I don't believe the exposition is written well (judged as either an exposition of linear algebra or just the graphical calculus the article develops). But this is perhaps a subjective point, and others here have already commented on this in detail.
More importantly, I would like to remark that string diagrams are not standard. The vast majority of mathematicians have never read the definition of a string diagram (or even a monoidal category), precisely because it is not standard and not needed for most (any?) useful mathematical work. They also have obvious disadvantages when compared to the usual notation, as I pointed out earlier.
When I read articles by category theory boosters, I get the sense (rightly or wrongly) that they think the world revolves around them and they have stumbled onto some deep and fundamental truths. This is not the case. It's a niche of a niche.
It's interesting to suddenly get all these hits.. I haven't touched the blog in a long time.
I'm sorry that you didn't find it well-written -- it wasn't written with you in mind. Originally I wanted to write about my research in a way that was understandable to a lay person, but I quickly abandoned that and went for the mythical "second year undergrad" level.
You have pretty strong thoughts about what is "useful mathematical work". Are you the high priest and decider of the usefulness of mathematics? To be honest, it almost sounds like some category theorist was super mean to you...
One more thing. Arguments like "niche of a niche" are sociological - just because something is niche doesn't mean it's not important (or, god forbid, fun!). Maths is one of the most conservative fields in this sense; what is and is not considered "standard" is extremely political.
"Are you the high priest and decider of the usefulness of mathematics? To be honest, it almost sounds like some category theorist was super mean to you..."
Interesting and totally not ad hominem response... I believe Kevin Buzzard (an actual mathematician) had a few words about this last year: https://youtu.be/Dp-mQ3HxgDE?t=1039
Note that Kevin Buzzard is talking in the context of convincing "mainstream" mathematicians to use a proof assistant. He specifically clarifies somewhere (not sure where in this video, but I've seen other videos where he clarifies it) that when he talks about "proper mathematics" he is doing so in kind of a tongue-in-cheek, British way and does not mean to suggest that other kinds of mathematics are not proper (don't remember the exact wording).
In the context of that video, his comments about category theorists and type theorists make a lot of sense: type theory people and (perhaps to a more limited extent) category theory people tend to be more easily sold on using a proof assistant since the kind of mathematics they do translate more easily into current proof assistants than, say, analysis or topology.
I definitely do not think that Kevin Buzzard is suggesting that type theorists and category theorists are not doing interesting and/or useful work (after all, the proof assistant he is advocating for is based on type theory). At best, he is making a sociological observation that there is a gap between the type theory/category theory community and "mainstream" mathematicians making the widespread adoption of proof assistants in mathematics more difficult.
I find it interesting that actual mathematicians working in Category Theory, such as Tom Leinster (who wrote a lovely little introductory text on Category Theory [not covering monads though] and made it freely available on arXiv: https://arxiv.org/abs/1612.09375) are able to engage in polite discussion about the contentious viewpoints on the use of CT without resorting to personal attacks (see 2-3 minutes from his talk from a few years back: https://youtu.be/UoutGluNVlI?t=410) ... which stands in sharp contrast to some of the evangelists, whose attitude in response to criticism often reeks of arrogance and puts people off taking CT seriously, which I think is a great shame.
> More importantly, I would like to point out that string diagrams are not standard.
They're standard in category theory, which is what the series is about.
> precisely because it is not standard and not needed for most (any?) useful mathematical work.
Uh oh, you better let everyone using it know, especially all those pesky type theorists.
> When I read articles by category theory boosters, I get the sense (rightly or wrongly) that they think the world revolves around them and they have stumbled onto some deep and fundamental truths. This is not the case. It's a niche of a niche.
I think this says more about you than the series. You seem upset that a category theorist is presenting their work in the language of category theory and that people are interested in it. Nobody mentioned anything about utility, depth, fundamentality or generality. In fact, he refers to some of his work as "hard and useless." [1]
> There has been intense political pressure on the academy to stop working on hard and useless things over the last 30 years or so. The result is a new generation of academics who do easy and useful things, things that impact the economy in the 5-10 year scale, and which bring in research funding. This is by design, because research funding is closely tied to academic career progression. Unfortunately, it’s not very hard to disguise easy and useless things as easy and useful. This has resulted in a totally out-of-control epidemic of easy and useless research. A symptom of this disease is the ever expanding use of increasingly ridiculous buzzwords. Easy and useless never leads to hard and useful. I’d much rather the government invest my tax money in the hard and useless.
> When I read articles by category theory boosters, I get the sense (rightly or wrongly) that they think the world revolves around them and they have stumbled onto some deep and fundamental truths.
(Professional mathematician here)
My sense, from talking to category theory boosters, is not typically that they regard category theory as "deep and fundamental truths". Rather, they often find it a useful way to declutter and describe their work, allowing them to more easily focus on deep truths without getting bogged down in details.
Indeed. A high-level programming language (e.g. Python) does not allow you to do _more_ than assembly but _less_. But it helps you declutter the presentation to such a great extent that realistically you could not write Python program in assembly.
More importantly, I would like to remark that string diagrams are not standard. The vast majority of mathematicians have never read the definition of a string diagram (or even a monoidal category), precisely because it is not standard and not needed for most (any?) useful mathematical work. They also have obvious disadvantages when compared to the usual notation, as I pointed out earlier.
When I read articles by category theory boosters, I get the sense (rightly or wrongly) that they think the world revolves around them and they have stumbled onto some deep and fundamental truths. This is not the case. It's a niche of a niche.