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The case against geometric algebra (2024)

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Re: The case against geometric algebra (2024)

#91

Post by chombier »

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…

> 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).

Re: The case against geometric algebra (2024)

#92

Post by zarzavat »

SonOfLilit wrote: post #70
zarzavat wrote: post #45

Earlier quoted context omitted.

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.

Author doesn't argue against the idea of choosing a new notation, he makes very detailed arguments about why this specific new notation is clumsy to work with.

Yes but my point is that their argument is undermined by the extreme clumsiness of standard mathematical physics notation. I don't believe GA is the best possible notation for physics but it could be a stepping stone. We need more people who explore such things rather than more people who call each other crackpots.

Re: The case against geometric algebra (2024)

#93

Post by jampekka »

srean wrote: post #78
jampekka wrote: post #76

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

> I don't care about proofs or theoretically problematic corner cases that "real mathematics" seems to be almost exclusively interested in. That is a rather strange take for a software engineer. When implementing something I do need to know what the corner cases are, whether the runtime can enter such a state. I need to think how to put in checks so that they cannot be reached, or alternatively, how to recover gracef…

I didn't mean corner cases like that. I mean stuff that has no practical consequences, at least for most practical use.

E.g. whether or not Navier-Stokes can form singularities doesn't really change how you analyze fluid dynamics in engineering. This doesn't mean it is not a mathematically important question worth extensive study, but it's not relevant for practitioners.

Re: The case against geometric algebra (2024)

#94

Post by BigTTYGothGF »

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…

> 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 no good reason, that is all. Please stop.

The absurdity of this claim is enough to call into question everything else in your post.

Re: The case against geometric algebra (2024)

#95

Post by bmacho »

NooneAtAll3 wrote: post #79
jsLavaGoat wrote: post #5

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

what is MTW?

https://en.wikipedia.org/wiki/Gravitation_(book)

Re: The case against geometric algebra (2024)

#96

Post by _alternator_ »

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.

Re: The case against geometric algebra (2024)

#97

Post by jmount »

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.

Re: The case against geometric algebra (2024)

#98

Post by eigenspace »

chombier wrote: post #91
eigenspace wrote: post #14

Earlier quoted context omitted.

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

> 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).

Both differential forms and geometric algebra are awkward for that sort of thing. I'd just stick with abstract index notation most of the time.

Re: The case against geometric algebra (2024)

#99

Post by srean »

srean wrote: post #55
Chinjut wrote: post #53

Earlier quoted context omitted.

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…

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 function…

@chinjut yes you are exactly right about sum to 1 bit.

For a moment I had got distracted by the exponential between Lie group and algebra.

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