One controversy that has settled into the foundations of math was over what it means to use infinite sets, and comparing their cardinalities. I was unsettled when I first encountered Cantor's diagonal argument to prove the reals are larger than the rationals, for example. Up until that point, infinite things were still just a kind of "and so on" instead of actual things.

https://en.wikipedia.org/wiki/Controversy_over_Cantor%27s_th...