Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Graphical Linear Algebra

graphicallinearalgebra.net

Post reply on HN
Print view View original

41–50 of 85 posts

Re: Graphical Linear Algebra

#41

Post by longtimegoogler »

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…

As someone who knows some Category Theory, Algebraic Topology, and a smidgen of Algebraic Geometry, I totally agree. Category Theory is a very useful and essential tool in those subjects.

However, to me it reminds me of String Theory, in that it has been sold as Theory of Everything for Mathematics and adjacent fields.

Category Theory might even be somewhat useful for language designers, but, I agree that it offers little benefit for programmers even with languages like Haskell.

Even Bartosz said that Adjoints aren't that interesting in Haskell because every functor is an endofunctor.

Re: Graphical Linear Algebra

#43

Post by feanaro »

stablenode wrote: post #36
graphlinalg wrote: post #23

Earlier quoted context omitted.

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 math…

"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

Given the tone of the poster who this reply was for, it seems entirely warranted.

Re: Graphical Linear Algebra

#44

Post by wenc »

Tainnor wrote: post #30

Regardless of the usefulness of category theory or not, the author seems to imply that "traditional linear algebra" is about matrices. This is not true. This may be true from an engineering perspective, or maybe if you've studied in the US, but over here, linear algebra starts with fields, vector spaces, homomorphisms. We establish pretty early on that matrices and linear transformations are in essence the same thing…

Just curious, what country would this be? I know many European countries favor a more theoretical approach.

I definitely learned linear algebra from the numerical and matrix perspective (though yes, we did cover the idea of vector spaces and linear transformations too). In engineering school, the focus tended to be on the craft rather than the theory, and in retrospect I think it was the right approach for engineering majors, whose primary concern was execution. It helped engineers use MATLAB to solve problems. The syllabus looked like this:

https://ocw.mit.edu/courses/mathematics/18-06sc-linear-algeb...

And this:

https://math.nyu.edu/media/math/filer_public/a4/00/a4008ffa-...

A syllabus for math majors might look quite different.

The US approach to linear algebra continued to be useful for me in grad school as an engineering major (I have an engineering Ph.D.). To be honest, for what I was/am doing, I don't think I would have derived much benefit from a more theoretical approach (but that's just me. People who work closer to theory may have a different opinion).

In my work, math has a retrospective role -- I first build something and later recognize patterns that fit a certain theory.

For theory-builders however, math has a constructive role. Theory builders however are relatively rare.

Re: Graphical Linear Algebra

#45

Post by wenc »

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 often go back to this post by John D Cook

Category theory can be very useful, but you don’t use it the same way you use other kinds of math. You can apply optimization theory, for example, by noticing that a problem has a certain form, and therefore a certain algorithm will converge to a solution. Applications of category theory are usually more subtle. You’re not likely to quote some theorem from category theory that finishes off a problem the way the selecting an optimization algorithm does.

and

I had been skeptical of applications of category theory, and to some extent I still am. Many reported applications of category theory aren’t that applied, and they’re not so much applications as post hoc glosses.

At the same time, I’ve seen real applications of categories, such as the design of LINQ mentioned above. I’ve been a part of projects where we used category theory to guide mathematical modeling and software development. Category theory can spot inconsistencies and errors similar to the way dimensional analysis does in engineering, or type checking in software development. It can help you ask the right questions. It can guide you to including the right things, and leaving the right things out.

https://www.johndcook.com/blog/applied-category-theory/

Re: Graphical Linear Algebra

#46

Post by stablenode »

feanaro wrote: post #43
stablenode wrote: post #36

Earlier quoted context omitted.

"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

Given the tone of the poster who this reply was for, it seems entirely warranted.

feanaro: fine, if you think so.

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.

Re: Graphical Linear Algebra

#47

Post by uglycoyote »

buddhiajuke wrote: post #31
Tainnor wrote: post #30

Regardless of the usefulness of category theory or not, the author seems to imply that "traditional linear algebra" is about matrices. This is not true. This may be true from an engineering perspective, or maybe if you've studied in the US, but over here, linear algebra starts with fields, vector spaces, homomorphisms. We establish pretty early on that matrices and linear transformations are in essence the same thing…

You’re not wrong. OTOH the other day I heard someone in the Netflix doc about Alberto Nisman’s murder that described a terrorist network as a “matrix”. I grumped as card-carrying pedants are duty-bound to, but then realized that if there’s a 1:1 correspondence between graphs and (adjacency, incidence) matrices, then there’s very little loss of meaning in referring to graphs as “matrices”. Maybe in some communication…

That's funny, I would have just chalked it up to "they are using a different sense of the word matrix", knowing that most dictionaries contain multiple definitions of the same word. But nice work making this mental leap, I have never thought about it!

But it's true, you can envision it as a NxN matrix of N terrorists, where the number in cell i,j represent's the strength of the relationship (it's a weighted graph!), and that might actually be useful for something. I don't doubt that "matrix" might actually exist in numeric form in some CIA or military computer somewhere.

Re: Graphical Linear Algebra

#48

Post by auggierose »

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 think category theory is a language you need to learn at some point if you are serious about math or computer science. I've refused to do so for quite a long time, but there is just too much interesting material out there that uses this language. The biggest problem is not falling asleep while learning it :-)

The most interesting part of category theory is topos theory, in my opinion. This is a generalisation of set theory, and can thus claim to be indeed useful for foundations.

Re: Graphical Linear Algebra

#49

Post by ianhorn »

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…

> ...the content is merely linguistic, not mathematical.

I disagree with the "merely" here. Half of mathematical practice seems to be finding the right way to think about a thing. If the author were claiming this was a new grand theory, I'd be right there with you, but they seem to be claiming that it's just another way to look at things. I don't see any claims of importance. There's always room for new viewpoints that are content-wise equivalent to old viewpoints but different in how you work with, interpret, and think about them.

> ...but end up just rehashing standard mathematics in opaque ways.

I'd say it's rehashing standard mathematics in strange interesting ways, making it a very fitting curiosity for a site where things like a history of methods of communicating with submarines gets a ton of upvotes.

It's a bit surprising to realist that it seems like we completely agree on what it is and what its contribution to existing standard material is, but for some reason I think it's interesting while you "wish it would stop."

Re: Graphical Linear Algebra

#50

Post by ianhorn »

Diagrammatic notations seem to pop up all over the place, but in my mind they all share a difficulty that interesting complex graphs tend to not be planar (can't be drawn without criss-crossing lines). That makes it hard to organize in a skimmable/glanceable way. I'd love to see what a future with ubiquitous 3D volumetric displays or VR or whatever could do for diagrammatic notations. You might suddenly be able to organize larger diagrams and I wouldn't be surprised if diagrammatic notations became more useful and started popping up more in that hypothetical future.
Post reply on HN