I just ran across the most incredible article that this thread has led me to.
I've argued that since the three body problem is not solvable by computer (although it can be arbitrarily well approximated given sufficient computing resources), the universe is not computable by a TM. Since the universe clearly does solve the n-body problem for an unfathomably large (but finite) value of n, either the universe violates the Church-Turing thesis, by being a "computation" of a type we haven't yet conceived; or the universe doesn't operate by any laws or principles at all. (I'm swapping out universe for mind here, but it's the same argument either way).
I've been percolating on all this for a while. Ever since I read Ivars Peterson's book,
Newton's Clock: Chaos in the Solar System. It turns out to be
not true that in Newtonian physics, "if you knew every particle and its motion you could predict the future perfectly." Once you realize that widely-held belief is
flat-out false, you begin to see the limits of computation itself. The universe is doing something that our idea of computation can not do. And this is important to understand.
Just now I ran across this article, which looks technical and will take a bit of a commitment beyond my usual short attention span. But finally I found someone who seems to be having the exact same thoughts. And a lot more. Here's the abstract.
Abtract of this paper said:
‘‘Church’s thesis’’ is at the foundation of computer science. We point out that with any particular set of physical laws, Church’s thesis need not merely be postulated, in fact it may be decidable. Trying to do so is valuable. In Newton’s laws of physics with point masses, we outline a proof that Church’s thesis is false; physics is unsimulable. But with certain more realistic laws of motion, incorporating some relativistic effects, the extended Church’s thesis is true. Along the way we prove a useful theorem: a wide class of ordinary differential equations may be integrated with ‘‘polynomial slowdown’’. Warning: we cannot give careful definitions and caveats in this abstract–you must read the full text—and interpreting our results is not trivial.
http://research.cs.queensu.ca/~akl/cisc879/papers/PAPERS_FROM_APPLIED_MATHEMATICS_AND_COMPUTATION/Special_Issue_on_Hypercomputation/smith[1].pdf
In particular this phrase made me feel so validated:
In Newton’s laws of physics with point masses, we outline a proof that Church’s thesis is false; physics is unsimulable.
Yes! That's exactly my point. Substitute mind for universe and it's the same argument.
To go further in my understanding I have to fill in my quantum ignorance, which is vast. This paper looks perfect as a starting point.
I'm going to take a run at this and if I find anything interesting I'll report back. If I'm a little less active online it's because I've summoned the discipline to work through the paper.
ps -- It's 183 pages and very technically dense. I'll have to change my objective from working through completely, to getting in there and finding the answer to my question:
How does quantum theory let the universe "compute" the n-body problem; what compute means in this context; and how this relates to Church-Turing. If I can begin to understand the outlines I'll have a much better understanding of the computationalist position. I might even become a convert, although I'd hate for that to happen!