It is perfectly valid in math to prove that something does not exist. For example, there are no positive integers {a,b,c} such that a^3 + b^3 = c^3
This is a direct result of Wiles (1994) proof of Fermats Last Theorem.
Now there are also a different class of assertions that cannot be proved either way. (Godels Incompleteness Theorem)
I am digressing a lot here, but if you find this sort of thing interesting, i suggest reading 'Godel, Escher, Bach' by Hofsteadter (spelling?)
...prepare to hurt your brain :)