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I always groan when I see posts on HN with grandiose claims about category theory, like this one. I think it is actively harmful to propagate pseudo-mathematic

I always groan when I see posts on HN with grandiose claims about category theory, like this one. I think it is actively harmful to propagate pseudo-mathematical claims like the those, for example, found in the slides of Guest Lecture 1: >>> It’s touched or greatly influenced all corners of mathematics. >>> It’s become a gateway to learning mathematics. And from the audio of the lecture (paraphrasing): >>> Category theory is the stem cell that differentiates into and lies at the root of all pure mathematics. ["All forms of pure math" is also written on the slides.] These statements are simply false. The vast majority of pure mathematics research done today does not involve category theory at all, and does not benefit from it. An even greater majority (like 99%+) of mathematics done in industry and in national labs does not involve category theory. Numerical analysis, probability, statistics, partial differential equations, dynamical systems, harmonic analysis, even lots of modern differential geometry – no category theory to be seen! Want proof? Pick up any introductory graduate textbook, or any major journal in these fields. Now, if you want to do research in number theory, or algebraic topology, or algebraic geometry – sure, you likely would benefit from categorical thinking. But those fields hardly have a monopoly on pure mathematics. Even in, for example, Hatcher's introductory graduate text on algebraic topology (perhaps the most widely used), category theory is stuck in a small appendix and you can read the entire thing without it, with no real loss [1]. I don't have the energy right now to explain why I find the lecture series misguided more generally, but I want to at least flag these obviously incorrect statements and urge caution. [And before anyone grabs a PDE book and tells me the use of cohomology groups in certain places means PDE uses category theory, please note that e.g. homological algebraic and category theory are different things.] [1] OK, I guess you need to know what e.g. a natural transformation is to read some parts of the last chapter, but no one does that in an introductory course anyway. A motivated teacher could easily present the material in such a way that this didn't matter.

Koshkin 6y ago
Personally, I wouldn't be that skeptical. By its very nature CT as a foundational theory and is relevant to, and has at least in that sense indeed touched, all corners of mathematics. (Mathematicians had the same skepticism about Set Theory when it first appeared.) Especially the "theoretical" (pure) math. So, sure, "you can read the entire thing without it, with no real loss", but this only says something about the particular textbook and not the subject itself. I assure you, the actual loss, whether you realize it or not, will be very real. Books like Aluffi's Algebra: Chaper 0 have a very good reason behind them. Category Theory is the chapter zero of the modern understanding of mathematics (and not only).
- spekcular 6y ago
  Can you please explain to me how learning category theory is relevant to, or would enhance the work of, a research mathematician who works on, say, harmonic analysis or probability theory? What is the actual loss to these mathematicians? Also, your claim that category theory is a foundational theory (in the sense that set theory is) is just mistaken. Homotopy type theory claims to be such a theory, but homotopy type theory should not be identified with category theory more broadly.
  - Koshkin 6y ago
    I wouldn't try, but Helemskii's Lectures And Exercises on Functional Analysis show all this in exquisite detail. The Convenient Setting of Global Analysis (freely available as a PDF) makes extensive use of the categorical notions and methods.
    - spekcular 6y ago
      This does not answer my question. The material in the first book is all well-known. A standard reference is Reed and Simon's Methods of Modern Mathematical Physics. Categories don't gain you anything there, and in any case it's tangentially relevant to most work in harmonic analysis and probabiility. The second book is not relevant at all. Just because you can produce books on analysis where someone uses the word "category," does not mean a working mathematician ought to care.
      - Koshkin 6y ago
        (Funny that you mentioned "a working mathematician.") Anyway, wouldn't it be nice to really understand, on some (higher, admittedly) level, what it is that you are actually doing? Functors and all...
        spekcular 6y ago
        But this is exactly my objection. Returning to your example, I don't think working through Helemskii's book helps one really understand functional analysis relative to a well-written standard treatment. What interesting problems does this viewpoint permit a probabilist or harmonic analyst to solve that the standard approach does not? What theorems does it enable? I find it bizarre that you (and others – I don't mean to pick on you) seem to think that a translation to category theoretic language is necessary (and sufficient?) to understand what one is actually doing. Do the many professional mathematicians who prove important theorems in functional analysis today without bothering to learn this language not actually understand what they are doing [0]? But the undergraduate who reads Helemskii does? This seems like an absurd notion of what it means to actually understand a subject. [0] See, for example, many of the papers noted here: https://en.wikipedia.org/wiki/Invariant_subspace_problem https://en.wikipedia.org/wiki/Invariant_subspace_problem
bcheung 6y ago
It's a bit of a white lie but I don't see it being harmful to claim that. It is increasingly becoming true. Lots of fields are adopting CT. And being taught CT earlier on in the process would provide a useful framework for building upon future knowledge. Mathematics seems to have 10 different names for the same concept depending on which field you are in. CT provides a common vocabulary.
- btilly 6y ago
  It's a bit of a white lie but I don't see it being harmful to claim that. It is increasingly becoming true. I do. Category theory is something that appeals to some mathematicians, and not to others. Those who it doesn't appeal to are likely to wind up in fields like combinatorics, functional analysis, numerical analysis, and so on. If you inflict category theory prematurely on the latter group, people who might have proven quite talented will be driven away from mathematics. And I feel this quite personally. I left mathematics for other reasons. But still, had I had to deal with category theory first, I'd have never gone into mathematics in the first place.
  - crawfordcomeaux 6y ago
    I consider your example revelatory, as it points to a different source of harm: Preferences. "appeals to some, and not others" - this speaks to the need to normalize the abandonment/minimization of preferences. The harm you describe is self-inflicted limitation due to clinging to preferences. I accidentally abandoned my preferences that lie beyond the meeting of my needs, ie. abandoning preferences for how to meet them. This has led me to realize most, if not all preferences, stem from a combination of arbitrary choices made when young and choices born out of trauma. While there exists preferences with biological/physiological origins, such as those that may have developed around allergies, any conditioned reactions to them are still likely unnecessary. I don't need to feel queasy from smelling something rotten to know it's rotten and to avoid it. Do you want help with disengaging these means of self-limitation?
    - btilly 6y ago
neonological 6y ago
>>> Category theory is the stem cell that differentiates into and lies at the root of all pure mathematics. ["All forms of pure math" is also written on the slides.] I would say this statement is not false. It does seem to lie at the root and is an alternative to set theory which means that mathematics can be perceived with category theory as it's foundation. Whether an actual written CT reformalization has been executed within all the pure mathematical sub fields or whether doing an actual formalization from the perspective of CT will be a benefit at all is another topic all together. I agree that category theory is not popular. However I feel software engineers should have interest in it as Category theory looks to intersect with an area of software engineering that is currently done ad-hoc style with just gut feelings. If you squint, Category theory looks to be a formalized theory for the design and complexity of modules. Fully understanding Category theory and how it applies to software could lead to a future where design and architecture is "calculated" rather then "designed."
nine_k 6y ago
I suspect such claims might grow from the idea that you can replace set-theoretic constructs with category-theoretic constructs in certain places, and see what happens. It must be pretty fundamental, in the sense that dependence on a set theory lies in the foundation of many branches of mathematics. I can't say how universally such an approach can be applied; likely not very much.
coldtea 6y ago
>These statements are simply false. The vast majority of pure mathematics research done today does not involve category theory at all, and does not benefit from it. An even greater majority (like 99%+) of mathematics done in industry and in national labs does not involve category theory. Numerical analysis, probability, statistics, partial differential equations, dynamical systems, harmonic analysis, even lots of modern differential geometry – no category theory to be seen! They might not think of it consciously, but they sure exploit and use properties of sets, rings, groups, and so on all the time in their calculations...
- btilly 6y ago
  The properties of sets, rings, groups and so on that they most often use were established prior to the invention of category theory in the 1940s. Therefore the fact that they use subjects that category theory tried to find abstractions in no way suggests that they are based on category theory.
villasv 6y ago
By chance, I had a sequence of professors that didn't care about category at all. As consequence, I went through the entirety of my Applied Mathematics Msc without ever, not even once using the bits of category theory I learned as undergrad. And I'm pretty sure that if I cherry-picked my credits right, the same would have happened in the Phd.
chobytes 6y ago
I went into my math undergrad having gotten hyped online about CT and HoTT. ...And basically the only time i've seen CT come up is in algebra (they use aluffi). And the only time Ive heard anything about HoTT was when I tried bringing it up with the logic graduate students and got some light teasing.

Koshkin 6y ago

Personally, I wouldn't be that skeptical. By its very nature CT as a foundational theory and is relevant to, and has at least in that sense indeed touched, all corners of mathematics. (Mathematicians had the same skepticism about Set Theory when it first appeared.) Especially the "theoretical" (pure) math. So, sure, "you can read the entire thing without it, with no real loss", but this only says something about the particular textbook and not the subject itself. I assure you, the actual loss, whether you realize it or not, will be very real. Books like Aluffi's Algebra: Chaper 0 have a very good reason behind them. Category Theory is the chapter zero of the modern understanding of mathematics (and not only).

spekcular 6y ago
Can you please explain to me how learning category theory is relevant to, or would enhance the work of, a research mathematician who works on, say, harmonic analysis or probability theory? What is the actual loss to these mathematicians? Also, your claim that category theory is a foundational theory (in the sense that set theory is) is just mistaken. Homotopy type theory claims to be such a theory, but homotopy type theory should not be identified with category theory more broadly.
- Koshkin 6y ago
  I wouldn't try, but Helemskii's Lectures And Exercises on Functional Analysis show all this in exquisite detail. The Convenient Setting of Global Analysis (freely available as a PDF) makes extensive use of the categorical notions and methods.
  - spekcular 6y ago
    This does not answer my question. The material in the first book is all well-known. A standard reference is Reed and Simon's Methods of Modern Mathematical Physics. Categories don't gain you anything there, and in any case it's tangentially relevant to most work in harmonic analysis and probabiility. The second book is not relevant at all. Just because you can produce books on analysis where someone uses the word "category," does not mean a working mathematician ought to care.
    - Koshkin 6y ago
      (Funny that you mentioned "a working mathematician.") Anyway, wouldn't it be nice to really understand, on some (higher, admittedly) level, what it is that you are actually doing? Functors and all...
      - spekcular 6y ago
        But this is exactly my objection. Returning to your example, I don't think working through Helemskii's book helps one really understand functional analysis relative to a well-written standard treatment. What interesting problems does this viewpoint permit a probabilist or harmonic analyst to solve that the standard approach does not? What theorems does it enable? I find it bizarre that you (and others – I don't mean to pick on you) seem to think that a translation to category theoretic language is necessary (and sufficient?) to understand what one is actually doing. Do the many professional mathematicians who prove important theorems in functional analysis today without bothering to learn this language not actually understand what they are doing [0]? But the undergraduate who reads Helemskii does? This seems like an absurd notion of what it means to actually understand a subject. [0] See, for example, many of the papers noted here: https://en.wikipedia.org/wiki/Invariant_subspace_problem https://en.wikipedia.org/wiki/Invariant_subspace_problem

bcheung 6y ago

It's a bit of a white lie but I don't see it being harmful to claim that. It is increasingly becoming true. Lots of fields are adopting CT. And being taught CT earlier on in the process would provide a useful framework for building upon future knowledge. Mathematics seems to have 10 different names for the same concept depending on which field you are in. CT provides a common vocabulary.

btilly 6y ago
It's a bit of a white lie but I don't see it being harmful to claim that. It is increasingly becoming true. I do. Category theory is something that appeals to some mathematicians, and not to others. Those who it doesn't appeal to are likely to wind up in fields like combinatorics, functional analysis, numerical analysis, and so on. If you inflict category theory prematurely on the latter group, people who might have proven quite talented will be driven away from mathematics. And I feel this quite personally. I left mathematics for other reasons. But still, had I had to deal with category theory first, I'd have never gone into mathematics in the first place.
- crawfordcomeaux 6y ago
  I consider your example revelatory, as it points to a different source of harm: Preferences. "appeals to some, and not others" - this speaks to the need to normalize the abandonment/minimization of preferences. The harm you describe is self-inflicted limitation due to clinging to preferences. I accidentally abandoned my preferences that lie beyond the meeting of my needs, ie. abandoning preferences for how to meet them. This has led me to realize most, if not all preferences, stem from a combination of arbitrary choices made when young and choices born out of trauma. While there exists preferences with biological/physiological origins, such as those that may have developed around allergies, any conditioned reactions to them are still likely unnecessary. I don't need to feel queasy from smelling something rotten to know it's rotten and to avoid it. Do you want help with disengaging these means of self-limitation?
  - btilly 6y ago

neonological 6y ago

>>> Category theory is the stem cell that differentiates into and lies at the root of all pure mathematics. ["All forms of pure math" is also written on the slides.] I would say this statement is not false. It does seem to lie at the root and is an alternative to set theory which means that mathematics can be perceived with category theory as it's foundation. Whether an actual written CT reformalization has been executed within all the pure mathematical sub fields or whether doing an actual formalization from the perspective of CT will be a benefit at all is another topic all together. I agree that category theory is not popular. However I feel software engineers should have interest in it as Category theory looks to intersect with an area of software engineering that is currently done ad-hoc style with just gut feelings. If you squint, Category theory looks to be a formalized theory for the design and complexity of modules. Fully understanding Category theory and how it applies to software could lead to a future where design and architecture is "calculated" rather then "designed."

nine_k 6y ago

I suspect such claims might grow from the idea that you can replace set-theoretic constructs with category-theoretic constructs in certain places, and see what happens. It must be pretty fundamental, in the sense that dependence on a set theory lies in the foundation of many branches of mathematics. I can't say how universally such an approach can be applied; likely not very much.

coldtea 6y ago

>These statements are simply false. The vast majority of pure mathematics research done today does not involve category theory at all, and does not benefit from it. An even greater majority (like 99%+) of mathematics done in industry and in national labs does not involve category theory. Numerical analysis, probability, statistics, partial differential equations, dynamical systems, harmonic analysis, even lots of modern differential geometry – no category theory to be seen! They might not think of it consciously, but they sure exploit and use properties of sets, rings, groups, and so on all the time in their calculations...

btilly 6y ago
The properties of sets, rings, groups and so on that they most often use were established prior to the invention of category theory in the 1940s. Therefore the fact that they use subjects that category theory tried to find abstractions in no way suggests that they are based on category theory.

villasv 6y ago

By chance, I had a sequence of professors that didn't care about category at all. As consequence, I went through the entirety of my Applied Mathematics Msc without ever, not even once using the bits of category theory I learned as undergrad. And I'm pretty sure that if I cherry-picked my credits right, the same would have happened in the Phd.

chobytes 6y ago

I went into my math undergrad having gotten hyped online about CT and HoTT. ...And basically the only time i've seen CT come up is in algebra (they use aluffi). And the only time Ive heard anything about HoTT was when I tried bringing it up with the logic graduate students and got some light teasing.