How did unsolvable math problems ‘start’? How can you have an answer but no idea how to get there in maths?
In: Mathematics
> How did unsolvable math problems ‘start’?
Usually somebody observes are particularity and wonders if that is always true:
– When printing an atlas somebody noticed that four colors are enough to print it such that neighbours never share the same color; is this true for absolutely every possible map?
– Every even integer beyond two and up to a million is the sum of two prime numbers; could this be the case for literally all of them?
– The sum of two cubes (third powers of positive integers) seems to never be a cube again; might that be true in general?
– Factoring numbers seems to be difficult and takes centuries on supercomputers even for most 1000 digit numbers; does an efficient method to factor any integer exist after all?
Then we set out to _prove_ this: find a formal, logical argument why it _must_ be so. Or find a _counterexample_ or _disproof_, something that demonstrates that our guess was wrong after all.
Until that we are uncertain what is truly going on. Maybe we have already checked all cases that we might ever see in real world problems by computer; or maybe we didn’t, as for the factorization example on which a lot of cybersecurity hinges on!
It sometimes even happens that we find questions where we can then prove(!) that it cannot ever be answered. We call those statements _independent_, ones where we cannot ever ascertain with our logic and from our assumptions whether they are correct. Kurt Gödel even proved that any non-silly system of rules (logics plus assumptions) always begets such statements, it is impossible to not have them.
> How can you have an answer but no idea how to get there in maths?
You can’t. Or rather, you then don’t know if the answer is actually correct. You can always take a _guess_ but without evidence in the form of a formal _proof_ it is just that. You don’t have an answer, just the guess.
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