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Poor Foundations in Geometric Algebra

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Re: Poor Foundations in Geometric Algebra

#41

Post by pvg »

ganzuul wrote: post #37
dahart wrote: post #27

Earlier quoted context omitted.

No need to assume that downvotes convey anger at all. They are given for lots of reasons, including style, rudeness, off-topic comments, irrelevance, incorrect or wrong comments, assumptions, negativity, etc. Sometimes downvotes (and upvotes) are nothing more than prioritizing. All that makes your edit and last sentence especially prone to downvotes, it’s multiple negative incorrect assumptions, and goes against HN g…

[flagged]

As you're fond of the guidelines, you have probably also read the bits where they ask you to avoid interrogating, to not to go on about votes and not to sneer, especially at your fellow co-commentators.

Re: Poor Foundations in Geometric Algebra

#42

Post by ganzuul »

gjm11 wrote: post #35
ganzuul wrote: post #26

Earlier quoted context omitted.

[flagged]

(I did not downvote the parent of this comment, nor am I angry. I can't speak for anyone else.) Seems a bit of a weird request, but sure. I assume the point of it is some sort of credential-test, so I'll focus on things relevant to that. My name's Gareth McCaughan. I'm a mathematician, though I've never worked specifically in the field of geometric algebra, nor in the foundations of mathematics, and I haven't been do…

(I know you it doesn't let you, unless you have an alt account.)

I must say that I am a little surprised that this is something you have not come across before and I take it you are surprised as well. Apparently foundations are not taught in every school.

When we are talking about foundations we are usually concerned with something that prompted Euclid to write a very long time ago. In his Book 2 he defines geometric algebra for the first time but it is his parallel postulate which gets the most scrutiny because it is an outlier amongst his other claims. Elements is widely regarded as instrumental for logic and "natural philosophy" which is today called science.

I'm the author of that paper construction I linked to. The thought experiment I am offering was something I invented 30 years ago and I have found much use for it in my studies of ontology. It is very simple and directly let me discover logics that do not rely on the law of the excluded middle and a rudimentary quantum logic. However its greatest strength is how it supports your intuition and gives legitimacy to your imagination.

Cauchy sequences are what shows us that we have a way to actually reach every Real number, with no gaps, and this is what lets us talk about the number line as a mathematical object. Without this we don't have a foundation for differentiation. Perhaps now you can see why we might want to construct alternative number-like objects?

Re: Poor Foundations in Geometric Algebra

#43

Post by Arelius »

rhelz wrote: post #22

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 circl…

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.

Re: Poor Foundations in Geometric Algebra

#44

Post by ganzuul »

pvg wrote: post #41
ganzuul wrote: post #37

Earlier quoted context omitted.

[flagged]

As you're fond of the guidelines, you have probably also read the bits where they ask you to avoid interrogating, to not to go on about votes and not to sneer, especially at your fellow co-commentators.

You can't be serious. Reconsider your assumptions about me. What you are saying in no way supports your perspective.

Re: Poor Foundations in Geometric Algebra

#45

Post by fouric »

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.

Re: Poor Foundations in Geometric Algebra

#46

Post by pvg »

ganzuul wrote: post #44
pvg wrote: post #41

Earlier quoted context omitted.

As you're fond of the guidelines, you have probably also read the bits where they ask you to avoid interrogating, to not to go on about votes and not to sneer, especially at your fellow co-commentators.

You can't be serious. Reconsider your assumptions about me. What you are saying in no way supports your perspective.

I don't have any assumptions about you, it's just stuff you should avoid in comments because it tends to trash the conversation and, like the guidelines point out, isn't interesting.

Re: Poor Foundations in Geometric Algebra

#47

Post by thechao »

nicf wrote: post #20
jdeaton wrote: post #5

> 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.

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 piece…

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.

Re: Poor Foundations in Geometric Algebra

#48

Post by sdenton4 »

lupire wrote: post #34
rhelz wrote: post #22

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 circl…

> 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

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."

Re: Poor Foundations in Geometric Algebra

#49

Post by nicf »

thechao wrote: post #47
nicf wrote: post #20

Earlier quoted context omitted.

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 piece…

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 a…

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.

Re: Poor Foundations in Geometric Algebra

#50

Post by obsoletehippo »

altairprime wrote: post #33
rhelz wrote: post #22

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 circl…

> 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.…

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.
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