It’s possible to decide which axioms are in effect from the inside of a sufficiently complex mathematical system (such as this universe), however.
For that matter, it would be possible to deduce the existence of a god, too; you just have to die. Granted, there are some issues with this, but nobody said deducing the axiom had to be convenient.
When you meet a god, how can you be sure it’s not a hallucination?
Assuming the entity in question is cooperative, try this:
Ask it if P=NP is true, and for a proof for its answer to that in a form that you can easily understand. There’s three possible outcomes:
It doesn’t comply. Time to get suspicious about its claims to godhood.
It hands you a correct proof, beautifully elegant and easy to grasp.
It hands you a lump of nonsense, which your mind is too damaged to distinguish from a proof.
If you get something that appears like an elegant proof, memorize it and recheck it every now and then. If your mind is sufficiently malfunctioning that it can’t distinguish an elegant proof for P=NP from something that isn’t, you may not be able to notice that from inside. There’s still a chance whatever is afflicting you will get better over time; hence, do periodic rechecks, and pay particular attention to any nagging doubts about the proof you get while performing those.
In the meantime, interpret the fact that you’ve gotten an apparent proof as significant evidence for the entity in question being real and very powerful.
If it really is undecidable, God must be able to prove that.
However, I think an easier way to establish whether something is just your hallucination or a real (divine) being is asking them about something you couldn’t possibly know about and then check if it’s true.
Ask again, with another famously unsolved math problem. Repeat until it stops saying that or you run out of problems you know.
If you ran out, ask the entity to choose a famous math problem not yet solved by human mathematicians, explain the problem to you, and then give you the solution including an elegant proof.
Next time you have internet access, check whether the problem in question is indeed famous and doesn’t have a published solution.
If the entity says “there are no famous unsolved math problems with elegant proofs”, I would consider that significant empirical evidence that it isn’t what it claims to be.
Depending on your definition of “elegant”, there are probably no famous unsolved math problems with elegant proofs. For example, I would be surprised if any (current) famous unsolved math problems have proofs that could easily be understood by a lay audience.
It could give a formally checkable proof, that is far from being elegant, but your own simple proof checkers that you understand well can plough through a billion steps and verify the result.
“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).
It is the latter (I’m an agnostic). However, I don’t see why the concept has no meaning. Would you say that axioms in math are meaningless?
It’s possible to decide which axioms are in effect from the inside of a sufficiently complex mathematical system (such as this universe), however.
For that matter, it would be possible to deduce the existence of a god, too; you just have to die. Granted, there are some issues with this, but nobody said deducing the axiom had to be convenient.
“It’s possible to decide which axioms are in effect from the inside of a sufficiently complex mathematical system (such as this universe), however.”
I don’t think I understand what you mean.
“For that matter, it would be possible to deduce the existence of a god, too; you just have to die.”
When you meet a god, how can you be sure it’s not a hallucination?
Assuming the entity in question is cooperative, try this:
Ask it if P=NP is true, and for a proof for its answer to that in a form that you can easily understand. There’s three possible outcomes:
It doesn’t comply. Time to get suspicious about its claims to godhood.
It hands you a correct proof, beautifully elegant and easy to grasp.
It hands you a lump of nonsense, which your mind is too damaged to distinguish from a proof.
If you get something that appears like an elegant proof, memorize it and recheck it every now and then. If your mind is sufficiently malfunctioning that it can’t distinguish an elegant proof for P=NP from something that isn’t, you may not be able to notice that from inside. There’s still a chance whatever is afflicting you will get better over time; hence, do periodic rechecks, and pay particular attention to any nagging doubts about the proof you get while performing those.
In the meantime, interpret the fact that you’ve gotten an apparent proof as significant evidence for the entity in question being real and very powerful.
Or: it says “This is undecidable in Zermelo-Fraenkel set theory plus the axiom of choice”. In the case of P=NP, I might believe it
I would not believe a purported god if it said all 9 remaining Clay math prize problems are undecidable.
If it really is undecidable, God must be able to prove that.
However, I think an easier way to establish whether something is just your hallucination or a real (divine) being is asking them about something you couldn’t possibly know about and then check if it’s true.
It says “There is no elegant proof”. Next?
Ask again, with another famously unsolved math problem. Repeat until it stops saying that or you run out of problems you know.
If you ran out, ask the entity to choose a famous math problem not yet solved by human mathematicians, explain the problem to you, and then give you the solution including an elegant proof. Next time you have internet access, check whether the problem in question is indeed famous and doesn’t have a published solution.
If the entity says “there are no famous unsolved math problems with elegant proofs”, I would consider that significant empirical evidence that it isn’t what it claims to be.
Depending on your definition of “elegant”, there are probably no famous unsolved math problems with elegant proofs. For example, I would be surprised if any (current) famous unsolved math problems have proofs that could easily be understood by a lay audience.
It could give a formally checkable proof, that is far from being elegant, but your own simple proof checkers that you understand well can plough through a billion steps and verify the result.
“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).