A certain problem is quantum mechanics has been shown to be uncomputable. This means that although it is in a certain sense making a prediction, there is no systematic way to determine what prediction this is, in the same way given a program, there is no systematic way to determine if it halts.

No, this does not mean that quantum mechanics will screech to a halt. No, this does mean that quantum mechanics will be discarded as metaphysics. I do feel though that this will affect the philosophy of mechanics, and science as a whole.

As for an analogy, Godel's Incompleteness Theorem had vast effects on the philosophy of mathematics. It did not wreck mathematics, but its philosophy, and even some parts of mathematics itself, where changed. The Halting Problem, which is related to this Theorem, has had closely related effects.

My question is, what effect does undecidable problems in Quantum Mechanics on the philosophy of science. What are the philosophical implications. In particular, part of the philosophy of science is that theories must be able to generate predictions. Although there technically is a prediction in this case, it is not effectively decidable in general.

Is it really making a prediction, if there is literally no way to find it? Is it falsifiable if there is no procedure with which to falsify it?

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    Possible duplicate of The death of reductionism? – Alexander S King Dec 28 '15 at 1:15
  • If true, this would disprove the Church Turing thesis, which to many computer scientists would be a big deal. But I do not believe that QM is uncomputable. Which part of QM do you believe is uncomputable? – Alex Flint Dec 28 '15 at 3:51
  • @AlexFlint I forgot to add the link. It's edited in now. (Also, I never thought to it this reversed way, the fact that this might mean we could use it for uncomputable computations.) – PyRulez Dec 28 '15 at 4:18
  • @AlexFlint the result that he is talking about doesn't disprove the Church-Turing thesis. People don't realize how fundamental the Church-Turing thesis is. – Alexander S King Dec 28 '15 at 4:30
  • There is this resource on the question: scottaaronson.com/blog/?p=2586 undecidability in the recent result only concerns specific problems with systems of infinite size. – Quentin Ruyant Dec 28 '15 at 10:35
up vote 3 down vote accepted

The problem described in that paper is about calculating the limiting behavior of a lattice as its size goes to infinity. Because the uncomputability only comes in when considering the limit, it is not possible based on the results in that paper to construct a real physical experiment with an uncomputable outcome.

If someone did find a real finite (time, space) physical experiment with uncomputable results, it would disprove the Church-Turing thesis.

There are already many well-known "limiting-behavior" physics questions that are undecidable. For example, the Wang tiling problem asks whether it is possible to tile the 2D plane using a certain set of 2D tiles, and it turns out that this question is undecidable. However, there is also no finite (time, space) physical experiment that could determine the answer to this question, so the fact that this question is undecidable does not imply that physics itself is uncomputable.

More here: http://www.scottaaronson.com/blog/?p=2586

QM is already 'undecidable'; the measurement of state collapses it into an eigenstate; this collapse is usually described as indeterministic, that is one can't determine or decide what the eigenstate will be given knowledge of the state and the measurement to be made.

This is usually considered part of the Copenhagen Interpretation - the mainstream interpretation of QM - where an (macroscopic and conscious) observor observes a measurement; but also in relational QM, which democratises observers by considering all subsystems to be capable of observations aka measurements (not though, in MWI which trades this for an always endlessly multiplying uncountable number of worlds, everywhere and almost all at once).

Philosophically, then it is opposed to at least to the classical notion of determinism of nature - the clockwork universe or the mechanistic philosophy of the 18th C; by making indeterminism a wholly integral part of nature.

It's worth now pointing out, that given other interpretations like MWI, or Bohmian Mechanics that try to do away with indeterminism in this fashion, are forced to admit it in through situations explored in the OPs question; there, all interpretations are forced to admit indeterminism

Also, too; that Aristotles later analysis of the concept of motion - provoked by Zeno - requires an ontology that is structured by potentialities and actualities.

Which might beg the question, why do away with it first to return to a classical conception of physical ontology?

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    +1 upvote. I consider this a clever answer based on the Copenhagen interpretation. – Jo Wehler Dec 28 '15 at 11:42
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    The unpredictability of measurement outcome doesn't require collapse. In the MWI, all you can know before doing a measurement is that there will be two versions of you. There is no way in principle to know which of those two versions you will be because there is no single fact of the matter about that issue. – alanf Dec 28 '15 at 11:48
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    This doesn't seem to address the OP's question: What are the philosophical implications of undecidability in QM? – Chris Sunami Dec 28 '15 at 14:34
  • @alanf: true, also in Bohmian Mechanics; I've added a qualifier naming the interpretation I'm using. – Mozibur Ullah Dec 28 '15 at 16:54
  • @sunami: ok, I've added a few lines. – Mozibur Ullah Dec 28 '15 at 17:02

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