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

The case against geometric algebra (2024)

alexkritchevsky.com

Post reply on HN
Print view View original

101–110 of 148 posts

Re: The case against geometric algebra (2024)

#101

Post by srean »

jampekka wrote: post #93
srean wrote: post #78

Earlier quoted context omitted.

> I don't care about proofs or theoretically problematic corner cases that "real mathematics" seems to be almost exclusively interested in. That is a rather strange take for a software engineer. When implementing something I do need to know what the corner cases are, whether the runtime can enter such a state. I need to think how to put in checks so that they cannot be reached, or alternatively, how to recover gracef…

I didn't mean corner cases like that. I mean stuff that has no practical consequences, at least for most practical use. E.g. whether or not Navier-Stokes can form singularities doesn't really change how you analyze fluid dynamics in engineering. This doesn't mean it is not a mathematically important question worth extensive study, but it's not relevant for practitioners.

Got you.

Re: The case against geometric algebra (2024)

#102

Post by [deleted] »

cherryteastain wrote: post #30

Not a fan of the article. It resorts to ad hominem attacks like > GA had gotten a bad reputation because of its tendency to attract bad mathematicians and full-on crackpots. Hestenes honestly sounds like one a lot of the time, and I’m not really sure whether he is or isn’t. It makes sense, really. > GA ended up appealing to a lot of fringes: people who only had undergraduate degrees, people who had dropped out of PhD…

[deleted]

Re: The case against geometric algebra (2024)

#103

Post by blauditore »

pphysch wrote: post #90
blauditore wrote: post #63

Earlier quoted context omitted.

Only tangenially relevant, but the exagerrated differentiation of universities, and levels of education (e.g. PhD vs. not) has always been bothering me. I only have experience in a different field (CS), and yes, those things can be indicators, but I've experienced so many outliers in both directions to know that degrees need to be taken with more than just a grain of salt.

This differentiation is amplified in (pure) mathematics, where two different subfields can have essentially zero overlap*, making peer review and general QA difficult to scale. * Physical sciences also have a lot of diversity, but at least you can go to their labs and see their equipment, reagents, data, etc cetera.

Yes, this is essentially because mathematics is such an old science and goes very deep into branches. That's btw also why getting a certain degree is much harder there than in any other fields I've seen (did a bunch of courses at university in various fields, and the diff between e.g. pure math and psychology is almost comical).

Re: The case against geometric algebra (2024)

#104

Post by ColinHayhurst »

cherryteastain wrote: post #30

Not a fan of the article. It resorts to ad hominem attacks like > GA had gotten a bad reputation because of its tendency to attract bad mathematicians and full-on crackpots. Hestenes honestly sounds like one a lot of the time, and I’m not really sure whether he is or isn’t. It makes sense, really. > GA ended up appealing to a lot of fringes: people who only had undergraduate degrees, people who had dropped out of PhD…

[dead]

Re: The case against geometric algebra (2024)

#105

Post by Steuard »

Certhas wrote: post #6

100 percent agree with the article. Wedge products are fundamental, GA is weird ideology. I had the bad fortune of reviewing some GA research articles once upon a time. It was almost embarrassing. Everything of substance had been published in a conceptually cleaner bivector language previously. The only "contribution" was writing everything in terms of weirder, more convoluted concepts that contributed neither techni…

Do you by chance have references to those conceptually cleaner bivector publications? I've spent a good bit of time looking for other people working in that space, and haven't found much other than an article by Jancewicz from 1980. (I like to imagine that my articles these past few years have been "conceptually clean", but they certainly aren't "previous" to much GA work.)

Re: The case against geometric algebra (2024)

#106

Post by CyLith »

This article captures so much of what I have felt but been unable to put into words about GA. I come from a computational physics background, and when translating theory into numerical algorithms, the dimensional analysis and units are very important (you have to be able to relate the simulation to something in the real world!). GA dispenses entirely with any notion of meaningful units, making the dimensional analysis and error checking extremely difficult. The geometric product has always seemed like some strange mathematical trick or coincidence that happens to maybe have some useful properties. Almost like how "new math" is perhaps easier to learn or understand at first, but you really just need to sit down and understand algorithmically what is going on with the basic arithmetic operations.

Re: The case against geometric algebra (2024)

#107

Post by oggreen »

I graduated with a Bachelors in Math in 2018 and there's an entire new math now?

For people who actually know the curriculum side: where does geometric algebra fit? Is it something that should come after Calc III / linear algebra, alongside linear algebra, or as part of a more geometric replacement for vector calculus?

Re: The case against geometric algebra (2024)

#108

Post by eigenspace »

oggreen wrote: post #107

I graduated with a Bachelors in Math in 2018 and there's an entire new math now? For people who actually know the curriculum side: where does geometric algebra fit? Is it something that should come after Calc III / linear algebra, alongside linear algebra, or as part of a more geometric replacement for vector calculus?

The idea is that it's an alternative way of talking about vectors, rotations, and geometry in general. I.e. a replacement for the vector notation you learned that makes it operate more like how complex numbers are used.

Re: The case against geometric algebra (2024)

#109

Post by hamish_todd »

I'm a GA researcher. I did this livestream of my reaction to the article

https://m.twitch.tv/videos/2282548167

TLDR it is quite a bad article. One of the closest thing he has to a real argument is "I don't like it when geometric objects are identified with operators, I want those to be separate things". But this is both anti-GA and anti-Lie-Theory. As he says, he is critical of mathematics as conventionally practiced. So be warned that if you find yourself disliking GA for anything like the reasons he dislikes GA, there's a lot of other (mainstream/prestigious) fields you dislike too.

Re: The case against geometric algebra (2024)

#110

Post by hamish_todd »

cherryteastain wrote: post #30

Not a fan of the article. It resorts to ad hominem attacks like > GA had gotten a bad reputation because of its tendency to attract bad mathematicians and full-on crackpots. Hestenes honestly sounds like one a lot of the time, and I’m not really sure whether he is or isn’t. It makes sense, really. > GA ended up appealing to a lot of fringes: people who only had undergraduate degrees, people who had dropped out of PhD…

The geometric product is transform composition.

TRANSFORM COMPOSITION!

(sorry, it's not your fault or even his that you didn't know this - GA textbooks should have it as the first thing they teach but they don't)

Post reply on HN