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/

My feeling on geometric algebra is that you should look too much into it until you exhaust the exterior algebra. That is (in my opinion): it isn't a good use of it to replace the cross product or specialized representations of 3-d geometric rotations. It is good for when you get a bit sick of the bookkeeping of the exterior algebra. From a computer scientist point of view it is sort of adding a bit of type information beyond just vector dimension and depth of product.

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 the Rust community. OK, so they're pushy. Nothing to do with the merits or demerits of the language (or of C for that matter).

If you're a baby duck about linear algebra and geometry, there's no need to care about different formalisms. Do whatever works. But it's interesting to see how all of this stuff comes together at different levels, whether it's the geometric product, differential forms, or just linear algebra.

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.

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 technical clarity nor conceptual parsimony.,

META: Pulling this out of its original context because I think more readers would find the code amusing. I am breaking the rules, but hopefully for a good/pardonable reason.

> 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.

I can give one counterexample.

I was asked to comment on a piece of code that did 2D geometry in Python. There was one piece that was a tangle of trigonometry to find the angular bisector of an angle subtended at the origin by two points.

Using the fact that points can be represented by complex numbers and that rotation is just multiplication one can make that function into a one liner.

      √(z1 * z2)

> That part is fine. But why, then, does multiplying zzˉ give a “magnitude” that works in a reasonable way?

Because the product of all Galois conjugates is a norm and the determinant of the linear operator defined by general field multiplication of a primitive element when viewing the field extension as a vector space of the extension field over the base field.

Although the geometric interpretation of norms in Galois theory really only works for the complex numbers because only the complex numbers are a field. Quaternions are not a field.

This seems like the pi vs tau argument on steroids. A lot of people who know a bit of math think that tau simplifies things enormously. Professionals are like "not really"; dropping a 2 in places simplifies a few formulas, makes others slightly more complex, and provides zero insight.

The hard problems in math are almost always still hard no matter the notation you choose to use. Sometimes notation makes transmitting ideas a bit easier, but usually faffing around with notation is a sign you aren't able to solve the real problems.

from a theoretical physicist point of view, i find GA don't add much to the standard tooling ppl use, i.e. Lie algebras, Clifford and (sometimes) differential forms. while it's always nice to have a formalism that "hides indices", in most cases (for (super-)gravitation at least) just writing tensor/clifford/lie indices is just much faster and less error prone.

i used to use differential form for gauge theories, einstein-cartan gravitation and ramond-ramond fields.

also, in a paper, we used O(D,D) clifford algebras/spinors to represent differential forms, which worked quite well in our very specific case (appendix A)

https://arxiv.org/pdf/1304.1472

ps: i had colleagues that worked on GA for ML in robotics but wasn't really impressed by what it accomplished

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 hear the other side of the argument too. I'm sure HN will get a long GA comment thread, so from their standpoint, what would it feel like? I agree that merging objects and operators is problematic, but I'm curious what the GA camp would say

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

Are these titles then the wrong avenue for learning math?

Projective Geometric Algebra: Illuminated (2024) (Not mentioned directly in the article [1]; including a quote from link [2].)

Algebraic Calculus (2016)

Divine Proportions: Rational Trigonometry to Universal Geometry (2005)

[1] https://terathon.com/blog/poor-foundations-ga.html

[2] "If you want solid foundations, this book is for you."

The basic issue with geometric algebra is that geometric vectors generally do not have a distinguished notion of unit magnitude (is unit magnitude 1 meter? 1 mile? 1 inch?), so it is silly to work in a framework that requires pretending they do (since the definition of the geometric product of two vectors is dependent upon this choice). Dimensional analysis (a very handy way of tracking mathematical symmetries and thus sanity checking results) goes out the window when working with mixed grade multivectors.

This is not an issue when working with non-mixed-grade multivectors, for which dimensional analysis works just fine in the ordinary way. As the linked article notes, exterior algebra/the wedge product is great. Thinking about exterior powers of vector spaces is great. It's the further move of forcing everything into a Procrustean bed of Clifford algebra that is misguided for almost any application other than some spinor stuff.

Is there another mathematician (likely an analyst) out there that finds this debate even more absurd with the existence of geometric measure theory? GMT bypasses all of these algebraic constructions; it finds very similar objects (currents and varifolds), but it just makes more sense to me. I never found the exterior algebra (or the Clifford algebra) to be a natural way of thinking geometrically. I do not agree that the exterior product is more natural than Jacobians and determinants. I was relieved to find that GMT cut through all of it at higher generality, at least for my purposes anyway. I don't think this belief is shared by many, since GMT is apparently notoriously incomprehensible, but hey, maybe there's someone else out there?

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.

Personally, I've got a line of mileage out of using GA to express animation rigs.

I don't know about the rest of the article—I'm not a mathematician—but I certainly enjoying using GA a lot more compared to linear algebra, I find it way more intuitive and being able to visualize intermediate products on my rig is like a super power.

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.

In my opinion, the article is very mistitled, because it does not contain even a single valid criticism against the geometric algebra theory, despite containing some perfectly valid criticism against some mistakes frequently made by geometric algebra proponents.

The author has completely failed to understand the meaning and the purpose of geometric algebras, though to be fair this is not entirely the author's fault, because there are a lot of bad presentations of the geometric algebra theory, many of which contain actual mathematical mistakes, as listed in an article by Eric Lengyel that is linked in the parent article.

The main correct criticism of the parent article is that the geometric product is an operation that is seldom useful in practice.

In practice, the important operations are the generalizations of the inner product and of the outer product. The inner product and the outer product have been defined by Hermann Grassmann in the 19th century and the publications of Grassmann together with the theory of quaternions by Hamilton have been the sources on which William Kingdon Clifford has created the theory of geometric algebras.

Unfortunately, today a lot of people use incorrectly the term "outer product", using it to name the product defined by Johann Georg Zehfuss, which is also called "tensor product". "Tensor product" is also not a really appropriate term, but at least it is not as ambiguous as "outer product" has become, so it should always be preferred for the Zehfuss product. For the outer product in the Grassmann sense, a non-ambiguous term is "wedge product" though it is rather meaningless.

While the geometric product does not have a practical importance, it has a great theoretical importance, because with it the geometric algebras can be defined with a small set of simple and natural axioms. Then the operations that are important in practice, i.e. the generalized inner and outer (wedge) products can be defined based on the geometric product.

The author is right that some geometric algebra proponents have tried to shoehorn the use of the geometric product in some applications for which it is not the right tool, but that has nothing to do with the theory of geometric algebras.

The theory of geometric algebras has a modest practical importance, but it has an immense theoretical importance, because it unifies many mathematical concepts that previously seemed to be unrelated and it illuminates the relationships between them and also the distinctions between things that were previously confused, even by the best mathematicians and physicists, for more than a century.

There is a high probability that the progress of physics has been delayed by many decades by the fact that both William Clifford and James Clerk Maxwell have died prematurely and almost simultaneously, before they could make order, based on the theory of geometric algebras, in the mess that was at that time the theory of vectors, complex numbers and quaternions. After their death, the theory of geometric algebras has been forgotten and a lot of mistaken theories of vectors have been created, by Josiah Willard Gibbs, Oliver Heaviside and others (because they did not understand the relationships between various physical quantities, like polar vectors, axial vectors, quaternions, complex numbers, pseudoscalars).

When I have first encountered the theory of geometric algebras, that was one of the most beautiful moments in my experience of learning mathematics, it was like turning the light on in a dark room full of previously hidden things. The only similar moments, have been when learning for the first time projective geometry, the theory of spatial symmetry groups and certain parts of topology, which are also theories that have unified a great number of seemingly unrelated concepts.

Like I have said, geometric algebras have very little importance for writing algorithms or the like, where the classic linear algebra with matrices is what matters most, but anyone who does not understand geometric algebras does not really understand physics and this lack of understanding will prevent the correct solution of many problems.

Background resource:

Comparison of vector algebra and geometric algebra - https://en.wikipedia.org/wiki/Comparison_of_vector_algebra_a...

It's a very fun framework when you're learning it. It constantly feels like you're learning something extremely profound and useful, but I've also found that feeling to be a bit of a mirage.

Despite trying many times to make greater use of it, I've found that it often just makes a lot of actual physics work less clear, and with very little practical benefit.

There's times where it affords quite pretty notation, but often you have to actually unpeel all that notation before you actually do something with it. And what's the point of nice notation if none of your colleagues can even read it? The only time I ever really found that GA was actually a benefit to me was performing rotations.

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 them.

Longer version:

I hear people bring up the conspiracy/crackpot side of GA a lot, but I learned about Geometric Algebra a few years ago and am currently learning it alongside standard linear algebra.

I think GA is pretty cool. The author seems to have some decent points about its limitations and some ontological smells (like, maybe there is a cleaner representation hiding somewhere). But a lot of the criticism is aimed at the social side of the movement, and maybe I am just blind to it, but I have not really run into that much.

The author says things like:

    Basically, GA is considered a kooky, crackpotty sideshow. And because it is so dubious and un-self-aware, the movement ends up alienating most people, except for a particular type of… zealous individual… who write about it with a sort of pseudoreligious zeal, and are prone to conspiracy, as if the only reason GA is not mainstream is that they are being oppressed by close-minded traditionalism.

   In practice GA always refers to the particular platform and social movement which descends from the work of David Hestenes from the 1960s. It specifically does not refer to the underlying material of Clifford Algebras

Trying to build a unifying framework seems pretty normal to me. Lots of math is trying to expose common structure across different domains. Category theory, abstract algebra, topology, and, to a much bigger extent, the Langlands program all have that flavor. Obviously some unifications are more successful than others, but “this gives a unified language for a bunch of things” does not seem like a red flag by itself.

Some of the actual technical criticisms of GA are interesting, e.g. the proliferation of operations, but at this point I'm more interested in a formal accounting of the complexity of both theories rather than opinions or vibes. It would be nice to have description-length / complexity-accounting comparison of the formalisms.

Disclaimer: I have not read Hestenes’s original work, so maybe I am missing some of the historical baggage. But the modern resources I have seen seem mostly grounded in their claims.

I'm also learning both GA and linear algebra at the same time, GA has definitely helped me understand the linear algebra more deeply. In my opinion, alternative representations like GA gives your brain more structure to grab onto, even if they aren't perfect.

Also... math pedagogy does have a lot of inertia that hurts students. Doesn't Lockhart's Lament famously resonate with anyone who fell in love with math?

[PDF Warning] https://worrydream.com/refs/Lockhart_2002_-_A_Mathematician%...