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Math isn't a completely formalized system. For example, a lot of work was done with real numbers before they were formalized, and I believe that happened relatively recently (late 1800s, early 1900s). It just would not have been possible to make statements about calculus prior to that time if you were to demand types.

There is an obvious benefit to the way math developed historically.

Interesting, It is funny how early 1900s is considered recent in math, While the first Computer Scientists are still alive.