7 ms·

The case against geometric algebra (2024)

137 points by Hbruz0 4mo ago

jdw64 4mo ago
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
- jdw64 3mo ago
  Reading this article, I think there are quite a few interesting points to consider further. C started as a DSL for the Unix kernel. JavaScript is also a DSL, and successful languages are often described as DSLs in certain respects. Then, as they grow and gain broader adoption, they evolve into general purpose languages. But if you think about it the other way around, since all programs are ultimately about data transformation, you could argue that UIs should essentially be drawn in SQL, but that would sound strange. That's because the tools we use have moved away from that mental model. (Though React's FRP premise does lean in that direction.) And when I think about why languages split apart, it seems to me that it's because the word 'programming' covers so many different things at once. Languages end up diverging because they serve different purposes. In fact, as a programmer, I see programming languages as a collection of tools that essentially decide what to give up. C gives you safety and low-level hardware access through its ABI. Python gives you expressiveness. They exist because their target goals are fundamentally different. In that sense, though I'm not an expert in this field, from my limited perspective this debate feels like it's just the noise that arises when Algebra tries to encompass too much and inevitably splits apart. I imagine these kinds of cases will only increase in the future. As things become more specialized, there will be more situations where existing frameworks don't fit, and new systems will be needed. Is there a term for this phenomenon? At that point, we might say we need to change the old system to fit the new one. Personally, I wonder if there isn't a general purpose language at the bottom that models the entire world, with other languages layered on top of it.
- eigenspace 3mo ago
  > this GA is not Clifford Algebra What makes you say that?
  - jdw64 3mo ago
    No, that's not my own argument — it's just how I understood the article's claim. I only know GA as something used for specific purposes. I was just sharing my thoughts on what the article was saying.
- jiggawatts 3mo ago
  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 the thing to work".[1] Certain kinds of perfect correctness are like pure and shining crystallised bits of refined knowledge created by the greatest wizards. "Parse, don't validate" or "Make invalid states unrepresentable." ought to be familiar to the better programmers here, the ones with decades of experience built on iterative, collaborative foundations with real consequences for error. Theoretical physics doesn't have those same consequences, because there is no real punishment for their equivalent of "spaghetti code". Perversely, there's cachet to be gained for gaining understanding of its unnecessarily esoteric knowledge, much like how biologists and lawyers spend half a decade or more studying... Latin.[2] Introducing Geometric Algebra to physics is like that wizard coder who sweeps away reams of spaghetti code and replaces it all with a call to a single standard library function. It's that "cheff's kiss" of cleanup. Meanwhile the juniors are screaming about how the senior "deleted all their hard work!" Meanwhile, I never understood where Pauli and Dirac matrices came from! It's like they were pulled from fat air. You've seen this in code, I bet. Some junior worked really hard on solving a problem and wrote a solid screen-filling wall of "a && b || c || !d && e && (f || g)..." continuing up to "ba, bc, bd", etc.. as they ran out single letters until they're well into the alphabet in double-character symbols.[3] That's what those matrices are. Someone's hacky attempt at "making things work". The problem is that we gave those people Nobel prizes and told everyone they're geniuses. They are, but they were like that brilliant junior. Brilliant.. but junior. Geometric Algebra sweeps all of that into one beautiful, consistent, crystal clear abstraction that is widely applicable. The magic matrix constants vanish. Bugs in 100-year-old textbook formulas suddenly come to light. Dozens of formulas, one set for each of the 1D, 2D, 3D, and 4D cases collapse into a single formula valid for any number of dimensions. It's like watching someone struggle with "catching every possible instance of JavaScript injection". No son, no. Just no. Stop enumerating badness. Stop. Just stop. Escape everything at the boundary instead, enforced by the type system. You'll thank me later. I know it might be obvious to you, and you always use properly parameterised SQL queries or whatever. This is not the norm everywhere! I still get arguments, long drawn out arguments from people convinced that this is unnecessary and just one more search & replace is all they need to be safe from the bad hackers. Physicists (and mathematicians) are still making that argument against GA. "It's isomorphic!" "That isn't the point!" [1] You can't convince someone to climb Everest if they struggled to hike up to the top of one of its foothills. [2] Let me be crystal clear: They're spending their precious time on this Earth learning a dead language instead of learning about the law or bugs. No amount of arguments will sway me. The bugs don't care what you call them. Criminals are guilty or innocent whether or not you speak funny in court. You've just made a simple thing harder for no good reason, that is all. Please stop. [3] Yes, I've seen this. Twice, from two different people whom have never met. Aliens are amongst us.
- TimorousBestie 3mo ago
  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 support this claim, so I suppose I’m a proponent of geometric algebra. I think it’s more or less been carried out for vector calculus by Spivak’s “classical” Calculus on Manifolds, which is somewhat widely taught. > Claim 2: That the Geometric Product (henceforth: GP) should be added to that list as the most fundamental operation, where by “fundamental” I mean that other operations should be constructed in terms of it, and theorems should be stated using it. Like the author, I also believe this claim is nonsense. “Rewriting classical linear algebra” is a honored pastime but it’s very difficult to make any headway doing it—the classical texts are classical for a reason, we more or less know how to teach them as an “80% solution” and it’s unclear that the investment in a new pedagogy would get us to an “81% solution.” Especially with today’s undergrads. If you’re not churning arithmetic, they’re not into it.
  - jdw64 3mo ago
    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!
  - eigenspace 3mo ago
    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.
- QuesnayJr 3mo ago
  GA is Clifford algebra plus a bunch of new terminology plus advocacy that it should replace linear algebra.
Ainaguade 4mo ago
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jsLavaGoat 3mo ago
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.
- Certhas 3mo ago
  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 much more niche and specialized subjects and not how the subject should be taught when first encountering it. https://en.wikipedia.org/wiki/Covariant_formulation_of_classical_electromagnetism https://en.wikipedia.org/wiki/Covariant_formulation_of_class...
  - jsLavaGoat 3mo ago
    OK, well, MTW is a pretty standard GR textbook and it is often cited as a useful text on differential forms for math.
    - Certhas 3mo ago
      Did you mean to reply to someone else? For the record, MTW by now also shows it's age. When I was doing research in GR adjacent fields the experts were rather recommending Wald. Might have just been my bubble of course...
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Certhas 3mo ago
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.,
- Steuard 3mo ago
  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.)
  - Certhas 3mo ago
    I am talking about some very specialised results in an application area. Not Electro Magnetism which I believe you are referring to? The formulation of EM with antisymmetric field tensors (which is the same thing as a bivector) is certainly very old and absolute standard in the physics curriculum. And I certainly was taught the definition in terms of wedge products that is also given here in undergrad: https://en.wikipedia.org/wiki/Electromagnetic_tensor https://en.wikipedia.org/wiki/Electromagnetic_tensor (See the section: Relationship to the classic fields). Not sure what you are looking for beyond that?
eigenspace 3mo ago
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.
- erichocean 3mo ago
  > The only time I ever really found that GA was actually a benefit to me was performing rotations. Maybe that's why I've found it so useful when doing rigging for animation—that's the entire job!
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hodgehog11 3mo ago
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?
- TimorousBestie 3mo ago
  Yeah, while I’ve never used GMT for anything substantial I certainly like currents and densities. That reminds me, I’ve been meaning to rewrite parts of Hormander’s epic with tools from GMT but never found the time.
Lapsa 3mo ago
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blurbleblurble 3mo ago
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.
aureate 3mo ago
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 https://en.wikipedia.org/wiki/Projective_geometry
- Twisol 3mo ago
  As you say, projections and rotations are easily accounted for in linear algebra. The issue is that translations are not a linear transformation. For instance, consider f(x) = 2 + x. It's certainly not the case that f is linear -- that is, that f(cx + y) = c f(x) + f(y) -- because on the one hand we'd expect 2 + cx + y, and on the other we'd expect (2 + cx) + (2 + y), which is 4 + cx + y. However, translation is an affine transformation, which is a particular case of a projective transformation [0]. It turns out that we can represent 3D affine (and general projective) transformations using a 4x4 matrix -- that is, as linear transformations in one dimension up, in a similar sense as how we can represent complex numbers as particular 2x2 matrices [1]. So yes, projective geometry is the right theoretical lens, even if we're usually able to forget about it (somewhat) when we use matrix representations. [0]: https://en.wikipedia.org/wiki/Affine_transformation#Representation https://en.wikipedia.org/wiki/Affine_transformation#Represen... [1]: https://en.wikipedia.org/wiki/Complex_number#Matrix_representation_of_complex_numbers https://en.wikipedia.org/wiki/Complex_number#Matrix_represen...
  - aureate 3mo ago
    Ah, interesting. I see "homogeneous coordinates" are covered later in the book I've just started reading (Projective Geometry, Coxeter) as a way of representing projective space. I think that's the link I couldn't see. Thanks!
gugagore 3mo ago
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.
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- ajkjk 3mo ago
  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!).
  - gugagore 3mo ago
    https://math.ucr.edu/home/baez/torsors.html https://math.ucr.edu/home/baez/torsors.html The distinction is whether zero is meaningful independent of a choice of origin. Zero displacement is meaningful. Zero position is arbitrary. Are you thinking of displacement as an operation? Because it is just as well a vector. I don't see the connection to section I highlighted from the article.
cherryteastain 3mo ago
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/ https://corde.phy.cam.ac.uk/
- groundzeros2015 3mo ago
  > 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
  - zarzavat 3mo ago
    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.
aeonik 3mo ago
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. and: 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 Maybe this is true in some parts of the internet or in some older discourse, but from the material I have read, people seem pretty explicit about the roots of Geometric Algebra. 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%27s_Lament.pdf https://worrydream.com/refs/Lockhart_2002_-_A_Mathematician%...
Chinjut 3mo ago
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.
GYLQ 3mo ago
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srean 3mo ago
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) The geometric mean of the two points as represented by complex numbers gives you the bisector. Python has native support for complex numbers so all the computation is handled by the runtime.
- Chinjut 3mo ago
  This is like how one often wants to distinguish the points of an affine space from the vectors representing displacements in that space (there is no distinguished origin for the physical world, but there is a distinguished concept of zero displacement). One can add a vector to a point to get a point, or a vector to a vector to get a vector, but cannot add a point to a point to get another point. Yet, it is meaningful to treat a linear combination of points in an affine space as yielding another point in the same space when the weights of the linear combination sum to 1. The exact same thing is happening here, only multiplicatively, where z1^(1/2) * z2^(1/2) is a combination with two weights of 1/2 (thus, summing to 1). It is geometrically meaningful to treat 2d vectors (displacements in a plane) as complex numbers, raise them to exponents summing to 1, and then multiply these together to get another vector in the same plane. But it is not generally geometrically meaningful to just multiply one vector by another vector to get a third vector in the same space (because this would require distinguishing some particular direction and magnitude as "1").
  - srean 3mo ago
    I agree with you on three dimensional vector products. It's too special, too cute and doesn't generalize to all dimensions, and as you said, you have to keep track of the two types of vectors. On complex multiplications though, I disagree. It's a great way to do Euclidean manipulations on the 2d plane. Rotations, translations and reflections (via conjugates) are simple. You rarely need calls to trigonometric functions. If you have runtime support, it's sorta criminal not to use complex multiplication when applicable. BTW there is another, equivalent, way of deriving the solution which to me seems more intuitive (and not limited to sum of powers to 1): The angular travel from z1 to z2 is z2 / z1. I want to travel half of that, so √(z2/z1). This half travel I apply to z1 like so √(z2/z1) * z1 done. If the need was to continue to travel angularly (rotate) beyond z2, say double the subtended angle, that's easy too. No need for the constraint the sum of powers be 1.
immmmmm 3mo ago
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 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
jordigh 3mo ago
> 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.
turtleyacht 3mo ago
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 https://terathon.com/blog/poor-foundations-ga.html [2] "If you want solid foundations, this book is for you."
- TimorousBestie 3mo ago
  Rational trigonometry is useful in some contexts (I’ve optimized trig computations with it in the past) but I wouldn’t call it GA, it’s a different kind of beast.
- impendia 3mo ago
  Math professor here. If you want to learn math, then for the most part I recommend choosing time-tested avenues, using popular materials. There are two reasons for this: (1) Popular materials are usually popular for a reason: they reflect an approximate consensus, across a significant fraction of the mathematical community, that their approaches are more-or-less the best. (2) If you learn the same way everyone else does, you'll have an easier time talking to others and finding materials on the internet. I know some very innovative books which I highly recommend, for example Visual Group Theory by Nathan Carter: https://bookstore.ams.org/clrm-32/ https://bookstore.ams.org/clrm-32/ But the innovation is pedagogical, in what Carter chooses to emphasize and how he presents everything. At the book's core, Carter agrees with everyone else about what the foundations of group theory are and should be. Even Sheldon Axler's Linear Algebra Done Right (another excellent book), with its hilariously provocative title, only differs in its choice of emphasis and order of presentation. His choices are quite compatible with everyone else's. https://linear.axler.net/LADR4e.pdf https://linear.axler.net/LADR4e.pdf
jampekka 3mo ago
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.
- NooneAtAll3 3mo ago
  > I did hit a wall quite soon with GA can you give an example of what's impossible/hard to do?
  - jampekka 3mo ago
    I don't recall the details anymore, and the work never amounted to anything, but generally it was about finding expressions for some properties of the visual flow field under curvilinear motion. I can't really say if the problem was with me or GA. Probably more like GA didn't end up providing tools for my level of math skills to solve the problem. But neither did the the traditional branches.
- srean 3mo ago
  > 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.
erichocean 3mo ago
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.
rramadass 3mo ago
Background resource: Comparison of vector algebra and geometric algebra - https://en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra https://en.wikipedia.org/wiki/Comparison_of_vector_algebra_a...
adrian_b 3mo ago
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.
_alternator_ 3mo ago
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.
jmount 3mo ago
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.
dang 3mo ago
Discussed at the time: The Case Against Geometric Algebra - https://news.ycombinator.com/item?id=39576214 https://news.ycombinator.com/item?id=39576214 - March 2024 (15 comments)
CyLith 3mo ago
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.
oggreen 3mo ago
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?
- eigenspace 3mo ago
  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.
hamish_todd 3mo ago
I'm a GA researcher. I did this livestream of my reaction to the article https://m.twitch.tv/videos/2282548167 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.
- jdshaffer 3mo ago
  I've become more and more curious about GA after bumping into it when learning a little 3D programming for fun. I'm looking to learn more, wondering if it would help my understanding of physics. Do you have any resources (books, videos, etc.) you would recommend to someone wanting to learn?
  - hamish_todd 3mo ago
    If you want to learn physics from a geometric viewpoint, GA is brilliant. Here's a playlist of my stuff, which is aimed at computer graphics folks: https://youtube.com/playlist?list=PL9a8DfUJQcuA9AXqvjxYolq5-87LkVsW9 https://youtube.com/playlist?list=PL9a8DfUJQcuA9AXqvjxYolq5-... Leading up to classical mechanics, you have the sibgraphi tutorials: https://youtube.com/playlist?list=PLsSPBzvBkYjxrsTOr0KLDilkZaw7UE2Vc https://youtube.com/playlist?list=PLsSPBzvBkYjxrsTOr0KLDilkZ... (I also recommend the bivector discord if you want a community) And from there, sudgy's videos are good, and the tome "geometric algebra for physicists" packs in a huge amount of stuff.
    - jdshaffer 3mo ago
      Thank you so very much, I'll definitely check them out!
- aeve890 3mo ago
  >TLDR it is quite a bad article You can write a rebuttal to address what's wrong with the article, from your point of view. Maybe I'm old but the whole "live reaction in twitch" thing doesn't help how the scientific community perceives your area of expertise.
markgall 3mo ago
Is this really a big debate? I am in the similarly-named (but apparently distant) field of algebraic geometry and have never even heard of geometric algebra. Certainly I know about Clifford and exterior algebras, but this debate has never reached me.
- hamish_todd 3mo ago
  In physics and especially computer graphics, yes. These are two fields that are culturally very different from algebraic geometry, in that mathematical techniques that appear at all "exotic" are "hard to sell", even if they're useful. Hence having to have a more marketable name for Clifford algebra ("geometric algebra" is just Clifford algebra), and having to have endless screaming from the rooftops for people to learn the slightest thing (like "multiplication is transform composition").

137 points by Hbruz0 4mo ago

jdw64 4mo ago

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

jdw64 3mo ago
Reading this article, I think there are quite a few interesting points to consider further. C started as a DSL for the Unix kernel. JavaScript is also a DSL, and successful languages are often described as DSLs in certain respects. Then, as they grow and gain broader adoption, they evolve into general purpose languages. But if you think about it the other way around, since all programs are ultimately about data transformation, you could argue that UIs should essentially be drawn in SQL, but that would sound strange. That's because the tools we use have moved away from that mental model. (Though React's FRP premise does lean in that direction.) And when I think about why languages split apart, it seems to me that it's because the word 'programming' covers so many different things at once. Languages end up diverging because they serve different purposes. In fact, as a programmer, I see programming languages as a collection of tools that essentially decide what to give up. C gives you safety and low-level hardware access through its ABI. Python gives you expressiveness. They exist because their target goals are fundamentally different. In that sense, though I'm not an expert in this field, from my limited perspective this debate feels like it's just the noise that arises when Algebra tries to encompass too much and inevitably splits apart. I imagine these kinds of cases will only increase in the future. As things become more specialized, there will be more situations where existing frameworks don't fit, and new systems will be needed. Is there a term for this phenomenon? At that point, we might say we need to change the old system to fit the new one. Personally, I wonder if there isn't a general purpose language at the bottom that models the entire world, with other languages layered on top of it.
eigenspace 3mo ago
> this GA is not Clifford Algebra What makes you say that?
- jdw64 3mo ago
  No, that's not my own argument — it's just how I understood the article's claim. I only know GA as something used for specific purposes. I was just sharing my thoughts on what the article was saying.
jiggawatts 3mo ago
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 the thing to work".[1] Certain kinds of perfect correctness are like pure and shining crystallised bits of refined knowledge created by the greatest wizards. "Parse, don't validate" or "Make invalid states unrepresentable." ought to be familiar to the better programmers here, the ones with decades of experience built on iterative, collaborative foundations with real consequences for error. Theoretical physics doesn't have those same consequences, because there is no real punishment for their equivalent of "spaghetti code". Perversely, there's cachet to be gained for gaining understanding of its unnecessarily esoteric knowledge, much like how biologists and lawyers spend half a decade or more studying... Latin.[2] Introducing Geometric Algebra to physics is like that wizard coder who sweeps away reams of spaghetti code and replaces it all with a call to a single standard library function. It's that "cheff's kiss" of cleanup. Meanwhile the juniors are screaming about how the senior "deleted all their hard work!" Meanwhile, I never understood where Pauli and Dirac matrices came from! It's like they were pulled from fat air. You've seen this in code, I bet. Some junior worked really hard on solving a problem and wrote a solid screen-filling wall of "a && b || c || !d && e && (f || g)..." continuing up to "ba, bc, bd", etc.. as they ran out single letters until they're well into the alphabet in double-character symbols.[3] That's what those matrices are. Someone's hacky attempt at "making things work". The problem is that we gave those people Nobel prizes and told everyone they're geniuses. They are, but they were like that brilliant junior. Brilliant.. but junior. Geometric Algebra sweeps all of that into one beautiful, consistent, crystal clear abstraction that is widely applicable. The magic matrix constants vanish. Bugs in 100-year-old textbook formulas suddenly come to light. Dozens of formulas, one set for each of the 1D, 2D, 3D, and 4D cases collapse into a single formula valid for any number of dimensions. It's like watching someone struggle with "catching every possible instance of JavaScript injection". No son, no. Just no. Stop enumerating badness. Stop. Just stop. Escape everything at the boundary instead, enforced by the type system. You'll thank me later. I know it might be obvious to you, and you always use properly parameterised SQL queries or whatever. This is not the norm everywhere! I still get arguments, long drawn out arguments from people convinced that this is unnecessary and just one more search & replace is all they need to be safe from the bad hackers. Physicists (and mathematicians) are still making that argument against GA. "It's isomorphic!" "That isn't the point!" [1] You can't convince someone to climb Everest if they struggled to hike up to the top of one of its foothills. [2] Let me be crystal clear: They're spending their precious time on this Earth learning a dead language instead of learning about the law or bugs. No amount of arguments will sway me. The bugs don't care what you call them. Criminals are guilty or innocent whether or not you speak funny in court. You've just made a simple thing harder for no good reason, that is all. Please stop. [3] Yes, I've seen this. Twice, from two different people whom have never met. Aliens are amongst us.
TimorousBestie 3mo ago
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 support this claim, so I suppose I’m a proponent of geometric algebra. I think it’s more or less been carried out for vector calculus by Spivak’s “classical” Calculus on Manifolds, which is somewhat widely taught. > Claim 2: That the Geometric Product (henceforth: GP) should be added to that list as the most fundamental operation, where by “fundamental” I mean that other operations should be constructed in terms of it, and theorems should be stated using it. Like the author, I also believe this claim is nonsense. “Rewriting classical linear algebra” is a honored pastime but it’s very difficult to make any headway doing it—the classical texts are classical for a reason, we more or less know how to teach them as an “80% solution” and it’s unclear that the investment in a new pedagogy would get us to an “81% solution.” Especially with today’s undergrads. If you’re not churning arithmetic, they’re not into it.
- jdw64 3mo ago
  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!
- eigenspace 3mo ago
  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.
QuesnayJr 3mo ago
GA is Clifford algebra plus a bunch of new terminology plus advocacy that it should replace linear algebra.

Ainaguade 4mo ago

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jsLavaGoat 3mo ago

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.

Certhas 3mo ago
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 much more niche and specialized subjects and not how the subject should be taught when first encountering it. https://en.wikipedia.org/wiki/Covariant_formulation_of_classical_electromagnetism https://en.wikipedia.org/wiki/Covariant_formulation_of_class...
- jsLavaGoat 3mo ago
  OK, well, MTW is a pretty standard GR textbook and it is often cited as a useful text on differential forms for math.
  - Certhas 3mo ago
    Did you mean to reply to someone else? For the record, MTW by now also shows it's age. When I was doing research in GR adjacent fields the experts were rather recommending Wald. Might have just been my bubble of course...
- 3mo ago

Certhas 3mo ago

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.,

Steuard 3mo ago
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.)
- Certhas 3mo ago
  I am talking about some very specialised results in an application area. Not Electro Magnetism which I believe you are referring to? The formulation of EM with antisymmetric field tensors (which is the same thing as a bivector) is certainly very old and absolute standard in the physics curriculum. And I certainly was taught the definition in terms of wedge products that is also given here in undergrad: https://en.wikipedia.org/wiki/Electromagnetic_tensor https://en.wikipedia.org/wiki/Electromagnetic_tensor (See the section: Relationship to the classic fields). Not sure what you are looking for beyond that?

eigenspace 3mo ago

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.

erichocean 3mo ago
> The only time I ever really found that GA was actually a benefit to me was performing rotations. Maybe that's why I've found it so useful when doing rigging for animation—that's the entire job!

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hodgehog11 3mo ago

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?

TimorousBestie 3mo ago
Yeah, while I’ve never used GMT for anything substantial I certainly like currents and densities. That reminds me, I’ve been meaning to rewrite parts of Hormander’s epic with tools from GMT but never found the time.

Lapsa 3mo ago

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blurbleblurble 3mo ago

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.

aureate 3mo ago

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 https://en.wikipedia.org/wiki/Projective_geometry

Twisol 3mo ago
As you say, projections and rotations are easily accounted for in linear algebra. The issue is that translations are not a linear transformation. For instance, consider f(x) = 2 + x. It's certainly not the case that f is linear -- that is, that f(cx + y) = c f(x) + f(y) -- because on the one hand we'd expect 2 + cx + y, and on the other we'd expect (2 + cx) + (2 + y), which is 4 + cx + y. However, translation is an affine transformation, which is a particular case of a projective transformation [0]. It turns out that we can represent 3D affine (and general projective) transformations using a 4x4 matrix -- that is, as linear transformations in one dimension up, in a similar sense as how we can represent complex numbers as particular 2x2 matrices [1]. So yes, projective geometry is the right theoretical lens, even if we're usually able to forget about it (somewhat) when we use matrix representations. [0]: https://en.wikipedia.org/wiki/Affine_transformation#Representation https://en.wikipedia.org/wiki/Affine_transformation#Represen... [1]: https://en.wikipedia.org/wiki/Complex_number#Matrix_representation_of_complex_numbers https://en.wikipedia.org/wiki/Complex_number#Matrix_represen...
- aureate 3mo ago
  Ah, interesting. I see "homogeneous coordinates" are covered later in the book I've just started reading (Projective Geometry, Coxeter) as a way of representing projective space. I think that's the link I couldn't see. Thanks!

gugagore 3mo ago

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.

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ajkjk 3mo ago
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!).
- gugagore 3mo ago
  https://math.ucr.edu/home/baez/torsors.html https://math.ucr.edu/home/baez/torsors.html The distinction is whether zero is meaningful independent of a choice of origin. Zero displacement is meaningful. Zero position is arbitrary. Are you thinking of displacement as an operation? Because it is just as well a vector. I don't see the connection to section I highlighted from the article.

cherryteastain 3mo ago

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/ https://corde.phy.cam.ac.uk/

groundzeros2015 3mo ago
> 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
- zarzavat 3mo ago
  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.

aeonik 3mo ago

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. and: 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 Maybe this is true in some parts of the internet or in some older discourse, but from the material I have read, people seem pretty explicit about the roots of Geometric Algebra. 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%27s_Lament.pdf https://worrydream.com/refs/Lockhart_2002_-_A_Mathematician%...

Chinjut 3mo ago

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.

GYLQ 3mo ago

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srean 3mo ago

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) The geometric mean of the two points as represented by complex numbers gives you the bisector. Python has native support for complex numbers so all the computation is handled by the runtime.

Chinjut 3mo ago
This is like how one often wants to distinguish the points of an affine space from the vectors representing displacements in that space (there is no distinguished origin for the physical world, but there is a distinguished concept of zero displacement). One can add a vector to a point to get a point, or a vector to a vector to get a vector, but cannot add a point to a point to get another point. Yet, it is meaningful to treat a linear combination of points in an affine space as yielding another point in the same space when the weights of the linear combination sum to 1. The exact same thing is happening here, only multiplicatively, where z1^(1/2) * z2^(1/2) is a combination with two weights of 1/2 (thus, summing to 1). It is geometrically meaningful to treat 2d vectors (displacements in a plane) as complex numbers, raise them to exponents summing to 1, and then multiply these together to get another vector in the same plane. But it is not generally geometrically meaningful to just multiply one vector by another vector to get a third vector in the same space (because this would require distinguishing some particular direction and magnitude as "1").
- srean 3mo ago
  I agree with you on three dimensional vector products. It's too special, too cute and doesn't generalize to all dimensions, and as you said, you have to keep track of the two types of vectors. On complex multiplications though, I disagree. It's a great way to do Euclidean manipulations on the 2d plane. Rotations, translations and reflections (via conjugates) are simple. You rarely need calls to trigonometric functions. If you have runtime support, it's sorta criminal not to use complex multiplication when applicable. BTW there is another, equivalent, way of deriving the solution which to me seems more intuitive (and not limited to sum of powers to 1): The angular travel from z1 to z2 is z2 / z1. I want to travel half of that, so √(z2/z1). This half travel I apply to z1 like so √(z2/z1) * z1 done. If the need was to continue to travel angularly (rotate) beyond z2, say double the subtended angle, that's easy too. No need for the constraint the sum of powers be 1.

immmmmm 3mo ago

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 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

jordigh 3mo ago

> 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.

turtleyacht 3mo ago

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 https://terathon.com/blog/poor-foundations-ga.html [2] "If you want solid foundations, this book is for you."

TimorousBestie 3mo ago
Rational trigonometry is useful in some contexts (I’ve optimized trig computations with it in the past) but I wouldn’t call it GA, it’s a different kind of beast.
impendia 3mo ago
Math professor here. If you want to learn math, then for the most part I recommend choosing time-tested avenues, using popular materials. There are two reasons for this: (1) Popular materials are usually popular for a reason: they reflect an approximate consensus, across a significant fraction of the mathematical community, that their approaches are more-or-less the best. (2) If you learn the same way everyone else does, you'll have an easier time talking to others and finding materials on the internet. I know some very innovative books which I highly recommend, for example Visual Group Theory by Nathan Carter: https://bookstore.ams.org/clrm-32/ https://bookstore.ams.org/clrm-32/ But the innovation is pedagogical, in what Carter chooses to emphasize and how he presents everything. At the book's core, Carter agrees with everyone else about what the foundations of group theory are and should be. Even Sheldon Axler's Linear Algebra Done Right (another excellent book), with its hilariously provocative title, only differs in its choice of emphasis and order of presentation. His choices are quite compatible with everyone else's. https://linear.axler.net/LADR4e.pdf https://linear.axler.net/LADR4e.pdf

jampekka 3mo ago

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.

NooneAtAll3 3mo ago
> I did hit a wall quite soon with GA can you give an example of what's impossible/hard to do?
- jampekka 3mo ago
  I don't recall the details anymore, and the work never amounted to anything, but generally it was about finding expressions for some properties of the visual flow field under curvilinear motion. I can't really say if the problem was with me or GA. Probably more like GA didn't end up providing tools for my level of math skills to solve the problem. But neither did the the traditional branches.
srean 3mo ago
> 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.

erichocean 3mo ago

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.

rramadass 3mo ago

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

adrian_b 3mo ago

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.

_alternator_ 3mo ago

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.

jmount 3mo ago

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.

dang 3mo ago

Discussed at the time: The Case Against Geometric Algebra - https://news.ycombinator.com/item?id=39576214 https://news.ycombinator.com/item?id=39576214 - March 2024 (15 comments)

CyLith 3mo ago

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.

oggreen 3mo ago

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?

eigenspace 3mo ago
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.

hamish_todd 3mo ago

I'm a GA researcher. I did this livestream of my reaction to the article https://m.twitch.tv/videos/2282548167 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.

jdshaffer 3mo ago
I've become more and more curious about GA after bumping into it when learning a little 3D programming for fun. I'm looking to learn more, wondering if it would help my understanding of physics. Do you have any resources (books, videos, etc.) you would recommend to someone wanting to learn?
- hamish_todd 3mo ago
  If you want to learn physics from a geometric viewpoint, GA is brilliant. Here's a playlist of my stuff, which is aimed at computer graphics folks: https://youtube.com/playlist?list=PL9a8DfUJQcuA9AXqvjxYolq5-87LkVsW9 https://youtube.com/playlist?list=PL9a8DfUJQcuA9AXqvjxYolq5-... Leading up to classical mechanics, you have the sibgraphi tutorials: https://youtube.com/playlist?list=PLsSPBzvBkYjxrsTOr0KLDilkZaw7UE2Vc https://youtube.com/playlist?list=PLsSPBzvBkYjxrsTOr0KLDilkZ... (I also recommend the bivector discord if you want a community) And from there, sudgy's videos are good, and the tome "geometric algebra for physicists" packs in a huge amount of stuff.
  - jdshaffer 3mo ago
    Thank you so very much, I'll definitely check them out!
aeve890 3mo ago
>TLDR it is quite a bad article You can write a rebuttal to address what's wrong with the article, from your point of view. Maybe I'm old but the whole "live reaction in twitch" thing doesn't help how the scientific community perceives your area of expertise.

markgall 3mo ago

Is this really a big debate? I am in the similarly-named (but apparently distant) field of algebraic geometry and have never even heard of geometric algebra. Certainly I know about Clifford and exterior algebras, but this debate has never reached me.

hamish_todd 3mo ago
In physics and especially computer graphics, yes. These are two fields that are culturally very different from algebraic geometry, in that mathematical techniques that appear at all "exotic" are "hard to sell", even if they're useful. Hence having to have a more marketable name for Clifford algebra ("geometric algebra" is just Clifford algebra), and having to have endless screaming from the rooftops for people to learn the slightest thing (like "multiplication is transform composition").