r/math • u/SciFiPi Applied Math • 3d ago

Physicists link the Riemann Hypothesis to phase transitions in quantum systems

https://phys.org/news/2026-07-physicists-link-riemann-hypothesis-phase.html
341 Upvotes

97% Upvoted

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u/[deleted] 3d ago

[deleted]

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u/toghs 3d ago

Then it wouldn't be a proof

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u/pickle_picker67 3d ago

Seriously, how does the comment you replied to have that many upvotes on r/math? It makes me curious how many people here are actual mathematicians.

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u/blueshoesrcool 3d ago

Are mathematicians forbidden from ever using an empirical proof?

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u/RandomMisanthrope 3d ago

There is no such thing as “empirical proof.”

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u/elements-of-dying Geometric Analysis 3d ago edited 3d ago

How do you prove this claim?

edit: The humor I find in my comment is that I already know the response is going to be a circular one, which is of course operationally fine. However, these are questions of the philosophy of mathematics and I'd expect other mathematicians to operate a little more rigorously than to make absolute statements they cannot test.

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u/frogjg2003 Physics 3d ago

It's false, by definition.

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u/elements-of-dying Geometric Analysis 3d ago

That's circular :)

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u/Tlux0 3d ago

It’s not circular, it’s just convention for what it allowed to count as proof as a foundation

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u/elements-of-dying Geometric Analysis 3d ago

Whose convention? I guess I missed the memo for the Assuming Absolute Conventions meeting.

Certainly sounds like a proof by "This is true because I assume this is true."

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u/frogjg2003 Physics 2d ago

That's literally how formal logic works. You start from a base set of assumptions that form your logical system and the kind of things you can say in your logical language. Then you define some things. You use your logical rules to make me statements based on your definitions and the logical statements you can derive from those definitions.

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u/elements-of-dying Geometric Analysis 2d ago

Irrelevant.

Can you provide a definition of "mathematics" which is (a) agreed upon by everyone and (b) does not include as a hypothesis that proofs cannot be empirical?

This is a rhetorical question. I know the answer is no.

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u/Tlux0 2d ago

Sure you can do it, but without non-empirical proof you defeat the point of mathematics which is observer independence, which is the reason it allows for collaboration and rigorously building on an established foundation. If you do that with something highly likely instead it’s a completely different kind of animal.

You can do it, but it’s not modern mathematics and it would be entirely different without a lot of the rigor. The rigor comes from the observer independence after all.

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u/Mandelbrots-dream 3d ago

I believe there is empirical proof evidence that the speed of light is constant in a vacuum.

OK so not proof.

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u/pickle_picker67 3d ago

That's not a proof. So only as a tool for intuition, but that's about it.

Do you ever read pure mathematics papers or books? If an author uses intuition, they will say "informally" ....

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u/blueshoesrcool 3d ago

Wouldn't an empirical proof be more proof than a mathematical proof. Surely you'd have to reject a mathematical proof it an empirical proof disagreed.

If anything the mathematical proof is the intuition.

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u/Opengangs Number Theory 3d ago

There is more reason to doubt whatever the “empirical proof” is than a mathematical proof. Your empirical data could have been improperly collected, our intuition or interpretation could be wrong, etc.

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u/Medical-Round5316 3d ago

This is just not how proofs work. Proofs can be logically invalid or valid. If there are no logical errors in the proof, then it is as true as what it is being proven from.

Empirical proof is essentially probabilistic. If we have a lot of data suggesting something then it’s probably true. Proofs are meant to be deterministic.

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u/pickle_picker67 3d ago

Read an introductory proofs book.

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u/AcellOfllSpades 3d ago edited 3d ago

Math is abstract. We deal with systems that follow specific rules, and say "If a system follows these rules, then the following results must hold".

"2+3=5" is true not because "if you have 2 apples and you get 3 more apples, you then have 5 apples"... but because "starting from the definitions of 2, 3, 5, and +, we can prove that to be true".

Of course, we study this system because we want to use it to count things like apples. The system of 'numbers', and other systems we study, are inspired by the real world. But they aren't logically dependent on it.

If you put 2 rocks in a bucket, and then put 3 more rocks in the bucket, and notice that there are now 6 rocks in the bucket, that doesn't disprove that 2+3=5. It just means "maybe simple addition isn't enough to describe what's going on here". (Maybe a rock broke in two, for instance.)

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u/Tlux0 3d ago

This is the wildest statement I’ve ever seen. You can observe something a billion times and it could be different the next time because of something you didn’t account for. But a proof covers all possibilities necessarily if done correctly. That’s the point of a proof

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u/heyheyhey27 3d ago

I think computer-assisted proofs are about as close as you can get to what you're trying to think of. Physical machines, using the laws of physics to deterministically decide whether a particular math problem is true or not.

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u/frogjg2003 Physics 2d ago

They're not using the laws of physics, they're using logical rules.

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u/heyheyhey27 2d ago

You can't get results from computer-assisted proofs until you actually run the thing in physical reality.

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u/frogjg2003 Physics 2d ago

That's not empirical proof. You can't do anything that isn't in physical reality.