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It's more helpful to view category theory as a language. Most undergraduates learn mathematics in the language of sets. If they're doing algebra, then the unde

It's more helpful to view category theory as a language. Most undergraduates learn mathematics in the language of sets. If they're doing algebra, then the underlying structures of their groups, rings, etc. are always sets. Same for topology and logic (e.g. classical model theory). An extreme example of this set-theoretic thinking is defining the real numbers in terms of Dedekind cuts. Category theory, very slowly, changes this perspective. If you study this for years, you will come to realise there is more to mathematics than sets. And your way of thinking will shift. You will learn to ask different questions. For example, instead of defining things in terms of their underlying set (often breaking symmetries), you will ask 'does it have a universal property?' and so on. Note, there are still tons of reasons to use sets / ZFC, even when working with categories mostly. But I do not want to get into that and it's irrelevant for the point I'm trying to make. Final note: the 'language of sets', i.e. the language of undergraduate mathematics, is very different from set theory. 'Set theory' is like the language talking about itself. The same goes for pure category theory.

occamrazor 6y ago
Very good point. To add on this, even category theory is often taught in the language of sets and classes (“A category is a class of objects with a set Hom(X,Y) of morphisms for each pair of objects, etc.”) It is possible to use categories as basic building blocks instead if sets but, in my anecdotal experience, this us not what the majority of graduate programs in Mathematics do. It will be interesting to see whether this will change in the next 20 yea.
- spekcular 6y ago
  They don't do it for good reason: set theory is basically a strictly better foundation for mathematics, and you have to ape all the set theoretic constructions when doing various things anyway (e.g. constructing the real numbers), so it doesn't buy you anything. This issue has been litigated extensively, and in my view successfully, by Harvey Friedman on the Foundations of Mathematics mailing list, if you want to check its archives.
jacobolus 6y ago
Set theory was developed to make analysis “rigorous” by the standards of 19th century mathematicians, who were probing the edge cases of the more intuitive (or maybe handwavey) assumptions from the 17th–18th century. Here’s Rudin to explain: https://www.youtube.com/watch?v=hBcWRZMP6xs https://www.youtube.com/watch?v=hBcWRZMP6xs In the 20th century it became trendy to (a) try to base everything on set theory, (b) write mathematics in a very dry and formal style, taking inspiration from the Bourbaki project. My impression is that undergraduates are taught in the language of sets partly because it is serviceable for describing the subjects they are trying to learn (esp. analysis) but also more importantly because mid-20th century mathematicians who set up the curriculum we are still using wanted to thoroughly teach (indoctrinate?) undergraduates the trendy style of the time. Personally I think it has been a mixed bag; the style is a big turn-off to many people, and ends up chasing people out of mathematics who could otherwise make valuable contributions. The people who remain seem to mostly like it okay though. YMMV.
- jrajav 6y ago
  Your observations strike a serious chord with me, because the first class I took where a professor wielded set theory like a ruler to the back of students' hands was exactly the point I realized that mathematics was not my calling. But I've come to understand that I was not realizing that I didn't like mathematics or thought it was too hard, but instead that if I made that my life's work, I'd be working with many more people like that, rather than the people I wanted to - bright and creative, yet kind and humble. I don't think set theory or any other set of tools is the problem behind potential academics getting turned off. It's the people, and the culture of glorified monkhood.
- spekcular 6y ago
  I have some problems with this comment. First, why the scare quotes around "rigorous"? Set theory literally did make the foundations of analysis more rigorous. Second of all, we don't use set theory because it's "trendy" or because we want to "indoctrinate" undergraduates, we use it because it's the best known foundation for mathematics. If there was something less complicated or annoying that worked, we'd use that instead. But as far as we know, there isn't, and there are good reasons to believe we won't find something better.
zozbot234 6y ago
Note that "sets" and ZFC are not the same thing - ZFC is simply one set theory among many. In fact, structural set theories like ETCS ("Elementary Theory of the Category of Sets") or SEAR ("Sets, Elements And Relations") are even more cleanly suited to typical undergrad mathematics than ZF(C), while also being easier to characterize categorically.
- coolgeek 6y ago
  New to me. Thanks. A jumping off point https://math.stackexchange.com/questions/331567/alternative-set-theories https://math.stackexchange.com/questions/331567/alternative-...
spekcular 6y ago
OK, but this doesn't answer the question. You intimate there are benefits but never say what they are. The vast majority of mathematicians go their entire lives without using the word "category" in a paper. What are they missing out on?
- wisnesky 6y ago
  Much of theoretical computer science uses categorical methods, for example, as semantics for type theory. In that field, such techniques are often more natural than set-theoretic ones due to issues of computability, decidability, etc. So at the very least, much of that research as originally written would be inaccessible to a non category theorist. Whether that field counts as mathematics and if it does whether it is worth missing out on depends on taste of course, but an example would be 'homotopy type theory' https://homotopytypetheory.org https://homotopytypetheory.org
  - spekcular 6y ago
    HoTT is a strictly worse foundation for mathematics than ZFC, and in the end has to end up copying a bunch of the usual constructions anyway (like defining the real numbers). So this is not convincing. One amusing problem is discussed here: https://mathoverflow.net/questions/289711/defining-sun-in-hott https://mathoverflow.net/questions/289711/defining-sun-in-ho.... But suppose HoTT were equally good. What compelling reason is there for a working mathematician to learn it? We already know about set theory, and it meets all of our needs.
    - wisnesky 6y ago
      Honestly; I'm not sure; I'm a working computer scientist, not a working mathematician, and I use a proof assistant (Coq) all the time to have confidence that the proofs I write are correct (and more and more this is a requirement for publication in CS conferences). I want HoTT to succeed because it would turn some the axioms I must assume in the current Coq proof assistant into theorems, with significant implications for 'proof engineering' at scale (eg, verifying an OS kernel or compiler or database). When I read the math over flow post my emotional response is gladness that I can't accidentally conflate a topological space with an infinity groupoid. Many of the HoTT researchers are mathematicians (including the late Vladimir Voevodsky); presumably, the HoTT book (which I understand is available online) can give you a better answer than I can from the perspective of a working mathematician.

occamrazor 6y ago

Very good point. To add on this, even category theory is often taught in the language of sets and classes (“A category is a class of objects with a set Hom(X,Y) of morphisms for each pair of objects, etc.”) It is possible to use categories as basic building blocks instead if sets but, in my anecdotal experience, this us not what the majority of graduate programs in Mathematics do. It will be interesting to see whether this will change in the next 20 yea.

spekcular 6y ago
They don't do it for good reason: set theory is basically a strictly better foundation for mathematics, and you have to ape all the set theoretic constructions when doing various things anyway (e.g. constructing the real numbers), so it doesn't buy you anything. This issue has been litigated extensively, and in my view successfully, by Harvey Friedman on the Foundations of Mathematics mailing list, if you want to check its archives.

jacobolus 6y ago

Set theory was developed to make analysis “rigorous” by the standards of 19th century mathematicians, who were probing the edge cases of the more intuitive (or maybe handwavey) assumptions from the 17th–18th century. Here’s Rudin to explain: https://www.youtube.com/watch?v=hBcWRZMP6xs https://www.youtube.com/watch?v=hBcWRZMP6xs In the 20th century it became trendy to (a) try to base everything on set theory, (b) write mathematics in a very dry and formal style, taking inspiration from the Bourbaki project. My impression is that undergraduates are taught in the language of sets partly because it is serviceable for describing the subjects they are trying to learn (esp. analysis) but also more importantly because mid-20th century mathematicians who set up the curriculum we are still using wanted to thoroughly teach (indoctrinate?) undergraduates the trendy style of the time. Personally I think it has been a mixed bag; the style is a big turn-off to many people, and ends up chasing people out of mathematics who could otherwise make valuable contributions. The people who remain seem to mostly like it okay though. YMMV.

jrajav 6y ago
Your observations strike a serious chord with me, because the first class I took where a professor wielded set theory like a ruler to the back of students' hands was exactly the point I realized that mathematics was not my calling. But I've come to understand that I was not realizing that I didn't like mathematics or thought it was too hard, but instead that if I made that my life's work, I'd be working with many more people like that, rather than the people I wanted to - bright and creative, yet kind and humble. I don't think set theory or any other set of tools is the problem behind potential academics getting turned off. It's the people, and the culture of glorified monkhood.
spekcular 6y ago
I have some problems with this comment. First, why the scare quotes around "rigorous"? Set theory literally did make the foundations of analysis more rigorous. Second of all, we don't use set theory because it's "trendy" or because we want to "indoctrinate" undergraduates, we use it because it's the best known foundation for mathematics. If there was something less complicated or annoying that worked, we'd use that instead. But as far as we know, there isn't, and there are good reasons to believe we won't find something better.

zozbot234 6y ago

Note that "sets" and ZFC are not the same thing - ZFC is simply one set theory among many. In fact, structural set theories like ETCS ("Elementary Theory of the Category of Sets") or SEAR ("Sets, Elements And Relations") are even more cleanly suited to typical undergrad mathematics than ZF(C), while also being easier to characterize categorically.

coolgeek 6y ago
New to me. Thanks. A jumping off point https://math.stackexchange.com/questions/331567/alternative-set-theories https://math.stackexchange.com/questions/331567/alternative-...

spekcular 6y ago

OK, but this doesn't answer the question. You intimate there are benefits but never say what they are. The vast majority of mathematicians go their entire lives without using the word "category" in a paper. What are they missing out on?

wisnesky 6y ago
Much of theoretical computer science uses categorical methods, for example, as semantics for type theory. In that field, such techniques are often more natural than set-theoretic ones due to issues of computability, decidability, etc. So at the very least, much of that research as originally written would be inaccessible to a non category theorist. Whether that field counts as mathematics and if it does whether it is worth missing out on depends on taste of course, but an example would be 'homotopy type theory' https://homotopytypetheory.org https://homotopytypetheory.org
- spekcular 6y ago
  HoTT is a strictly worse foundation for mathematics than ZFC, and in the end has to end up copying a bunch of the usual constructions anyway (like defining the real numbers). So this is not convincing. One amusing problem is discussed here: https://mathoverflow.net/questions/289711/defining-sun-in-hott https://mathoverflow.net/questions/289711/defining-sun-in-ho.... But suppose HoTT were equally good. What compelling reason is there for a working mathematician to learn it? We already know about set theory, and it meets all of our needs.
  - wisnesky 6y ago
    Honestly; I'm not sure; I'm a working computer scientist, not a working mathematician, and I use a proof assistant (Coq) all the time to have confidence that the proofs I write are correct (and more and more this is a requirement for publication in CS conferences). I want HoTT to succeed because it would turn some the axioms I must assume in the current Coq proof assistant into theorems, with significant implications for 'proof engineering' at scale (eg, verifying an OS kernel or compiler or database). When I read the math over flow post my emotional response is gladness that I can't accidentally conflate a topological space with an infinity groupoid. Many of the HoTT researchers are mathematicians (including the late Vladimir Voevodsky); presumably, the HoTT book (which I understand is available online) can give you a better answer than I can from the perspective of a working mathematician.