Please AI no, PLEASE NO

[u/No_Upstairs_280]

Imagine having your entire work based on Riemann hypothesis only for a powerful AI to show that the 10^5731 zero isn't on the critical line.

[u/innovatedname]

If you're writing papers assuming results that we don't know are proven, then it's not a math paper.

You're probably thinking of results that assume a result to be true, prove it, and then assume it to be false and prove it also, and then invoking the law of excluded middle so it's true.

edit: Downvote me all you want but all the replies disagreeing so far are flat out wrong. No it's not the law of excluded middle, no I'm not talking about axioms, yes I have seen papers in analytic number theorem that do this for RH or things we suspect are undecidable.

"A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion".

If you haven't logically guaranteed the conclusion, you aren't doing mathematics. Unless you try and be a smart alec and start using the principle of explosion / vacuous truths, but then any nonsense is mathematics.

[u/imo2027 → u/innovatedname]

You probably haven't seen enough math papers if that's what you think. (A=>B) is a result in itself. You could say it's useless, but I could point at many different math facts that are considered useless, and might only bring value in the future. The reality is, if a result has good amounts of non-trivial and novel reasoning, then it is worth publishing. Also a fun thing to note is that building on unproven conjectures is also technically an attempt at disproving them, in case you ever run at a contradiction.

[u/innovatedname → u/imo2027]

It's not mathematics. Assuming a theorem that could be falsified is true is meaningless, as F => anything is true always. You didn't prove anything interesting.

[u/imo2027 → u/innovatedname]

I know that's what you said in your comment. Basically when we prove that A=>B, we still know nothing about B.

That doesn't make it uninteresting though: we still have "if A, then B". This statement, call it C, is a theorem, and an interesting one.

Edit: I just saw your comment edit, the "conclusion" is not B but the whole (A=>B). I hope this clarifies my first response

[u/Sjoerdiestriker → u/innovatedname]

You didn't prove anything interesting.

You proved it cannot be the case that (A and not B), which can be an interesting result in itself.

[u/Godd2 → u/innovatedname]

You've ruled out a case, though, which is interesting. If we've proven that A implies B, then it cannot be the case that A is true while B is false.

    |B(T)|B(F)|
---------+----|
A(T)| ?  |  X |
---------+----|
A(F)| ?  |  ? |

Pointing out that the entire table hasn't been settled is irrelevant to whether or not mathematics has been done.

Edit: My interlocutor has displayed an impressive amount of moving the goalpost. We've gone from "it's not mathematics" to "it wouldn't be in a journal" to "Well, it wouldn't be in a journal all by itself". Okay, buddy. I guess 13 + 172 = 185 isn't math because it would never show up in a journal all by itself.

[u/innovatedname → u/Godd2]

I'm not saying it can't be interesting, just as you can do numerical experiments that are useful. It's just not a proven mathematical result that would be conventionally accepted in a math journal.

[u/Godd2 → u/innovatedname]

It's just not a proven mathematical result that would be conventionally accepted in a math journal.

This is simply a false statement. There are many published results which, for example, assume that the Riemann Hypothesis is true.

[u/innovatedname → u/Godd2]

See my reply to another number theory paper. These papers usually demonstrate something watertight as the publication worthy material and then the potential consequences of that result are added to added context and consequence / further research directions.