Radical empiricism(the philosophy endorsed by this article) is one of the worst philosophies of math I've ever heard of. It completely ignores the point of doing math: it's not to be useful or even to say what's true and false about the real world. It's because we want to know why, for instance, 1 + 1 = 2. No empirical proof can ever tell you why that is. Only by divorcing ourselves from reality and making up our own rules can we meaningfully answer that question, and while we're there why don't we prove some other stuff?