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Poor Foundations in Geometric Algebra

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Re: Poor Foundations in Geometric Algebra

#111

Post by hgomersall »

koolala wrote: post #83
hkmaxpro wrote: post #64

Earlier quoted context omitted.

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…

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.

Re: Poor Foundations in Geometric Algebra

#112

Post by bmacho »

mkl wrote: post #98
koolala wrote: post #96

Earlier quoted context omitted.

How is complex / imaginary numbers not what they are doing? Numbers that square to -1, 0, and 1 are the bread and butter of the GA I know. Exploring different combinations of types of imaginary numbers and their products and space describing algebras. (including naturally quaternions, duel quaternions, i-rotation, nilpotent, ect)

You seem to be calling complex numbers imaginary numbers, but they're not the same thing. Imaginary numbers are a subset of the complex numbers consisting of the imaginary axis without 0, e.g. i, 2i, -3.1i. Complex numbers also include the real numbers and all combinations of real and imaginary.

People should stop calling that set "imaginary numbers". It's net bad. That set isn't really worth it to be called anything. Maybe "pure imaginary numbers". Also "imaginary numbers" is used for the complex numbers already.

Re: Poor Foundations in Geometric Algebra

#113

Post by koolala »

hgomersall wrote: post #111
koolala wrote: post #83

Earlier quoted context omitted.

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.

Re: Poor Foundations in Geometric Algebra

#114

Post by [deleted] »

koolala wrote: post #107
mkl wrote: post #98

Earlier quoted context omitted.

You seem to be calling complex numbers imaginary numbers, but they're not the same thing. Imaginary numbers are a subset of the complex numbers consisting of the imaginary axis without 0, e.g. i, 2i, -3.1i. Complex numbers also include the real numbers and all combinations of real and imaginary.

I'm glad you seem to know what I mean. I love imaginary numbers like the Square Root of -1 or the Square Root of 0. They can only be used like: nil²=0 i²=-1 j²=-1 And combining them into complex forms like: (i + j) (nil + i) What can I call this general idea of using imagination to determine new number rules and combining them together? If I call this GA or Clifford Algebra in a math community will it trigger rigor a…

[deleted]

Re: Poor Foundations in Geometric Algebra

#115

Post by koolala »

bmacho wrote: post #112
mkl wrote: post #98

Earlier quoted context omitted.

You seem to be calling complex numbers imaginary numbers, but they're not the same thing. Imaginary numbers are a subset of the complex numbers consisting of the imaginary axis without 0, e.g. i, 2i, -3.1i. Complex numbers also include the real numbers and all combinations of real and imaginary.

People should stop calling that set "imaginary numbers". It's net bad. That set isn't really worth it to be called anything. Maybe "pure imaginary numbers" . Also "imaginary numbers" is used for the complex numbers already.

People also like colloquially saying "The Square Root of -1". Technically wrong but personally meaningful. GA is mainstream once normal people start joking about the square root of zero. We need to imagine and teach even more imaginary numbers one day.

Re: Poor Foundations in Geometric Algebra

#116

Post by ajkjk »

gugagore wrote: post #110
ajkjk wrote: post #75

Earlier quoted context omitted.

Those concepts are from exterior algebra which GA is built on. You do not need GA (the geometric product, etc) for them.

By the way, I love your blog post https://alexkritchevsky.com/2024/02/28/geometric-algebra.htm... I didn't understand this part: > I strongly believe that if GA would make this distinction they would lose a lot fewer people. It is a completely interesting and useful thing to talk about “a representation of a particular class of operations that makes composition and inversion easy”, and completely offputting when you…

So, I've amended the article some since posting it because the main objection I got from a few GA enthusiasts was this idea of the GP as being used to compose operators. Which I had barely noticed the importance of because it's so hard to identify in the texts! Although once they mentioned it I began to appreciate that, when the GP works and is useful, this is why. So I changed things to address that point more directly... but I didn't really find the energy to do a perfect job of it (like researching all the examples I would need to make the point more clearly), and it's a bit muddy at the moment. Like at the spot you pointed out. I need to figure out a clearer way to say this stuff...

What I am getting at is that if you go read, say, the Doran/Lasenby book, they start out talking about multivectors for areas and volumes and etc---and they do all this with the GP. Which makes no sense! Ever calculation they do leaves you think "huh?" The GP makes no sense at all if you're talking about units of length, area, volume, etc. Its transformation laws, its composition laws... you end up having to undo it all afterwards with a bunch of janky other operations.

But if you talk about the GP for composing reflections to make rotations, it's fine, that makes sense. I just really want this distinction to be made more clearly. I'm only interested in the GP when it corresponds to an explicit geometric operation. Nobody makes this distinction as clear as I want; I hope to eventually find a really sound version of the argument and then write it out as another article.

Roughly speaking it's equivalent to conflating the sense of a complex number as a vector with a complex number as an operator on vectors. Yes, they're isomorphic, but given a vector in R^2 there's no intrinsic sense in which you should be able to interpret it as also being the operation of multiplying by r e^(iθ) on other vectors. Pretending like they're the same thing is just bewildering: that identification between vectors and operations should be something you have to explicitly construct. For starters, if you change bases for (x,y) the vector should rotate but the rotation operation shouldn't change. That sort of thing. GA is making this same confusion but on a larger scale.

Re: Poor Foundations in Geometric Algebra

#117

Post by ajkjk »

gugagore wrote: post #109
ajkjk wrote: post #105

Earlier quoted context omitted.

Well by "taught in linear algebra" I mean "taught to undergraduates in their first or second year". It is totally bizarre that people learn about determinants, matrix minors, curl, magnetic fields, angular momentum, etc... but not about wedge products, in which all of these things are much easier to understand.

Totally agree! And this is why I sympathize with GA fanatics. Although it might not really make sense to do the rebranding or be maximalist about the geometric product, I do think there is a shared goal of making these things easier to understand.

Ya, as I write in my article, I'm very on board with the "change the curriculum to make more sense" project. I just want to be talking about how to do it in a critical way, rather than just going with the GA way, which requires, in my opinion, a lot of justification that I haven't seen.

Re: Poor Foundations in Geometric Algebra

#118

Post by at_compile_time »

Geometric algebra was, for me, an easier on-ramp to more advanced topics in mathematics, leveraging the geometric intuition built into these objects. I learned vector algebra in university, so extending the idea of a direction with magnitude and orientation into areas and volumes with direction and magnitude was a manageable step. Projective algebras similarly allow the construction of points, lines, and planes with orientation and magnitude. Learning that you can generate transformations by dividing one object by another was also really neat. Then you can take the logarithm of that transformation to perform linear interpolation. Geometric algebra is a tool that I use to manipulate geometric objects as easily as I would manipulate real numbers.

The hard part was getting there amidst a host of confusing and conflicting source materials, as this post highlights, and as its author helps proliferate. I used Eric Lengyel's materials a lot in my journey, but I really dislike his decision to represent points as vectors. Notice how every transformation in his poster [1] uses the antigeometric product and antireverse? These operations are only necessary because he's defined everything in the point-based dual space and has to bring everything back to the plane-based space to perform transformations. But he has a wiki and posters, and I'm just here doing my own thing, so I guess that's that.

Alan MacDonald's two books [2] were great, and I would recommend them as an introduction to geometric algebra. Work through the examples and you will learn the material.

[1] - http://projectivegeometricalgebra.org/projgeomalg.pdf

[2] - http://www.faculty.luther.edu/~macdonal/index.html#geometric...

Re: Poor Foundations in Geometric Algebra

#119

Post by elengyel »

generationP wrote: post #65
kevinventullo wrote: post #58

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.

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/~lundh…

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.

Re: Poor Foundations in Geometric Algebra

#120

Post by elengyel »

auggierose wrote: post #90
nkhodyunya wrote: post #85

The claim that geometric algebra is not rigorous made me confused. But the lack of proper references is much more confusing to me. Back in the days when I was doing research in Chevalley groups the definitive and most cited source was (and still is) Geometric algebra by E. Artin. And there are some more recent treatments recognized in math community, e.g. classical groups and geometric algebra by L. Groove and the ge…

They are talking about different stuff. Geometric algebra is here the name used for the popularisation of a brand of Clifford algebra. The popularisation books are often written by physicists and engineers, and their maths is shaky, because they care more about applications than about proofs and rigour. There are exceptions, for example I like the two books by Alan Macdonald about the topic: http://www.faculty.luther…

I think Macdonald's book is very concise and clean compared to others, but it does have some of the same issues that I wrote about in my post. In particular, Definition 6.15 gives one of the problematic definitions of the inner product, and Definition 6.23 gives the same broken definition of dual that has an inconsistent orientation and fails to extend to the degenerate metrics of projective algebras. Has also says, at the bottom of page 111, that the Hodge dual can't be defined directly in the exterior algebra, which is not correct.
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