Modern math generally follows Cantor when it comes to stochastic calculus (financial and engineering modelling) and some other things.. that doesn't make Kronecker any less correct and his ideas (including the circle cross product thingy) were very valuable contributions to mathematics.
Your point gets to the whole crux of the issue which is that mathematics is a philosophy and not a science and it's entirely based on man made rules. It just happens to be a very useful tool for enabling us to solve all kinds of problems as well as to help us do Science. But it's no more a "science" than my hammer.