Proving a negation is not a proof by contradiction, that's just the proof of an (absurd) implication. Logic people actually write blog posts about this: https://math.andrej.com/2010/03/29/proof-of-negation-and-pro...

> I am discovering that mathematicians cannot tell the difference between “proof by contradiction” and “proof of negation”.

I am glad to learn, that I just as dumb as the average mathematician lol.

> Proving a negation is not a proof by contradiction

did I say that it is, though?

In P/NP case I would need to assume that I can solve a problem in polynomial time, and then find a contradiction. To me it is really hard to see a useful path forward from there.

That's what's called a "non-constructive" proof in some circles. It /doesn't/ have a useful path forward, which makes it frustrating. There are some weird sandwiches of complexity classes - like, iirc, P-SPACE, EXP, NP-SPACE - which we know to have exactly 2 subsets plus the total subset, but which constellations? Dunno.

This is different from a constructive proof. "Here we have a problem, and all algorithms so far are in O(n^3). We can demonstrate the existence of a substructure in all instances of this problem which lowers the complexity to O(n^2.978)" would be a huge constructive proof in some fields. Such a breakthrough could lead to a large number of follow-up improvements.

A non-constructive breakthrough is like "Yup, this is a barrier. No clue why though, but it's hard."

> I would need to assume that I can solve a problem in polynomial time, and then find a contradiction

iirc that's actually how many of these proofs work.

> To me it is really hard to see a useful path forward from there.

yes, CS is hard.

there are resources online, that explain the general idea, e.g. [1]. But understand the specifics, you really do need a solid foundation in theoretical math and theoretical CS

[1] https://www.quora.com/What-is-a-proof-by-contradiction-in-co...

If you see a path, you'll be one million USD richer. Don't forget about us. :-)