Three body problem is computable for any finite amount of time (like all conservative systems). To get problems the end state must be reached in a finite time. This can happen in dissipative systems.
There are many cases where you can’t even get approximate solutions, though you can get probabilities of various solutions. For example, Mercury is in a 3/2 rotation to revolution rate around the Sun. It was expected to be 1:1 like the Moon around the Earth. A bit of a surprise, since the 1-1 ratio is the lowest energy one. However, everything near the 3/2 state is higher energy, so it is stable. Now the interesting thig is that the boundaries between the attractors are such that there are regions in which any two points in one attractor has a point in the other attractor between them. So no degree of accuracy of measurement can allow predicting which attractor the system is in. So Frances Darwin’s explanation of why the Moon always faces the Earth is incomplete, and can never be fully completed. There is about 50% likelihood of 1-1 capture, 33% for 3-2 capture, and the rest take up the remaining chances. Note that the end states aren’t just a little bit different, but a lot different. Things get much more complicated in evolution and development, where more factors are involved. I argue that information dissipation (e.g., through death eliminating genetic information) works the same way. I first published on this as the first paper in the journal Biology and Philosophy n 1986. The main point is the problem is not one of our limited calculation capacity. It holds in principle. Even Laplace’s demon, if they are like a regular computer, but arbitrarily large, could not do the calculations. Basically, there are far more functions that are not Turing computable than are, and many of these give widely different possible solutions. It’s really just another case of the number of theorems being aleph 1, but the number of possible proofs is only aleph 0. I call systems like the Mercury –Sun system reductively explainable, but not reductive. Physicalism is not violated, but reduction is not possible. But we can get a good idea of what is going on, after the fact (though our first guess in the Mercury-Sun case was wrong). Personally, I think all thirds are of this nature, which is why they can’t be reduced to dyads. I have never found Pierce’s arguments convincing about the irreducibility. John From: Clark Goble [mailto:cl...@lextek.com] Sent: Wednesday, 12 April 2017 1:47 PM To: Peirce-L <PEIRCE-L@list.iupui.edu> Subject: Re: [PEIRCE-L] Laws of Nature as Signs On Apr 12, 2017, at 11:21 AM, John Collier <colli...@ukzn.ac.za<mailto:colli...@ukzn.ac.za>> wrote: Some reductions are impossible because the functions are not computable, even in Newtonian mechanics. Are you talking about the problem in mathematics of solving things like the three body problem? That’s not quite what I was thinking of rather I was more thinking that any solution is approximate and the errors can propagate in weird ways. But that’s true of almost any real phenomena which is more complex than we can calculate. It’s not just an issue of reduction although it clearly manifests in the problem of reduction and emergence.
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