Is there more than one form of logic in mathematics? I would be inclined that mathematics only cover one type of formal logic, but I would be interested to know if there are variants thereof or completely different logics than what we're taught in university. In philosophy, there are several logics, but in mathematics it seems there's only one, or at least only one that's prevalent.
Yes, on a fundamental level, much simpler than fleshed out arithmetic, we have classical, non-classical, dialetheistic logic, just to name a few.
You can choose either one of those systems and we can build complex mathematical structures, like fields, etc, using it - though it would be tedious, for example, if you don't have proof by contradiction at your disposal. So, yes, we most definitely have more than one form of logic in mathematics.