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Graphical Linear Algebra

graphicallinearalgebra.net

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Re: Graphical Linear Algebra

#31

Post by buddhiajuke »

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 contexts it’s even clearer.

Re: Graphical Linear Algebra

#32

Post by impendia »

spekcular wrote: post #17
guerrilla wrote: post #13

Earlier quoted context omitted.

> "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…

> 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.

Re: Graphical Linear Algebra

#33

Post by [deleted] »

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…

[deleted]

Re: Graphical Linear Algebra

#34

Post by dmch-1 »

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…

These types of notations have been around for a long time. In fact they fist started to study geometric objects (such as knots and braids) with algebraic tools. The graphical notation does the opposite - represents algebraic objects by geometric ones. These may have some benefit for human eyes, but for machines they are quite useless indeed.

Incidentally, I partially agree with you on category theory, especially on computer science. In abstract maths, however, category theory has more significance than pure language. It can serve more or less like an algebraic tool. There are abstract mathematicians who see deeper meaning in it too.

Re: Graphical Linear Algebra

#35

Post by nerdponx »

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…

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 (up to choice of basis), and then we go on to prefer coordinate-free proofs. But, of course, we do discuss the computational aspect (i.e. the Gauss algorithm) because this is exactly what makes linear algebra so useful: many questions can be answered extremely efficiently.

This is how I was taught linear algebra in the USA too. Almost to a fault, where I lacked practical intuition for how matrices and vectors of real numbers work and had to re-learn that stuff later.

Re: Graphical Linear Algebra

#36

Post by stablenode »

graphlinalg wrote: post #23
spekcular wrote: post #17

Earlier quoted context omitted.

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…

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

Re: Graphical Linear Algebra

#37

Post by chadcmulligan »

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…

Have you ever done much electronics? An electronic circuit diagram is a wonderful thing - a huge page of paper that abstracts the entire operation of a circuit that you can trace through. I'd be very excited to have something similar for mathematics, I don't know if this implementation is that, but its certainly worth striving for.
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