The case against geometric algebra (2024) • Hacker News Board

Re: The case against geometric algebra (2024)

#21

Post by jdw64 » Sun, Jun 21, 2026, 1:12 PM UTC

TimorousBestie wrote: post #16

jdw64 wrote: post #2

With my limited knowledge, I read through it stumbling along, and from what I gather, this GA is not Clifford Algebra, and the argument is that the GA movement itself is misguided, and that combining operators and geometric objects without distinguishing between them is problematic. From a programmer's perspective, it seems like they're saying it's a flawed abstraction, while the GA stance is different. I'd like to h…

Mathematician here. > As I see it, GA is not so much a subject as an ideological position, consisting of basically two ideological claims about the world: > Claim 1: That the concepts of EA (so, wedge products, multivectors, duality, contraction) are incredibly powerful and ought to be used everywhere, starting at a much lower level of math pedagogy—basically rewriting classical linear algebra and vector calculus. I…

Interesting. To summarize your argument: the current state of Algebra is like an 80 point solution, but to push it a few points higher requires an enormous cognitive load, and the question is whether that's really worth it, even from an educational perspective. As mentioned in another comment, this is exactly the kind of issue that comes up in Rust discussions. It seems the argument from the GA camp is that top tier mathematicians are already using these tools just fine without needing to talk about it in that way, so there's no reason for it to become general purpose. Thank you for explaining it in a way that's easy to understand. But on the other hand, maybe anomalies like these could actually become generally useful concepts. Thanks for the comment. upvoted!

Re: The case against geometric algebra (2024)

#22

Post by eigenspace » Sun, Jun 21, 2026, 1:16 PM UTC

TimorousBestie wrote: post #16

jdw64 wrote: post #2

With my limited knowledge, I read through it stumbling along, and from what I gather, this GA is not Clifford Algebra, and the argument is that the GA movement itself is misguided, and that combining operators and geometric objects without distinguishing between them is problematic. From a programmer's perspective, it seems like they're saying it's a flawed abstraction, while the GA stance is different. I'd like to h…

Mathematician here. > As I see it, GA is not so much a subject as an ideological position, consisting of basically two ideological claims about the world: > Claim 1: That the concepts of EA (so, wedge products, multivectors, duality, contraction) are incredibly powerful and ought to be used everywhere, starting at a much lower level of math pedagogy—basically rewriting classical linear algebra and vector calculus. I…

More or less agreed. I think though that one reason the geometric product is so tempting is that if you take matrix representations of all of these objects, then the geometric product is literally just straightforward matrix multiplication.

Because of that, it just becomes so tempting to try and phrase everything you can in terms of this geometric product. I'm very sympathetic to the temptation, and I even think the geometric product has some great uses (it shows up a lot in some physics I do), and using it makes writing rotations a treat, but I think it's still vastly overemphasized by GA people.

I still don't really know what my favoured notation for differential geometry is, I find myself switching around so much.

Re: The case against geometric algebra (2024)

#23

Post by blurbleblurble » Sun, Jun 21, 2026, 1:19 PM UTC

Those quadratic forms loop in some nice structure for modeling all kinds of geometric problems with high level control that's hard to articulate so concisely otherwise. Conformal geometric algebra is awesome to work with, have you tried it?

But mostly the broad strokes points about the community are exactly the kind of hostility that makes geometric algebra communities so refreshing for curious young people. Geometric algebra is a welcoming pedagogy and community as much as it is a mathematical framework. If only mathematics as a whole was more welcoming.

I started out on with shaky linear algebra despite years of undergraduate education, but plenty of curiosity and intuition. The geometric algebra community schooled me and me prepared me for all kinds of "real math".

Yes the attitude that geometric algebra is the best language for everything is misguided and welcomes a lot of confusion, but most serious geometric algebra people I've met don't actually think that or say that. They're just off doing cool stuff.

Re: The case against geometric algebra (2024)

#24

Post by aureate » Sun, Jun 21, 2026, 1:21 PM UTC

Tiny nit / check of my understanding:

> It was already widely understood that projective geometry allowed one to represent rotations and translations in R^3 with a single linear operator on R^4.

I think it's projection operators (in linear algebra) that allow one to do that, not projective geometry [1]. The latter, AIUI, studies projective spaces and projective transformations on them (which differ from vector spaces and their transformations by including "points at infinity"), contains no concepts of length or angle (and therefore no equivalent of translations and rotations) and is in some sense "geometry with only the straightedge, no compass".

Curious if I'm just missing something there, though. I'm no expert on any of this.

[1] https://en.wikipedia.org/wiki/Projective_geometry

Re: The case against geometric algebra (2024)

#25

Post by jsLavaGoat » Sun, Jun 21, 2026, 1:24 PM UTC

eigenspace wrote: post #14

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…

> 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. While it's neat to write them all as one equation, I disagree that it's an enlightening perspective…

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.

Re: The case against geometric algebra (2024)

#26

Post by jsLavaGoat » Sun, Jun 21, 2026, 1:25 PM UTC

Certhas wrote: post #10

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…

The space time approach with E as t wedge x and B as x wedge y is purely linear algebra, not differential forms. As opposed to the weird GA form it actually makes the physically most meaningful symmetry (Lorentz transformations) explicit. That's why it's actually used in Physics. Anti symmetric space time tensors are the absolute standard . Further formulations that reveal other aspects, dualities, symmetries are muc…

OK, well, MTW is a pretty standard GR textbook and it is often cited as a useful text on differential forms for math.

Re: The case against geometric algebra (2024)

#27

Post by TimorousBestie » Sun, Jun 21, 2026, 1:28 PM UTC

eigenspace wrote: post #22

TimorousBestie wrote: post #16

Earlier quoted context omitted.

Mathematician here. > As I see it, GA is not so much a subject as an ideological position, consisting of basically two ideological claims about the world: > Claim 1: That the concepts of EA (so, wedge products, multivectors, duality, contraction) are incredibly powerful and ought to be used everywhere, starting at a much lower level of math pedagogy—basically rewriting classical linear algebra and vector calculus. I…

More or less agreed. I think though that one reason the geometric product is so tempting is that if you take matrix representations of all of these objects, then the geometric product is literally just straightforward matrix multiplication. Because of that, it just becomes so tempting to try and phrase everything you can in terms of this geometric product. I'm very sympathetic to the temptation, and I even think the…

> I still don't really know what my favoured notation for differential geometry is, I find myself switching around so much.

Yep, me too. Maybe someday the HoTT folks will get around to formalizing it and standardizing the notation. /j

Re: The case against geometric algebra (2024)

#28

Post by gugagore » Sun, Jun 21, 2026, 1:28 PM UTC

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 we actually we want them in both roles at the same time. So it is not very natural to equate the two objects, as opposed to finding a correspondence between them.

> So GA ends up being very stuck because it equates “vectorial objects” and “operators that act on vectorial objects”. It would be better to express all the geometric objects you care about in their most natural forms, and then find isomorphisms between them when it’s necessary to do so. Otherwise all the meanings get blurred together and it’s very confusing. So that’s another problem with geometric algebra: eliding the distinction between vectors and operators is undesirable, confusing, and disingenuous.

Re: The case against geometric algebra (2024)

#29

Post by eigenspace » Sun, Jun 21, 2026, 1:29 PM UTC

jsLavaGoat wrote: post #25

eigenspace wrote: post #14

Earlier quoted context omitted.

> 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. While it's neat to write them all as one equation, I disagree that it's an enlightening perspective…

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.

I guess I'd say my point though is that the gauge structure is the mathematically interesting part of Maxwell's equations. (i.e. the fact that `F` is a closed differential form).

Without it, I think it'd be of significantly less mathematical interest because it'd lose almost all of its geometric properties.

Re: The case against geometric algebra (2024)

#30

Post by cherryteastain » Sun, Jun 21, 2026, 1:29 PM UTC

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 PhDs, people with PhDs from unrigorous programs, people who had been good at math but were perhaps going a bit senile, random passerbies from engineering or computer programming, run-of-the-mill circle-squarers, people who had a bone to pick with establishment mathematics and felt like all dissenting views were being unfairly suppressed

> It didn’t help that a lot of the texts by the actually-competent GA people, like the Cambridge group, tended to say things that sounded and still sound kind of crackpotty as well.

After reading the article, the main "case against geometric algebra" I could find in there was that the author does not like the people using/doing research in geometric algebra, such as the ostensibly failed academics from a Cambridge research group [1] which the article links to.

I was expecting in the "An Actual Case Against GA" section that the author would demonstrate something like "Geometric Product actually does not work if you apply it to xyz domain". Rather, the section just ended up being mostly about the type of bikeshedding you see about naming of variables in programming.

There is I guess merit to the core "there is no good general interpretation or usage for the geometric product or mixed-grade multivectors" thesis of the article but calling other academics crackpots really subtracts from that message.

[1] https://corde.phy.cam.ac.uk/