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The case against geometric algebra (2024)

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Re: The case against geometric algebra (2024)

#131

Post by chombier »

ajkjk wrote: post #124
chombier wrote: post #91

Earlier quoted context omitted.

> pull out Hodge Duals every time you want to do something that involves the metric, but I'm also unconvinced that geometric algebra is the answer here. I don't know, I recently tried to work out how the metric on vectors/1-forms induces a metric on higher-degree forms, and if the geometric product magically gives this for free I'd say it's a win (same for the Hodge star).

That comes from exterior algebra on its own, it's the k'th exterior power of the metric. Best not to conflate that with GA (unless I'm misunderstanding what you're talking about).

IIRC there's a fairly natural positive definite quadratic form on GA (used as the canonical norm) that takes the scalar part of the geometric product of a multi-vector and its reverse.

On the other hand, there's the k-th exterior power of the metric where one asks that wedge/interior products be adjoint in order to extend the metric to higher-degree forms.

I was under the impression that these metrics are the same, but maybe I'm completely wrong? Assuming I'm not, then the GA approach seems more natural to me.

Re: The case against geometric algebra (2024)

#132

Post by gugagore »

ajkjk wrote: post #72
gugagore wrote: post #28

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

One finds in regular vector algebra that "position vectors" and "displacement vectors" are sort of two distinct types of objects, and that it is never physically valid to add two position vectors together unless you create an affine combination like (a+b)/2. A position vector 'a' is really 'O + a', so [(O+a) + (O + b)]/2 = O + (a+b)/2, another position... but a+b on its own would really be (O +a) + (O + b) = 2O + a +…

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.

Re: The case against geometric algebra (2024)

#133

Post by gugagore »

jaen wrote: post #87
gugagore wrote: post #28

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

This is like a type theory question - do you like it untyped or typed? In physics, values have units too. Analogously, you could say - why incorporate units into the algebra in physics (as is often done)? Why not just add scalars etc. and not bother carrying around the units everywhere? Well, because doing anything else is mostly nonsensical - it does not make sense to add meters and seconds together. Using unit alge…

I follow your connection, though I think your unit analogy is a strawman.

Do you want to give two different types to complex numbers, depending on whether a given complex number represents a point versus a transformation (an amplitwist)?

Re: The case against geometric algebra (2024)

#134

Post by oddthink »

eigenspace wrote: post #14
jsLavaGoat wrote: post #5

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

> From a mathematician's point of view, yes, you should write the Maxwell field equations, at least to see it once, that way because you're showing a very low-level symmetry that even the differential forms approach doesn't get all the way to. Differential forms is a standard approach for general relativity, e.g. MTW. While it's neat to write them all as one equation, I disagree that it's an enlightening perspective…

I've found differential forms to be more useful than GA, but that might just be that I was brought up in the MTW tradition and don't quite get GA.

Whenever I look at GA, I try to figure out where the metric comes in, and I just don't see it.

For context, way back when I did astro theory and wanted to do things like figure out things like the magnetic field structure in the curved spacetime near highly-magnetized rotating conducting spheres, and then do some basic plasma physics in that environment.

The differential geometry approach at least gives the structure to think about that, then you can go down to the index-style notation to actually get the differential equations you need to solve. The GA approach, I'm not even sure how to frame the problem.

Re: The case against geometric algebra (2024)

#135

Post by jaen »

gugagore wrote: post #133
jaen wrote: post #87

Earlier quoted context omitted.

This is like a type theory question - do you like it untyped or typed? In physics, values have units too. Analogously, you could say - why incorporate units into the algebra in physics (as is often done)? Why not just add scalars etc. and not bother carrying around the units everywhere? Well, because doing anything else is mostly nonsensical - it does not make sense to add meters and seconds together. Using unit alge…

I follow your connection, though I think your unit analogy is a strawman. Do you want to give two different types to complex numbers, depending on whether a given complex number represents a point versus a transformation (an amplitwist)?

Ideally, yes. Depends on the context. Same way you don't use unit algebra when doing trivial everyday math. At some level of complexity, it becomes worthwhile.

Re: The case against geometric algebra (2024)

#136

Post by kortex »

jiggawatts wrote: post #13
jdw64 wrote: post #2

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

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…

I find it really fascinating that you use the metaphor to GA of a senior dev sweeping all the cruft away into a single clean abstraction, is that my read/smell of TFA (as a layperson for this depth of math) is that GA runs the exact same risk of leaky abstraction. It's really general and elegant and covers all these cases with a single abstraction, but in doing so it sometimes conflates very similar things (precisely because they are so similar) when they really ought to be separate things. Like a complex number and the rotation operator it performs. My seat-of-pants take is GA is just a bit too DRY.

My understanding is too shallow to get why we don't just go straight for EA/Clifford Algebra when the "lower" systems like cross product are insufficient.

I share the author's intuition that there ought to exist some mathematical object that begets Clifford Algebras and multivectors and GA and all the like that we have yet to discover.

> They're spending their precious time on this Earth learning a dead language instead of learning about the law or bugs

I know this is hyperbole, but it's my opinion Latin/Greek emerged so dominantly in law/bio/medical fields is that it allows at the same time semantic bleaching and composition. "Jargon is a DSL" if you will. Sure, you could say "heart muscle no worky cause insufficient oxygen" but "myocardial infarction" is a) more concise b) comprises reusable composable pieces of meaning (myo + card + -ial, in + farcire + -ion) c) most importantly, is extremely precise. It's like the trouble of using English + LLM to define a program, vs just writing damn code. Sure you can do the former, but it's lossy, and that lossiness causes issues.

Re: The case against geometric algebra (2024)

#137

Post by BigTTYGothGF »

jiggawatts wrote: post #115
BigTTYGothGF wrote: post #94

Earlier quoted context omitted.

> biologists and lawyers spend half a decade or more studying... Latin.[2] > [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 n…

Hyperbole as literary device, not sworn testimony. Argumentum ad literalismum: dismissed.

I'll just assume all the rest of your claims are hyperbole then.

Re: The case against geometric algebra (2024)

#138

Post by hamish_todd »

ajkjk wrote: post #123
hamish_todd wrote: post #110

Earlier quoted context omitted.

The geometric product is transform composition. TRANSFORM COMPOSITION! (sorry, it's not your fault or even his that you didn't know this - GA textbooks should have it as the first thing they teach but they don't)

I completely agree, incidentally... And I have come to see this more clearly since writing the article, in part due to talking to you. Maybe I should update it accordingly. I've been meaning to write a followup where I explain this viewpoint but have had trouble finding the mental energy for it.

Ah! If you do it would probably would need to be a followup rather than addendum

Re: The case against geometric algebra (2024)

#139

Post by ajkjk »

hamish_todd wrote: post #138
ajkjk wrote: post #123

Earlier quoted context omitted.

I completely agree, incidentally... And I have come to see this more clearly since writing the article, in part due to talking to you. Maybe I should update it accordingly. I've been meaning to write a followup where I explain this viewpoint but have had trouble finding the mental energy for it.

Ah! If you do it would probably would need to be a followup rather than addendum

Well, I think my points in the parent article here do stand (I'm aware that you do not). The fact that there is a good interpretation of the geometric product in some cases does not obviate the fact that everyone's writing crappy intuitive things about it. Anyway it has always been my stance that there is a _good_ version of GA, and people need to figure out what it is and write about that instead of bloviating about how good the current version is. Treating the GP as a composition of operators is a start, but it's not the whole picture --- why do operators compose in that particular way? why are these your operators in the first place? My hunch is that GA is really a subset of a larger and more intuitive algebra that has very obvious answers to these questions, probably from a representation-theory perspective.

(I meant to do this followup soon after the first article but I've been been having a lot of difficulty focusing / constant brain fog for the past few years so it's been on the to-do list instead; part of the problem is that to do it right I need to go read and digest everyone's different treatises / expositions on GA ... and that has felt taxing, to say the least.)

Re: The case against geometric algebra (2024)

#140

Post by ajkjk »

gugagore wrote: post #132
ajkjk wrote: post #72

Earlier quoted context omitted.

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 +…

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.

I think of displacement vectors and position vectors as having different "types". Position vectors are geometric objects whose meaning comes from whatever physical system you're studying (this way of thinking is generally useful even if you're not doing physics; for instance if you're talking about a manifold you would think of that as the physical system independent of the coordinates you put on it).

In the same way I think of "vectors as operators" (rotations/scaling) as a displacement vector / torsor, but of a different type than their sense as translations. As far as I can tell, the geometric product between two displacement vectors is not so meaningful, whereas the geometric product between two "operator" vectors is (because it composes them as operators, in some sense). But in practice you're often rapidly switching between these representations so it's hard to tell which object you're actually talking about. For this reason I find it useful to distinguish their types explicitly.

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