What's the difference between deciding if a mathematical statement is true VS proving it? Isn't a statement only true if you can prove it?
Edit: To elaborate, when reading about foundations of math, there seems to be concepts of completeness and decidability that seems to suggest they are proving and deciding if true are different...
 A: What the authors are getting at is that provability and truth are fundamentally different things. Provability means that we have some rigorous argument for it from some assumptions that we believe are true. Truth means... well... that it is true. The latter can in principle hold without the former: there can be a statement that is true and yet there is no proof of it. (The former holding without the latter shouldn't happen provided our methodologies for proving things are only capable of proving true statements, which is something we should demand.)
The fact that we may not be able to prove every true statement is called incompleteness, and as you have no doubt heard, this really happens in mathematics. What further cements the distinction between truth and provability further is that there are statements that are unprovable in a certain system, and yet we can prove them true in a more powerful system. For instance we can't prove the consistency of arithmetic using arithmetical methods, and yet we can prove it using more powerful set-theoretical methods.
But then there are statements like the continuum hypothesis (CH) that we know for a fact that we can't prove true or false by any currently accepted mathematical methods/assumptions, even in principle. This presents a problem since the only way of 'deciding a statement is true' is giving a proof by some accepted method/assumptions.$^*$ 
What does this mean for the CH? Well, there are (at least) two ways forward. One is to search for more powerful method that are still "self-evidently valid" but can decide things like the CH. This program is being pursued in the form of finding stronger axioms of set theory, but even if it is eventually successful in the eyes of the pursuers, it seems unlikely to generate broad consensus, at least not quickly. 
The other way is to give up on the idea of the CH, or mathematical statements more generally, always having a well-defined truth value. As others have remarked, truth (which I could only define circularly above) is something that only makes sense relative to a structure. So giving up on a well-defined truth value means giving up on the idea of there only being one unique mathematical universe we are describing about with the axioms of set theory. But then we are challenged to justify how for a seemingly concrete question about the seemingly concrete real numbers, there isn't one right answer.

$^*$ Note there is another distinction to be made here. It's one thing to have a proof, it's another to believe it is a valid argument from valid assumptions. So even give that proof is the only accepted way of generating definitive mathematical knowledge, there is some wiggle room if some people have different precepts as to what are valid inferences and assumptions. One framing is that we have many different formal deductive systems out there (a zoo of acronyms like PA, PRA, HA, ATR$_0$, ZFC, MK, etc.) and different mathematicians will sometimes differ on which systems' theorems they accept as true.
A: My opinion is that there is sometimes theorems that work everytime but we still don't have a specific reasonable proof for (and they are quite rare). Also there is something called axioms that are agreed on to be true with no proof, just by convention (I know this is a little it bit different than "deciding" if it is true or not, but I am just giving some ideas that may help)
