Why does mathematics use 0 instead of +0 and -0?

We have no problem thinking of infinity as a number that's constantly expanding, so why isn't zero treated as a number that's constantly collapsing into itself? Why isn't it given a positive or negative sign like other values?

And I understand that the direct answer is "The math checks out", but just because an answer works in mathematics doesn't mean it has to be 100% correct right? Neutral zero being a substitute for positive zero and negative zero might work in certain cases, but in others, such as divided by zero, it does not. Zero might be such a small number that you can essentially say that it is worth nothing, but is that actually correct? Or just a hotfix solution?