31 March 2008

Kids are better at algebra than you think

Word problems take place in a graded ring, from (the recently relocated) Mathematics under the Microscope (Alexandre Borovik), via The Unapologetic Mathematician (John Armstrong).

In short, Borovik claims that elementary school word problems take place over $\mathbb{Q}[x_1, x_1^{-1}, x_2, x_2^{-1}, \ldots, x_n, x_n^{-1}]$, where the xi represent different things that could be added. In this formalism, it makes sense to add, say, apples and oranges, going against the usual rule that you're only allowed to add quantities with the same "dimension". (Indeed, Borovik illustrates the idea with an example of this nature.)

I'm reminded of "dimensional analysis" as taught in, say, introductory physics classes, where we only allow monomials to have meaning, namely that the monomial x ma kgb secc measures some physical quantity with dimensions LaMbTc where L, M, T stand for length, mass, and time. (For example, in the case of speed, a = 1, b = 0, c = -1.) I can't think of situations in physics where one deals with a quantity of the form, say, a kg + b m. Is this because they don't exist, or because I don't know as much physics as some people?


CarlBrannen said...

Mixed metaphors occur in Clifford algebra, but they're kind of subtle. I've always thought it was kind of weird.

For example, consider Dirac's gamma matrices used to model electrons and positrons. There are four gamma matrices which I will call x, y, z, and t. These are the vectors, and as such carry units of distance (or time, depending on where you put the c).

The gamma matrices anticommute and square to +1, +1, +1 and -1 (at least they do in my favorite signature).

The "Dirac bilinears" consist of all the products of the vectors. There are 16 of these and they come in 5 blades: scalar, vector, bivector, psuedo vector (or axial vector), and psuedo scalar.

In the theory of elementary particles, the weak force is modeled with "V-A" that combines objects from the vector and axial vector blades.

So I don't know if you'd count this. But you can see more of the same stuff by looking at what David Hestenes writes about geometric algebra, which is Clifford algebra with a physics interpretation, and related to the spacetime manifold.

Anonymous said...

"blades"? "mixed metaphors"? Either you've got a completely different source for your studies than I've ever seen or you're just making up terminology as you go along, Carl.

Now, if by "blades" you mean "grades", then yes, there is a grading on a Clifford algebra. However, in the physical interpretations, one still only ever uses homogenous elements. Isabel's point is that in algebra we add elements of different grades all the time. In physics, we forbid adding elements of different grades by positing magical "superselection rules". The Clifford algebra formulations are no different.

CarlBrannen said...

If you want to know what a "blade" is in physics, do a google search. And whether you call them grades or blades, A-V mixes them.

Anonymous said...

Dimensional analysis is useful for answering the "Hey, I've got lots of stuff, how do I [do math operations on them to] get what I need? Plus, if it works, it's Occam-compliant out of the box [up to constants].

If you have a physical quantity in kg+m units, there is a kg*m equivalent [yay, Pythagoras].

[sigh] meetings...