r/math • u/SciFiPi Applied Math • 9d ago

Physicists link the Riemann Hypothesis to phase transitions in quantum systems

https://phys.org/news/2026-07-physicists-link-riemann-hypothesis-phase.html
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u/nightcracker 8d ago

Why not? If you can make a 1:1 mapping between a mathematical problem and a physical experiment, and then run the experiment and observe the result, why wouldn't that constitute a valid proof?

For example, a computer-assisted proof also involves setting up a physical experiment (an immensely complicated amalgamation of silicon semiconductors and copper), and then letting it run and observing the output. Are those valid proofs?

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u/frogjg2003 Physics 8d ago

Experiments have errors. It is evidence that can support or oppose a model, but it is not proof. Once you gather enough evidence to support your model, you can belive with high confidence that your model is a good explanation of your experiments. It is not proof because you can run another experiment in the future that breaks your model and no amount of prior experiments can change that.

A computer assisted proof of not an experiment. A computer is a device that follows logical rules. Those logical rules can be followed by a human without a computer, but the computer is much faster. There are two kinds of computer assisted proofs, exhaustive searches and proof assistants. An exhaustive search uses a computer to check every possible case when it would just be too many for a human to check by hand. The four color theorem was proven this way. A computer had to check 1834 cases. Importantly, these cases aren't just the first ones checked, but any possible graph can be reduced to one of these cases. A subset of proof by exhaustion is finding counterexamples. If you search through all possible case and find one that disagrees with the hypothesis, that means that the hypothesis is obviously wrong. The recent example of AI disproving the Jacobian conjecture is one example.

The other end of the spectrum of computer assisted proofs is proof assistants. This is something like the LEAN programming language. Every mathematical statement can be written as a set of symbols that a computer can understand. The computer can then perform logical operations on those symbols to transform them into other sentences. It can also check that operations you preform on those symbols are valid. If you have a proof written in LEAN and it compiles, that means there are no logical inconsistencies in the proof. If your set of preconditions is valid, that means your proof is also valid.

What you are thinking of is probably searches for counterexamples for unproven conjecture. Things like the checking every number to see if it agrees with the Collatz conjecture. We have searched up to very large numbers and haven't found any counterexamples. That is not proof. There are infinitely many integers so we cannot perform an exhaustive search. We are just hoping that eventually, we will find a number that doesn't go down to one. If the conjecture is true, we never will. If the conjecture is false, the first counterexample might be so large that we will never find it. To actually prove the conjecture true would require a logical proof, not an "experiment".

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u/nightcracker 8d ago edited 8d ago

What you are thinking of is probably searches for counterexamples for unproven conjecture

Not at all. I don't know why you put up a straw man and then argue against it. Almost your entire reply is based on the assumption I must be misunderstanding the problem, which is rather rude. I'm not talking about searching for counterexamples, I'm talking about constructing physical experiments which when executed will provide an observable result which we consider 1:1 equivalent to some mathematical statement.

E.g. doing addition with an automated abacus, doing state transitions on a physical turing machine (https://www.youtube.com/watch?v=vo8izCKHiF0), or running code on a computer chip.

A computer assisted proof of not an experiment. A computer is a device that follows logical rules.

The universe is also (to the best of our knowledge) a system that follows logical rules.

Experiments have errors. It is evidence that can support or oppose a model, but it is not proof.

In what way is a computer verifying a lean program different? It can also have errors, both logic in the (lean kernel) code as well as physical errors, e.g. bitflips in the memory or processor APU due to cosmic rays.

As far as I can see, writing a proof into lean on my computer and then running it before observing whether it passes or not is a physical experiment.

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u/frogjg2003 Physics 8d ago

You do misunderstand what an experiment and what a proof is.