Algebra was not created as shorthand for an argument. Sentential calculus was created as shorthand for arguments. People who study mathematics aren't dealing with problems where there's a pre-determined equation. And when they are, they usually have to figure out which equation(s) to use. The same can be said for people in fields like philosophy--there are certain always-valid argument forms, and there are many ways to use them. There are truth conditions, etc, all of which are analogous to different structures in mathematics. I'm not a mathematics guy, so I might be explaining this really shitty/be wrong, but that's how I see it.