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

71–80 of 148 posts

Re: The case against geometric algebra (2024)

#71

Post by groundzeros2015 »

zarzavat wrote: post #45
groundzeros2015 wrote: post #33

Earlier quoted context omitted.

> I could find in there was that the author does not like the people using/doing research in geometric algebra The start of the article makes a specific technical claims: > Hestenes’ Geometric Product is not a very good operation and we should not be rewriting all of geometry in terms of it Later he explains why: > there is no good general interpretation or usage for the geometric product or mixed-grade multivectors

How is the geometric product any less motivated than any other notation? Ultimately the value of a notation is how easy it makes it to work and think. I'm not sure if GA achieves that or not, but what's the harm in trying a new approach? AFAIK nobody is proposing to replace all of geometry with GA, only 3+1 spacetime.

I’m just refuting the claim that the article only focuses on ad hominem arguments.

Re: The case against geometric algebra (2024)

#72

Post by ajkjk »

gugagore wrote: post #28

The part in this that I most question / deviate from is what I've quoted below about having distinctions (syntactically?) between objects and operations. Conceptually, it's a good distinction. But is it so clearly wise to bake in that distinction into the formal framework when doing calculations or proof? > Most of the time we think of complex numbers as vectors in R2 or as rotation+scaling operators, but rarely do w…

One finds in regular vector algebra that "position vectors" and "displacement vectors" are sort of two distinct types of objects, and that it is never physically valid to add two position vectors together unless you create an affine combination like (a+b)/2. A position vector 'a' is really 'O + a', so [(O+a) + (O + b)]/2 = O + (a+b)/2, another position... but a+b on its own would really be (O +a) + (O + b) = 2O + a + b, which is not geometrically meaningful. So positions and displacements might both be elements of R^2, mathematically speaking, but there is something physically different about them, which physical applications/geometry forces you to contend with. I think it is something like a historical accident that there's not a great notation for expressing this in normal mathematics (or at least, I'm not aware of one!).

Re: The case against geometric algebra (2024)

#73

Post by aeonik »

ajkjk wrote: post #60
aeonik wrote: post #44

I found this article pretty confusing. And my comment ended up being pretty long, so I will TL;DR it: 1. The social critique doesn’t match my experience and seems under-supported? 2. The technical critique is interesting, looks like a mix of good points, and some that need more work put into it. I think GA is legitimately cool in my opinion, but if there are better abstractions, we should find/define them and use the…

When I wrote " GA is considered a kooky, crackpotty sideshow..." I didn't mean I consider it to be... , I mean, it is the case that it is considered to be... . I guess I'm surprised if you haven't run into this? I'm not sure, but it's am impression I've gotten online for a long time. And if you read many of the older Hestenes-era writings you can't help but get it yourself. I agree about the importance of alternative…

I hear authors mention it sometimes. But, I don't see the examples or evidence.

Maybe I'm just not on the math departments enough.

Re: The case against geometric algebra (2024)

#74

Post by itishappy »

skybrian wrote: post #54
jiggawatts wrote: post #13

Earlier quoted context omitted.

As someone who studies physics and then went into a long IT career (but kept reading papers casually), my view is that this whole GA saga is very reminiscent of how after decades of experience, I still can't convince juniors of the benefits of what I now consider obvious best practices. No amount of demonstrations of the blindingly obvious improvement of some better technique seems to work on someone who "finally got…

As a programmer I’m wondering how you get a decent graphics library out of it. If it’s conceptually better, shouldn’t it make writing code to do calculations easier?

It can write some fascinating stuff, but you have to learn to think in it:

https://enkimute.github.io/ganja.js/examples/coffeeshop.html

A major problem is that its a very general theory. Most calculations turn into very large but very sparse matrix multiplications. To make them work fast requires code generation and an optimization pass.

These types of optimization problems show up all over graphics programming though:

* Representing rotations with matrices takes more space than quaternions.

* Sacrificing a dimension to projective geometry actually makes representing things like projections (duh) but also translations more efficient.

Re: The case against geometric algebra (2024)

#75

Post by SonOfLilit »

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…

I think this is unfair to the article.

Those paragraphs are in the background section, clearly labeled as "this is what other people think", and are followed with a high effort explanation of (presumably) the substance of the theory and why the author considers some of their ideas to be good and others to just increase the confusion.

The technical arguments are less like variable naming discussions and more like arguments against teaching logic circuit design with only nand (without naming the and/or/not operators) or using untyped lamba calculus (with Church numerals, e.g. `3 := λf.λx.f (f (f x))`) to do calculations on numbers.

At the least, the five bolded statements summarizing 5 of the 7 highly technical arguments should count as substantial claims.

Of course, having learned of the subject only from the author, it's hard to know whether it's a good representation of GA or a strawman, but the theory that he teaches as GA indeed seems quite flawed as a tool for thought.

Re: The case against geometric algebra (2024)

#76

Post by jampekka »

I tried to solve some engineering problems with PGA few years ago. Seemed to work OK up to a point, and at least for me was easier to approach than say Lie algebra or differential geometry.

TFA denigrates papers and websites that are "non-theoretical" or "trivial". As a user of the formalisms, these kinds of materials are exactly what I need. I don't care about proofs or theoretically problematic corner cases that "real mathematics" seems to be almost exclusively interested in.

I did hit a wall quite soon with GA, and got a feel that it may indeed be overhyped, but at least the scene seems to be interested about applied use.

There seems to be similar debate about nonstandard calculus. For my modest use it has provided some tools that can give me results that I don't know how to get with epsilon-delta etc. I don't really care if I don't "really understand" it because the underlying proofs need some heavy machinery. I don't understand those for standard calculus either, and in applied use you either manipulate infinitesimals without any proper algebra, or just hope what you need is in some table.

I can't comment on deeper theoretical or philosophical questions about these, and I don't really care about them. But to me maths communication often seems analogous to making people learn turing machines and lambda calculus before they are allowed to program in Javascript.

I don't think the author necessarily disagrees with me much, but this is maybe a kinda mini rant from a perspective of someone who is just an "end user" of mathematics.

Re: The case against geometric algebra (2024)

#77

Post by NooneAtAll3 »

jampekka wrote: post #76

I tried to solve some engineering problems with PGA few years ago. Seemed to work OK up to a point, and at least for me was easier to approach than say Lie algebra or differential geometry. TFA denigrates papers and websites that are "non-theoretical" or "trivial". As a user of the formalisms, these kinds of materials are exactly what I need. I don't care about proofs or theoretically problematic corner cases that "r…

> I did hit a wall quite soon with GA

can you give an example of what's impossible/hard to do?

Re: The case against geometric algebra (2024)

#78

Post by srean »

jampekka wrote: post #76

I tried to solve some engineering problems with PGA few years ago. Seemed to work OK up to a point, and at least for me was easier to approach than say Lie algebra or differential geometry. TFA denigrates papers and websites that are "non-theoretical" or "trivial". As a user of the formalisms, these kinds of materials are exactly what I need. I don't care about proofs or theoretically problematic corner cases that "r…

> 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 gracefully. That's my job after all, why would anyone pay me if I didn't.

Perhaps a topical example is a gimbal lock. I need to be aware that it can happen and I need to know how to prevent it.

Re: The case against geometric algebra (2024)

#79

Post by NooneAtAll3 »

jsLavaGoat wrote: post #5

From a mathematician's point of view, yes, you should write the Maxwell field equations, at least to see it once, that way because you're showing a very low-level symmetry that even the differential forms approach doesn't get all the way to. Differential forms is a standard approach for general relativity, e.g. MTW. I guess the people pushing this are a little pushy, but this reminds me of the whole pie fight over th…

what is MTW?

Re: The case against geometric algebra (2024)

#80

Post by simpaticoder »

lrasinen wrote: post #49
jsLavaGoat wrote: post #25

Earlier quoted context omitted.

What interests a mathematician isn't 100% the same as what interests the physicist. All I'm saying is there is some math there that's interesting and people should see it once for the math.

And then there are us engineers. I don't care much either way whether Maxwell's equations are ∇F = J or some other form, as long as it makes the problem easier to solve. If I were in the GA Marketing Committee I'd publish a paper with suitably hand-picked worked examples where the vector approach is long and tedious, and GA version is short and sweet.

I like this idea but I get the sinking feeling GA proponents don't really solve problems with GA. Like how Haskell advocates don't write programs and modular synth enthusiasts don't write music.
Post reply on HN