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Given that there is no exact general solution to the $N$-body problem, can it be concluded that the Universe is non-deterministic, even for the Newtonian case (ignoring relativistic and quantum effects)?

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    $\begingroup$ Not having a closed-form solution is not the same as not having a unique future. $\endgroup$ – dmckee Apr 8 '14 at 15:46
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    $\begingroup$ But there does exist an exact, unique solution for given initial data. The fundamental theorem of ODEs guarantees this. $\endgroup$ – Robin Ekman Apr 8 '14 at 15:47
  • $\begingroup$ There is a difference between closed-form expressibility and determinism. The $N$-body problem is not closed-form expressible, but it is still deterministic. $\endgroup$ – DumpsterDoofus Apr 8 '14 at 19:10
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No.

The n-body problem is a problem of calculation, not of determinism. There is nothing non-deterministic about Newtonian mechanics, it is just hard to calculate.

By contrast, a quantum system is generally accepted to be non-deterministic by its nature; Einstein disagreed with this idea, hence his statement that 'God does not play dice'.

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  • $\begingroup$ is it generally accepted that QM is non-deterministic? surely many-worlds etc believe in nothing but deterministic evolution of the wave-function by schrodinger's equation. $\endgroup$ – innisfree Apr 8 '14 at 16:12
  • $\begingroup$ I would say it is generally accepted. As we can only falsify non-determinism (i.e. we cannot prove that something has no cause, only that none has been found), there will always be some who dislike the idea of something unseen deciding whether the cat lives or not. However, unless you are in the field of many-worlds theories and so on, it is immaterial what you believe; the equations give you the non-determinism you will have to deal with. The most broadly accepted interpretation (Copenhagen) is that we do not know what happens in the midst of an interaction, only that it follows our equations $\endgroup$ – Phil H Apr 8 '14 at 16:18
  • $\begingroup$ on second thoughts, i am being pedantic. yes perhaps it is generally accepted. $\endgroup$ – innisfree Apr 8 '14 at 16:41
  • $\begingroup$ I'm not confusing them, they are the same thing. We see from the double slit experiment that making the result deterministic changes the very probabilities of the evolution of the system. I'm probably not explaining it very well. Many worlds just says that the mechanism of multiple states evolving simultaneously is deterministic; i.e. the universe bifurcates at every instant. But it does not give an answer on how or why recombination happens. Yet some prefer it because it takes one of the question marks away (at the cost of adding an infinitely bifurcating universe). $\endgroup$ – Phil H Apr 8 '14 at 16:42
  • $\begingroup$ sorry phil, i deleted the above comment :) i was splitting hairs, your answer is good +1 $\endgroup$ – innisfree Apr 8 '14 at 16:44

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