First of all, I disagree that the negative time solution can be removed using math; the math will tell you that the solution is perfectly valid.

But you could, say, write a computer program that gave you the right answer to classical mechanics problems, right? In order to write this program, the knowledge you have that tells you that when you want a length of time, you want a positive number would have be translated into "computer language," i.e. math.

That is, when I say "you can remove nonsense solutions by using math" I mean "all you have to do is make the theory already contain your knowledge of what's a nonsense solution."