precondition: Either A is true or it is false.
question: is A false?
solution:
1. assume that A is true
2. <math stuff>
3. find a contradiction
4. ==> A can not be true
5. ==> therefor A is false
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?
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...
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."