The case against geometric algebra

Every 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.