(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 doing proper pure-mathematics research for several decades. I have a BA degree, a master's, and a PhD in mathematics from the University of Cambridge. After that I worked for a couple of years as a junior research fellow (i.e., the lowliest type of academic staff member) at one of the colleges there, and then moved into the Real World where the mathematics is easier and the pay is better. I've worked for a number of startups doing broadly mathematical (often but not always geometrical) things, and after a chain of acquisitions I'm now working for HP.
The post you linked to claims to be philosophical as well as mathematical. I have no particular qualifications in philosophy, unless you count the irrelevant fact that PhD stands for "philosophiae doctor". (I've read a fair bit, but obviously that proves nothing much.)
Of course, whether it's true that your earlier post has nothing to do with geometric algebra doesn't depend on my qualifications or abilities or knowledge or whatever; it depends on what that post is about and what geometric algebra is about. So I'll say a few words about those.
Geometric algebra is a quite specific thing in mathematics, related to differential geometry and exterior algebra and the like. It involves a construction in which one starts with a vector space (say, R^3, which one can think of as the place where ordinary three-dimensional things live), and embeds this in a larger structure called a Clifford algebra whose elements can be multiplied together in a meaningful way. This turns out to be a useful framework for various things in physics and computer graphics.
The linked post is titled "An exploration of the foundations of logic and philosophy". I remark that if it were about that then it would be very difficult for it also to be about geometric algebra, since geometric algebra doesn't have much to do with the foundations of logic and philosophy. In fact it doesn't seem to me that the linked post has much to do with the foundations of logic and philosophy either.
It invites us to consider a straight line on a piece of paper, and then a construction where a differently-coloured piece of paper is overlaid on the first one with an edge along that straight line, and then one where instead the two pieces of paper are the same colour. It then starts talking about "an object which we compare with itself it as it is represented on each side of our line", and -- to my mind, though of course it may be that I'm just not clever enough to understand the author's point(s) -- devolves further and further into word salad from there. "If our 0 width line is a number line and the parallel postulate is a statement about Cauchy completeness, what do our extended logics construct as number-like objects?" It means nothing to say that some specific geometrical line "is a number line"; the parallel postulate is not really a statement about Cauchy completeness, and the usual constructions of geometries where it fails aren't ones where anything strange is going on with the real numbers; the situations involving pieces of paper etc. are not "extended logics" in any useful sense I can see.
Anyway: none of that makes any reference to vector spaces (other than, implicitly, one particular vector space of dimension two), nor to Clifford algebras (even implicitly), nor to any sort of "algebra" construction (in the sense of a vector space with a multiplication operation defined on it). The only overlap between it and "geometric algebra" is that both have something to do with geometry. And even that seems kinda dubious, since the author's intention with the linked post is apparently to say something about "the foundations of logic" and that's taking things in an entirely different direction from anything to do with geometric algebra.
(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?
If your thought experiment involving pieces of paper has helped you to think of useful or interesting things that you wouldn't otherwise have thought of, that's great. It hasn't so far done anything of the kind for me. The fault could of course be mine.
What I said was not (as you claimed in another comment in this thread) that I don't see how constructivism is relevant -- though in fact I don't think constructivism is relevant -- but that I don't see how your thought experiment about pieces of paper is relevant.
I regret that it wasn't immediately apparent to me that your thought experiment was about constructivism. Now that you say it is, I can kinda see how it kinda relates, but for me thinking about your pieces of paper doesn't conjure up any useful insights about constructivism, and if I didn't already know about constructivism I don't think it would particularly point me in that direction. Again, if it works that way for you, good for you, but I think that when you present it to other people you are forgetting that the others (who unlike you haven't been thinking about this picture for 30 years) don't have the same associations with your imaginary pieces of paper that you have.
I have no problem with the idea that one might want to construct alternative number-like objects. (And yes, I know what Cauchy sequences are and what one does with them. And no, they really don't have very much to do with the parallel postulate. There's an analogy between the parallel postulate and, say, the LEM, but that's all.) But (1) again, this has nothing much to do with Eric Lengyel's critique of how geometric algebra is presented -- when Lengyel talks about "foundations" he isn't talking about going down to the level of set theory, logic, HOTT, or whatever, he's talking about getting the basics of geometric algebra right when one already has notions like "real numbers", "vector space", etc., in hand -- and (2) your thought experiment happens not to do anything for me to clarify, inspire, etc., ideas about alternative number-like objects. (Also, it seems to me that if you want to use that thought experiment to say something about number lines then you want your 0-width lines, half-planes, etc., to lie across the number line, not along it.)
It is simply not true that book 2 of Euclid "defines geometric algebra for the first time" or indeed defines it at all. Euclid is doing geometry but is not doing geometric algebra, which is something more specific.
"What can we do with the black-on-white paper construction to extract information from the system?"
I will need to leave this discussion with you here. I only mean to hold open the door for you and to give you two more variations of the tools that were used to define this long tradition of rational inquiry.
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 guidelines.
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.
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.
