Poor Foundations in Geometric Algebra

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.

> 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?

(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.

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