There is a whole body of mathematics which takes these things seriously (constructive math, intuitionistic foundations) and it's, if anything, even more heavily

There is a whole body of mathematics which takes these things seriously (constructive math, intuitionistic foundations) and it's, if anything, even more heavily formalized. I think the author has some technical points, but his desire for "make math more like science" is outmoded in many areas. Mathematicians went toward formalism for good reasons. They might not have made all the right decisions, but that doesn't mean that upending the table and starting fresh makes sense.