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Ralph Sarkis (Aug 27 2023 at 14:57):I've heard and believed for some time that category theory is overrepresented online meaning we see it pop up more often than it should compared to the actual proportions of mathematicians studying category theory. This is obviously my experience too, but I am clearly biased. I looked at the submissions for this year's SoME, and there is only one video on CT, last year's had three. So I am not convinced category theory has a bigger place than it should.
Have you seen evidence of CT being overrepresented or underrepresented online?
Have you felt that CT is overrepresented or underrepresented online?
Dylan Braithwaite (Aug 27 2023 at 15:39):I get the impression that category theory has a level of hype/evangelism that other areas of research maths don't receive, I think partially due to the popularity online of Haskell, and the fact that the basics of category theory are easier to learn than other "advanced" topics in maths (or at least are more standalone). That said, I don't know how much this translates into actual novel category theory content - for example because the amount of category theory one encounters when learning the main concepts of Haskell is fairly surface level.
I would guess this is what we see reflected in the SoME submissions: it's more reflective of the proportion of people who work deeply enough with category theory that they have the depth of knowledge to make novel expository content as opposed to the proportion of people who have dipped their toes in category theory and are talking about it online.
John Baez (Aug 27 2023 at 15:58):One of my theories about this is that category theorists, much more than say analysts, are able to state exciting snappy results that fit in a tweet, which become obvious when you just think about them, if you understand the relevant concepts.
Ivan Di Liberti (Aug 28 2023 at 09:34):I think some people more than others are responsible for this (toxic) culture.
Mike Shulman (Aug 28 2023 at 16:29):I'm curious what you think is toxic about it. If this is true, then I'm inclined to view it as a welcome correction from the days of the late 20th century when category theory tended to be deprecated (especially in the U.S.) as useless abstraction or "just a language". Maybe an overcorrection, but (as a category theorist) it doesn't seem intrinsically bad. In my own opinion, there's a lot more room for categorical concepts to usefully infuse, say, an undergraduate math curriculum than has yet happened across the board.
Steve Huntsman (Aug 28 2023 at 17:49):CT gets a bad rap not least because of the astonishing amount of jargon that obscures things when deployed by a majority of its proponents (!= experts).
For example I remember joking with some folks I know about an episode (can't recall precise details, I think it was actually on Zulip) wherein some categorical construct was discussed and someone asked something to the effect of "well what is this in the case of Set" and it turned out that the answer was "it's a function".
There are very few or zero nontrivial examples of constructions in many cases and the culture is not geared towards producing or exploring them. Arnol'd of course would say that (paraphrasing again) one's understanding of a concept is bounded by the examples that can be exhibited, and many folks (myself included) are deeply sympathetic to that viewpoint.
This is all manifested online more than in personal interactions, which heightens perceptions.
The net result is a sense--at times justified by experience and at other times ironically buttressed by seeing what can be done with the tools in the hands of a master that also has facility in "ordinary" mathematics--that many proponents of CT don't understand very much at all and that it is a grift.
FWIW, I don't share this opinion but I understand it as a reasonable conclusion for folks who haven't been immersed in the subject. Still I understand much CT at all but find it extremely useful in some infrequent and narrow circumstances.
Mike Shulman (Aug 28 2023 at 17:58):Part of that is just a conflict of cultures and ways of thinking. As John famously said, category theorists are dual to ordinary people: they get more confused when you surround an abstract concept with lots of distracting specifics. But CT does definitely tempt a certain sort of person to be overly dismissive of examples, which can certainly lead to bad mathematics and bad exposition. I think this is part of what led to the original backlash against CT that I mentioned.
However, I would tend to think that more online exposition of category theory ought to help alleviate this problem rather than exacerbate it.
Mike Shulman (Aug 28 2023 at 18:00):By the way, this:
the basics of category theory are easier to learn than other "advanced" topics in maths (or at least are more standalone).
is an interesting perspective to hear. I generally think of category theory as requiring more background than most other fields of higher math, because in order to really understand what it's doing and why, you tend to need to already know about multiple examples from multiple other subjects.
Patrick Nicodemus (Aug 28 2023 at 19:39):Steve Huntsman said:
CT gets a bad rap not least because of the astonishing amount of jargon that obscures things when deployed by a majority of its proponents (!= experts).
For example I remember joking with some folks I know about an episode (can't recall precise details, I think it was actually on Zulip) wherein some categorical construct was discussed and someone asked something to the effect of "well what is this in the case of Set" and it turned out that the answer was "it's a function".
I recall this, you might have been thinking of "what is an adjunction in the bicategory of spans"
https://categorytheory.zulipchat.com/#narrow/stream/229199-learning.3A-questions/topic/Adjunction.20of.20spans/near/351698870
Steve Huntsman (Aug 28 2023 at 20:30):Mike Shulman said:
But CT does definitely tempt a certain sort of person to be overly dismissive of examples, which can certainly lead to bad mathematics and bad exposition.
There is a famous urban legend about this sort of bad mathematics: https://blog.computationalcomplexity.org/2010/09/mathematical-urban-legend.html
Joe Moeller (Aug 28 2023 at 21:33):I spent some time in 2020/2021 trying to insert myself into the online communities for representation theorists. I found almost nothing. I have several rep theory friends, and nobody could point me to anything.
Todd Trimble (Aug 28 2023 at 22:01):Steve, I'm not so sure that CT has an astonishing amount of jargon more so than other fields. What I often see in CT is a few basic concepts being used and reused in an astonishing number of ways. To me, this is where a lot of its power comes from. I think I use the Yoneda lemma just about every day of my life.
I hope, after the laughter died down, some of those folks might be prepared to believe that a statement like "functions are left adjoints in the bicategory of relations" might have a point to it after all, even a substantial point, as opposed to being an exercise in intellectual wankery.
Incidentally: more urban legends here.
Steve Huntsman (Aug 29 2023 at 01:49):Todd Trimble said:
Steve, I'm not so sure that CT has an astonishing amount of jargon more so than other fields.
I am. Maybe my sense of number is off but the ratio of theorems to definitions seems to me to be lower than in any other field I can think of (for the other extreme, consider areas like numerical analysis, PDE, or probability). The nLab seems to exemplify this.
Mike Shulman (Aug 29 2023 at 02:47):I agree that CT probably has a lower ratio of theorems to definitions, but I would tend to say that's because it has fewer theorems rather than because it has more definitions. And the reason it has fewer theorems, I would say, may be partly because it just has fewer results, but als partly because the most important facts tend to be so easy to prove that people don't bother calling them theorems.
David Michael Roberts (Aug 29 2023 at 03:11):Another thing to keep in mind is that a bunch of results in category theory are constructions, and even definitions are results. How often do you see a paper where the main contribution is a novel definition that manages to re-prove a known result?
Further, areas like numerical analysis, PDE etc, the results are often techniques, and one might in a parallel universe make a formal definition that outlines a technique, but that's now how those fields work.
Algebraic geometry surely has much more jargon than category theory, in any case. But it is a lot older, and that's a field where there are a large number of concrete examples. One could similarly make a formal definition of an example (this is true for some, eg the Hilbert scheme, or even as low-brow as projective space), but this doesn't happen.
David Michael Roberts (Aug 29 2023 at 03:16):The representation of category theory online was no doubt helped by the attitude of the category theory community to the internet, which had the mailing list, do doubt because of its geographic sparseness, and then TAC started in the early days - and people like John B who was chronically online before that was a phrase, and willing to talk about categories to anyone. Being a younger subject no doubt helped (another early electronic-journal-adopting community was that of combinatorics, to some extent, and that is also a relatively young field).
The over-representation of CT online balances out the over-representation of certain traditional fields in some of the fancy "generalist" journals that nevertheless somehow focus largely on relatively few topics. :upside_down:
Eduardo Ochs (Aug 29 2023 at 03:17):Shameless plug: I've been working on techniques for studying general cases and particular cases - usually in Set - in parallel, using diagrams "with the same shape", and I've been trying to find other people who are also interested in that... but they seem to be very few. My best paper about that is the one in the link below, and if you find it interesting, drop me a line! Cheers and sorry for noise! =P
http://anggtwu.net/math-b.html#2022-md
Steve Huntsman (Aug 29 2023 at 03:31):Mike Shulman said:
I agree that CT probably has a lower ratio of theorems to definitions, but I would tend to say that's because it has fewer theorems rather than because it has more definitions. And the reason it has fewer theorems, I would say, may be partly because it just has fewer results, but als partly because the most important facts tend to be so easy to prove that people don't bother calling them theorems.
Now imagine you don’t know CT but you see that sort of stuff whenever you glance at a CT paper. You might think nothing much is being done. (You might think the same thing about EGA too, or Spivak’s Calculus on Manifolds as a youngster.)
Mike Shulman (Aug 29 2023 at 03:49):Sure. But that doesn't mean that category theory intrinsically has more jargon than any other field.
Todd Trimble (Aug 29 2023 at 04:18):Steve Huntsman said:
Todd Trimble said:
Steve, I'm not so sure that CT has an astonishing amount of jargon more so than other fields.
I am. Maybe my sense of number is off but the ratio of theorems to definitions seems to me to be lower than in any other field I can think of (for the other extreme, consider areas like numerical analysis, PDE, or probability). The nLab seems to exemplify this.
Of course the nLab would exemplify this, that's to be expected where almost every page is titled by a name of a concept, and not by the announcement of a theorem. I'm pretty familiar with the nLab, having made many contributions to it, and I'm familiar with how the pages get generated. I doubt the nLab is all that representative.
For a more balanced assessment, I'd suggest looking at journal articles, and examine the ratio of assertions (theorems, propositions) to definitions there.
John Baez (Aug 29 2023 at 07:38):I think any branch of math you're just learning seems to have an annoyingly large amount of jargon. Later, once you've mastered a lot of the jargon and are digging deeper, you discover it has an annoyingly large number of theorems. :upside_down:
John Baez (Aug 29 2023 at 07:41):I'm at the second stage with category theory: I can remember enough definitions to know what the theorems are saying, but I can't remember all the theorems. The theorems I remember best are only the ones I've actually used.
John Baez (Aug 29 2023 at 07:43):Here's a theorem I learned yesterday from Tom Leinster. Take the definition of monoid written out using commutative diagrams, where the associative law becomes a commutative square. What's a monoid where this commutative square is a pullback? It's a group.
John Baez (Aug 29 2023 at 07:45):I may forget this, because even though it's utterly amazing (and it gives a way to define a group in any monoidal category), I don't have any need for it - at least, not now.
John Baez (Aug 29 2023 at 07:45):But it is the kind of result that would make a good tweet: it's short, snappy and ends in a surprising punchline.
John Baez (Aug 29 2023 at 07:47):Now, someone who doesn't know category theory might think this result epitomizes the pointlessness of category theory: "oh, he's defining a group in terms I don't understand".
John Baez (Aug 29 2023 at 07:49):But if you know category theory you see right away that's not what's going on here: the amazing thing is that we're defining a group without using the fact that Set has finite products, just its monoidal structure and pullbacks. This is the sort of thing you tend to think is impossible!
John Baez (Aug 29 2023 at 07:55):So if you dug into the proof of this result - which I haven't - you might learn a lot. What else can actually you do in monoidal categories that we tend to think requires cartesianness?
David Michael Roberts (Aug 29 2023 at 08:17):Lawvere's fixed point theorem, for one :-)
Ivan Di Liberti (Aug 29 2023 at 09:15):Mike Shulman said:
I'm curious what you think is toxic about it. If this is true, then I'm inclined to view it as a welcome correction from the days of the late 20th century when category theory tended to be deprecated (especially in the U.S.) as useless abstraction or "just a language". Maybe an overcorrection, but (as a category theorist) it doesn't seem intrinsically bad. In my own opinion, there's a lot more room for categorical concepts to usefully infuse, say, an undergraduate math curriculum than has yet happened across the board.
This is going to be a very long reply, because your comment is wrong on a multitude of levels in my opinion, and I will try to address at least some of them.
The following is a quote from Rota 1993 on the proceedings of the Category Theory conference held in 1991:
It is good to know that category theory is alive and well after all these years. We were turned off to category theory by the excesses of the sixties, when a small but loud crowd pretended to reduce all of mathematics to the language of categories. Now at least they have toned down their claims, and category theory has taken its modest place in the mathematical spectrum side by side with lattice theory, more pretentious perhaps than the latter, but with a good pedigree. [...]
What stems from the quote is that the CT community always had a kind of evangelic attitude, and like many minorities, it had to work in order to assess/conquer its relevance and importance. For the lack of counterfactuals, it is hard to claim that this behavior has brought success to the community. Maybe the community has gained some success, despite this behavior?! It's hard to say. It is a fact though that between 1945 and the late 2000s the community has grown in size and prestige. Some pieces of evidence of this are JPAA, which has found its place among the most relevant specialist journals in algebra, Street joining the EB of Advances, and some prestigious publications, say by Moerdijk, in journals of the level of Compositio. These three examples, which are far from being cherry-picked, all happened before what I would call "the blog post era", and ages before the "Twitter era".
Since 2010, if anything, the community has lost in prestige and academic relevance, despite (I totally admit it) having reached a much broader audience in popularization and pop culture. The number of prestigious journals we have access to has drastically declined and is now totally restricted to Advances, JPAA is losing quite some prestige and is less and less controlled by CT-sts, it would be unimaginable nowadays to see a CT paper appear on Compositio. Yes, the community has had the HoTT sprint for a while, but that has lost quite some of its initial momentum too and -- I think --, overall, if we compare generation by generation, yours (Mike) is less prestigious than its parents, and mine is even less prestigious than yours.
Now, on a more "pop-culture" side of the story, I agree we have gained a lot in visibility, and this is mostly due to the twitting/blogging thing I touched on previously. Has this truly benefitted our community?! That's unclear. And let me stress, this is the behavior I was calling toxic, not the one of the elder generation. In order to discuss it, let me describe it better.
John, who is the person I was responding to, wrote "category theorists, much more than say analysts, are able to state exciting snappy results that fit in a tweet". This is indeed what happens all the time, a somewhat young and often not very expert PhD student that was nourished in mottos and nlab stories writes on Twitter that whatever all mathematicians are doing is, in fact, all trivial and reduces to the bright and somewhat magical fact that a monad is a monoid in the category of endofunctors. The result? A number of narcissist crackpots enter this very zulip for a couple of weeks and then leave it, a number of young PhD students in other areas think that the PhD student is just an annoying prick (deservedly), and the PhD student grows with a strange complex of superiority which is incredibly hard to heal later in their studies. How is this not supposed to be toxic, I wonder. Now, I have mentioned PhD students here, but I honestly believe that a number of more senior searchers are not only themselves part of what I described, but also responsible for the behavior of their PhD students, who just repeated their attitude.
All in all, am I saying that we should not try to make category theory broader in scope and more impactful on the math community? Of course not! We should! And I think my actions show that I am very much invested in that. Twitting "ah-ah, you are all my pawns, because universal algebra is just a monoid in the category of endofunctors and I rule them all. Am I not the smartest?", instead, may not be the best way to actively do healthy popularization, let alone dissemination of category theory.
Todd Trimble (Aug 29 2023 at 10:52):John Baez said:
Here's a theorem I learned yesterday from Tom Leinster. Take the definition of monoid written out using commutative diagrams, where the associative law becomes a commutative square. What's a monoid where this commutative square is a pullback? It's a group.
Oh, nice! That may be the first time I've heard it, but it resonates. When you think of the Beck-Chevalley condition derived from a pullback, it looks like it's going to be connected to Frobenius equations (as in Frobenius algebras).
John Baez (Aug 29 2023 at 11:24):Wow, I hadn't thought of that. I may want to reprove this result myself sometime. Btw, Tom said when you define a "group object in a monoidal category" this way, the group object automatically gets its own "personal symmetry" . I'm just noting that in case anyone wants to dig into this.
John Baez (Aug 29 2023 at 11:25):I asked Tom if this was really a symmetry or just a braiding and he said he forgets.
Steve Huntsman (Aug 29 2023 at 12:13):Todd Trimble said:
Steve Huntsman said:
Todd Trimble said:
Steve, I'm not so sure that CT has an astonishing amount of jargon more so than other fields.
I am. Maybe my sense of number is off but the ratio of theorems to definitions seems to me to be lower than in any other field I can think of (for the other extreme, consider areas like numerical analysis, PDE, or probability). The nLab seems to exemplify this.
Of course the nLab would exemplify this, that's to be expected where almost every page is titled by a name of a concept, and not by the announcement of a theorem. I'm pretty familiar with the nLab, having made many contributions to it, and I'm familiar with how the pages get generated. I doubt the nLab is all that representative.
For a more balanced assessment, I'd suggest looking at journal articles, and examine the ratio of assertions (theorems, propositions) to definitions there.
Guess I should have explicitly couched this in the thread title's context: viz., online
John Baez (Aug 29 2023 at 12:25):@Ivan Di Liberti wrote:
John, who is the person I was responding to, wrote "category theorists, much more than say analysts, are able to state exciting snappy results that fit in a tweet". This is indeed what happens all the time, a somewhat young and often not very expert PhD student that was nourished in mottos and nlab stories writes on Twitter that whatever all mathematicians are doing is, in fact, all trivial and reduces to the bright and somewhat magical fact that a monad is a monoid in the category of endofunctors.
That's not what I was talking about: I was talking about how category theorists trade insights on Twitter, in a quite fun and useful way. This particular result is like : it's indeed exciting when you first learn it, but then you have to suffer through the excitement of a thousand people who learn it after you, and you begin to hate it.
I don't think there's much to do about the puppy-like enthusiasm of beginners, or their occasional arrogance. I suppose we could slap them down. But that would come across as arrogant in an even worse way.
John Baez (Aug 29 2023 at 12:28):Now, I have mentioned PhD students here, but I honestly believe that a number of more senior searchers are not only themselves part of what I described, but also responsible for the behavior of their PhD students, who just repeated their attitude.
If you're complaining about certain specific people it's probably good to name them. Otherwise we'll all nod and think you're talking about someone else. :upside_down:
Tim Hosgood (Aug 29 2023 at 13:48):I think that one of Ivan's points was "to whom are these Twitter insights useful? is it just other category theorists? and at what level?" (or at least this is what I gained from reading his reply)
Todd Trimble (Aug 29 2023 at 13:51):nourished in mottos and nlab stories
What are "nlab stories"? I've heard of the nLab, but they sound really awful, the way you say it. (Seriously, what examples do you have in mind?)
The nLab is both a great resource, and a rather skewed resource, being in very large part the creation of a single very energetic individual with a decidedly individual point of view and agenda (using "agenda" not in a negative sense). A lot of it is, to me, quite hard to read, and not exactly "nourishing" unless I invest a major effort. Perhaps it's quoted a lot, but is it representative of online category theory? I'd think the Cafe is more indicative of what online interactions between category theorists look like. (You'll find theorems stated there, too.)
I'm hardly on Twitter ever, so can't comment on the smart-ass PhD student or his adviser who sound like walking caricatures. My own form of wasting time is mostly on YouTube. There's some pretty good online representation there in the form of talks. I tend to avoid watching the intro stuff.
David Egolf (Aug 29 2023 at 14:43):Are there online communities like this one focused on other areas of math? If not - or even if there are a few others, but not many - I think that makes category theory pretty special in terms of how it is represented online. If there are other similar communities, I would be very interested to learn about them!
In terms of what I've been able to find online so far, this community is exceptional for (1) the presence of many subject-matter experts, (2) having a high level of activity, (3) having good moderation, and (4) being welcoming to questions from beginners. I also feel that this community is really good at facilitating extended open-ended conversations, something that is difficult to do on many other websites, including stackexchange and reddit.
Mike Shulman (Aug 29 2023 at 15:11):@Ivan Di Liberti, I'm not sure in what way your comment is a disagreement with mine. My understanding is that the "small but loud crowd" you mention is the major reason why category theory tended to be viewed badly in the 20th century, as I mentioned. And the gradual generational displacement of those folks, and of the folks who were originally turned off by them, is I think a major factor in the growth of CT through the late 20th century and early 21st that you mention. I don't know if I agree that CT has declined in prestige over the past decade -- I'm doubtful of the usefulness of "number of category theorists on the board of hoity-toity journals" as a metric of that -- but even if that's just your definition of "prestige", I think the pop-culture visibility is more relevant to what I was talking about and responding to.
Steve Huntsman (Aug 29 2023 at 15:26):John Baez said:
I think any branch of math you're just learning seems to have an annoyingly large amount of jargon. Later, once you've mastered a lot of the jargon and are digging deeper, you discover it has an annoyingly large number of theorems. :upside_down:
I have never felt this in disciplines wherein one actually computes things. Take enumerative combinatorics as an extreme opposite case: there are certainly a lot of concepts and terms, but at a basic level there are not many definitions to wade through for any particular sort of problem. If I want to enumerate all the Dyck paths of a certain size then there are quite a few ways to do it, and perhaps some machinery to learn (say a generating function), but not a lot of concepts without associated machinery, results, or examples.
Ralph Sarkis (Aug 29 2023 at 15:44):David Egolf said:
Are there online communities like this one focused on other areas of math?
This community only has 3k users with 225 active (in the last 15 days). I don't know how much it contributes to the CT representation online.
Jean-Baptiste Vienney (Aug 29 2023 at 15:50):Ivan Di Liberti said:
Mike Shulman said:
I'm curious what you think is toxic about it. If this is true, then I'm inclined to view it as a welcome correction from the days of the late 20th century when category theory tended to be deprecated (especially in the U.S.) as useless abstraction or "just a language". Maybe an overcorrection, but (as a category theorist) it doesn't seem intrinsically bad. In my own opinion, there's a lot more room for categorical concepts to usefully infuse, say, an undergraduate math curriculum than has yet happened across the board.
This is going to be a very long reply, because your comment is wrong on a multitude of levels in my opinion, and I will try to address at least some of them.
The following is a quote from Rota 1993 on the proceedings of the Category Theory conference held in 1991:
It is good to know that category theory is alive and well after all these years. We were turned off to category theory by the excesses of the sixties, when a small but loud crowd pretended to reduce all of mathematics to the language of categories. Now at least they have toned down their claims, and category theory has taken its modest place in the mathematical spectrum side by side with lattice theory, more pretentious perhaps than the latter, but with a good pedigree. [...]
What stems from the quote is that the CT community always had a kind of evangelic attitude, and like many minorities, it had to work in order to assess/conquer its relevance and importance. For the lack of counterfactuals, it is hard to claim that this behavior has brought success to the community. Maybe the community has gained some success, despite this behavior?! It's hard to say. It is a fact though that between 1945 and the late 2000s the community has grown in size and prestige. Some pieces of evidence of this are JPAA, which has found its place among the most relevant specialist journals in algebra, Street joining the EB of Advances, and some prestigious publications, say by Moerdijk, in journals of the level of Compositio. These three examples, which are far from being cherry-picked, all happened before what I would call "the blog post era", and ages before the "Twitter era".
Since 2010, if anything, the community has lost in prestige and academic relevance, despite (I totally admit it) having reached a much broader audience in popularization and pop culture. The number of prestigious journals we have access to has drastically declined and is now totally restricted to Advances, JPAA is losing quite some prestige and is less and less controlled by CT-sts, it would be unimaginable nowadays to see a CT paper appear on Compositio. Yes, the community has had the HoTT sprint for a while, but that has lost quite some of its initial momentum too and -- I think --, overall, if we compare generation by generation, yours (Mike) is less prestigious than its parents, and mine is even less prestigious than yours.
Now, on a more "pop-culture" side of the story, I agree we have gained a lot in visibility, and this is mostly due to the twitting/blogging thing I touched on previously. Has this truly benefitted our community?! That's unclear. And let me stress, this is the behavior I was calling toxic, not the one of the elder generation. In order to discuss it, let me describe it better.
John, who is the person I was responding to, wrote "category theorists, much more than say analysts, are able to state exciting snappy results that fit in a tweet". This is indeed what happens all the time, a somewhat young and often not very expert PhD student that was nourished in mottos and nlab stories writes on Twitter that whatever all mathematicians are doing is, in fact, all trivial and reduces to the bright and somewhat magical fact that a monad is a monoid in the category of endofunctors. The result? A number of narcissist crackpots enter this very zulip for a couple of weeks and then leave it, a number of young PhD students in other areas think that the PhD student is just an annoying prick (deservedly), and the PhD student grows with a strange complex of superiority which is incredibly hard to heal later in their studies. How is this not supposed to be toxic, I wonder. Now, I have mentioned PhD students here, but I honestly believe that a number of more senior searchers are not only themselves part of what I described, but also responsible for the behavior of their PhD students, who just repeated their attitude.
All in all, am I saying that we should not try to make category theory broader in scope and more impactful on the category theory community? Of course not! We should! And I think my actions show that I am very much invested in that. Twitting "ah-ah, you are all my pawns, because universal algebra is just a monoid in the category of endofunctors and I rule them all. Am I not the smartest?", instead, may not be the best way to actively do healthy popularization, let alone dissemination of category theory.
You talk of (1) the place of category theory in the mathematical community and (2) the popularity of category theory on the net but don't say anything about (3) the place of category theory in computer science. The reason why I've been going into category theory is because together with type theory it provides a fundational language that can speak at the same time of computer science and mathematics. By the way, I don't have any problem with saying that category theory is a language because language is important. I see the mathematical community often hesitant to adopt this language, because they are used to the set-theory + classical logic foundation and don't want to change their habits and being told that they are not working in the right way. I see the computer science community much more open to adopt this language, because contrary to the former community, they don't already have a strong foundational language they are used to and they have only to gain into adopting this language. Some people in engineering or other applied sciences too have an open mindset towards category theory because they don't already have another foundational language they are used to and thus have mostly to gain with adopting this one.
There is probably some link between the popularity of CT on the net and its use and perhaps importance in computer science. Functional progamming language like Haskell are strongly inspired by lambda calculus (ie. cartesian closed categories), or other things like functors, or monads. There are big journals and conference in computer science, including the most prestigious ones, where a lot of work on category theory is published. I don't think the popularity of CT on the net is only a consequence of the attitude of some mathematicians on Twitter. The relation between category theory and computer science probably also plays a role. And this relation has nothing to do with the behavior of some people but is just a fact in science, human knowledge or whatever you call something that includes these two fields at the same time.
Ivan Di Liberti (Aug 29 2023 at 16:21):Mike Shulman said:
@ I don't know if I agree that CT has declined in prestige over the past decade -- I'm doubtful of the usefulness of "number of category theorists on the board of hoity-toity journals" as a metric of that
Mike, really, how can you say this? Just think about it for a second! Please!
Jean-Baptiste Vienney (Aug 29 2023 at 16:30):Lol, CT is not going to die out of your desperation
Tim Hosgood (Aug 29 2023 at 16:47):I mean, for CT to "survive" it needs new people, and for new people to get positions they need to publish things in good journals — this is a fact, irrelevant of how unsatisfying or uncomfortable to admit it might be, so I think I see the point that, if you want the community to survive (whatever this means), prestige is important, purely because it's how funding committees decide who to give money to to hire new people :shrug:
Tim Hosgood (Aug 29 2023 at 16:53):I really dislike getting involved in these sorts of conversations, but I really was rather taken aback by what seems to be a common thread of logic here, namely that one can dispute the reputation of one's field. If people outside of CT say "oh CT people seem very X", then people inside of CT just cannot dispute this — not because X is right or wrong (this makes no difference!) but just because by very definition members of a community cannot say how they do appear to outsiders. They can say all they like about how they should appear, or how they intend to appear, but not how other people perceive them.
Steve Huntsman (Aug 29 2023 at 17:13):People can and do say lots of things that are simply false because they trust their feelings more than they should. (I'm not claiming to be an exception to this but I try to be.)
Mike Shulman (Aug 29 2023 at 17:38):I think we're just talking about different things. Sure, if that's what you mean by "prestige," then yes. Maybe I should have figured that was what you meant by prestige. But I didn't think that was what we were talking about. The original subject of this thread was about the representation of category theory in online conversations, and that's what I was still talking about and what I assumed you were also talking about. That doesn't have much to do with the presence of category theorists on editorial boards.
Mike Shulman (Aug 29 2023 at 17:41):Tim Hosgood said:
If people outside of CT say "oh CT people seem very X", then people inside of CT just cannot dispute this — not because X is right or wrong (this makes no difference!) but just because by very definition members of a community cannot say how they do appear to outsiders. They can say all they like about how they should appear, or how they intend to appear, but not how other people perceive them.
This is true. But I think it's also true that "people outside of CT" (or inside CT for that matter) are not monolithic, and any individual person expressing a view like "CT people seem very X" can only be basing that on their personal experience, and others might have different experiences.
Paolo Perrone (Aug 31 2023 at 06:51):John Baez said:
Here's a theorem I learned yesterday from Tom Leinster. Take the definition of monoid written out using commutative diagrams, where the associative law becomes a commutative square. What's a monoid where this commutative square is a pullback? It's a group.
In case anyone is interested, we have recently generalized this result here.
A possible intuition is that the pullback allows you to "complete parallelograms", as in an affine space over a vector space, or as in a torsor over a group.
Paolo Perrone (Aug 31 2023 at 15:54):(Maybe it can be considered an instance of the microcosm principle, since the pullback is a sort of "completed parallelogram" too.)
Ryan Wisnesky (Sep 02 2023 at 20:19):one way CT is different than most areas of math is that because of "commercial category theory", CT concepts participate in marketing campaigns. For example, we have marketing copy (some of which is on my web site) likening kan extensions to the silmarils from lord of the rings. but hopefully that's just harmless fun and doesn't interfere in using online tools to get math done. I suppose this is also seen a bit in type theory where you still see "lambda" used in marketing (e.g. "lambda conf" for programmers, etc)
fosco (Sep 25 2023 at 16:48):There's a number of things I don't understand about this conversation: for starters, why the opening message asks about "overrepresentation of category theory" and the subsequent replies focus instead on the usual pitch "category theory is difficult to learn", "category theorists are asshole gung-hos", "category theory is very abstract as opposed to numerical analysis"?
I interpret the question as follows: there are many resources to learn category theory online, there are many communities for category theorists to discuss, and there are many different communities using category theory in very different ways. But the offline community is scattered and (over the number of all mathematicians) small. Why?
fosco (Sep 25 2023 at 16:55):In the above form, I heard the same comment from an algebraic geometer, my office mate at 3rd and 4th year of PhD.
As far as I remember from the conversation I had, the reason for my colleague's surprise was that category theorists seem to form a quite compact community (little he knew about the quibbles that we can raise!), and one who is quite vocal about how category theory is cool, you should learn it, and join us.
Plus, a conference like the CT, where almost the entire community gathers together to do category theory, looked unthinkable to my colleague (different algebraic geometers have different sub-channels).
fosco (Sep 25 2023 at 17:03):I think my colleague saw a pattern that until that day I was unaware of. And I believe it's true that there is an inverse correlation with how few we are (and how little political influence we have in the grand scheme of things) compared with other larger communities. We feel the urge to stick together because our peers are quite far, geographically. Collaboration becomes more difficult, staying updated on what other people are working on is more difficult (and thus, learning new things).
An example of a lively, goal-oriented, well-trained group of category theorists is the Macquarie school. I can think of very few others -more precisely: I can't think of any equally well-established, similar reality in my surroundings (or in the surroundings of the place where I lived, or where I could move: let me also remark the obvious fact that if you live in a different continent than Oceania, it's rarely an easy step to relocate in Australia; some people can't, some people don't want to.
Ralph Sarkis (Sep 25 2023 at 17:04):I agree with your assessment of the conversation and your understanding of my question. Although there is one thing missing from your message: I still don't know if my feeling (our collective feeling it seems) is justified. Is there data to support that the field of CT is different in those ways than other fields? Do all fields of mathematics have peculiarities (if all do, then I guess they are not peculiar) in how they are practiced online and offline that we do not observe?
fosco (Sep 25 2023 at 17:04):So, staying in touch with the people that do the math we like, despite the distance, is a matter of survival that people working in PDE or numerical analysis do not know as much as we do.
(let me address @Ralph Sarkis question above -and at the same time, sorry for not getting all my replies in a single message: my train of thoughts works that way, I don't know if your feeling is justified: I would say yes, but my data points are anecdotal).
My experience is also anecdotal, but I can try to outline it shortly. When I decided that I wanted to study category theory, thirteen years ago, the situation was quite different than it is today. It was nearly impossible to find someone with which you could study category theory as an unknown student from another university. Sure, the basics can be learned no matter what -be it with logic, algebra or else- but from MSc on things got harder and harder, to the point that finding an advisor to do a thesis "in category theory" was impossible -and mine was a compromise between what I could do given where I was, and what I wanted to do.
MSc in Padova had a course in representation theory, and a course in rings and modules. I was mocked by the teacher because I phrased a question I had in the language of monoidal categories. "Mocked" is not an exaggeration: I was literally exposed to the class as the weird guy (so, pretty low in the iceberg meme...)
PhD then was pure hell, and I reserve to the majority of the people I endured during those years nothing but endless hatred and disdain. It is not an exaggeration to say that I earned a PhD degree despite them.
At each step of this headed-down stairway it became more and more important to rely on the online community of category theorists to (maintain my mental health and) keep in touch with people doing math I cared about. I didn't succeed in the first, so I became a sociopath with an extremely high tendency to rip the jugular vein of whoever puts other people in the same situation I was in, but in order to do the second, the nLab, the n-café and other resources outside of handbooks and lecture notes and papers were nothing less than essential to survive as a mathematician, despite my sheer isolation.
I consider my experience an exception, but I wasn't the only victim of isolation; I find miracolous that I remained in the system despite isolation, that I didn't break under that pressure. But this is another story, for another time.
I feel this aspect of the situation is changing, more and more as category theory permeates more and more the outside world, and gets "tainted" with other things. There is a price to pay for this, but this is another story for a different day.
Your perception of the situation is, then, true in my opinion; we're few people. We struggle with recognition (CT is young, and got many people stoned -or it was something they cut it with? in its infancy; then everyone was a bit nauseated). We find refuge where we can when we are ostracized. Internet is good because it shortens the distance between people.
Fabrizio Genovese (Sep 25 2023 at 17:15):Ralph Sarkis said:
I've heard and believed for some time that category theory is overrepresented online meaning we see it pop up more often than it should compared to the actual proportions of mathematicians studying category theory. This is obviously my experience too, but I am clearly biased. I looked at the submissions for this year's SoME, and there is only one video on CT, last year's had three. So I am not convinced category theory has a bigger place than it should.
Have you seen evidence of CT being overrepresented or underrepresented online?
Have you felt that CT is overrepresented or underrepresented online?
I'm really surprised by the fact that no one openly mentioned this could merely be a form of bias.
Fabrizio Genovese (Sep 25 2023 at 17:18):I mean I am interested in CT. Unsurprisingly I follow some category theorists, I read wiki pages about cats, the ncatlab, I search arxiv for papers about CT etc.
In the age of surveillance capitalism all this data is shared and marketed across platforms and guess what, twitter started suggesting me more category theorists to follow, more category theory video popped up on my youtube homepage and so on and so forth.
I never ever cared about number theory. Like, ever. Guess what? No number theorists in my timeline, no number theory videos suggested, etc.
Fabrizio Genovese (Sep 25 2023 at 17:20):This became starkly evident to me when I left academia. The more I got involved with industry the more my twitter timeline began to shift towards industry oriented things. As soon as one realizes that our online life is tailor made and customized predicting what we are interested in from a few data points, it becomes evident how some things end up being overrepresented in our online life.
fosco (Sep 25 2023 at 17:45):Fabrizio Genovese said:
Ralph Sarkis said:
I've heard and believed for some time that category theory is overrepresented online meaning we see it pop up more often than it should compared to the actual proportions of mathematicians studying category theory. This is obviously my experience too, but I am clearly biased. I looked at the submissions for this year's SoME, and there is only one video on CT, last year's had three. So I am not convinced category theory has a bigger place than it should.
Have you seen evidence of CT being overrepresented or underrepresented online?
Have you felt that CT is overrepresented or underrepresented online?I'm really surprised by the fact that no one openly mentioned this could merely be a form of bias.
I avoided mentioning twitter and other social networks precisely because of this: my timeline is essentially all category theory. Yours, isn't. There is no objective sense of "overrepresentation" as a consequence. I can only think of overrepresentation as "given how loud and scary category theorists are online, I would expect them to be way more present in the landscape of modern mathematics"
And here's where I am surprised Mike can be a bit shortsighted: "how many category theorists are in the hoity toity journals" is a metric I wouldn't want to adopt, but that I consider faithful to describe how category theory is perceived by non-category theorists. Same with "how many category theorists are invited speakers at the ICM?" and "how many category theorists receive prestigious mathematical prizes?", and "how many category theorists won a Fields medal?" In all these cases, the answer gets more and more depressing as years pass. And people talk shit about category theory because they measure what branch of math is cool, influential or deep using exactly these parameters.
fosco (Sep 25 2023 at 17:49):I could talk about this (which in my opinion is the heart of the matter) for hours, but I won't. In short: people who think that "cool math is ICM math / Fields medal stuff only" are idiots whose head is filled with tar. The responsibility to change the situation is not of a twitter star. It's ours, and it's the responsibility to write good books, teach well (and sneak category theory inot everything we teach), but most of all the responsibility to stay in the system, get tenured and hire our colleagues.
Mike Shulman (Sep 25 2023 at 17:51):In this message I explained that I was just talking about something different.
fosco (Sep 25 2023 at 17:55):It is difficult to follow the conversation...
That doesn't have much to do with the presence of category theorists on editorial boards.
I disagree, but maybe it's better if you rephrase your answer. afaicu Ivan is saying: " people do category theory, cat theorists are politically influential in editorial boards of top tier journals, and online conversations happen to be about category theory, how to learn it, and where, and what it's good for, and how to understand concept X, and ..."
This thread can be summarized as the following experimental fact: .
Discuss: why?
Ralph Sarkis (Sep 25 2023 at 17:58):I am also interested in comparing the ratios and with other fields.
For example, do people in combinatorics share the same feeling?
Steve Huntsman (Sep 25 2023 at 18:03):Fabrizio Genovese said:
I'm really surprised by the fact that no one openly mentioned this could merely be a form of bias.
Seems unlikely to me. I have heard more than a few folks who don't engage with CT remark on its overrepresentation.
Mike Shulman (Sep 25 2023 at 18:06):The original message that started the thread made no mention of , and so in my response to it I was only talking about and . Their relationship to is also interesting and important, but Ivan was the first one to bring into the conversation.
Steve Huntsman (Sep 25 2023 at 18:08):fosco said:
I could talk about this (which in my opinion is the heart of the matter) for hours, but I won't. In short: people who think that "cool math is ICM math / Fields medal stuff only" are idiots whose head is filled with tar. The responsibility to change the situation is not of a twitter star. It's ours, and it's the responsibility to write good books, teach well (and sneak category theory inot everything we teach), but most of all the responsibility to stay in the system, get tenured and hire our colleagues.
I humbly suggest that a higher responsibility than what sounds to me like forming a mafia is to solve problems that others--and especially others in different fields-- regard as meaningful. I am not sure of any instances of this besides perhaps the cobordism hypothesis but would welcome mentions of other instances for edification.
fosco (Sep 25 2023 at 19:06):Steve Huntsman said:
fosco said:
I could talk about this (which in my opinion is the heart of the matter) for hours, but I won't. In short: people who think that "cool math is ICM math / Fields medal stuff only" are idiots whose head is filled with tar. The responsibility to change the situation is not of a twitter star. It's ours, and it's the responsibility to write good books, teach well (and sneak category theory inot everything we teach), but most of all the responsibility to stay in the system, get tenured and hire our colleagues.
I humbly suggest that a higher responsibility than what sounds to me like forming a mafia is to solve problems that others--and especially others in different fields-- regard as meaningful. I am not sure of any instances of this besides perhaps the cobordism hypothesis but would welcome mentions of other instances for edification.
"You should paint a hyperrealist picture instead of a cubist one, because the people who look at it find the latter meaningful."
Do you see how ridiculous and absurd it sounds if you express the same concept like this?
The point you're making implicitly assumes that category theory has something to prove to other mathematicians before being accepted as a legitimate field of study. After it has been "solving problems that others -especially others in different fields- regard as meaningful" for 78 years. Surprise though: the same is not required from those who deal with the theory of finite groups, or with improving of epsilon/3 an inequality that, when you consider the heat equation in 6 dimensions, says that...
So no: I neither want to be "accepted" by non-category theorists, nor for category theory to be "recognized as important in itself as other areas of mathematics": instead,
I equally humbly submit to you the fact that you're asking me to adapt to work on stuff that other people find interesting; why can't it be the other way round, also considering that category theorists know way better than other mathematicians how to recognize a deep/interesting question? The answer is one and one only: the complement set of the CT community has more soldiers, more power, more money, more influence than us.
And this opens the real, huge can of snake.
fosco (Sep 25 2023 at 19:27):Now, lest we fight over some petty political / philosophical divergence, let me conclude peacefully with a meme I made in 2020
Steve Huntsman (Sep 25 2023 at 19:48):fosco said:
Steve Huntsman said:
I humbly suggest that a higher responsibility than what sounds to me like forming a mafia is to solve problems that others--and especially others in different fields-- regard as meaningful. I am not sure of any instances of this besides perhaps the cobordism hypothesis but would welcome mentions of other instances for edification.
"You should paint a hyperrealist picture instead of a cubist one, because the people who look at it find the latter meaningful."
Do you see how ridiculous and absurd it sounds if you express the same concept like this?
The point you're making implicitly assumes that category theory has something to prove to other mathematicians before being accepted as a legitimate field of study. After it has been "solving problems that others -especially others in different fields- regard as meaningful" for 78 years. Surprise though: the same is not required from those who deal with the theory of finite groups, or with improving of epsilon/3 an inequality that, when you consider the heat equation in 6 dimensions, says that...
Fields and people who solve problems that others--and especially others in different fields-- regard as meaningful get more respect than those that don't. This is true across the board. The respect may come from different places (pay, prestige, tenure, etc.) but if you want respect from others, do things they respect. Nothing is respected by onlookers more than solving problems (theory-building is far behind). If self-respect suffices for you, then great.
Steve Huntsman (Sep 25 2023 at 20:07):fosco said:
I equally humbly submit to you the fact that you're asking me to adapt to work on stuff that other people find interesting; why can't it be the other way round, also considering that category theorists know way better than other mathematicians how to recognize a deep/interesting question? The answer is one and one only: the complement set of the CT community has more soldiers, more power, more money, more influence than us.
I'm not asking anyone to adapt. I am saying that if folks want their work to be valued and taken seriously, then their work should address things that other people will value and take seriously. If you think your work will be valued and taken seriously then this problem is as solved as it can be.
Steve Huntsman (Sep 25 2023 at 20:10):If a category theorist recognizes an interesting question and nobody else appreciates it, is it really interesting? Maybe they are like Grassmann and the world will come around in decades. But most people are not like Grassmann.
Steve Huntsman (Sep 25 2023 at 20:12):Furthermore, most people who care what anyone else thinks would probably not want to be like Grassmann.
John Baez (Sep 25 2023 at 21:15):I think this current phase of the conversation is acting like there's a sharp divide between category theorists and other mathematicians, and some sort of extreme lack of appreciation of category theory on the part of outsiders. Maybe I'm weird but I just don't see that. There are plenty of big-shot mathematicians - Fields medalists, etc. - who use category theory. Voevodsky's difficulty with n-categories let him to develop homotopy type theory, and now that's been revealed to have deep connections to higher topos theory. Alain Connes is advocating the use of topos theory in his work on the Riemann Hypothesis. Peter Scholze's work on condensed mathematics is of great interest to category theorists, and he's even commented on the n-Category Cafe about this stuff. There are also other rock stars who aren't Fields Medalists, notably Jacob Lurie at MIT, who revolutionized work on higher categories and higher topoi. All this stuff is of interest to a lot of mathematicians.
John Baez (Sep 25 2023 at 21:20):Meanwhile on the more applied side of things there's the Topos Institute in Berkeley, which has a lot of people from many subjects going through, giving talks, and talking to people there. @Bob Coecke and @Ryan Wisnesky have companies that use category theory, and there are others too. I managed to get funding for a 6-week program at the International Centre for Mathematical Sciences to apply category theory to agent-based models in epidemiology. And so on.
So while category theorists may well have once been a somewhat embattled, misunderstood, embittered community - and some probably still are - I don't find it wise to keep focusing on that.
Steve Huntsman (Sep 25 2023 at 21:23):Echoing this: if I didn't appreciate category theory then I wouldn't use it.
Bob Coecke (Sep 25 2023 at 21:32):John Baez said:
Meanwhile on the more applied side of things there's the Topos Institute in Berkeley, which has a lot of people from many subjects going through, giving talks, and talking to people there. Bob Coecke and Ryan Wisnesky have companies that use category theory, and there are others too. I managed to get funding for a 6-week program at the International Centre for Mathematical Sciences to apply category theory to agent-based models in epidemiology. And so on.
So while category theorists may well have once been a somewhat embattled, misunderstood, embittered community - and some probably still are - I don't find it wise to keep focusing on that.
I am increasingly noting some kind of renewed excitement about category theory, mostly from outside of mathematics, and also in particular in AI. Academic culture is notoriously conservative so it will take a while before mainstream mathematics gets back on board, but the world is moving on, either with or without their approval. Don't want to do a spoiler, but got some really big news soon to be announced. Say no more for now...
Fabrizio Genovese (Sep 25 2023 at 22:19):fosco said:
Steve Huntsman said:
fosco said:
I could talk about this (which in my opinion is the heart of the matter) for hours, but I won't. In short: people who think that "cool math is ICM math / Fields medal stuff only" are idiots whose head is filled with tar. The responsibility to change the situation is not of a twitter star. It's ours, and it's the responsibility to write good books, teach well (and sneak category theory inot everything we teach), but most of all the responsibility to stay in the system, get tenured and hire our colleagues.
I humbly suggest that a higher responsibility than what sounds to me like forming a mafia is to solve problems that others--and especially others in different fields-- regard as meaningful. I am not sure of any instances of this besides perhaps the cobordism hypothesis but would welcome mentions of other instances for edification.
"You should paint a hyperrealist picture instead of a cubist one, because the people who look at it find the latter meaningful."
Do you see how ridiculous and absurd it sounds if you express the same concept like this?
The point you're making implicitly assumes that category theory has something to prove to other mathematicians before being accepted as a legitimate field of study. After it has been "solving problems that others -especially others in different fields- regard as meaningful" for 78 years. Surprise though: the same is not required from those who deal with the theory of finite groups, or with improving of epsilon/3 an inequality that, when you consider the heat equation in 6 dimensions, says that...
So no: I neither want to be "accepted" by non-category theorists, nor for category theory to be "recognized as important in itself as other areas of mathematics": instead,
- I want to be left alone while I do the mathematics that I consider important, important because I say so, and this judgment must be considered unquestionable as it is for many other folks;
- I want those who think like me to be able to do the same, without wasting a considerable part of our life, and pollute our talks with the usual pitch on you wouldn't believe how useful this adjunction is to cure cancer;
- I want to be part of a critical mass of individuals (like in other parts of mathematics, older or more popular) making 1 and 2 possible.
I equally humbly submit to you the fact that you're asking me to adapt to work on stuff that other people find interesting; why can't it be the other way round, also considering that category theorists know way better than other mathematicians how to recognize a deep/interesting question? The answer is one and one only: the complement set of the CT community has more soldiers, more power, more money, more influence than us.
And this opens the real, huge can of snake.
I can understand where you are coming from, but I couldn't disagree more. Simply put, as a taxpayer, I would get _very_ angry in knowing that part of my money is being spent to let someone 'being left alone while I do the mathematics that I consider important, important because I say so'.
You do not exist in a vacuum. Especially you do not exist in a vacuum when your bills are paid out of tax coming from other people's work.
Cole Comfort (Sep 25 2023 at 22:23):Fabrizio Genovese said:
fosco said:
Steve Huntsman said:
fosco said:
I could talk about this (which in my opinion is the heart of the matter) for hours, but I won't. In short: people who think that "cool math is ICM math / Fields medal stuff only" are idiots whose head is filled with tar. The responsibility to change the situation is not of a twitter star. It's ours, and it's the responsibility to write good books, teach well (and sneak category theory inot everything we teach), but most of all the responsibility to stay in the system, get tenured and hire our colleagues.
I humbly suggest that a higher responsibility than what sounds to me like forming a mafia is to solve problems that others--and especially others in different fields-- regard as meaningful. I am not sure of any instances of this besides perhaps the cobordism hypothesis but would welcome mentions of other instances for edification.
"You should paint a hyperrealist picture instead of a cubist one, because the people who look at it find the latter meaningful."
Do you see how ridiculous and absurd it sounds if you express the same concept like this?
The point you're making implicitly assumes that category theory has something to prove to other mathematicians before being accepted as a legitimate field of study. After it has been "solving problems that others -especially others in different fields- regard as meaningful" for 78 years. Surprise though: the same is not required from those who deal with the theory of finite groups, or with improving of epsilon/3 an inequality that, when you consider the heat equation in 6 dimensions, says that...
So no: I neither want to be "accepted" by non-category theorists, nor for category theory to be "recognized as important in itself as other areas of mathematics": instead,
- I want to be left alone while I do the mathematics that I consider important, important because I say so, and this judgment must be considered unquestionable as it is for many other folks;
- I want those who think like me to be able to do the same, without wasting a considerable part of our life, and pollute our talks with the usual pitch on you wouldn't believe how useful this adjunction is to cure cancer;
- I want to be part of a critical mass of individuals (like in other parts of mathematics, older or more popular) making 1 and 2 possible.
I equally humbly submit to you the fact that you're asking me to adapt to work on stuff that other people find interesting; why can't it be the other way round, also considering that category theorists know way better than other mathematicians how to recognize a deep/interesting question? The answer is one and one only: the complement set of the CT community has more soldiers, more power, more money, more influence than us.
And this opens the real, huge can of snake.
I can understand where you are coming from, but I couldn't disagree more. Simply put, as a taxpayer, I would get _very_ angry in knowing that part of my money is being spent to let someone 'being left alone while I do the mathematics that I consider important, important because I say so'.
You do not exist in a vacuum. Especially you do not exist in a vacuum when your bills are paid out of tax coming from other people's work.
luckily government expenditures are not made by referrendum, otherwise academia would not be funded (among many other things)
Fabrizio Genovese (Sep 25 2023 at 22:24):For me, this is a resource allocation problem. A state (or university, or group, or non profit, or company) has some resources to allocate to various things. Ideally, you would like people providing the resources to express some kind of preference, and then you would like to allocate said resources to maximize collective welfare.
We can make an argument about investing resources into a 'riskier' enterprise - that is, something that may deliver very big but only on a very long timescale. This is what investing money on the majority of mathematical research is about. And still mathematics, like everything else, does not exist in a vacuum.
So going from 'justifying why something is needed' to 'pretending funding and recognition because I say so' is really something I cannot justify.
Fabrizio Genovese (Sep 25 2023 at 22:27):Cole Comfort said:
Fabrizio Genovese said:
fosco said:
Steve Huntsman said:
fosco said:
I could talk about this (which in my opinion is the heart of the matter) for hours, but I won't. In short: people who think that "cool math is ICM math / Fields medal stuff only" are idiots whose head is filled with tar. The responsibility to change the situation is not of a twitter star. It's ours, and it's the responsibility to write good books, teach well (and sneak category theory inot everything we teach), but most of all the responsibility to stay in the system, get tenured and hire our colleagues.
I humbly suggest that a higher responsibility than what sounds to me like forming a mafia is to solve problems that others--and especially others in different fields-- regard as meaningful. I am not sure of any instances of this besides perhaps the cobordism hypothesis but would welcome mentions of other instances for edification.
"You should paint a hyperrealist picture instead of a cubist one, because the people who look at it find the latter meaningful."
Do you see how ridiculous and absurd it sounds if you express the same concept like this?
The point you're making implicitly assumes that category theory has something to prove to other mathematicians before being accepted as a legitimate field of study. After it has been "solving problems that others -especially others in different fields- regard as meaningful" for 78 years. Surprise though: the same is not required from those who deal with the theory of finite groups, or with improving of epsilon/3 an inequality that, when you consider the heat equation in 6 dimensions, says that...
So no: I neither want to be "accepted" by non-category theorists, nor for category theory to be "recognized as important in itself as other areas of mathematics": instead,
- I want to be left alone while I do the mathematics that I consider important, important because I say so, and this judgment must be considered unquestionable as it is for many other folks;
- I want those who think like me to be able to do the same, without wasting a considerable part of our life, and pollute our talks with the usual pitch on you wouldn't believe how useful this adjunction is to cure cancer;
- I want to be part of a critical mass of individuals (like in other parts of mathematics, older or more popular) making 1 and 2 possible.
I equally humbly submit to you the fact that you're asking me to adapt to work on stuff that other people find interesting; why can't it be the other way round, also considering that category theorists know way better than other mathematicians how to recognize a deep/interesting question? The answer is one and one only: the complement set of the CT community has more soldiers, more power, more money, more influence than us.
And this opens the real, huge can of snake.
I can understand where you are coming from, but I couldn't disagree more. Simply put, as a taxpayer, I would get _very_ angry in knowing that part of my money is being spent to let someone 'being left alone while I do the mathematics that I consider important, important because I say so'.
You do not exist in a vacuum. Especially you do not exist in a vacuum when your bills are paid out of tax coming from other people's work.
luckily government expenditures are not made by referrendum, otherwise academia would not be funded (among many other things)
I agree with you that not everything should be decided by direct vote, indeed I do not believe direct democracy is inherently a good idea. You're preaching to the choir here. But 'justifying my existence to whoever has to decide resource allocation (it can be the university, a DAO, a bureaucrat behind a desk, does not matter)' is still needed. If you want to do mathematics (or whatever, it doesn't matter) without being bothered by justifying anything to society, then create an autarchical commune somewhere and do not rely on society at all.
Cole Comfort (Sep 25 2023 at 22:30):Fabrizio Genovese said:
Cole Comfort said:
Fabrizio Genovese said:
fosco said:
Steve Huntsman said:
fosco said:
I could talk about this (which in my opinion is the heart of the matter) for hours, but I won't. In short: people who think that "cool math is ICM math / Fields medal stuff only" are idiots whose head is filled with tar. The responsibility to change the situation is not of a twitter star. It's ours, and it's the responsibility to write good books, teach well (and sneak category theory inot everything we teach), but most of all the responsibility to stay in the system, get tenured and hire our colleagues.
I humbly suggest that a higher responsibility than what sounds to me like forming a mafia is to solve problems that others--and especially others in different fields-- regard as meaningful. I am not sure of any instances of this besides perhaps the cobordism hypothesis but would welcome mentions of other instances for edification.
"You should paint a hyperrealist picture instead of a cubist one, because the people who look at it find the latter meaningful."
Do you see how ridiculous and absurd it sounds if you express the same concept like this?
The point you're making implicitly assumes that category theory has something to prove to other mathematicians before being accepted as a legitimate field of study. After it has been "solving problems that others -especially others in different fields- regard as meaningful" for 78 years. Surprise though: the same is not required from those who deal with the theory of finite groups, or with improving of epsilon/3 an inequality that, when you consider the heat equation in 6 dimensions, says that...
So no: I neither want to be "accepted" by non-category theorists, nor for category theory to be "recognized as important in itself as other areas of mathematics": instead,
- I want to be left alone while I do the mathematics that I consider important, important because I say so, and this judgment must be considered unquestionable as it is for many other folks;
- I want those who think like me to be able to do the same, without wasting a considerable part of our life, and pollute our talks with the usual pitch on you wouldn't believe how useful this adjunction is to cure cancer;
- I want to be part of a critical mass of individuals (like in other parts of mathematics, older or more popular) making 1 and 2 possible.
I equally humbly submit to you the fact that you're asking me to adapt to work on stuff that other people find interesting; why can't it be the other way round, also considering that category theorists know way better than other mathematicians how to recognize a deep/interesting question? The answer is one and one only: the complement set of the CT community has more soldiers, more power, more money, more influence than us.
And this opens the real, huge can of snake.
I can understand where you are coming from, but I couldn't disagree more. Simply put, as a taxpayer, I would get _very_ angry in knowing that part of my money is being spent to let someone 'being left alone while I do the mathematics that I consider important, important because I say so'.
You do not exist in a vacuum. Especially you do not exist in a vacuum when your bills are paid out of tax coming from other people's work.
luckily government expenditures are not made by referrendum, otherwise academia would not be funded (among many other things)
I agree with you that not everything should be decided by direct vote, indeed I do not believe direct democracy is inherently a good idea. You're preaching to the choir here. But 'justifying my existence to whoever has to decide resource allocation (it can be the university, a DAO, a bureaucrat behind a desk, does not matter)' is still needed. If you want to do mathematics (or whatever, it doesn't matter) without being bothered by justifying anything to society, then create an autarchical commune somewhere and do not rely on society at all.
Yeah, I think this whole thread is just ragebait. Of course people have to justify their work to be funded. Whether or not people are funded too much or too little is stupid to argue about. No one is going to give up their funding to give it to other disciplines, and everyone will continue advocating that they get more funding.
Fabrizio Genovese (Sep 25 2023 at 22:31):This is very likely. But what @Steve Huntsman was saying, on which I agree, is that this is a game won by who's smarter and proves to be more useful, not by whoever rages more (unless you rage enough to cause a revolution, but that's unlikely).
David Michael Roberts (Sep 26 2023 at 01:57):I always felt that one justification for being paid to do research in pure mathematics was the fact I fronted up and taught large classes of engineers first-year calculus, and they can go on and do useful things.
David Michael Roberts (Sep 26 2023 at 01:59):And it's worth the thought experiment of viewing funding for pure mathematics in much the same way that governments subsidize the arts. How many artists/musicians/etc are supported by government funding schemes (in countries that have them), and to the tune of what amount of money, and how many pure mathematicians/how much? Not to knock artists etc, but mathematicians at the least can teach people things that they generally require to go and work for a company and contribute to the economy.
But this is getting a long way from the original topic!
John Baez (Sep 26 2023 at 09:45):Getting even further from the original topic:
For some reason governments have decided that people need to know some basic math but don't need to know anything about making art or music. So funding for art and music education has been slashed. This idea that the arts and music don't deserve to be part of education is a fairly recent cultural decision; I suspect over the course of history few cultures have felt this way. Maybe art and music are so necessary in our lives that we don't need help to learn about them, but it's still strange how sometime around the 1980s these were decreed unimportant in the US.
John Baez (Sep 26 2023 at 09:46):The US has also decided that "foreign languages" are unimportant.
John Baez (Sep 26 2023 at 09:48):Since my wife teaches classical Chinese and Greek philosophy, I never whine about lack of state support for pure math. There's a huge amount of state support for pure math, compared to most arts and humanities.
Fabrizio Genovese (Sep 26 2023 at 11:03):David Michael Roberts said:
And it's worth the thought experiment of viewing funding for pure mathematics in much the same way that governments subsidize the arts. How many artists/musicians/etc are supported by government funding schemes (in countries that have them), and to the tune of what amount of money, and how many pure mathematicians/how much? Not to knock artists etc, but mathematicians at the least can teach people things that they generally require to go and work for a company and contribute to the economy.
But this is getting a long way from the original topic!
Indeed don't let me started on funding contemporary art projects. Lol
Fabrizio Genovese (Sep 26 2023 at 11:05):The discussion about funding for arts is very complicated, because in the last 200 years it has become increasingly difficult even to answer basic questions such as 'what is art?' and 'who is art for?'
John Baez (Sep 26 2023 at 11:08):I feel it really helps kids' cognitive development to do things like sing, play instruments, maybe learn to read music a bit, draw, paint, do art and music on computers, etc. But I'm no expert so this is just my feeling.
John Baez (Sep 26 2023 at 11:09):Maybe they should spend all their time learning how to be useful cogs in the economic machine. :upside_down:
Fabrizio Genovese (Sep 26 2023 at 11:25):In any case, I'd just like to point out how back in the days of mecenatism market forces were even more important than today. It is my view that artists (and mathematicians) have pretty much never been free to do whatever they wanted. If anything, the situation is now better than it was before, when these people were at the whims of whoever protected and paid for them.
Indeed, by looking at the lives of great artists like Michelangelo, Leonardo etc, one realizes quickly how they were very aware of the context in which they lived (what to do and not to do, who to not piss off, which rules would be cool to break and which ones weren't). Here's an excerpt from Michelangelo's biography by Condivi (https://archiv.ub.uni-heidelberg.de/artdok/714/1/Davis_Fontes34.pdf pg 16 ):
[...] avvenne che un giorno fu dal Granacci menato al giardin de Medici à san Marco, il qual [...] havea di varie statue antiche et de figure ornato. [...] Tra le altre considerando un giorno la Testa d’un Fauno in vista già vecchio, [...] et piacendogli oltre à modo, si propose di ritrarla in marmo. [...] con tanta attentione et studio si pose a ritrarre il Fauno, che in pochi giorni lo condusse a’ perfettione, di sua fantasia suplendo tutto quello, che nel antico mancava, cioè la bocca aperta [...] con tutti i denti. In questo mezzo venendo il Magnifico per vedere à che termine fusse l’opera sua [...] considerata primieramente l’eccellenza del opera, et havuto riguardo al età di lui, molto si maravigliò: et avenga che lodasse lopera, non dimeno mottegiando con lui, [...] oh tu hai fatto questo Fauno vecchio, et lasciatigli tutti identi. Non sai tu che à vecchi di tal età sempre ne mancha qualchuno? [...] Michelagnolo [...] restato solo, cavò un dente al suo vecchio di quei di sopra, trapanando la gingiva, come se ne fusse uscito colla radice, aspettando l’altro giorno il Magnifico, con gran desiderio. Il quale venuto, et vista la bontà et simplicità del fanciullo, molto se ne rise, ma poi stimata seco la prefettione* della cosa, et l’età di lui, come padre di tutte le virtù, si deliberò d’aiutare et favorire tanto ingegno [...]
Roughly summarized: Young Michelangelo is making a marble copy of an old faun. He sculpted the mouth with all the teeth in it. Lorenzo de Medici is impressed but remarks that old people usually lack some teeth. As soon as he walks away Michelangelo takes some teeth out of the statue, further impressing Lorenzo, which decides to give him patronage.
Fabrizio Genovese (Sep 26 2023 at 11:28):John Baez said:
Maybe they should spend all their time learning how to be useful cogs in the economic machine. :upside_down:
Call me a reactionary if you want to, but yes, people should spend all their time learning how to be useful and how to contribute to the social environment they're in. This is how art, literature and science have been carried out since antiquity, and the idea of people being free to do whatever they like without having to answer in any way is a rather recent (and in my opinion worse) development, which probably started with romanticism and reached its peak with the hippy movement and rock and roll idols.
John Baez (Sep 26 2023 at 11:36):There are lots of things to talk about here. When I was bemoaning the decline in arts and music education, I was not thinking about the 1500s, nor was I suggesting that most people will or should become full-time artists and musicians, and somehow expect to survive. I don't think someone should expect to survive doing whatever they want - simply because it's not at all true.
I was comparing pre-1980s education (in the US, and maybe elsewhere) to education today. Starting at some point - I don't know when, but certainly by the 1940s - and before the 1980s, it was widely expected that students in elementary school and high school would take some classes in art and music. For example my mother was a high school art teacher in Michigan in the early 1950s, and this was not considered unusual. But at some point the powers that be decided that teaching kids art and music was a waste of time. And I think this coincides with a general trend toward the concentration of wealth.
John Baez (Sep 26 2023 at 11:38):I think that if the government were really smart, it would realize that even to be truly useful cogs in the machine, people's brains should be exposed to a lot of music and art when young. But the government is definitely not very smart: even though it claims to value science, technology, engineering and mathematics, it is not very good at getting kids to be good in these subjects.
Fabrizio Genovese (Sep 26 2023 at 11:39):John Baez said:
There are lots of things to talk about here. When I was bemoaning the decline in arts and music education, I was not thinking about the 1500s, nor was I suggesting that most people will or should become full-time artists and musicians, and somehow expect to survive. I don't think someone should expect to survive doing whatever they want - simply because it's not at all true.
I was comparing pre-1980s education (in the US, and maybe elsewhere) to education today. Starting at some point - I don't know when, but certainly by the 1940s - and before the 1980s, it was widely expected that students in elementary school and high school would take some classes in art and music. For example my mother was a high school art teacher in Michigan in the early 1950s, and this was not considered unusual. But at some point the powers that be decided that teaching kids art and music was a waste of time. And I think this coincides with a general trend toward the concentration of wealth.
Yes, I suspect that this is a result of the libertarian turn that came with Reagan, which is terrible. Still, there is difference between 'receiving an education in something' and 'working in that field'.
John Baez (Sep 26 2023 at 11:40):Yes, it's a huge difference! I'm glad we agree about some things.
Fabrizio Genovese (Sep 26 2023 at 11:48):I'm a strong believer that instruction, healthcare and other things are fundamental rights and should be guaranteed no matter what. What irks me is that since 1968 the overall political discourse seems to be centered around the concept of 'right' (fighting for rights, restricting rights), whereas the concept of 'duty', which is its dual, has been neglected completely. Even the very word 'duty' today has some militaristic vibe and is considered to be evocative of urfascistic thought. Which to be honest is bewildering to me.
John Baez (Sep 26 2023 at 12:00):The concepts of right and duty are both quite slippery. People make up 'rights' when they think we should have them, but then they try to enforce these rights by sort of pretending they're like laws of nature... but laws which, unlike actual laws of nature, need to be enforced. I haven't thought as much about how people make up 'duties', in part because - yes - the whole concept of 'duty' has been deprecated by many people. (I guess we at least have a duty to let people have their rights, though. :upside_down: )
Steve Huntsman (Sep 26 2023 at 12:13):If too many folks goof off too much and don't support utility then this is the Pandora's box that will open:
I don't think anyone here would want that, and I think that we all value beauty in mathematics as worthy in its own right. But the privilege to work on beautiful mathematics without obvious utility is a treasure that is in scarce supply and can be easily squandered.
Steve Huntsman (Sep 26 2023 at 12:22):If CT folks vociferously advocate for Hardyism (and I claim that on balance, they very much do) then they should not be surprised when folks in PDE or probability who can at least pay lip service to utility use that advocacy as a cudgel to grab a bigger piece of the pie.
Fabrizio Genovese (Sep 26 2023 at 12:24):The focus here is on 'utility'. I think 'utility' as it is defined in the excerpt you quoted is a very myopic concept. My definition of 'utility' is, instead, 'something that may not be of immediate utility for the masses, but that is considered useful and worthy of pursuit by a big enough, non-insular community of specialists'.
Fabrizio Genovese (Sep 26 2023 at 12:24):I think everyone here can agree that studying PDEs or Number Theory is a useful pursuit. This is totally independent from how boring each of us may find research in those fields.
Steve Huntsman (Sep 26 2023 at 12:38):We have the same definition of utility. But it is not the same definition held by most of any present society that I know of. Myopic or not, a society's definition is the one that is invoked to pay the bills.
Fabrizio Genovese (Sep 26 2023 at 12:53):Nah, it's a matter of risk management. Invest less in high-risk activities, more in low-risk activities. If the budget to research is 100, then the less applicable something is, the smaller the slice of the budget allocated to that thing. I'm pretty sure this is how many societies do their academic budget.
Fabrizio Genovese (Sep 26 2023 at 12:54):In the end to figure out how much you have to allocate to mathematicians, you should ask engineers, mathematicians , physicists, and other people proximal to mathematics 'how important funding maths is'. Say the answer is 5% of the total budget. How do you allocate that 5% within the mathematical research fields? you ask mathematicians how important each field is...
Todd Trimble (Sep 26 2023 at 12:56):I think Lawvere would say one of the main reasons CT is useful because it simplifies and clarifies so much, making it easier to pass on our knowledge and insights to generations to come. I agree with him. The concept of universal property which permeates traditional category theory, simple though it is, can be applied to cover huge swaths of mathematics, and I don't think we're close to the end of that exploration.
Toward the end of education, it helps that CT is beautiful. I think more and more young people are discovering that.
Steve Huntsman (Sep 26 2023 at 13:09):Fabrizio Genovese said:
How do you allocate that 5% within the mathematical research fields? you ask mathematicians how important each field is...
Actually I (and the NSF/DoD/etc) would ask the engineers, physicists, etc. In my experience theorists from other fields know a lot of mathematics better than most mathematicians in nearby subfields, e.g. information theory. And this is also is why statisticians and ML theorists have better prospects and pay than (other) mathematicians.
Fabrizio Genovese (Sep 26 2023 at 13:13):These are details. The point is that you ask a crowd of specialists from liminal fields (justifying even more the idea that no one lives in a vacuum)!
Steve Huntsman (Sep 26 2023 at 13:25):Fabrizio Genovese said:
These are details. The point is that you ask a crowd of specialists from liminal fields (justifying even more the idea that no one lives in a vacuum)!
It's like Yoneda: a discipline (or anything) is best understood by considering its interactions with external stuff.
Fabrizio Genovese (Sep 26 2023 at 13:56):I was thinking this precise thing yesterday. Yoneda is the ultimate proof that nothing exists in a vacuum.
fosco (Sep 26 2023 at 14:56):Steve Huntsman said:
If CT folks vociferously advocate for Hardyism (and I claim that on balance, they very much do) then they should not be surprised when folks in PDE or probability who can at least pay lip service to utility use that advocacy as a cudgel to grab a bigger piece of the pie.
I never said I am surprised.
Steve Huntsman (Sep 26 2023 at 14:59):Fair enough
Ralph Sarkis (Feb 22 2024 at 19:16):I just discovered that you can ignore tags on mathoverflow. I conjecture that, because CT has a bad reputation (and because people sometimes act out of spite), the tag ct.category-theory will be the most ignored (out of all "big" tags, I don't know if there is a formal hierarchy of tags on MO). Is it possible to check this?
David Michael Roberts (Feb 22 2024 at 21:10):Hmm, I'll see/ask if there's a way...