GoKawiilEvery once in a while the internet gets talking about Geometric Algebra (henceforth GA) and how it’s a new theory of math that fixes everything that’s wrong with linear algebra and multivariable calculus. When I come across this stance I am compelled to respond with something like: “wait wait, it’s not true! GA is clearly onto something but there’s also a lot wrong with it. What you probably want is just the concepts of multivectors and the wedge product!” Which is not very effective, because it takes a long time to convince anyone why, and it’s also not very productive, because this just keeps happening over and over without anything changing.
Many people agree with me on this, but they deal with it by mostly ignoring GA instead of complaining about it. But I actually like what GA is trying to do and I want it to succeed. So today I’m going to actually make those points in a longer article that I can link to instead.
Specifically what I have a problem with is that the subject is pretty clearly flawed and needs serious work, and especially that the culture around it does not seem to realize this or be interested in addressing those flaws. In particular: Hestenes’ Geometric Product is not a very good operation and we should not be rewriting all of geometry in terms of it. For some reason GA is obsessed with the geometric product, and it’s causing all sorts of problems. They act like this is clearly the way that geometry should be done and everyone else can’t just see it yet, and they have this weird religious zeal about it that is problematic and offputting. It’s also just ineffective: treating certain models as if they are somehow canonical and obvious is wrong, mathematically and socially, and it puts people off right from the start. There probably is a place for the geometric product in a grand theory of geometry, but it’s not front-and-center like GA has it today, and as a result the theory is a lot less compelling than it could be.
If something like GA is to succeed, it will need to be improved. It will need to fix the problems with establishment mathematics better than it does now, in a way that everyone can get behind. Today it helps sometimes but often misses the mark, and people who can see that are alienated by the lack of self-awareness about this. As a result GA’s relationship to mainstream mathematics is tenuous.
Basically, GA is considered a kooky, crackpotty sideshow. And because it is so dubious and un-self-aware, the movement ends up alienating most people, except for a particular type of… zealous individual… who write about it with a sort of pseudoreligious zeal, and are prone to conspiracy, as if the only reason GA is not mainstream is that they are being oppressed by close-minded traditionalism. If that’s what you think, let me be the first to inform you: no, that’s not it. GA is interesting, but it’s just not very compelling at the moment.
GA continues to find more enthusiasts every year, because it really does address some actual problems. And it will take those new people a while to realize what’s going on—that the thing they’ve discovered is not as solid and revolutionary as they think it is. In the meantime they will go on selling other people on GA, creating the next years’ converts and repeating the cycle.
My opinion is that this dynamic is causing GA to be stuck in a sort of perpetual mediocrity, where everyone’s defending the surface-level philosophy because they think they’re part of a revolution, but nobody’s bothering to criticize or improve the underlying structural problems. My purpose in writing this is to push it to improve and address those problems. They’re very fixable, but first you have to notice that something is wrong.
The rest of this article substantiates my stance. It is very long, because I decided to include every argument I could think of. But I want to emphasize that, although this is my own long and opinionated rant with lots of individual parts that nobody else is really saying, I am far from the only person who believes the big picture (quite a few have emailed me to agree on this). The state of things is that lots of credible people just roll their eyes at GA and then move on and ignore it. But I actually do believe in GA’s philosophical project: math should be changing in this direction. So I would like to see that change, and to do that we need to establish what’s wrong with it now.
Big disclaimer: I’m not a professional mathematician, and this is not going to be the case that a serious mathematician would make, which would probably be something like “GA doesn’t prove anything new so who cares?” (Fine, but the goals of research mathematics are rather unrelated to the goals of people who use mathematics for practical purposes.) Also, since I do more-or-less subscribe to the underlying program of GA, I am at least slightly on the crank side of the fence as well. Take me seriously at your own risk.
First I am going to describe my understanding of what GA is, how it got to be that way, and where it lives in relation to the rest of math and physics. This will be useful in order to pinpoint exactly what we’re disagreeing about here. As far as I know the whole story isn’t really documented anywhere else, so I’m just going off what I’ve picked up over the years. But I am no historian and don’t really know how to check it against reality; I’d be happy to be corrected on anything in here.
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