“Would you say that axioms in math are meaningless?”
They distinguish one hypothetical world from another. Furthermore, some of them can be empirically tested. At present, Euclidean geometry seems to be false and Riemannian to be true, and the only difference is a single axiom.
At present, Euclidean geometry seems to be false and Riemannian to be true
I think the words “true” and “false” have some connotation that you might not want to imply? Perhaps it would clearer to phrase this as “At present, it seems like the geometry of our universe is not Euclidean and that the geometry of our universe is Riemannian.”.
They distinguish one hypothetical world from another.
It’s a subtle distinction, but I think it’s more accurate and useful to say that the axioms define a mathematical universe, and that a mathematical universe cannot be true or false but only a better or poorer model of the physical universe.
Euclidean geometry isn’t a theory about the world, and therefore cannot be falsified by evidence from the world. The primitives (e.g. “line” and “point”) do not have unambiguous referents in the world.
You can associate real-world things (e.g. patterns of graphite, or wooden rods) to those primitives, and to the extent that they satisfy the axioms, they will also satisfy the conclusions.
Riemannian geometry is not an axiomatic geometry in the same way that Euclidean geometry is, so it is not true that “the only difference is a single axiom.” I think you are thinking of hyperbolic geometry. In any case, the geometry of spacetime according to the theory of general relativity is not any of these geometries, but it is instead a Lorentzian geometry. (I say “a” because the words “Riemannian” and “Lorentzian” both refer to classes of geometries rather than a single geometry—for example, Euclidean geometry and hyperbolic geometry are both examples of Riemannian geometries.)
First i’ve heard of this, super interesting. Hmm. So what is the correct way to highlight the differences while still maintaining the historical angle? Continue w/ Riemannian geometry? Or just say what you have said, Lorentzian.
Special relativity is good enough for most purposes, which means that (a time slice of) the real universe is very nearly Euclidean. So if you are going to explain the geometry of the universe to someone, you might as well just say “very nearly Euclidean, except near objects with very high gravity such as stars and black holes”.
I don’t think it’s helpful to compare with Euclid’s postulates, they reflect a very different way of thinking about geometry than modern differential geometry.
“They distinguish one hypothetical world from another.”
Just like different religions.
“Furthermore, some of them can be empirically tested. ”
Empirical tests do not prove a proposition, but increase the odds of its being correct (just like “miracles” would raise the odds in favor of religion).
“Would you say that axioms in math are meaningless?”
They distinguish one hypothetical world from another. Furthermore, some of them can be empirically tested. At present, Euclidean geometry seems to be false and Riemannian to be true, and the only difference is a single axiom.
I think the words “true” and “false” have some connotation that you might not want to imply? Perhaps it would clearer to phrase this as “At present, it seems like the geometry of our universe is not Euclidean and that the geometry of our universe is Riemannian.”.
They distinguish one hypothetical world from another.
It’s a subtle distinction, but I think it’s more accurate and useful to say that the axioms define a mathematical universe, and that a mathematical universe cannot be true or false but only a better or poorer model of the physical universe.
Euclidean geometry isn’t a theory about the world, and therefore cannot be falsified by evidence from the world. The primitives (e.g. “line” and “point”) do not have unambiguous referents in the world.
You can associate real-world things (e.g. patterns of graphite, or wooden rods) to those primitives, and to the extent that they satisfy the axioms, they will also satisfy the conclusions.
Math is not physics.
“Math is not physics.”
It’s made out of physics. I think perhaps you mean that math isn’t about physics.
To the degree that axioms aren’t being used to talk about potential worlds, I would say that they’re meaningless.
Riemannian geometry is not an axiomatic geometry in the same way that Euclidean geometry is, so it is not true that “the only difference is a single axiom.” I think you are thinking of hyperbolic geometry. In any case, the geometry of spacetime according to the theory of general relativity is not any of these geometries, but it is instead a Lorentzian geometry. (I say “a” because the words “Riemannian” and “Lorentzian” both refer to classes of geometries rather than a single geometry—for example, Euclidean geometry and hyperbolic geometry are both examples of Riemannian geometries.)
First i’ve heard of this, super interesting. Hmm. So what is the correct way to highlight the differences while still maintaining the historical angle? Continue w/ Riemannian geometry? Or just say what you have said, Lorentzian.
Special relativity is good enough for most purposes, which means that (a time slice of) the real universe is very nearly Euclidean. So if you are going to explain the geometry of the universe to someone, you might as well just say “very nearly Euclidean, except near objects with very high gravity such as stars and black holes”.
I don’t think it’s helpful to compare with Euclid’s postulates, they reflect a very different way of thinking about geometry than modern differential geometry.
“They distinguish one hypothetical world from another.”
Just like different religions.
“Furthermore, some of them can be empirically tested. ”
Empirical tests do not prove a proposition, but increase the odds of its being correct (just like “miracles” would raise the odds in favor of religion).