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The case against geometric algebra (2024)

alexkritchevsky.com

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Re: The case against geometric algebra (2024)

#111

Post by amelius »

hamish_todd wrote: post #110
cherryteastain wrote: post #30

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 PhD…

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)

Is that linear composition or affine?

Re: The case against geometric algebra (2024)

#112

Post by hamish_todd »

amelius wrote: post #111
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)

Is that linear composition or affine?

Can do either! If your multivectors represent linear transformations then their products will also be linear transformations. If they're affine transformations then their products will be affine transformations. Euclidean, conformal, etc!

Re: The case against geometric algebra (2024)

#113

Post by hamish_todd »

groundzeros2015 wrote: post #33
cherryteastain wrote: post #30

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 PhD…

> 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

He's distressingly wrong about this, the geometric product is transform composition

(Apologies to folks who have seen me repeat this a million times; but it's very important folks be aware of it)

Re: The case against geometric algebra (2024)

#114

Post by markgall »

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.

Re: The case against geometric algebra (2024)

#115

Post by jiggawatts »

BigTTYGothGF wrote: post #94
jiggawatts wrote: post #13

Earlier quoted context omitted.

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 n…

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

Re: The case against geometric algebra (2024)

#116

Post by jiggawatts »

SonOfLilit wrote: post #59
jiggawatts wrote: post #13

Earlier quoted context omitted.

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…

(This is a nitpick and does not argue against your main claim that GA is a better abstraction to represent and solve physics problems with, that I have no way to evaluate because I don't speak GA, though now I'm curious and will maybe spend an afternoon trying to figure out) I mean, come on, lawyers and biologists don't really spend half a decade studying Latin. You can tell because smart people that spend a year or…

To be honest I was struggling to phrase my argument in a cohesive narrative without it turning into a ten page blog post.

The point I’m trying to make is that there are necessarily complexities inherent in all areas of study, and there are incidental complexities because of historical reasons, “culture” within certain fields, or juniors putting out their fields’ equivalent of spaghetti code.

Geometric Algebra sweeps away a lot of the rather messy parts of now century-old physics, but the work of doing that substitution is decidedly non-trivial and thankless, so other than Hestenes, nobody seems to be pushing for it.

It’s like the 2pi versus tau fad on the internet.

Mathematicians argue that they’re “the same”, so it doesn’t matter, and ramble on about their equivalent of “learn the Latin to be smart like me”.

No. It’s stupid. It was an error. Tau is the correct circle constant and eliminates magic constants that don’t belong from literally hundreds of famous formulas!

I and many others simply failed to understand radians until I learnt to treat 2pi as a single ligature instead of “two of something”.

Re: The case against geometric algebra (2024)

#117

Post by Certhas »

jsLavaGoat wrote: post #26
Certhas wrote: post #10

Earlier quoted context omitted.

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

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

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...

Re: The case against geometric algebra (2024)

#118

Post by Certhas »

Loquebantur wrote: post #57
Certhas wrote: post #10

Earlier quoted context omitted.

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

One should teach the next generation the best way possible, and not turn them into conformists. "Standards" are things to be overcome when they've outlived their prime. Disparaging new ideas as "niche" and "specialised" when their explicit aspiration is to be better foundations is motivated reasoning.

The ideas of GA aren't new. As the article explains, the ideas of Clifford algebras are common place throughout research and deserve to be introduced earlier. Understanding bivectors and wedge products is much more important than, for example, Euler angles in my opinion.

The geometric product on the other hand obscures much of the structure, and serves no pedagogical or fundamental purpose. Not everything that's new is better, just by virtue of being new.

You might have misunderstood my point about what's niche, or I misunderstood which formulation of Maxwell the post I was replying to was referring to. Either way, it feels this discussion went off the rails rather immediately...

Re: The case against geometric algebra (2024)

#119

Post by SonOfLilit »

zarzavat wrote: post #92
SonOfLilit wrote: post #70

Earlier quoted context omitted.

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.

Author is not calling them crackpots, and _is_ strongly advocating for some of their new notation, and explicitly encouraging readers to find better notation where he dnsagrees with theirs.

Re: The case against geometric algebra (2024)

#120

Post by SonOfLilit »

jiggawatts wrote: post #116
SonOfLilit wrote: post #59

Earlier quoted context omitted.

(This is a nitpick and does not argue against your main claim that GA is a better abstraction to represent and solve physics problems with, that I have no way to evaluate because I don't speak GA, though now I'm curious and will maybe spend an afternoon trying to figure out) I mean, come on, lawyers and biologists don't really spend half a decade studying Latin. You can tell because smart people that spend a year or…

To be honest I was struggling to phrase my argument in a cohesive narrative without it turning into a ten page blog post. The point I’m trying to make is that there are necessarily complexities inherent in all areas of study, and there are incidental complexities because of historical reasons, “culture” within certain fields, or juniors putting out their fields’ equivalent of spaghetti code. Geometric Algebra sweeps…

I tried to make it clear that I wasn't arguing against your main point, that was made very clearly, just against a comparison you used that I think was a bit slanderous (tongue in cheek). Yes, obviously Tau is correct, and that's a better comparison to use.

Having dived deeper into the essay, author claims that some of the new notation is obviously better (clifford algebras) and the rest is overzealous unification that obscures rather than clarifies because it mixes types in a weird way (geometric product).

I've never heard of any of this before, but author's second point looks rather convincing. Can you give counterexamples, ideas that are much clearer to think about once represented using GP? I'd love to dive a bit deeper.

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