical realm here, we are not talking about physics in reality. Where the mechanics becomes nondeterministic is that (to continue with the heuristic analogy from above) 1/S(X) where X=0 is nondeterministic. Just like in number theory The literal math states that the system is undefined at the singularity. In other words the math DEFINES the solution to be UNDEFINED at the singularity. However the math itself makes zero statement about the system AFTER the particle exits the singularity. Hence we can say certain things about the particle if/when it exits the singularity even though many things are unknown. This is the reason why Norton is able to arrive at his conclusion of bounded nondeterminism. Certain things can be derived but newtons laws about the particle at 1/S(X) but not enough to arrive at a unique nondeterministic solution. The exact same thing applies to the n-body problem as it does to Nortons Dome. Also take a look at the paper I linked above and read the "Space Invaders" section. It addresses another possibility of in-determinism in a Newtonian n-body system based on how we arbitrarily choose to define it. This topic alone will initiate another angle in which to approach the question of determinism of an N-body system. >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. I am serious. The problem with experts on this topic is they all have different opinions on the topic. What if I pick the expert and author of the paper I linked above? There is no consensus among experts. I'll think about it but I don't want to be in a situation where the expert picks a solution and I remain unconvinced only to later find another expert who has the opposite opinion. I will acknowledge that there's an obvious $1000 bias on my side here and that a 3rd party levels the playing ground but unfortunately there is heavy bias on the "expert" side as well. I mean aren't you an "expert"? Hence your confidence on this topic. I think for now, unfortunately, we're going to have to settle with you convincing me and you'll have to take my word that if you can convince me I'll concede and try to look past my biases. Either way I'll think about your proposal. Also you should note my initial bet post was flagged so this entire thread is basically dead except for a couple people still responding to me.