It’s not a proof. But it might make RH more plausible.
Mathematics is full of situations where an “empirical logical fact” is held to affect the plausibility of some other mathematical proposition we cannot yet decide. I cannot offhand think of an example of a physical observation being used that way, but it may have occurred. E.g. some quantum system might have equations of motion we don’t know how to solve, but empirically we can see that it’s stable, and from this we can infer something about solutions to those equations.
I’m not entirely sure this is what you’re talking about...
...but the use of dynamic similitude) in engineering can be justified by referring to the modeling equations and showing they are equivalent for the scale model and the full-size system. Because both sets of equations are identical, their solutions are identical, and parameters derived from those solutions—such as drag and lift coefficients—must also be identical. Effectively, you solve the equation experimentally.
Edit: I can write out the algebra for the Navier-Stokes equation if you would like an example.
Nice example. Actually, whenever you rely on a computer to perform a calculation, you are likewise assuming that the physical structure of the object and the logical structure of the math problem have a degree of isomorphism.
But let’s try to get a little clearer regarding the validity of anthropic reasoning in mathematics… The anthropic principle can never be the reason why a mathematical statement is true, whereas it can be the reason why a particular contingent physical fact is true (true ‘locally’ or ‘indexically’). However, faced with a mathematical uncertainty, it may be that certain possibilities are friendlier to the existence of observers. We might therefore regard our own existence as at least consistent with one of those possibilities being the mathematical truth, and even as favoring those possibilities. This appears to be a form of inductive reasoning.
So to sum up, you cannot anthropically prove a mathematical proposition; but in the absence of more decisive considerations, anthropic induction may provide a reason to favor the hypothesis.
It’s not a proof. But it might make RH more plausible.
Mathematics is full of situations where an “empirical logical fact” is held to affect the plausibility of some other mathematical proposition we cannot yet decide. I cannot offhand think of an example of a physical observation being used that way, but it may have occurred. E.g. some quantum system might have equations of motion we don’t know how to solve, but empirically we can see that it’s stable, and from this we can infer something about solutions to those equations.
I’m not entirely sure this is what you’re talking about...
...but the use of dynamic similitude) in engineering can be justified by referring to the modeling equations and showing they are equivalent for the scale model and the full-size system. Because both sets of equations are identical, their solutions are identical, and parameters derived from those solutions—such as drag and lift coefficients—must also be identical. Effectively, you solve the equation experimentally.
Edit: I can write out the algebra for the Navier-Stokes equation if you would like an example.
Nice example. Actually, whenever you rely on a computer to perform a calculation, you are likewise assuming that the physical structure of the object and the logical structure of the math problem have a degree of isomorphism.
There is actually a quantum cosmological model that starts with chaos, and a quantum-chaotic encoding of the Riemann hypothesis, so I’m wondering if Roko’s starting point was more than just a whimsical example. :-)
But let’s try to get a little clearer regarding the validity of anthropic reasoning in mathematics… The anthropic principle can never be the reason why a mathematical statement is true, whereas it can be the reason why a particular contingent physical fact is true (true ‘locally’ or ‘indexically’). However, faced with a mathematical uncertainty, it may be that certain possibilities are friendlier to the existence of observers. We might therefore regard our own existence as at least consistent with one of those possibilities being the mathematical truth, and even as favoring those possibilities. This appears to be a form of inductive reasoning.
So to sum up, you cannot anthropically prove a mathematical proposition; but in the absence of more decisive considerations, anthropic induction may provide a reason to favor the hypothesis.
“Listen to water flowing—the Navier-Stokes equations are well-posed.”