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.
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.