Lambda the Ultimate

Wikipedia has not one, but four articles about Clifford algebras (three of the articles there are not really about them).

I thought when I read the post I was going to read about the algebraic theory of geometric logic, which does not have a WP article.

C'mon, I was hoping to at least see a rationale for the posting which involved connecting programming languages to algebras and then noting an isomorphism from symbolic representations into the geometric domain. Where's category theory when you need it?

GA is one of those subjects that make me scream like a 12 year old girl at a Backstreet Boy concert (are they still around? I'm not really up to date...) Anyway, as a first year physics undergrad, it's been a real pain to find any decent, really decent lecture courses, despite being at Cambridge, where there is an actual Clifford Algebra research group. The only lecture course available is a Part III (fourth-year, Masters-level) course, which clashes with the more mundane E&B and MSM lectures. I've got the books by Hestenes and Doran/Lasenby, but they're pretty hard going, if you're trying to to really get to grips with special-relativity and finding that geometry goes further than you ever thought, what with null vectors and other esoteric metrics. The last few chapters from Doran/Lasenby are just incomprehensible unless you're already familiar with gauge theory and general relativity.

*Sigh*

(rant over)

Shamelessly abusing LtU even further, my copy of Divine Proportions has just arrived from Australia.

I ordered it after being intrigued by a write-up on Slashdot, but don't let that put you off. It would certainly be useful for any computer scientist (or game programmer?) hoping to speed up and improve the accuracy of various geometric calculations.

Ok, you asked for wacky ideas. I've been wanting to model the lambda calculus or the pi calculus in GA, so that I can write functional programs as GA expressions and equations. This is so I can start thinking of physical processes as computations. I haven't gotten far at all, but perhaps thinking of vectors as lambda abstractions would be a start. So if a and b are vectors qua lambda abstractions, then ab is the geometric product qua function application. Not sure I can get very far with this idea, since the most obvious model for lambda's variables is basis vectors and there aren't enough of them.

If we note that every n-dim GA has a matrix representation of dim 2^n, then we can think in linear algebra, instead, where we can write lambda variables as column vectors and lambda abstractions as matrices, that is, as linear transformations in the 2^n-dim vector space of GA multivectors. So, now, we have three things floating around: GA, lambda, and linear algebra. But, for the moment, put the GA aside, and let D=2^n so we don't have to keep writing it over and over.

So, if the lambda calculus looks like this
T ::= x | \x.T | T T
then our linear-algebra version of it would look like this (I'll resist writing it in Haskell, yet)
T ::= x (of type DVector)
| M (DxD matrix, of type DVector -> DVector)
| M M (function application, matrix-to-matrix)
| M x (function application, vector-to-vector)

At this point I run out of gas because I don't have a straightforward transcription for lambda's substitution model. (EDIT (thinking out loud): M x scrambles the elements in x, it doesn't substitute x for any other vectors that are hiding in the matrix M, perhaps as columns or rows. Now, it occurs to me that M-as-function is a dead idea... there is just a finite amount of information inside a matrix. Waving my hands wildly, matrices might just be able to model classes of primitive recursive functions, so would never be Turing-complete i.e. Godellable. Ok, so now, what ... there just MUST be a way to model arbitrary computations as matrix computations, then backmodel them as GA computations)

Part of me is reminded of Ed Fredkin, who made NOR gates for billiard balls, so certainly I could just model some low-level hardware component with GA, then Church-Turing my way back into the lambda calculus, but this would be a defeat. I really want a way to do a higher-level model of the lambda in GA or LA.

EDIT: oh by the way, one application of such a thing would be as a natural, composable, functional programming language for quantum computing, since quantum mechanics fits GA like a talcum-dusted medical glove (spinors are for free!). Most QC researchers work in the Turing world since they care about performance and need to count instruction executions. I haven't seen any QC in the Church world ... doesn't mean it isn't there, I just haven't seen it.