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Vaniver comments on Edward Nelson claims proof of inconsistency in Peano Arithmetic - Less Wrong
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Comments (115)
I am curious why this received downvotes. Is it because people expected me to rot13 my answer?
I didn't downvote, but if I had it would have been because you seem to be missing the point a bit.
It may well be the case that we live in a discrete universe without infinities (I assume that's what you meant by integer) but this is not the case for Newtonian Mechanics, which asserts that both space and time remain continuous at every scale. Thomas never claimed to have a paradox in actual physics, only in Newtonian mechanics, so what you are saying is irrelevant to his claim, you might was well bring up GR or QM as a solution.
Appealing to the third law also doesn't help, his whole point is that if you do the calculations one way you get one answer and if you do them another way you get another answer, hence the use of the word paradox.
His point is if you only do some of the calculations, you can come up with the wrong answer. I fail to see how that implies a paradox rather than just sloppiness.
If I can do a calculation and stop and then do another calculation and get a contradiction that's still a contradiction. it doesn't matter that I can do other calculations that would lead to a non-contradiction.
If he put a single equal sign in his 'proof', I would be more charitable. As it is, it's not clear to me that he actually did any calculations or showed any contradictions.
I'm not sure he did, but I have done the calculations and it seems to check out (although I may have made a mistake). The only laws I used were F=ma and F=Gm_1m_2/(r^2), of which the third law should emerge as an immediate consequence rather than needing to be added in on top.
The reason they "check out" is because you calculate the force caused by N+1 particles on N particles. Because your calculation has an external particle, the CoM has an acceleration. This is entirely an artifact of how the limit is taken, and is thus a sign of sloppiness and incompleteness.
If you did the calculations for the system of N particles, then took the limit as N approached infinity, you would get no CoM acceleration. This really has nothing to do with Newtonian physics.
I don't think I do this.
Obviously the problem is with an infinity not taken as a limit. If you had said that, instead of saying other irrelevant things, then I doubt anyone would have objected.
Does your leftmost particle have a rightward acceleration which makes the weighted average of acceleration (i.e. CoM acceleration) 0?
I have edited the ancestral post to say that. Hopefully, the superior articulation will cause its karma to rise into the positives.
Since I wasn't using a limit I didn't have a leftmost particle.
I didn't downvote you, however both your objections were rather besides the point. One doesn't deal with the issue of Newtonian mechanics, and whether they were indeed as obviously flawed from the beginning -- it feels a bit like explaining the paradox of Achilles reaching the turtle by saying we're living in an integer universe. That may be true, but it feels a rather weak dodging-the-issue explanation.
As for the second answer, it doesn't really state where the calculations in the presentation of the paradox are wrong or are missing something. It just says that they are, so it's a non-solution.
Maybe this is because I was a physics student, but to me the missing pieces implied by the second answer are so obvious as to make it not worth my time to type them and yours to read them. Apparently I was mistaken, so here they are.
By the principle of superposition, we can break down a N-body problem into (N choose 2) 2-body problems. Consider the mass M and the mass m, a distance R away from each other (with m lighter and at the leftward position x). Their CoM is at x+RM/(M+m). The force FMm, leading to the acceleration am, is GMm/R^2, leading to GM/R^2. The force FmM, leading to the acceleration aM, is -GMm/R^2, leading to -Gm/R^2 (these are negative because it is being pulled leftward). To determine the acceleration of the center of mass, we calculate m*am+M*aM=GMm/R^2-GMm/R^2=0. The CoM of that pair will not move due to forces exerted by that pair. This is independent of M, m, and R. When we add a third mass, that consists of adding two new pair systems- each of which has a CoM acceleration of 0. This can be continued up to arbitrarily high N.
Whenever there's a paradox that mentions infinity, delete the infinity and see if the paradox still exists. Odds are very high it won't. A lot of diseased mathematical thinking is the result of not being clear with how you take limits, and so when I find a "paradox" that disappears when you get rid of infinity, that's enough for me to drop the problem.
This seems like an unproductive attitude. If everyone took this attitude they would never have hammered out the problems in calculus in the 19th century. And physicists would probably not have every discovered renormalization in the 20th century.
A better approach is to try to define rigorously what is happening with the infinities. When you try that, either it becomes impossible (that is there's something that seems intuitively definable that isn't definable) or one approach turns out to be correct, or you discover a hidden ambiguity in the problem. In any of those cases one learns a lot more than simply saying that there's an infinity so one can ignore the problem.
I will make sure to not spread that opinion in the event that I travel back in time.
I agree with you that this is a better approach. However, the problem in question is "find the error" not "how conservation of momentum works," and so as soon as you realize "hm, they're not treating this infinity as a limit" then the error is found, the problem is solved, and your curiosity should have annihilated itself.