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Comment by kevinventullo | Hacker News Reader

As a former mathematician from one of the more elitist subfields of math, I have to say this is the first time I’ve heard anyone claim geometric algebra lacks rigor.

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The subject doesn't lack rigor, but all of the popular and quite a few of the mathematically minded introductions to it do. Bourbaki and Chevalley did a good job introducing Clifford algebras properly some 80 years ago, but even this has been forgotten by the modern bloggy expositors.

This here seems to be one of the few good modern texts that are elementary yet rigorous: https://www.mathematik.uni-muenchen.de/~lundholm/clifford.pd...

Unfortunately, the Lundholm and Svensson text you've cited suffers from the same problems I wrote about in my post. Definition 2.7 connects the interior products to the scalar product, not the inner product. Definition 2.8 gives the same broken definition of dual that has an inconsistent orientation and fails to extend to the degenerate metrics of projective algebras.

It fixes the worst problem, which is the well-definedness of the interior product, since it defines the product explicitly on the standard basis rather than through a bunch of not-obviously-consistent axioms. It is too basis-dependent for my tastes (it should depend on the symmetric bilinear form, not on the basis). As for the sign conventions, I think they're ultimately matters of taste, and while I agree with you about the inner/scalar product, I disagree about the derivation-ness of a contraction (I like it to satisfy the Koszul sign rule, which allows signs to be inserted only when some factors move past each other).

My ideal approach is along the lines of Chevalley's "The Algebraic Theory of Spinors and Clifford Algebras", Chapter III, but he doesn't get to much geometric algebra.

This is the second time I’ve heard geometric algebra lacks rigor. The first time:

https://news.ycombinator.com/item?id=39576214

> even by the 90s/00s, GA had gotten a bad reputation because of its tendency to attract bad mathematicians and full-on crackpots

> those reasons disproportionately attract people who are not actually capable of rigorous mathematics, or are slightly prone to conspiratorial thinking, or are otherwise slightly deranged

> if you look around for papers that explicitly talk GA, they are very disproportionately (a) non-theoretical, (b) poorly-written, (c) trivial, i.e. restating widely-known results as if they’re novel, (d) only citing other GA papers, and of course (e) just plain crackpotty

In the linked blog, the author describes himself: “I am not a mathematician or physicist and am definitely not credible at all”.

The author responded to this exact statement at this Reddit comment:

https://old.reddit.com/r/math/comments/1b5s32x/the_case_agai...

You may also read other discussions by the author and others at the Reddit post:

https://old.reddit.com/r/math/comments/1b5s32x/the_case_agai...

A few Reddit comments appear to agree at that non-mathematicians (e.g. amateurs and game programmers) use geometric algebra non-rigorously.

I suspect the issue is not about mathematicians who know how to be rigorous. Both posts are complaining about how mathematical outsiders are using and teaching geometric algebra non-rigorously.

I see, that makes sense. Thank you!

Well the real issue is that the GA proponents do this weird bait-and-switch where they talk about caring a lot about being intuitive and simple, and then they do a bunch of nonsense with the geometric product that makes no sense. It's all at least possibly rigorous (depends on who you're reading), but the thing I want people to question is: is it good? Is the GA way better?

When GA has clean formulas for calculating stuff, sure. But should you write all your linear algebra formulas in terms of the geometric product? heck no. For most people GA is their first exposure to wedge products, though, and those actually are great, so they think that's GA. No, that's just ordinary well-known material that should be in linear algebra classes already. What GA adds on top of that is mostly really weird, although there is a kernel of quality inside it somewhere.

What's it even mean to use Imaginary Numbers or Quaternions "rigorously"? It's Geometry. The point is points, lines, applying transformations. If anything the field needs less rigor and more friendly intuition.

Right, complex numbers and the delta function lacked rigour for years and physicists and engineers didn't care because it turned out it captured the logic sufficiently well that useful stuff could be gained with it. Then the mathematicians did the rigour bit and everyone was happy again.

1 + 1 didn't need rigor for years. then people did it in the Pricipia Mathematica and no one was happy.

i + j is just as silly as 1 + 1.

yeah, oops, that was supposed to be tongue-in-cheek. No, I'm just a guy who likes math, and I try to write things that make seem correct. I just don't want anyone, like, citing me on something... like, check the work yourself if you want to use it, I'm not a reliable source.

Anyway, go read a bunch of GA books and papers and you'll see exactly what I'm talking about there. At some point in the past (like ~8 years ago) I think I had read or at least skimmed everything that had ever been published on the subject. And some of it's good! Although at times, like, unnecessary. The rest, though... yikes.

This person is selling "rigor"...

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