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The Exterior Algebra and Central Notions in Mathematics (2015) [pdf]
- unao 9y agoThe neglect of the exterior algebra is the mathematical tragedy of our century. Why has it been neglected?
- xyzzyz 9y agoI don't know what the quotation author meant. Exterior algebra has definitely been well understood and frequently used tool in 20th century.
- unao 9y agoYou are most likely right, but it is still kind of a niche. If you take a look at curriculum of Computer Science studies it is pretty non-present. From what I know Exterior / Geometric Algebra is superior in many ways to Linear Algebra: first it is backwards compatible and second (apparently) it is much more intuitive. But still Linear Algebra is pretty standard.
- xyzzyz 9y agoExterior algebra is not particularly useful in Computer Science, and that's probably it's non-present. There are other things competing for place in curriculum, and they probably deserve it more. Exterior algebra is not any more superior to linear algebra than multiplication is superior to addition. Both are important, there are important connections between the two, and you definitely need to understand addition first before you understand multiplication.
- auggierose 9y agoIf you look at the paper you will see that it uses a relatively advanced mathematical language. Did you understand the significance and meaning of exterior and geometric algebra from that? If not, then there you have a partial explanation for it. It is much easier to just understand vectors and matrices and maybe tensors for practical and applied work.
- yiyus 9y agoI do not agree with this. The simplest mathematical concepts can look very complicated when presented in a formal way. Learning the geometric product in high school wouldn't be more difficult than learning the dot and cross products, and would make obvious difficult to grasp concepts as complex numbers and even quaternions. There are historical reasons for which we do not learn this from another point of view, and in my opinion it is, indeed, a great tragedy. A tragedy that I hope will be remedied some day. Disclaimer: I deal everyday with 3D rotations. Euler angles have been traditionally used in my field, but they present many problems. Everybody knows we could do better with quaternions, but very few people understand them. I have shown many people how to interpret what quaternions are from geometric algebra concepts and I have not yet found anybody who doesn't think it is much more approachable that way.
- auggierose 9y agoI do not disagree with you! For example the book http://faculty.luther.edu/~macdonal/laga/ http://faculty.luther.edu/~macdonal/laga/ gives an elementary introduction to the subject. I have learnt about geometric algebra just this year, and applied it to compute the graphics in an app I wrote for a customer. It was a real eye opener, concepts that I struggled with before were really simplified by using geometric algebra. BUT: I would never have guessed the usefulness of it for me from this paper.
- zardo 9y agoSecond recommendation for laga.
- unao 9y agoThe formalism is pretty hard to grasp indeed, but I guess it concerns any mathematical theory. From what I know Exterior / Geometric Algebra is much simpler and more intuitive than let say Linear Algebra.
- auggierose 9y ago
- tw1010 9y agoThis stuff really isn't that hard once it is described well (which doesn't need to involve any simplification). I can't wait to see what happens to the programming community once these notions are embraced with the love and respect that it deserves. If anything is going to make a big dent in the field, on a 5-10 year horizon, it'll be this. What we need to do is not to think of it (modern concepts in pure mathematics, especially from algebra) as something separate from programming, but to allow it to mix and skew the fundamental language that we use to reason about coding.
- Retra 9y agoI don't think this will make a big dent on account of geometry being fundamentally simpler and easier to reason about than language in general, and thus mathematicians dealing in geometry have a much more workable framework for employing an algebraic approach than do computer scientists. I think if these ideas were going to be revolutionary, they'd already have been. They're not new, unfamiliar, or esoteric, they are in standard curricula.
- nextos 9y agoExterior algebra is elegant. It simplifies some things. But it's mainstream mathematics. For example, it's covered in the little Spivak (not to be confused with his elementary Calculus book). As a pure CS person, I'm more interested in advances that push forward constructive mathematics (which has a huge overlap with CS). That's why Martin-Löf's type theory and its further development into HoTT as a new alternative foundation for mathematics is something I'm following with great interest.
- jacobolus 9y agoThe way people interact with differential forms and tangent bundles and so on is somewhat different (and takes much more conceptual machinery) than the way multivectors and geometric algebra (“Clifford algebra”) would best be presented to novices or non-mathematicians.
- trashtoss 9y ago
- adamnemecek 9y agoI’ve been looking into this stuff as of late (I bought like 400 bucks worth of books on it). I always kinda liked math but always felt like some parts don’t make any fucking sense (cross product only in 3d? Nice algebra bro). Also wtf are complex numbers really about? Geometric algebra resolves all this and it has insane applications esp for any space related reasoning. Space time algebra is very much related and it’s also balls to the wall insanity. Maxwells four equations reduce to a single equation, a simple fraction. It’s beyond me we aren’t going hard on this.
- tw1010 9y agoDon't start with geometric algebra. Start with abstract algebra. It is the assembly language, if you will, of geometric algebra (and of a lot of other stuff). It'll also give you a new (from first principles) understanding of complex numbers. And don't read explanations by programmers of abstract algebra, those tend to be very shallow and only touch on a few definitions. Also, don't just read these books. The real gold comes after you've struggled with proofs and a mountain of exersizes.
- m00n 9y agoThis ^. If you wonder, 'what complex numbers are all about', then chances are you can't give the universal property of the tensor product, or the definition of a Riemannian manifold, or a complex Lie group, either. I suggest to look at the literature listings of any undergrad math curriculum to get suggestions on what to start with. There are reasons 'linear algebra' is usually taught before 'Applications of the representation theory of Lie groups in special relativity'. Or maybe I'm too conservative in my view on math education?
- pqh 9y agoWhile I agree with the spirit of your post, starting with abstract algebra is probably too formal and abstract. The best way to learn something is by learning the concepts first and progressively formalizing, always with some link in the chain connecting back to tangible experience. I would possibly throw this out as an interesting way to learn a bit (not really a whole lot, but a fun start) about abstract algebra: https://www.crypto101.io/ https://www.crypto101.io/
- daniel-levin 9y agoFor a gentle appetiser, see Spivak's Calculus on Manifolds. It introduces exterior algebra as a means to and end in doing computations on (embedded) manifolds. At first, the definitions seem opaque and the formalism clunky. At the end, Spivak recovers Stokes' theorem as a computation. This was a breath of fresh air for me after I took a warped version of Calc III.
- tobbe2064 9y agoDoesn't the example show that (ii): e1□e2 = -e2□e1 and (iii): (e1□e2)□e3 = e1□(e2□e3) contradict each other? In the end we end up with (e1□e3)□e5 = e5□(e1□e3) which seems to go against (ii)
- aquamongoose 9y agoNo: (ii) only holds when e1, e2 come from the original vector space (not the product).