> very real toxicity within the geometric algebra community. I can’t do much about
I was hoping the article would be about this instead. OP wondering if you have any elaborations for us to hear.
I was hoping the article would be about this instead. OP wondering if you have any elaborations for us to hear.
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.
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.
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.
Or normal vs tangent vectors transforming differently?
I didn't understand this part:
> 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.
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.
But, pedagogical treatment is a separate question from what is linear algebra.
Should these all be the same wikipedia page?
- https://en.wikipedia.org/wiki/Exterior_algebra
- https://en.wikipedia.org/wiki/Multilinear_algebra
- https://www.georgehart.com/research/multanal.html (okay, not mainstream enough to have a relevant page, but it is extremely relevant in any engineering practice of linear algebra)