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In a talk Gödel and the End of Physics by Steven Hawking, he argues all mathematical problems are also physical problems for example:

  • Given an even number of wood blocks, can you always divide them into two piles, each of which cannot be arranged in a rectangle?

which is of course a physical problem equivalent to the Goldbach Conjecture.

I'm not sure whether this is really a physical problem, and I'd love to hear some thoughts on this, though my actual question isn't so philosophical.

  • Are there any actual physical questions that use unsolved (not necessarily undecidable) mathematics?

For example, if X is true then Y should be the case, and if X is false then Y should not be the case. Where X is an unsolved (not necessarily undecidable) mathematical statement and Y is a truly physical statement like

  • "Whats the maximum density of electrons?" or "Do black holes exhibit such-and-such radiation" etc.
user203652
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  • There is an equivalence between the question "Is 2483952475209437598 a sum of two primes?" and a question about wood blocks. It does not follow that there is an equivalence between the question "Is every even number a sum of two primes?" and some question about wood blocks. – WillO Dec 28 '15 at 18:21
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    I don't see how to separate open math problems from open physics problems. E.g., if we ask if Yang-Mills theories have a mass gap, is this a physics or a math problem? – Norbert Schuch Dec 28 '15 at 22:42
  • I think you are suffering from severe misunderstanding about both Goedel's theorem and physics. 1) When mathematicians hit an undecidable theorem in one set of axioms, they look for a different set of axioms in which it is decidable and they move on with the synthesis of the two. It creates a CHOICE. 2) All undecidable problems involve infinite sets, which do not exist in physics, hence they are irrelevant. 3) Physics is not ruled by math but by empirical observations. Math is just a tool to order those observations. It has no deciding power, at all. – CuriousOne Dec 29 '15 at 00:11
  • Given the "wood block" criterion of what counts as physical, of course there are physical problems that use unsolved mathematics. Any Diophantine equation can be translated into a question about piles of wood blocks, and there are Diophantine equations that have never been solved. – WillO Apr 08 '17 at 15:03

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Out of fairness, there is a recent paper on the topic http://dx.doi.org/10.1038/nature16059 by Toby S. Cubitt, David Perez-Garcia, and Michael M. Wolf. Undecidability of the spectral gap. Nature, 2015; 528: 207-211. And that paper claims there are undecidable problems in physics.

But it does so by the exact same hack as the examples you give, i.e. taking an infinite number of problems, grouping them together as one problem and then arguing that you can't make a single solution that works for them all.

I'm not sure how that's physics. Sure, it's nice to have one solution that works for an infinite number of different problems. So it's nice to have when you can have it. But it's hardly an issue when you can't.

And note, the example of the spectral gap, isn't a mere unsolved math problem, it's an undecidable one. Which means no single standard algorithm can solve all of the infinitely many cases.

A truly physical, experimentally falsifiable, question is a specific prediction. It takes a specific setup and generates a specific prediction. One situation. Not infinitely many.

Sure, it can be nice to group a whole bunch of individual situations into a single group, especially if they can be tackled in one go. But they should be grouped together exactly if they can be solved together in one go.

If you can't solve them in one go, then you shouldn't have grouped them together into allegedly one problem. And if you can't solve a specific problem enough to make a prediction, then it isn't really science.

So truly science doesn't have any predictions based on undecidable results. And if a theory in progress depends on an unfinished result, then it is an unfinished theory. In science, a theory is the predictions. You can tell because that's how a theory is rejected, based on its predictions. So until you have the predictions you aren't a theory yet.

Timaeus
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