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The main problem I have with geometric algebra is that it's an alternative framework to express exactly the same things. Some things are expressed or derived e

The main problem I have with geometric algebra is that it's an alternative framework to express exactly the same things. Some things are expressed or derived easier in G.A. e.g. the Kerr solution of General Relativity, electron Zitterbewegung, etc. The problems with G.A. as I see it... 1) It's a specialized tool that does not generalize well to simple everyday physics problems. Don't get me wrong, I like G.A. and learned it from Hestenes' papers almost 20 years ago. I just don't see the point in trotting out G.A. to solve certain problems or illustrate some theory any more than I would abandon Gibbs vector calculus and insist on doing all calculations in 4-vector notation using field tensors and differential forms. I wouldn't even want to use it to teach differential geometry. Once someone knows these subjects I'd say go for it and learn G.A. It's actually very easy once you know the more conventional stuff. 2.) I don't know how things are now but in the past G.A. had the feeling of a cult theory. Unlike most cults, this one had substance. The problem learning the material first with G.A. is that you won't be able to understand other papers, books or solved problems of others. You'll be dependent on using G.A. and unable to make sense of Goldstein or Spivak. 3.) I'm all for alternative ways of doing things. If you have time to burn there's a "Bird Tracks" method of doing calculations in Lie algebras which is very different than the traditional Cartan matrix, Dynkin diagrams. I advise you to avoid learning these topics the first time around in a weird mathematical language. After you know you're way around these topics go for it.

adamnemecek 9y ago
1.) I think that therein lies the reason it's not more popular. You can achieve the same things in other formalisms but GA generally makes the most sense to me. Simplification to the point of obviation is a big win for me. 2.) Diversify. I picked it up because I felt like I can mold this better for my problems than LA. And I'm more about publishing code than publishing papers. So I care more about making it "my own" than explaining things to people. 3.) Coincidentally Lie algebras are also easier in GA. Look at this https://upload.wikimedia.org/wikipedia/commons/thumb/1/14/E8Petrie.svg/180px-E8Petrie.svg.png https://upload.wikimedia.org/wikipedia/commons/thumb/1/14/E8... and don't tell me that the symmetries seem more circular than linear.
vanderZwan 9y ago
Sorry, but pretty much every argument you make looks like a different flavour of "I already put a lot of effort into mastering other ways, and now that I have they work fine for me, I don't see why we should switch to this one." Because you seem to suffer from survivor bias, that's why. By comparison, I am a physics drop-out. Allow me to give you the view from below. First, I know this pdf is about GA in relation to physics, but you're throwing all kinds of names and terms around that only make sense to people who already have studied physics; conscious or not, that's basically gatekeeping via jargon. Claiming that we'll be unable to make sense of Goldstein and Spivak is also directly contradicting your claim that it's easier once you know the "more conventional stuff" - so learning one thing after the other is easier, but not the other way around? Why would it not be easier to learn the more conventional stuff after you have a solid GA fundation? Both my own experience and that of many commentators here seem to indicate GA is inherently more intuitive to grasp than the disparate mathematics it connects, precisely because it all seems more logically structured and connected. And perhaps the people who didn't manage to get through have a better idea of what is more intuitive to learn that the people who did here. Similarly, given that GA connects so many things together, I have a hard time believing that it is a "specialised tool", but since I never had to deal with most of the stuff you mention I guess I can't be a good judge of that (did I mention the gatekeeping?). Anyway, why should people first have to slog through all the other stuff, sometimes learning it more by rote than insight? We don't force kids to learn assembly or punch-cards before we let them play with a higher level programming language either. I'm not suggesting we throw all the old physics books out the window, but we also managed to stop doing science in Latin. That removed a huge barrier of exclusion, has not lead to a loss of knowledge, and making the field more accessible to everyone has only benefited humankind.

adamnemecek 9y ago

1.) I think that therein lies the reason it's not more popular. You can achieve the same things in other formalisms but GA generally makes the most sense to me. Simplification to the point of obviation is a big win for me. 2.) Diversify. I picked it up because I felt like I can mold this better for my problems than LA. And I'm more about publishing code than publishing papers. So I care more about making it "my own" than explaining things to people. 3.) Coincidentally Lie algebras are also easier in GA. Look at this https://upload.wikimedia.org/wikipedia/commons/thumb/1/14/E8Petrie.svg/180px-E8Petrie.svg.png https://upload.wikimedia.org/wikipedia/commons/thumb/1/14/E8... and don't tell me that the symmetries seem more circular than linear.

vanderZwan 9y ago

Sorry, but pretty much every argument you make looks like a different flavour of "I already put a lot of effort into mastering other ways, and now that I have they work fine for me, I don't see why we should switch to this one." Because you seem to suffer from survivor bias, that's why. By comparison, I am a physics drop-out. Allow me to give you the view from below. First, I know this pdf is about GA in relation to physics, but you're throwing all kinds of names and terms around that only make sense to people who already have studied physics; conscious or not, that's basically gatekeeping via jargon. Claiming that we'll be unable to make sense of Goldstein and Spivak is also directly contradicting your claim that it's easier once you know the "more conventional stuff" - so learning one thing after the other is easier, but not the other way around? Why would it not be easier to learn the more conventional stuff after you have a solid GA fundation? Both my own experience and that of many commentators here seem to indicate GA is inherently more intuitive to grasp than the disparate mathematics it connects, precisely because it all seems more logically structured and connected. And perhaps the people who didn't manage to get through have a better idea of what is more intuitive to learn that the people who did here. Similarly, given that GA connects so many things together, I have a hard time believing that it is a "specialised tool", but since I never had to deal with most of the stuff you mention I guess I can't be a good judge of that (did I mention the gatekeeping?). Anyway, why should people first have to slog through all the other stuff, sometimes learning it more by rote than insight? We don't force kids to learn assembly or punch-cards before we let them play with a higher level programming language either. I'm not suggesting we throw all the old physics books out the window, but we also managed to stop doing science in Latin. That removed a huge barrier of exclusion, has not lead to a loss of knowledge, and making the field more accessible to everyone has only benefited humankind.