Are all mathematical statements either true or false?


I think I have to go to Kurt Godel, the mathematician/logician to answer this question properly. And in agreement with him, I'm going to say no; mathematical statements may be true, false, or undecidable. Godel defined such undecidable statements as those that may be true, but cannot be proven true or false.

Godel published his work Principia Mathematica in which he arrived at the "incompleteness theorem." The incompleteness theorem is often compared to the old philosophers' trick: "This statement is false." Is that statement true or false? Of course, if it were true, then it would be false. If it were false, then it would be true... It's an undecidable paradox. Godel applied this idea to mathematics with the incompleteness theorem that basically said the following:

"Godel essentially constructed a formula that claims that it is unprovable in a given formal system. If provable, it would be false, which contradicts the fact that in a consistent system, provable statements are always true. Thus there will always be at least one true but unprovable statement."
(Don't judge me, but that's Wikipedia's surprisingly concise description of the Incompleteness Theorem.)

As a result, mathematics is not simply statements that are either true or false. It is, as Godel revealed, a "logical mess," like language or anything else.