How can a math problem be unsolvable?
In: Mathematics
I’ll assume that “unsolvable” and “unsolved” are effectively the same, as I am not aware of any problem for which we have a proof that it is solvable, but not a solution.
First of all, what is done in school as “maths” would be better described as calculating. While in higher years you might do some proofs, most of it is applying rules that you were told are correct.
In mathematics, you start with a set of Axioms, statements and rules that you consider true, and derive everything else from those. If you follow the rules correctly, than any statement you manage to arrive at is also true.
In this case, a “problem” is finding a proof for a statement or proving a stament is not true.
One example would be the [Collatz conjecture](https://en.wikipedia.org/wiki/Collatz_conjecture): You start with a any positive, whole number, if it is even, divide it by 2, if it is odd, multiply by three and add one. Then apply this rule to the resulting number. [relevant XKCD](https://xkcd.com/710/)
For all number we have tested, you will eventually end up at one, so it looks like this might be true for all numbers, but there are infinitly many numbers we have not tested yet (and there always will be).
So just continuing to test numbers might solve it, if we find one where this is not the case. Then we know for a fact that there is one number for which it isn’t true, therefore it cannot be true for all numbers.
The other option is that clever mathematicans figure out a way to apply the rules of maths to things we know are true in such a way that they end up at the statement above. Then we have the proof that it is true. Or they apply rules to that statement until they end up at a statement we know is wrong. Then we know that the statement we started at was wrong (proof by contradiction).
Until then, it is unsolved. And, as I said in the beginning, I do not know if we have a way of knowing wheter or not it can be solved, at all.
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