• • Thread: Undecidability and Collapse of the Wavefunction

1. Undecidability and Collapse of the Wavefunction

 It was shown about a century ago that there are some mathematical problems which it is impossible to prove true or false. A famous example is Turing's halting problem. The problem is simply this; will a given Turing machine (/program) ever stop? The only way you can be sure that it will stop is by following out the instructions and seeing if they stop. However, you could be doing that until the end of time. Even then, you would have no idea if it was ever going to stop or not. Interestingly this mathematical fact has physical consequences. All you have to do is physically act out a Turing machine. For example, you could use a model railway. There are two points, one at the start of the line, representing the start of the program, and one at the end, representing STOP. There are also multiple points and tracks in between, representing a program. As the train passes through each stop, it is able to change the direction of the point (memory). Now, this got me thinking. I'm not sure if it has been proposed before. What people then say is that 'there is no method by which you can work out if the train will ever stop. Obviously it either stops or it doesn't, but you can't work out which'. I see a clear analogy with quantum mechanics here: take for example the electron double slit experiment. Until the electron actually goes through one of the slits, there is no way of knowing which way it will go; only when you observe what it does can you know which of the two options occurs. Is this not exactly the same mysterious situation that we have with the undecidable train track?  Reply With Quote

2.  Not in my view. Quantum mechanics is just fundamentally probabilistic. That may be unintuitive to humans, but there's no reason to think that it's philosophically unnatural. On the other hand, the halting problem is a case of a deterministic system, so yeah it's different.  Reply With Quote

3.  Well, my point was that the deterministic system is essentially probabilistic. There's no way of knowing what the outcome will be, even though it is deterministic, and it is hard to see what else probability is other than not possibly knowing what an outcome will be. Perhaps probability and determinism are not so antithetical? That's what I'm suggesting.  Reply With Quote

4. Originally Posted by Xei Well, my point was that the deterministic system is essentially probabilistic. There's no way of knowing what the outcome will be, even though it is deterministic, and it is hard to see what else probability is other than not possibly knowing what an outcome will be. Perhaps probability and determinism are not so antithetical? That's what I'm suggesting. So a related question would be, are chaotic systems probabilistic? Yes and no. They are sensitive to initial conditions, so in the real world, if you're using a floating point system, yes they're probabilistic. But in a purely mathematical sense, there are ways to predict behaviour given an exact starting point. However, quantum systems don't have that property. Even in principle, even if the starting point is exact, you can't predict their behaviour.  Reply With Quote

5. Originally Posted by Xei A famous example is Turing's halting problem. The problem is simply this; will a given Turing machine (/program) ever stop? The only way you can be sure that it will stop is by following out the instructions and seeing if they stop. However, you could be doing that until the end of time. Even then, you would have no idea if it was ever going to stop or not. Now I just read about this the first time so this might be totally off, but I'm under the impression that that's not the point of the proof. Given a program and an input it is possible to decide if the program will ever stop for lots of cases, just not all. For those that you can't decide, it is essentially a logical contradiction.  Reply With Quote

6.  What do you mean by 'logical contradiction'? And no, not all of them are ambiguous. For example, if it loops, then clearly it will never terminate. But lots of them are. So a related question would be, are chaotic systems probabilistic? Yes and no. They are sensitive to initial conditions, so in the real world, if you're using a floating point system, yes they're probabilistic. But in a purely mathematical sense, there are ways to predict behaviour given an exact starting point. However, quantum systems don't have that property. Even in principle, even if the starting point is exact, you can't predict their behaviour. I'd say that's something different. Chaotic systems are still completely predictable given the initial conditions.  Reply With Quote

7. Originally Posted by Xei I'd say that's something different. Chaotic systems are still completely predictable given the initial conditions. The problem with that is that it's impossible to give the initial conditions exactly, i.e. there will always be error. Particles obey Heisenberg uncertainty principle, which means they don't sit in space with a exact position or momentum, if you knew the precise momentum you would not know the precise position. So if you have a choatic system that has momentum and position you can't predict anything from it. The universe is not a Turing machine. Quantum mechanics doesn't act like a classical computer it acts like a Quantum computer.  Reply With Quote Posting Permissions

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