A blog series that teaches linear algebra through string diagrams and category theory, building up from matrices to monoids.
Commenters call it a fantastic introduction and recommend starting at the beginning, as a bridge between algebra-style and analysis-style math.
The language they introduce, Π, is essentially a simple concatenative programming language where all of the primitive combinators happen to be reversible; the graphical depiction is essentially the same as graphical linear algebra¹, which is no coincidence. Concatenative combinator calculus is related to combinatory logic—specifically, it’s a variation that replaces application of combinators with composition, leading to a bunch of nice mathematical properties. Both of these logics have the nice feature that you can syntactically restrict the substructural rules of contraction (dropping), exchange (swapping), and weakening (copying): for example, a well-formed program that doesn’t use the K combinator to drop a value is guaranteed to be linear, i.e., not destroy any information. This isn’t true in ordinary lambda calculus, where you can use a variable zero or many times, so you need additional checks on the uses of a variable to determine whether a function is linear, affine, &c.
I like the theory of concatenative programming languages because they provide an elegant compositional/dataflow-oriented style of programming, have deep connections to linear logic, Hughes’ “arrows”, effects & coeffects, and quantum/reversible computation, while also admitting very efficient implementation on classical hardware.
I’m (very slowly) working on a statically typed low-level concatenative language that’s meant to provide all the nice high-level typed functional language features like…
> 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…
I think it's worth linking to the 'Notation as a tool of thought' thread from the other day, as it gave me quite a lot to read, watch and think about [0].
It also reminds me of Graphical Linear Algebra [1] which I occasionally see mentioned here. And, as included in my comment in that thread, the notion of using Tau as the circle constant in equations [2].
Notation is a weird topic to tackle. Like with new technologies or languages on HN, there seem to be those who get [a new notation when it is proposed] and evangelise it, and those who see it as pointless and vocally dismiss it. Posts like the article where you're weighing up and exploring benefits and limitations of notation seem rare - and even those that do exist seem to be pitching for their new notation to be a global replacement rather than as a pedagogical or epistemological tool.
[0] https://news.ycombinator.com/item?id=25249563
[1] https://graphicallinearalgebra.net/about/
[2] https://tauday.com
If you feel comfortable with Math A but not Math B you might enjoy _Graphical Linear Algebra_[1], which is specifically that bridge!
As someone who is decent at Math B but mostly incompetent at Math A I suspect it comes down to the old analysis vs algebra opposition — being better at thinking about things visually/spatially vs linguistically. Both are trainable though.
I think the approach outlined here works well enough. When teaching us about adjunctions, our category theory lecturer used to have us recite the definition of a left adjoint at the start of every lesson, and he'd draw the diagram as we spoke. I can't say I can still recite it by heart but I do feel like I have a decent intuition for adjoint functors.
[1]: https://graphicallinearalgebra.net/
I kind of agree with you (especially on "math-lite category theory" being upvoted by people who read it and feel smart for understanding it), but I think the point has slightly been missed. The blog is a sort of side project, a popular treatment of the linear algebra stuff coming out of the "real research", which is applying the string diagrams to find graphical, compositional axiomatisations of concurrent systems. The programme has had some success with signal flow graphs (even making it onto the blog: https://graphicallinearalgebra.net/2016/09/07/31-fibonacci-and-sustainable-rabbit-farming/ ) and they were starting to look into Petri nets as my study was wrapping up.
(source: I was a postgrad supervised by Pawel)
If you're the type of person who read comments first, you might want to access the site via this link:
http://graphicallinearalgebra.net/2015/04/23/makelele-and-linear-algebra/
It's a text book written as a blog and currently features 24 chapters / posts and for some reason the website really wants you to start at the end, instead of the beginning.
The above link is to the introductory post, where they introduce the course as a new way of looking at an under appreciated discipline, that is usually taught in an uninspired and counterintuitive manner, linear algebra.
More interesting is that the website is a call for phd students, and they have research position available, both paid and scholarships.