A comment I often here from non-mathematicians in the sciences: "I wish I had learned more mathematics."1 This is usually followed by a story about how they realized that math shows up a lot more often than they'd thought, compared to their initial training, in their discipline.
Fortunately, I didn't have this problem. I continued to take mathematics courses in college past the usual stopping point for science majors (which, for some reason, is Calculus II, as if that's the apex of mathematics2), mostly thanks to this man. At first he encouraged me to take discrete mathematics, then linear algebra, and at that point I was hooked. He got me through the door to see that mathematics is more than just computing integrals and derivatives.
While I did take more mathematics courses than the usual science major (I did, after all, end up majoring in math, after switching from chemistry), I did not take more mathematics courses than the usual mathematics major. I have never had a course in complex analysis, partial differential equations, topology, number theory, or 'advanced' analysis (i.e. measure theory).
Which makes me wonder: how far will I get in my career before I begin to say, "I wish I had learned more mathematics!"
I know enough about a lot of mathematical topics to speak roughly about them. I have an intuition about the Lebesgue measure (chop up the range instead of the domain), and can spout off that a Banach space is a complete normed vector space. And I've certainly heard enough about certain mathematical topics to know I should know more about them. It's a running joke with my housemates that they should run a directed reading program (something usually for undergraduates) to teach me measure theory.
But on the flip side of the coin, I've seen how far people can get without knowing the finer details of the mathematics underlying their field (a lot of physicists working in essentially stochastic processes who don't know measure theory, for example, or computer scientists working in machine learning who don't really have a firm background in elementary probability theory). For the record, I think this is bad, but I also think that these people publish more than I do and currently have a bigger impact than I do, so I should pay attention to this.
Of course, this entire discussion is a little silly. It's not as if I can't continue to learn more math3. My education didn't stop when I finished taking courses4. I'm more or less self-taught in modern statistics, except for what I learned in undergrad from Roger Coleman. This indicates that with enough time and attention, I should be able to teach myself about other subject as well.
There's no doubt in my mind that knowing more math can only help me. But I only have a finite amount of time on this planet, and a finite amount of focus in any given day, so I have to pick and choose the things that I spend time learning. That means, from any given discipline, I can only go so deep5. How deeply do I need to go into mathematics? I probably should learn the rudiments of measure theory. I probably don't need to learn much category theory.
I don't think there's a 'correct' answer to this question. Different computational scientists will have differing success with differing degrees of mathematical education. I guess what it comes down to is what I want. And what I want is to continue learning mathematics.
Now if I could just convince myself of that.
It sounds hokey, but I really think I 'learned how to learn' in graduate school. To quote Mark Reid, "[learning mathematics] involves sitting alone in a room being confused almost all the time." Graduate school teaches you to push through this confusion and reach the understanding on the other side.↩
Though self-directed learning is hard. Not only because the more advanced material is more difficult, but also because I have to find the motivation to sit down and teach myself something. Even if I really want to learn the subject, this can be more difficult than it should be.↩