Well, I guess you can prove that there are no odd numbers in set (2,4,6), so some negatives you can prove, but in general case, I do not know how to prove that something does not exist and will not exist ever.
Don't worry you're not the first one mentioning it so that's something that must be some kind of colloquialism somewhere but I was asking to understand what people may have meant.
To top it all, you proved a negative (by providing a counterexample) :o)
Claim: There is no odd number in {2, 4, 6}.
Proof: 2 is not an odd number, 4 is not an odd number, 6 is not an odd number, there are no other elements in {2, 4, 6}. Therefore, there is no odd number in {2, 4, 6}.
If there was simply an absence of proof, we would be forced to conclude that we don't know whether there's an odd number in {2, 4, 6}. That's clearly not the case.
The claim "S is a set with no odd number" is equivalent to the claim "S is a set where all numbers are even", which is something I assume you agree that we can prove.
(I even wrote that he proved it himself at the end)
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
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."
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?