Are all proofs direct proofs or proof by contradiction? If I prove something can't be proven by direct proof or by contradiction does that mean it can't be proven?
I don't know a lot of the jargon, so simpler explanations would be nice if possible.
 A: The logic of mathematics works as follows:
There are a number of axioms, which are defined to be true. From these axioms, we deduce other properties and results. From these, we then get more and more results.
Taking all the results that are already proven to be true (assuming the axioms to hold) and deducing a new result from that is what is called "to prove something".
Now as you surely know, there are two basic ways to conclude one thing from another. Given that $A$ is true, you can show that $B$ is true either by showing
$$A \Rightarrow B$$
or by showing
$$ \overline{B} \Rightarrow \overline{A}.$$
The first method is called direct proof, the second one is called indirect.
One well-known example of such a system would be a vector space. We have a list of axioms (the vector space properties for addition, etc.) and from that, we deduce whole textbooks full of linear algebra.
Regarding your question: If you show that something can neither be proven directly nor indirectly from the current knowledge base, then you are right, you have shown that it can't be proven at all. However, I assume such things to be either really trivial (e.g. "we can't decide this given that information") or almost impossible to show that no proof exists. At least for "real world" mathematics, e.g. linear algebra or analysis. If you are working in a closed logical system, containing only a few axioms and deduced properties, a non-provable proof might be easier.
