I'm not the OP, I'm not a part of this community, and I don't know if the thing I'm about to complain about is what the author was thinking of with this comment, but as someone who was trained as a mathematician and who has read some of the popularizations of geometric algebra that sometimes get posted to HN, there is a tone that some (though probably a minority) of them take that I find pretty obnoxious.
These pieces are the ones that take the position that geometric algebra is this super secret anti-establishment mathematical samizdat that *they* don't want you to know about. They'll pit themselves against "mainstream mathematics" and say things like, "in differential geometry you do X, but you shouldn't do differential geometry; you should do geometric algebra where we do Y, which is so much better than X."
My reaction is always, "My friend, you are doing differential geometry!" Clifford algebras --- the objects that the geometric algebra people study --- are firmly within the "mainstream" of mathematics; there's simply no conflict here, at least not of the sort that these writers often seem to be imagining. It's great that people are enjoying learning about Clifford algebras. I think Clifford algebras are really fun! But we can all just come together and enjoy them together, and I think this "join me in taking down the cabal of gatekeepers who are suppressing the truth" attitude is unnecessary and turns off a lot of people who might otherwise be fun to engage with.
If you're into this stuff and feel like this doesn't describe you or the people you know, then that's great, keep doing what you're doing! But it does exist and I wish it didn't.
I used GA as a way to bootstrap into 'real' Clifford algebras, and a way to get over a "reader's block" when it came to Lie algebras, tensors, and (finally) algebraic geometry. I'm not sure GA is great math, but it was really great way to learn "advanced math concepts" for "basic..ish math". Personally, I like Alan MacDonald's GA books — they're a great way to learn more complicated concepts, but couched in a very approachable geometry/visual learning style.
That sounds like a fun and satisfying process! I realize my comment could be taken as denigrating all the people who write about this stuff, but that's certainly not my intention; I've also enjoyed a lot of the visualizations and geometric explanations that people writing under this heading have come up with. My complaint is really just about the ones who take this oppositional attitude, and a big part of why I think it's such a shame when that happens is that there really are some very cool ideas here, so it's sad to see walls being raised for no good reason.
Mathematicians will take a moment denigrate Geometric Algebra as "linear algebra with a uselessly nonstandard notation", ignoring that we should prefer a less awkward way of structuring linear algebra than "pseudoscalars" and "pseudovectors".
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Mathematicians will take a moment denigrate Geometric Algebra as "linear algebra with a uselessly nonstandard notation", ignoring that we should prefer a less awkward way of structuring linear algebra than "pseudoscalars" and "pseudovectors".
I have never heard a mathematician using the terms "pseudoscalar" and "pseudovector". These rather seem to be common terms among physicists.
> "linear algebra with a uselessly nonstandard notation"
Let me chime in that as a physicist (who does use the "pseudo" stuff occasionally) I very much share this opinion.
The notation may be really cool and compact, but I just do not see the benefit - for example, d*F = j and dF = 0 is compact enough for me.
It is all fine if people use this language to learn linear algebra or differential geometry. And maybe it has a use for numerics or computer science. But I am quite sure that the geometric algebra formalism will not be widely adopted in physics any time soon. Sorry.
> I strongly believe that if GA would make this distinction they would lose a lot fewer people. It is a completely interesting and useful thing to talk about “a representation of a particular class of operations that makes composition and inversion easy”, and completely offputting when you blur the distinction between operators and geometric objects themselves, and write every operation in terms of the geometric product when only a few of them are really compositions of operators.
I can't tell what "a few of them" refers to. What is this potential distinction between operators and geometric objects? Sounds like the the distinction between a group action and a group object?
I am willing to believe that GA is an unnecessary renaming of other simpler things, and also that it has these kind of culty vibes, but I'm focusing on the claim that (I understood as) "unifying the operators and geometric objects" is a bad feature rather than a good feature.
So, I've amended the article some since posting it because the main objection I got from a few GA enthusiasts was this idea of the GP as being used to compose operators. Which I had barely noticed the importance of because it's so hard to identify in the texts! Although once they mentioned it I began to appreciate that, when the GP works and is useful, this is why. So I changed things to address that point more directly... but I didn't really find the energy to do a perfect job of it (like researching all the examples I would need to make the point more clearly), and it's a bit muddy at the moment. Like at the spot you pointed out. I need to figure out a clearer way to say this stuff...
What I am getting at is that if you go read, say, the Doran/Lasenby book, they start out talking about multivectors for areas and volumes and etc---and they do all this with the GP. Which makes no sense! Ever calculation they do leaves you think "huh?" The GP makes no sense at all if you're talking about units of length, area, volume, etc. Its transformation laws, its composition laws... you end up having to undo it all afterwards with a bunch of janky other operations.
But if you talk about the GP for composing reflections to make rotations, it's fine, that makes sense. I just really want this distinction to be made more clearly. I'm only interested in the GP when it corresponds to an explicit geometric operation. Nobody makes this distinction as clear as I want; I hope to eventually find a really sound version of the argument and then write it out as another article.
Roughly speaking it's equivalent to conflating the sense of a complex number as a vector with a complex number as an operator on vectors. Yes, they're isomorphic, but given a vector in R^2 there's no intrinsic sense in which you should be able to interpret it as also being the operation of multiplying by r e^(iθ) on other vectors. Pretending like they're the same thing is just bewildering: that identification between vectors and operations should be something you have to explicitly construct. For starters, if you change bases for (x,y) the vector should rotate but the rotation operation shouldn't change. That sort of thing. GA is making this same confusion but on a larger scale.
My position is that the geometric product and antiproduct are good for one thing, performing transformations with sandwich products q ⟑ p ⟑ q̃ or q ⟇ p ⟇ q̰ and composing those transformations. Literally everything else (join, meet, contraction, expansion, projection, inner product, norm, ...) can and should be done in the exterior algebra without any geometric products.
agree but I am still trying to grok the divine truth as to why exactly that is. What's up with the sandwich products? Why do they work? I guess it is like a change-of-basis for a matrix (PAP^{-1)) but I still don't quite see why, and why it works as a change-of-basis on multivectors, not just vectors.
Well by "taught in linear algebra" I mean "taught to undergraduates in their first or second year". It is totally bizarre that people learn about determinants, matrix minors, curl, magnetic fields, angular momentum, etc... but not about wedge products, in which all of these things are much easier to understand.
Totally agree! And this is why I sympathize with GA fanatics. Although it might not really make sense to do the rebranding or be maximalist about the geometric product, I do think there is a shared goal of making these things easier to understand.
Ya, as I write in my article, I'm very on board with the "change the curriculum to make more sense" project. I just want to be talking about how to do it in a critical way, rather than just going with the GA way, which requires, in my opinion, a lot of justification that I haven't seen.
These pieces are the ones that take the position that geometric algebra is this super secret anti-establishment mathematical samizdat that *they* don't want you to know about. They'll pit themselves against "mainstream mathematics" and say things like, "in differential geometry you do X, but you shouldn't do differential geometry; you should do geometric algebra where we do Y, which is so much better than X."
My reaction is always, "My friend, you are doing differential geometry!" Clifford algebras --- the objects that the geometric algebra people study --- are firmly within the "mainstream" of mathematics; there's simply no conflict here, at least not of the sort that these writers often seem to be imagining. It's great that people are enjoying learning about Clifford algebras. I think Clifford algebras are really fun! But we can all just come together and enjoy them together, and I think this "join me in taking down the cabal of gatekeepers who are suppressing the truth" attitude is unnecessary and turns off a lot of people who might otherwise be fun to engage with.
If you're into this stuff and feel like this doesn't describe you or the people you know, then that's great, keep doing what you're doing! But it does exist and I wish it didn't.