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I've tried to learn category theory off and on for many years and have gotten pretty much nowhere. From my perspective it's the worst area of mathematics as far

by kens 14y ago

I've tried to learn category theory off and on for many years and have gotten pretty much nowhere. From my perspective it's the worst area of mathematics as far as effort in vs benefit/understanding out. Have other people encountered this or am I just a bonehead?

j2kun 14y ago
OP here. I personally had a lot of trouble when trying to learn category theory on my own. Part of the reason for this is that the need for category theory comes from a lot of other abstract mathematics being unified under the category theoretic lens. I didn't have much of that background my first few times, and I think that was one of the major reasons I didn't succeed in understanding it. After a few years of graduate school, with lots of colleagues working on the same material as me, it has gradually become much easier. The best text I've seen so far is Paolo Aluffi's "Algebra: Chapter 0". It's an algebra textbook meant for first year graduate students in pure mathematics, but it was the major factor in clarifying category theory for me. Of course, the book also contains pretty much all topics in abstract algebra from basic groups to abelian categories. So if you're not pegging to be a research mathematician, chances are the book is not worth your time. Part of my goal with this series is to demystify the subject as it was demystified for me, and to incorporate as many concrete applications to programming concepts as I can. I hope you find some value in what I produce :)
- kens 14y ago
  Thanks for the book recommendation. I'll take a look at your series, and maybe read Algebra: Chapter 0 when I'm feeling ambitious.
btilly 14y ago
Mathematicians can be roughly divided into algebraists and analysts. My biased impression is that algebraists focus on kinds of manipulations that have proven effective in the past, while analysts focus on specific mental models of a system, and then the necessary calculations flow from those models. There are a lot of areas of math that have no obvious connection to abstract algebra or real analysis (for instance combinatorics, logic, number theory...) yet the vast majority of mathematicians will know what you mean when you ask whether they are an algebraist or an analyst, and will know how they should be classified. In fact if I see a mathematician eating corn on the cob, I can generally tell which they are without asking. (Analysts eat in spirals, analysts in rows. My best guess for why is that analysts pay attention to the cue from how their teeth eat corn, while algebraists follow the visual cue from the lines on the corn.) This classification is a matter of taste. Any professional mathematician is adept at both areas of mathematics, but will have a clear preference. Category theory serves as an excellent litmus test. It seems to have a magnetic attraction for algebraists, and a similar repulsion for analysts. At its heart, category theory is an abstraction of common patterns of algebraic manipulation where the details of what you are manipulating become entirely irrelevant. (Indeed you can often replace one type of thing with an apparently unrelated type of thing. Such as a geometric problem about toruses with an algebraic question about pairs of integers.) Speaking personally, I learned enough category theory to pass all of the courses I had to take, but words like "category", "monad" "cohomology" and "functor" are red flags warning me that I'm about to be uninterested in the rest of the discussion. But YMMV.
- gruseom 14y ago
  Great comment. You make it sound like analysts have more imagination—was that intentional? When I studied math I was definitely, in your taxonomy, an algebraist. But if I went back to it now I'm pretty sure I would be an analyst. I say that because my approach to software has evolved in an analogous way. Also I eat corn-on-the-cob in rows :) I'd love to properly re-learn all the math I learned before. The way I did it at the time was mostly just symbol manipulation, and that way of thinking no longer interests me.
  - btilly 14y ago
    You make it sound like analysts have more imagination—was that intentional? Not intentional. I was trying to be fair, but my interests are clearly on the analysis side, and that shows.
- jules 14y ago
  An analyst starts with an object and then looks for things he can prove about the object. An algebraist starts with the proof, and then looks for objects for which the proof is valid. Both methods have advantages and disadvantages. An analyst can get bogged down in the details of the object he studies. An algebraist has the tendency to make trivial statements complicated. It is rare to take a result from the algebraic side and apply it in your concrete situation that you get something that wasn't already immediately obvious. Personally I lean to the analytic side, although the algebraic approach can be helpful if you apply it judiciously and take care to only generalize when this has an advantage and not just for the sake of generalization. If you look at the history of mathematics, most of the good stuff has analyst roots, and was only later incorporated in an algebraic framework. Though this could easily be my bias for what is good stuff. Even traditionally algebraic subjects can be often approached analytically. For example if you do group theory by starting with group actions, it gets an analytic feel (and in my opinion, everything becomes much clearer). The reverse is also true, that's how many of the algebraic subjects initially started: take a bunch of proofs from the analytic side, and see for which classes of objects they remain valid. In programming you have something similar. Programmers that try to generalize their code as much as possible are algebraists. Dependency injection is an example. Programmers that say "YAGNI" are analysts. If you like interfaces and design patterns you're probably an algebraist. If you like bottom up programming you're probably an analyst. If you like Haskell, you're probably an algebraist. If you like Lisp you're probably an analyst.

by kens 14y ago

j2kun 14y ago

OP here. I personally had a lot of trouble when trying to learn category theory on my own. Part of the reason for this is that the need for category theory comes from a lot of other abstract mathematics being unified under the category theoretic lens. I didn't have much of that background my first few times, and I think that was one of the major reasons I didn't succeed in understanding it. After a few years of graduate school, with lots of colleagues working on the same material as me, it has gradually become much easier. The best text I've seen so far is Paolo Aluffi's "Algebra: Chapter 0". It's an algebra textbook meant for first year graduate students in pure mathematics, but it was the major factor in clarifying category theory for me. Of course, the book also contains pretty much all topics in abstract algebra from basic groups to abelian categories. So if you're not pegging to be a research mathematician, chances are the book is not worth your time. Part of my goal with this series is to demystify the subject as it was demystified for me, and to incorporate as many concrete applications to programming concepts as I can. I hope you find some value in what I produce :)

kens 14y ago
Thanks for the book recommendation. I'll take a look at your series, and maybe read Algebra: Chapter 0 when I'm feeling ambitious.

btilly 14y ago

Mathematicians can be roughly divided into algebraists and analysts. My biased impression is that algebraists focus on kinds of manipulations that have proven effective in the past, while analysts focus on specific mental models of a system, and then the necessary calculations flow from those models. There are a lot of areas of math that have no obvious connection to abstract algebra or real analysis (for instance combinatorics, logic, number theory...) yet the vast majority of mathematicians will know what you mean when you ask whether they are an algebraist or an analyst, and will know how they should be classified. In fact if I see a mathematician eating corn on the cob, I can generally tell which they are without asking. (Analysts eat in spirals, analysts in rows. My best guess for why is that analysts pay attention to the cue from how their teeth eat corn, while algebraists follow the visual cue from the lines on the corn.) This classification is a matter of taste. Any professional mathematician is adept at both areas of mathematics, but will have a clear preference. Category theory serves as an excellent litmus test. It seems to have a magnetic attraction for algebraists, and a similar repulsion for analysts. At its heart, category theory is an abstraction of common patterns of algebraic manipulation where the details of what you are manipulating become entirely irrelevant. (Indeed you can often replace one type of thing with an apparently unrelated type of thing. Such as a geometric problem about toruses with an algebraic question about pairs of integers.) Speaking personally, I learned enough category theory to pass all of the courses I had to take, but words like "category", "monad" "cohomology" and "functor" are red flags warning me that I'm about to be uninterested in the rest of the discussion. But YMMV.

gruseom 14y ago
Great comment. You make it sound like analysts have more imagination—was that intentional? When I studied math I was definitely, in your taxonomy, an algebraist. But if I went back to it now I'm pretty sure I would be an analyst. I say that because my approach to software has evolved in an analogous way. Also I eat corn-on-the-cob in rows :) I'd love to properly re-learn all the math I learned before. The way I did it at the time was mostly just symbol manipulation, and that way of thinking no longer interests me.
- btilly 14y ago
  You make it sound like analysts have more imagination—was that intentional? Not intentional. I was trying to be fair, but my interests are clearly on the analysis side, and that shows.
jules 14y ago
An analyst starts with an object and then looks for things he can prove about the object. An algebraist starts with the proof, and then looks for objects for which the proof is valid. Both methods have advantages and disadvantages. An analyst can get bogged down in the details of the object he studies. An algebraist has the tendency to make trivial statements complicated. It is rare to take a result from the algebraic side and apply it in your concrete situation that you get something that wasn't already immediately obvious. Personally I lean to the analytic side, although the algebraic approach can be helpful if you apply it judiciously and take care to only generalize when this has an advantage and not just for the sake of generalization. If you look at the history of mathematics, most of the good stuff has analyst roots, and was only later incorporated in an algebraic framework. Though this could easily be my bias for what is good stuff. Even traditionally algebraic subjects can be often approached analytically. For example if you do group theory by starting with group actions, it gets an analytic feel (and in my opinion, everything becomes much clearer). The reverse is also true, that's how many of the algebraic subjects initially started: take a bunch of proofs from the analytic side, and see for which classes of objects they remain valid. In programming you have something similar. Programmers that try to generalize their code as much as possible are algebraists. Dependency injection is an example. Programmers that say "YAGNI" are analysts. If you like interfaces and design patterns you're probably an algebraist. If you like bottom up programming you're probably an analyst. If you like Haskell, you're probably an algebraist. If you like Lisp you're probably an analyst.