"x + 1" and "Is 7 prime?" are both rejected as mathematical statements, for what looks like the same reason. Why does mathematics insist on a definite truth value, when ordinary language manages without one?

A. Because logic operates on truth values, not on meanings.

The whole apparatus runs on truth values — 'and' is a table over true and false, modus ponens is a rule about preserving truth — so an expression with none gives the machinery nothing to compute with. The negation of 'Is 7 prime?' is not a false statement; it is not a statement. Variables are welcome: quantify one and you get a statement directly.