> Virtually all authors define the dual of A […] and there seems to be no consensus as to which one is best. It doesn’t matter because every single one of these definitions is crap.
Love it :)
Poor Foundations in Geometric Algebra
61–70 of 131 posts
Re: Poor Foundations in Geometric Algebra
#62A lot of the (valid) criticism can be summarized by the "No scalar product necessary. Get rid of it" point (early in one of the tables). The next point on inner product is good, and related.
The point is: there is no natural or distinguished isomorphism from a finite vector space to the vector space of linear functionals over it. We know they are isomorphic- but there are many such isomorphisms with little to distinguish between them. The note's "arbitrary multivector, G" fix for inner products is good. The idea is most of us are using the same G that looks a lot like an identity matrix. And as long as you always use the same G you uniquely identify the inner product by equality (as the note does). But if you accidentally combine work that is using different G you run into trouble. You can end up with different Gs if you don't worry enough about naming your basis.
A lot of this keeps getting re-invented as variations of tensors, the exterior algebra, wedge product and so on. I think most of these end up being more bookkeeping than consumers want.
Re: Poor Foundations in Geometric Algebra
#63Earlier quoted context omitted.
It goes both ways. Mathematicians will take a moment denigrate Geometric Algebra as "linear algebra with a uselessly nonstandard notation", ignoring that we should prefer a less awkward way of structuring linear algebra than "pseudoscalars" and "pseudovectors".
> "linear algebra with a uselessly nonstandard notation" Let me chime in that as a physicist (who does use the "pseudo" stuff occasionally) I very much share this opinion. The notation may be really cool and compact, but I just do not see the benefit - for example, d*F = j and dF = 0 is compact enough for me. It is all fine if people use this language to learn linear algebra or differential geometry. And maybe it has…
What other way is there around avoiding polar vs axial vectors looking the same but behaving differently?
Or normal vs tangent vectors transforming differently?
Re: Poor Foundations in Geometric Algebra
#64Earlier quoted context omitted.
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 o…
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.
Re: Poor Foundations in Geometric Algebra
#65As 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.
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...
Re: Poor Foundations in Geometric Algebra
#66Earlier quoted context omitted.
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…
I see, that makes sense. Thank you!
Re: Poor Foundations in Geometric Algebra
#67As a fellow game dev this article should be targeted at readers like me. But my eyes can't help but *immediately* glaze over as soon as I read all the math notation.
I'm pretty ok at 3d video game math. I do lots of work with matrices, quaternions, vectors, and friends. It's not particularly difficult.
I can't for the life of me read mathy math. Wikipedia Math is inscrutable hieroglyphics. It's quite frustrating.
I wish someone would write a "foundations" article or book that spoke in language understandable by normal humans. Or at minimum had a bloody legend that explained what all the %&(#%& symbols meant.
Re: Poor Foundations in Geometric Algebra
#68As a fellow game dev this article should be targeted at readers like me. But my eyes can't help but *immediately* glaze over as soon as I read all the math notation. I'm pretty ok at 3d video game math. I do lots of work with matrices, quaternions, vectors, and friends. It's not particularly difficult. I can't for the life of me read mathy math. Wikipedia Math is inscrutable hieroglyphics. It's quite frustrating. I w…
Honestly I'm surprised no one has done the programming/CS version of it where all the symbols and formula are explained with code. I'm rusty as hell on math but seems like it should be pretty reasonable to translate a lot of them to functions or similar per symbol.
Re: Poor Foundations in Geometric Algebra
#69Earlier quoted context omitted.
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 o…
In the linked blog, the author describes himself: “I am not a mathematician or physicist and am definitely not credible at all”.
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.
Re: Poor Foundations in Geometric Algebra
#70As a fellow game dev this article should be targeted at readers like me. But my eyes can't help but *immediately* glaze over as soon as I read all the math notation. I'm pretty ok at 3d video game math. I do lots of work with matrices, quaternions, vectors, and friends. It's not particularly difficult. I can't for the life of me read mathy math. Wikipedia Math is inscrutable hieroglyphics. It's quite frustrating. I w…
Honestly I'm surprised no one has done the programming/CS version of it where all the symbols and formula are explained with code. I'm rusty as hell on math but seems like it should be pretty reasonable to translate a lot of them to functions or similar per symbol.
Yes please.
This image makes the rounds semi-regularly: https://twitter.com/FreyaHolmer/status/1436696408506212353