I see philosophy as a place to make nonrigorous arguments.
It's the other way around: math is where you just ignore questions about what makes sense, what knowledge is, what truth is, what a proof is, how scientific consensus is reached, what the scientific method should be, and so on. Instead, you just handwave and assume it will all work out somehow.
Philosophy of mathematics is were these questions are treated rigorously.
Of course, serious mathematicians are often philosophers at the same time.