Graphical Linear Algebra • Hacker News Board

Re: Graphical Linear Algebra

#11

Post by siraben » Tue, Aug 18, 2020, 9:17 AM UTC

spekcular wrote: post #5

The short version appears to be that the author has managed to replace standard linear algebra notation with an alternative (but mathematically equivalent) graphical notation, which is harder to do computations and write programs in. This submission is the latest in a trend of what I would classify as "math-lite" category theory and Haskell articles that reach the HN front page, which purport to explain something int…

This graphical notation for linear algebra isn't like the usual diagram notation category theory at all.

> Is category theory a useful organizational tool in certain abstract branches of mathematics?

Yes. Abstracting math in a very general way lets you explore patterns and a result in category theory (e.g. Yoneda lemma) generalizes immediately to other instances of categories.

> I also find any claims that category theory is relevant to the average working programmer to be dubious at best.

I would say there is value in knowing some category theory, especially if the programmer is working in a strongly-typed functional language. Knowing what a functor is, what a catamorphism is and so on, can be useful ways to write more declarative code and teach about abstractions in the same way OOP uses Design Patterns to teach programmers about reusuable code.

Re: Graphical Linear Algebra

#12

Post by jshholland » Tue, Aug 18, 2020, 9:28 AM UTC

spekcular wrote: post #5

The short version appears to be that the author has managed to replace standard linear algebra notation with an alternative (but mathematically equivalent) graphical notation, which is harder to do computations and write programs in. This submission is the latest in a trend of what I would classify as "math-lite" category theory and Haskell articles that reach the HN front page, which purport to explain something int…

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-a... ) and they were starting to look into Petri nets as my study was wrapping up.

(source: I was a postgrad supervised by Pawel)

Re: Graphical Linear Algebra

#13

Post by guerrilla » Tue, Aug 18, 2020, 9:30 AM UTC

spekcular wrote: post #5

The short version appears to be that the author has managed to replace standard linear algebra notation with an alternative (but mathematically equivalent) graphical notation, which is harder to do computations and write programs in. This submission is the latest in a trend of what I would classify as "math-lite" category theory and Haskell articles that reach the HN front page, which purport to explain something int…

> "math-lite"

The series is intended for a general audience and starts by laying the groundwork necessary for exposition of this paper: [1]. Article 4 [2] explains why it is introduced this way.

> end up just rehashing standard mathematics in opaque ways

It's just an explanation of string diagrams [3][4], pretty transparent and standard.

I find it funny that your response to this is so similar to that of the abacists to Fibonacci.[5]

[1]. https://arxiv.org/abs/1403.7048

[2]. https://graphicallinearalgebra.net/2015/04/29/dumbing-down-m...

[3]. https://ncatlab.org/nlab/show/string+diagram

[4]. https://en.wikipedia.org/wiki/String_diagram

[5]. https://graphicallinearalgebra.net/2015/04/26/adding-part-1-...

Re: Graphical Linear Algebra

#14

Post by eru » Tue, Aug 18, 2020, 9:31 AM UTC

CharlesMerriam2 wrote: post #4

Meh. Sorry. Start with [Essence of Linear Algebra]( https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x... ) by [3 Blue 1 Brown]( https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw ) for an intuitive overview of all the major concepts. While working through a full Linear Algebra course, it is important to develop your own software to do things. That is, some sort of command tool to enter matrices and r…

> While working through a full Linear Algebra course, it is important to develop your own software to do things. That is, some sort of command tool to enter matrices and run operations on them. As you go through a course, you will keep adding operations.

That's definitely possible, but not everyone approaches linear algebra from such a number oriented perspective.

In fact, my linear algebra prof in Uni tried to avoid coordinates (ie numbers) as much as possible. Eg talking about abstract linear transformations instead of emphasising that you can represent them you can represent them as a rectangle of numbers.

Re: Graphical Linear Algebra

#15

Post by _f9cu » Tue, Aug 18, 2020, 9:52 AM UTC

spekcular wrote: post #5

The short version appears to be that the author has managed to replace standard linear algebra notation with an alternative (but mathematically equivalent) graphical notation, which is harder to do computations and write programs in. This submission is the latest in a trend of what I would classify as "math-lite" category theory and Haskell articles that reach the HN front page, which purport to explain something int…

> I really wish this would stop > Getting excited about category theory is like getting excited about matrix notation – useful, sure, but not where the meat is

Patience. These kind of work is not worth for verbal readers, but for visual learners and visual creators this is very educative. It's a work of visual learner and these kind of work will not stop.

> replace standard linear algebra notation with an alternative (but mathematically equivalent) graphical notation.

Yes. Modern computation is based on AI which is based on matrices which runs on GPU which does computation on graphical input. This work is a masterpiece for simple solutions for complicated computations.

Re: Graphical Linear Algebra

#16

Post by _f9cu » Tue, Aug 18, 2020, 9:53 AM UTC

guerrilla wrote: post #13

spekcular wrote: post #5

The short version appears to be that the author has managed to replace standard linear algebra notation with an alternative (but mathematically equivalent) graphical notation, which is harder to do computations and write programs in. This submission is the latest in a trend of what I would classify as "math-lite" category theory and Haskell articles that reach the HN front page, which purport to explain something int…

> "math-lite" The series is intended for a general audience and starts by laying the groundwork necessary for exposition of this paper: [1]. Article 4 [2] explains why it is introduced this way. > end up just rehashing standard mathematics in opaque ways It's just an explanation of string diagrams [3][4], pretty transparent and standard. I find it funny that your response to this is so similar to that of the abacists…

Funny thoughts have truth in them. Can you share what truth did you realise ?

Re: Graphical Linear Algebra

#17

Post by spekcular » Tue, Aug 18, 2020, 10:03 AM UTC

guerrilla wrote: post #13

spekcular wrote: post #5

The short version appears to be that the author has managed to replace standard linear algebra notation with an alternative (but mathematically equivalent) graphical notation, which is harder to do computations and write programs in. This submission is the latest in a trend of what I would classify as "math-lite" category theory and Haskell articles that reach the HN front page, which purport to explain something int…

> "math-lite" The series is intended for a general audience and starts by laying the groundwork necessary for exposition of this paper: [1]. Article 4 [2] explains why it is introduced this way. > end up just rehashing standard mathematics in opaque ways It's just an explanation of string diagrams [3][4], pretty transparent and standard. I find it funny that your response to this is so similar to that of the abacists…

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.

Re: Graphical Linear Algebra

#18

Post by spekcular » Tue, Aug 18, 2020, 10:10 AM UTC

siraben wrote: post #11

spekcular wrote: post #5

The short version appears to be that the author has managed to replace standard linear algebra notation with an alternative (but mathematically equivalent) graphical notation, which is harder to do computations and write programs in. This submission is the latest in a trend of what I would classify as "math-lite" category theory and Haskell articles that reach the HN front page, which purport to explain something int…

This graphical notation for linear algebra isn't like the usual diagram notation category theory at all. > Is category theory a useful organizational tool in certain abstract branches of mathematics? Yes. Abstracting math in a very general way lets you explore patterns and a result in category theory (e.g. Yoneda lemma) generalizes immediately to other instances of categories. > I also find any claims that category t…

I realize that, obviously, that the diagrams presented in this article are different from the ones found in category theory textbooks. Moreover, my later comments were not really aimed at this article so much as others that are posted here. The criticisms just happen to fit this article, too.

Also, I don't really see how learning the abstract definition of a functor is going to help someone write better OCaml programs. But it is possible I am just not imaginative enough. Do you have a concrete example of this?

Re: Graphical Linear Algebra

#19

Post by enriquto » Tue, Aug 18, 2020, 10:12 AM UTC

spekcular wrote: post #5

The short version appears to be that the author has managed to replace standard linear algebra notation with an alternative (but mathematically equivalent) graphical notation, which is harder to do computations and write programs in. This submission is the latest in a trend of what I would classify as "math-lite" category theory and Haskell articles that reach the HN front page, which purport to explain something int…

> an alternative (but mathematically equivalent) graphical notation, which is harder to do computations and write programs in.

But this does need to be the case. Sometimes a graphical notation is a great help. For example, many tensor computations in differential geometry become straightforward using Penrose's graphical notation. Is that the case here? It is difficult to say, as the authors fail to present that notation separately to the results.

Re: Graphical Linear Algebra

#20

Post by spekcular » Tue, Aug 18, 2020, 10:13 AM UTC

jshholland wrote: post #12

spekcular wrote: post #5

The short version appears to be that the author has managed to replace standard linear algebra notation with an alternative (but mathematically equivalent) graphical notation, which is harder to do computations and write programs in. This submission is the latest in a trend of what I would classify as "math-lite" category theory and Haskell articles that reach the HN front page, which purport to explain something int…

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. T…

What "success" has this research program had with signal flow graphs? What problems has it helped solved?

My impression is that it ends up being largely some kind of linguistic translation project, or abstraction for the sake of abstraction, but I'm happy to be corrected on this point.