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> 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

> 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
This has grown too long for me to reply to fully, but I want to make one technical clarification. When I say "singularity," I mean that the coefficients blow up (aren't continuous because they shoot toward infinity in finite time). When author of the pdf file you link to writes "singularity," he means "C^1, but not C^2." You might want to re-read my posts above with this in mind. You seem to have in mind some regularization procedure to get past the singularity. I don't agree this is really a valid solution. It wouldn't satisfy the original ODE! (Note that one can regularize e.g. simple collisions of two of the bodies by changing coordinates, see http://www.math.tifr.res.in/~publ/ln/tifr42.pdf http://www.math.tifr.res.in/~publ/ln/tifr42.pdf.) I think you're getting too bogged down in trying to think physical/heuristically here. There's a well-defined math problem, a system of ODEs. It's a theorem that, in the case of a collision, there's a solution on some interval [0, t_0), and as the system approaches t_0, the masses collide and at least one coefficient in the equations blows up, rendering it undefined. You cannot give a solution there (in the original coordinates) because the problem isn't even defined at that point.
- deltasixeight 5y ago
  >When I say "singularity," I mean that the coefficients blow up (aren't continuous because they shoot toward infinity in finite time). When author of the pdf file you link to writes "singularity," he means "C^1, but not C^2." You might want to re-read my posts above with this in mind. This is just pedantry. It doesn't matter in what space or what variable the singularity occurs if one exists the solution is undefined at time t=T. This is common knowledge. Additionally in your own words you said there wasn't a singularity and that is definitively false both technically and in terms of common mathematical parlance, there IS a singularity. So this is more of a technical correction rather than a clarification. >You seem to have in mind some regularization procedure to get past the singularity. I don't agree this is really a valid solution I don't see it as a valid solution EITHER. regularization is just making shit up. I don't think you understood my number theory example. Put it this way. With number theory 1/t is undefined at t = 0. But at the exact left side and right side of t = 0 everything is defined again. There is no mathematical property preventing this from happening with numbers. Same with the ODEs. Undefined at the point of collision with no explicit mathematical rule saying it's not defined everywhere else. So really it's [0, t_0) and (t_0, infinite]. Either way there is a singularity at the top of the dome so what occurs on the dome is isomorphic to a system of n-bodies at the singularities. Simply look what happens at t > T for nortons dome. For some arbitary time T the system has an infinite but bounded amount of directions to roll off the dome. So infinty possible T's with infinite possible directions. It can at any time emerge from the singularity and roll in any direction. That is nondeterminism. Not even worth fully deriving the motion mathematically because of so many possibilities. This is THE SAME thing for the singularity in the n body problem, which is probably why the standard procedure to "solve" this sort of problem past the singularity (and without regularization) is uncommon. At an arbitrary time T the particle may emerge from the singularity with arbitrary unknown properties. But we do know that arbitrary laws still hold like the conservation of mass and energy, we know the particle isn't going to teleport 20 light years away or anything like that. So there are an infinite amount of possible outcomes but those outcomes are bounded by the laws of newton JUST like what happens to the particle on top of Nortons Dome. I think what's going on here is that you're bogged down by the procedural math routines to solve these types of problems. You have to think a bit outside of the box in order to deal with paradoxes and singularities, yet you must stay within the logical realm of axioms and theorems. There is a proof that shows a contradiction that basically explains why certain things are undefined at certain points. There is no proof that says the ODE is only defined on an interval up to a singularity. The proof only demonstrates it is undefined AT the singularity. Every possible state of the particle emerging from the singularity at every possible time t > T is a potential solution.

spekcular 5y ago

This has grown too long for me to reply to fully, but I want to make one technical clarification. When I say "singularity," I mean that the coefficients blow up (aren't continuous because they shoot toward infinity in finite time). When author of the pdf file you link to writes "singularity," he means "C^1, but not C^2." You might want to re-read my posts above with this in mind. You seem to have in mind some regularization procedure to get past the singularity. I don't agree this is really a valid solution. It wouldn't satisfy the original ODE! (Note that one can regularize e.g. simple collisions of two of the bodies by changing coordinates, see http://www.math.tifr.res.in/~publ/ln/tifr42.pdf http://www.math.tifr.res.in/~publ/ln/tifr42.pdf.) I think you're getting too bogged down in trying to think physical/heuristically here. There's a well-defined math problem, a system of ODEs. It's a theorem that, in the case of a collision, there's a solution on some interval [0, t_0), and as the system approaches t_0, the masses collide and at least one coefficient in the equations blows up, rendering it undefined. You cannot give a solution there (in the original coordinates) because the problem isn't even defined at that point.

deltasixeight 5y ago
>When I say "singularity," I mean that the coefficients blow up (aren't continuous because they shoot toward infinity in finite time). When author of the pdf file you link to writes "singularity," he means "C^1, but not C^2." You might want to re-read my posts above with this in mind. This is just pedantry. It doesn't matter in what space or what variable the singularity occurs if one exists the solution is undefined at time t=T. This is common knowledge. Additionally in your own words you said there wasn't a singularity and that is definitively false both technically and in terms of common mathematical parlance, there IS a singularity. So this is more of a technical correction rather than a clarification. >You seem to have in mind some regularization procedure to get past the singularity. I don't agree this is really a valid solution I don't see it as a valid solution EITHER. regularization is just making shit up. I don't think you understood my number theory example. Put it this way. With number theory 1/t is undefined at t = 0. But at the exact left side and right side of t = 0 everything is defined again. There is no mathematical property preventing this from happening with numbers. Same with the ODEs. Undefined at the point of collision with no explicit mathematical rule saying it's not defined everywhere else. So really it's [0, t_0) and (t_0, infinite]. Either way there is a singularity at the top of the dome so what occurs on the dome is isomorphic to a system of n-bodies at the singularities. Simply look what happens at t > T for nortons dome. For some arbitary time T the system has an infinite but bounded amount of directions to roll off the dome. So infinty possible T's with infinite possible directions. It can at any time emerge from the singularity and roll in any direction. That is nondeterminism. Not even worth fully deriving the motion mathematically because of so many possibilities. This is THE SAME thing for the singularity in the n body problem, which is probably why the standard procedure to "solve" this sort of problem past the singularity (and without regularization) is uncommon. At an arbitrary time T the particle may emerge from the singularity with arbitrary unknown properties. But we do know that arbitrary laws still hold like the conservation of mass and energy, we know the particle isn't going to teleport 20 light years away or anything like that. So there are an infinite amount of possible outcomes but those outcomes are bounded by the laws of newton JUST like what happens to the particle on top of Nortons Dome. I think what's going on here is that you're bogged down by the procedural math routines to solve these types of problems. You have to think a bit outside of the box in order to deal with paradoxes and singularities, yet you must stay within the logical realm of axioms and theorems. There is a proof that shows a contradiction that basically explains why certain things are undefined at certain points. There is no proof that says the ODE is only defined on an interval up to a singularity. The proof only demonstrates it is undefined AT the singularity. Every possible state of the particle emerging from the singularity at every possible time t > T is a potential solution.