[HN Gopher] Poor Foundations in Geometric Algebra
___________________________________________________________________
Poor Foundations in Geometric Algebra
Author : ibobev
Score : 114 points
Date : 2024-08-24 08:51 UTC (2 days ago)
(HTM) web link (terathon.com)
(TXT) w3m dump (terathon.com)
| mgarym wrote:
| Oh man, I didn't know there was going to be a special episode of
| Friday Night Smackdown!
| user070223 wrote:
| Note don't to be confused with Algebraic Geometry they are
| different;
|
| When I first came across this topic it was eye opening,
| especially the fact that you could squeeze Maxwell's equations
| into one and the fact that pseudovector create by cross product
| from physics is just a bivector which in 3d could be represented
| like a vector orthogonal to the plane created by the two vectors
| in the product
|
| Primer on the topic https://www.youtube.com/watch?v=60z_hpEAtD8
| (And other videos on his channel) Another greate playlist is
| https://www.youtube.com/watch?v=0VGMxSUDBH8&list=PLLvlxwbzkr...
|
| BTW the author has the following page
| https://projectivegeometricalgebra.org/ with great infographics
| and references
| BenoitP wrote:
| Hah! I knew Sudgy would be your first link. There's relatively
| little content about GA, but once you're hooked you begin to
| consume it all.
|
| It's like you never got to study Relativity, conformal
| geometry, electron spins, quaternions, etc then someone comes
| with a simple cheat code which introduces you to these topics
| gently. It's like what category theory wants to do for
| mathematics, but simple.
|
| Here are additional resources:
|
| GA playground:
| https://enkimute.github.io/ganja.js/examples/coffeeshop.html...
|
| A physics engine in 100 lines (Gravity, Hook, and damping laws
| are just one line each!), and you can go from 2D to 3D to 4D by
| changing a single parameter:
| https://enki.ws/ganja.js/examples/pga_dyn.html
|
| Other resources: https://bivector.net/
|
| And anything by David Hestenes:
| https://worrydream.com/refs/Hestenes_2002_-_Reforming_the_Ma...
| gowld wrote:
| The absolute first video to watch should be the (ironically
| titled) _Why can 't you multiply vectors?_
|
| https://www.youtube.com/watch?v=htYh-Tq7ZBI
| carterschonwald wrote:
| This explains very nicely why despite working through lots of GA
| resources, I never quite grasped it. It's logically incoherent!
| adrian_b wrote:
| This is not a critic of all geometric algebra and especially
| not of its more basic parts. Therefore it is not an excuse for
| not grasping e.g. what Hestenes has written about GA, or what
| Eric Lengyel himself has written, e.g. in his new book that is
| advertised in this article.
|
| It is a critic of many books about geometric algebra, which
| have made attempts to expand and further develop some parts of
| its theory, but those attempts have not been thought carefully
| and they have produced various inconsistent or useless
| definitions.
|
| It is also a critic of attempts of presenting geometric algebra
| as preferable for applications where in fact it is not optimal,
| by showing misleading "benchmarks". Unfortunately this tactic
| is not at all specific to geometric algebra, but it is
| frequently encountered for almost any kind of algorithm known
| to mankind when it accumulates for one reason or another some
| kind of fan base.
| wudangmonk wrote:
| The author has books on Geometric Algebra so it makes no sense
| to assume that he is going against GA as a whole.
| joe_the_user wrote:
| Calculus had poor foundation and was thus logically
| incoherent from Newton/Leibniz' discovery to roughly the
| middle of 19th Century. None-the-less it was a powerful tool
| and most of the key theorems were discovered then.
|
| The basic situation, I think, is a set of tools can be
| consistent in the way mathematicians use them but in the way
| the mathematicians explain them. And the tools can be very
| useful despite this.
|
| So my guess is saying "it has poor foundations" isn't saying
| "I'm against it, it's worthless"
| catgary wrote:
| Yeah I'm pretty well-versed in free constructions of various
| algebraic objects and how this would interact with things like
| a quadratic form, etc, but couldn't sort out GA (I think the
| authors of "GA4CS" had a very different sort of computer
| scientist in mind). When I saw the geometric algebra
| constructions clash with constructions I was already familiar
| with, I generally got suspicious and lost interest.
|
| I'm actually quite interested in checking out Lengyel's book.
| It looks rock solid.
| jdeaton wrote:
| > very real toxicity within the geometric algebra community. I
| can't do much about
|
| I was hoping the article would be about this instead. OP
| wondering if you have any elaborations for us to hear.
| nicf wrote:
| I'm not the OP, I'm not a part of this community, and I don't
| know if the thing I'm about to complain about is what the
| author was thinking of with this comment, but as someone who
| was trained as a mathematician and who has read some of the
| popularizations of geometric algebra that sometimes get posted
| to HN, there is a tone that some (though probably a minority)
| of them take that I find pretty obnoxious.
|
| These pieces are the ones that take the position that geometric
| algebra is this super secret anti-establishment mathematical
| samizdat that *they* don't want you to know about. They'll pit
| themselves against "mainstream mathematics" and say things
| like, "in differential geometry you do X, but you shouldn't do
| differential geometry; you should do geometric algebra where we
| do Y, which is so much better than X."
|
| My reaction is always, "My friend, _you are doing differential
| geometry_! " Clifford algebras --- the objects that the
| geometric algebra people study --- are firmly within the
| "mainstream" of mathematics; there's simply no conflict here,
| at least not of the sort that these writers often seem to be
| imagining. It's great that people are enjoying learning about
| Clifford algebras. I think Clifford algebras are really fun!
| But we can all just come together and enjoy them together, and
| I think this "join me in taking down the cabal of gatekeepers
| who are suppressing the truth" attitude is unnecessary and
| turns off a lot of people who might otherwise be fun to engage
| with.
|
| If you're into this stuff and feel like this doesn't describe
| you or the people you know, then that's great, keep doing what
| you're doing! But it does exist and I wish it didn't.
| gowld wrote:
| 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".
| aleph_minus_one wrote:
| > 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".
|
| I have never heard a mathematician using the terms
| "pseudoscalar" and "pseudovector". These rather seem to be
| common terms among physicists.
| prof-dr-ir wrote:
| > "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 a use
| for numerics or computer science. But I am quite sure that
| the geometric algebra formalism will not be widely adopted
| in physics any time soon. Sorry.
| thechao wrote:
| I used GA as a way to bootstrap into 'real' Clifford
| algebras, and a way to get over a "reader's block" when it
| came to Lie algebras, tensors, and (finally) algebraic
| geometry. I'm not sure GA is great _math_ , but it was really
| great way to learn "advanced math concepts" for "basic..ish
| math". Personally, I like Alan MacDonald's GA books --
| they're a great way to learn more complicated concepts, but
| couched in a very approachable geometry/visual learning
| style.
| nicf wrote:
| That sounds like a fun and satisfying process! I realize my
| comment could be taken as denigrating all the people who
| write about this stuff, but that's certainly not my
| intention; I've also enjoyed a lot of the visualizations
| and geometric explanations that people writing under this
| heading have come up with. My complaint is really just
| about the ones who take this oppositional attitude, and a
| big part of why I think it's such a shame when that happens
| is that there really _are_ some very cool ideas here, so it
| 's sad to see walls being raised for no good reason.
| MarkusQ wrote:
| Ouch.
|
| I love it when people say what they mean and don't beat around
| the bush.
| mncharity wrote:
| So much education content is so very poor. And even the best of
| it... A first-tier physics professor at a munch was delighted -
| he regaled believing he had found an error in a highly-regarded
| introductory textbook! But, upon many day's of thought, and
| several close reads of the text, he had realized it had been very
| carefully worded so as to be not incorrect. And so he was so
| delighted - yay! He thought of this as a good state of affairs,
| reflecting well on the text, and associated instruction. I...
| afterwards wished I'd pointed out that the target audience for an
| intro textbook, was perhaps not well modeled as first-tier
| experts with a week to wrestle with and closely ponder a
| paragraph in order to avoid being misled.
|
| I'm unclear on how we get better at this. I've seen OER texts
| with open errata databases still struggle. Perhaps a github-like
| fine-grain (Xanadu-like transclusion) wikipedia? Or "nLab all the
| fields"? Or... ??
| selimthegrim wrote:
| I tried to report an open calculus textbook from Rice
| University's talking about relativistic mass as an error (It's
| pretty well-established as a bad concept in physics education
| at this point as opposed to the momentum energy 4 vector) and
| they wouldn't accept my feedback.
| mncharity wrote:
| Yeah - I've seen "but it's on lists of most common
| misconceptions" closed wontfix. Errata are good for
| "author:oops,tnx", but work much less well for confused
| authors and bad calls, and not at all for judgment calls and
| alternate approaches. Some other mechanisms are needed.
| meroes wrote:
| I don't know the solution either. My stats professor
| religiously attended and espoused these "teaching stats"
| conventions. But the end result was him always deferring to how
| the committee did things. The entire pedagogy including how he
| answered questions. I really didn't like this solution and it
| made me hate stats until some reacquaintance with it in
| discrete math.
|
| But then if you're at the mercy of a professor who does things
| their own way, you can have cases like you give.
|
| One thing that helped was getting syllabi from future potential
| classes and comparing which textbooks they used. My advisor
| helped me do this and I credit it with making my senior year
| more tolerable.
| ganzuul wrote:
| If you don't like their geometric algebra you can try mine:
|
| https://news.ycombinator.com/item?id=41344163
|
| Then you can try stuff like folding these spaces to make your own
| multiplication and division with numbers you don't have to
| explain to anyone!
| gjm11 wrote:
| So far as I can see, this has absolutely nothing whatsoever to
| do with geometric algebra in the sense being discussed here.
| rhelz wrote:
| I'm a big fan of Eric Lengyel, and I never would have ever gotten
| my arms around Geometric Algebra if it hadn't been for his books
| and articles. How many people can do theoretical math AND code a
| state-of-the-art game engine? The guy is a walking, coding
| miracle.
|
| So if I'm a little critical of the tone of the article, it comes
| from a place of love. There _has_ been a very toxic, clickish
| vibe in Geometric Algebra circles, which have lead to some
| pseudo-disputes among those who should be natural allies.
|
| One such is that Gunn, et al, prefer to represent a 3D vector
| using a dual basis (e.g. [a1, a2, a3]^T = a1*e32 + a2*e31+
| a3*e12) whereas Lengyel prefers to just represent them as a1*e1 +
| a2*e2 +a3*e3. Some really unfortunately hostile back and forth
| arguing about which one is "the right way"--when in reality, it's
| a big-endian vs little-endian thing. One of the best parts of
| projective geometric algebra is that you can flip back and forth
| to the dual representation whenever you want to, according to
| what makes sense to you--and what makes the problem at hand
| easier to solve. Moreover, if you look at the actual calculations
| doing it one way vs doing it the other it's the same damn exact
| numbers being multiplied/added in the same damn way. It's not
| _quite_ as silly as arguing about what font numbers should be
| printed with, but it 's pretty close.
|
| The tone of the article reflects wounds which are still pretty
| sore from these sorts of battles. So I understand. But sometimes
| the invective goes a bit to far.
|
| An example of this is his critique of Gunn's initial cut of
| dualizaiton for PGA. The fact that e0 is not invertible is a big,
| fat wart, for sure. And frankly, I spent _weeks_ trying to
| understand Gunn 's workaround and its wierd lingo. J-map? What in
| the world is a J-map? I finally understood the concept, but I've
| never found out what "J" stands for :-) And Lengyel's treatment
| is much smoother and more coherent.
|
| Yet, I don't think Gunn should be criticized for it at all. It
| was an act of courage for Gunn to come up with his janky J-map,
| and not let its janky-ness stop him and the rest of the field
| from moving forward. Sometimes that is exactly what is required
| in mathematics. For example, infinitesimals. For centuries,
| mathematicians from Archimedes to Newton found them
| indispensible, even though they had absolutely no coherent
| mathematical foundation. In point of fact, if you listen to, say,
| a Feynman lecture, you'll find that they are still indispensible
| today. But they didn't have any kind of mathematical foundation
| until the 1960's, when Robinson found a way to coherently
| axiomitize them.
|
| I saw a video of Freeman Dyson once, where he was talking about
| how he was able to prove something which was a longstanding open
| problem. He described his proof as "very ugly" and then went on
| to say (with tongue partly in cheek) that you can judge how great
| a mathematician is by how many ugly proofs he creates :-) Because
| the first time something is proved, the proof is almost always
| very ugly. It's not until other mathematicians come in and find
| connections with other branches of math, and start being able to
| come up with more elegant proofs.
|
| So let's celebrate the ugly, messy, janky-ness which is the
| reality of how mathematics is actually created, and the courage
| of the mathematicians to not rat-hole and bike-shed.
|
| Eric Leyngel's presentation of projective geometric algebra is,
| IMHO, far more coherent and elegant than any other presentation.
| His books (and his source code) are a joy to read. For a newb
| like me, it is far easier and quicker to absorb. Isn't that good
| enough? Did he really have to go on to flame everybody else to a
| crisp? _sigh_ like I said, he has been subjected to very unfair
| and toxic pillorying, and the wounds are still fresh, so like I
| said, I understand. But its very regrettable nevertheless.
| altairprime wrote:
| > _Yet, I don 't think Gunn should be criticized for it at
| all._
|
| The article dedicates itself to critiquing the techniques, but
| spends no time that I can remember talking about the human
| beings and their feelings. That's how I write professionally,
| and I've found it can really upset people who infer that
| critique of some thing is therefore critique of some person --
| even when no such inference is intended by the author.
|
| > _Did he really have to go on to flame everyone else to a
| crisp?_
|
| No one _person_ is flamed to a crisp here, but some people's
| mathematical works are absolutely set on fire. If this had been
| a critique about the _people_ in geometric algebra, I never
| would have read it at all, because that's not what interests
| me. Critiquing cargo-curled erroneous hearsay as invalid
| resonates strongly, especially with the focus on the math
| instead of the people.
|
| As a newb, I learned a great deal about mathematics from
| reading this, but I still don't know who any of the people
| involved are. Isn't that the holy grail of professional
| critique: it's about the work, not the worker?
|
| Or, am I missing something where this article is personally
| attacking people _rather rhan_ people's work output?
| obsoletehippo wrote:
| Well, he describes some folks' work as "crackpot-level
| bullshit" and repeatedly states that other authors "have no
| idea what they're talking about". These seem to me to be
| personal critiques. Generally, I'm not sure the distinction
| you draw is quite so clear cut -- if I say somebody is great
| but all their work is hot garbage, that sounds fairly
| personal.
| andrewflnr wrote:
| He repeatedly makes assertions that "the authors" have "no
| idea what they're talking about". The difference between
| flaming an individual person and "the authors" is thin at
| best, arguably only different in that it flames more than one
| person at once. And honestly, to the extent he's correct, I
| think that's fine. But let's not kid ourselves about what
| we're reading.
| lupire wrote:
| > I've never found out what "J" stands for
|
| > _janky_ J-map
|
| There you go
|
| > Infinitesimals
|
| Infinitesimals are completely well-founded. It's just that
| Archimedes and Newton didn't know a foundation for them.
|
| https://en.m.wikipedia.org/wiki/Surreal_number
| sdenton4 wrote:
| Yes, that's exactly the point...
|
| "But they didn't have any kind of mathematical foundation
| until the 1960's, when Robinson found a way to coherently
| axiomitize them."
| Arelius wrote:
| So, I've shipped a game with his engine, so have a unique
| relation ship with him, and he's a complicated individual.
|
| To be clear, he is a very talented, intelligent individual. But
| he form strong opinions, has a very hard time taking criticism,
| or understanding other people's differing context, and has a
| hard time not taking disagreements personally.
|
| This is really nothing new from him. I think the best way to
| interact with him, is hear out his points, and use them to
| synthesize your own viewpoint. And be careful to take his views
| as your own. I think if you do that, you can learn a lot from
| him, but to be careful as his strong disagreements are often
| more nuanced then he makes them out to be.
| gowld wrote:
| Where are the mathematicians who could be describing Geometric
| Algebra correctly?
|
| For a professional working in linear algebra, supervising a
| graduate student, it should be straightforward to translate these
| "higher order" linear algebra objects into the geometric algebra
| model, without publishing technically incoherent mistakes.
| fouric wrote:
| I'm currently _very_ slowly making my way through _Geometric
| Algebra for Physicists_ by Doran and Lasenby. The book is a
| delight to read, but I 'm not a mathematician, and this article
| is showing me that my small amount of understanding is...not
| nearly as deep, and especially not nearly as _rigorous_ , as I
| would like. I should try to re-read with Eric's criticisms in
| mind.
___________________________________________________________________
(page generated 2024-08-26 23:00 UTC)