Though I am no mathematician, I deeply respect mathematics, and appreciate your defense of it as a humanity, and the importance of its rules for choosing rules to suit the situation.

That said, as I understand it, mathematics in the deep sense you describe gives great tools to specify patterns and how they repeat, but it breaks down somewhat when specifying exceptions to those patterns. For example, consider the debates around 'natural kind' in modal logic (Is a cat with three legs still a cat? How about Lewis Carroll's Cheshire Cat?). Wrestling with what becomes part of the 'data analysis within a framework' is essential. As it turns out, philosophy has lately turned toward the tools of mathematical logic to specify these things, while its concerns remain essentially human. That's at least one reason why I think the author preferred philosophy to mathematics in the choice of majors suggested.

Without recognizing the limits of mathematics and its abilities to describe what occurs in the real world, mathematicians are prone to being caricatured as the physicist is in the XKCD cartoon the author links.