Eugenia Cheng may be my mathematical opposite. She did a great interview with Peter Rowlett in which she bubbles over with enthusiasm for category theory. She explains that she couldn’t stand applied math, but stuck with math because she believed there was something there she could love. The further she moved from applicable math, the happier she became. Abstract algebra was a big improvement, but still too concrete. When she discovered category theory, she was home.
Category theory is a sort of meta-mathematics. It aims to identify patterns across diverse areas of math the way a particular area of math may identify patterns in nature. I like the idea of category theory, but I get that deer-in-the-headlights look in my eyes almost immediately when I look at category theory in any detail.
I enjoy pure math, though I prefer analysis to algebra. I even enjoyed my first abstract algebra class, but when I ran into category theory I knew I’d exceeded my abstraction tolerance. I moved more toward the applied end of the spectrum the longer I was in college. Afterward, I moved so far toward the applied end that you might say I fell off the end and moved into things that are so applied that they’re not strictly mathematics: mathematical modeling, software development, statistics, etc. I call myself a very applied mathematician because I actually apply math and don’t just study areas of math that could potentially be applied.
I appreciate Eugenia Cheng’s enthusiasm even though I don’t share her taste in math. I have long intended to go back and learn a little category theory. It would be great mental exercise precisely because it is so foreign to my way of thinking. Cheng’s interview inspired me to give it one more try.
Update: After this post was written, I did give category theory another try, and gave up again. Then a couple years later I finally committed to digging into category theory and have found practical uses for it.
I enjoy pure math, though I prefer analysis to algebra.
Back as an undergraduate at UofT, I preferred analysis too. Part of the reason was how much my education stressed analysis (starting with calculus). Back then, I did not know much at about Computer Science (though I knew elementary programming).
In my mind, discrete mathematics was an approximation to reality, which was itself continuous. Well. It is not that simple, is it? I have now flipped entirely backward and I think that analysis (continuous math.) is an approximation to the discrete nature of the universe. It is not called quantum mechanics for naught: fundamentally, our universe is discrete… it is just that the quanta are too small for us to see.
Thus, discrete mathematics is the really important stuff. It just took me 10 years to come to my senses.
Also, I could not stand combinatorics. For one thing, I always “counted” wrong in my head. I also thought that probabilities were too hard for the same reason… I still think it is a very hard subject… and it does not help that at UofT, we never had any class on probabilities…
Yet, probabilities and combinatorics are *extremely* useful. (They are almost the same subject too!)
As for being a “very applied mathematician”, I am also with you. Though I have a B.Sc., a M.Sc. and Ph.D. in Math, though I was assistant prof. in Math. (like yourself!), I have branched out in a totally different direction where my motivation is much stronger. There must be many of us out there, we should create a club.
Now, I look at what my math. friends are publishing and I roll my eyes… what nonsense! (No doubt, they say the same thing about my work…)
However, I still struggle with probabilities… very much… I wish I had spent my undergraduate on probabilities instead of limits, derivatives and integrals…
Thanks for this post. In trying to get back into category theory (I actually did my MSc. thesis in a category theoretic topic), I was helped by Lawvere and Schanuel’s book “Conceptual Mathematics,” you might like it also. In it, I think they are pretty successful in presenting CT in a way that is appealing to CT outsiders (although I can’t say how appealing it would be to applied mathematicians). It seems to me that someone with an applied sensibility might have to approach CT from a completely different mindset – it seems to be more about ‘theory building’ and organizing concepts than solving problems. Although real CT proponents could probably point to some key problems it has solved, most of the time CT is credited with ‘clarifying’ or providing a better language for describing structures. The fact that CT provides a better way to talk about Algebraic Topology is not a big motivator for applied mathematicians to learn it. The problems that CT solves seem (to me) often to be motivated by CT itself.
A while ago, Tim Gowers wrote a paper on “The Two Cultures” of mathematics, and Freeman Dyson wrote a paper on “Birds and Frogs” – both papers talked a bit about the cultural divisions between the problem solvers and the theory builders. They are both helpful in pointing out how the different approaches to what is aggregated into “mathematics” is just an indication of its richness and diversity.
Here’s the Conceptual Math book:
http://www.amazon.com/Conceptual-Mathematics-First-Introduction-Categories/dp/0521478170
The “Two Cultures” article is here:
http://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf
The “Birds and Frogs” article is here:
http://www.ams.org/notices/200902/rtx090200212p.pdf
Dan, thanks for the links. I’ve had a little algebraic topology. I understand that’s what motivated CT, but algebratic topology is very tangible compared to CT. I’m sure CT experts would tell me that if I can understand algebraic topology then I’m almost there, but it doesn’t seem that way to me. I can easily visualize things like a fundamental group, but I get totally lost when I read “A foo is an object such that for every map from bar to baz there exists a map phi so that the following diagram commutes.”
In this I am with what V.I. Arnold said on his speech ‘On Teaching Mathematics’, which begins:
“Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap. ”
Übertheoretical mathematicians be warned before reading on: you will very likely feel insulted…
http://pauli.uni-muenster.de/~munsteg/arnold.html
Nice post. Glad to hear you were inspired.
If you are a software developer are there parallels between category theory and object-oriented programming (I certainly thought so when I listened to the interview)? Aha, typing this into Google reveals “Category Theory for the Java Programmer“. Does that help?
Additionally, you might be amused by xkcd cartoon #435.
Being in high school still, I have yet to hit my abstraction limit, but I can definitely imagine hitting one some day.
You will find some fun characters and lots of category theory related posts at then-category cafe
You said in one of the links
> I’ve been a part of projects where we used category theory to guide
> mathematical modeling and software development.
Could you give one-two examples? It’d really help for those just starting out.