Math, Algebra: on the Phraseology of X Over K, and What's Group Theory?

By Xah Lee. Date: . Last updated: .

Xah's rumination extempore! on the math phraseology of X over k. (xah's rumination extempore is when i type as fast as things on my mind and i dunno what am talking about or going, but you rip my brain.)

one of the most annoying phraseology in math is X over k, where the X and k are algebraic structures.

to wit:

An n-ary quadratic form over a field K is a homogeneous polynomial of degree 2 in n variables with coefficients in K…

what's with the “over”?

for a long time, in the 1990s, i couldn't understand it, couldn't get over it still. The issue is that, if you are learning math, it presents a significant illogical hurdle on what it is. I'm guessing, that most math students don't have that problem. But for me, a logic obsessive minded person, couldn't get over it.

look and repeat it slowly after me: X over k.

what is with the “over” thing? you mean flying over? you mean covering up? you mean it's ALL OVER me? like in every place or it's THE END? End of the world? Game Over!

ok, now, this issue is quite interesting. I speak of my mind, which i presume is of a particular perhaps idiosyncratic logical. I'm guessing, most mathematicians don't have a problem with the phrase, but actually i am not too sure. As in most things, when you dig into the detail, you find out that what you presumed isn't so. (in short, i do think this phraseology is problematic, and i do not doubt there's a sizable mathematician or sub-community are against it, perhaps even with lots literature)

anyway, i was saying, this issue is quite interesting. It involves: linguistics. That is, in linguistics, one thing they study is idioms. This is one particular idiom, and specialized among mathematicians.

the other interesting thing is that, in math, there are quite a lot such rather colorful and linguistically abnormal phraseologies . Let's say you just started to study math as a math major in college. At first, you'll be puzzled by them, but after a while, you get used to, as the non-sensical english “okay”!

the other interesting thing is that, i've been, kinda now and then thought about what is a proper replacement for such phrase. By the way, a short explanation of the term is in order for those of you programers who have no idea what it means:

in algebra, there's a “structure”. Algebra is about “structure”. One simplest example is “group” as in group theory. Basically, the most dry and precise definition goes like this:

A group is:

This function ƒ and the set G must have these properties:

so, for your programers out there, this is what group means. Group of group theory, which is the most basic and useful structure of algebra. In algebra, there's also “field” and “ring”. They are like group, except there are 2 functions instead of one, and each satisfies some properties as do in group.

you see, what you've just read is basically all algebra is about. Very dry, and very abstract. as a coder, you may ask, why would this be useful for? as it seems very arbitrary and just bunch of meaningless logic statements. Well, the thing is, group theory and ring and fields came from eventual abstraction of numbers and real numbers, and the trying to understand solutions of equation such as x^3+ 44 x^2 - 5 = 0. Try to solve such arbitrary equations, and after a hundred years, and you'll have lots issues, and eventually, you find pattern and theories about it, and, that is: group theory! ring theory! field theory! Abstract Algebra! Rather very interesting and uttermost satisfying mind trip. (what the above is basically the first few hundred years history of abstract algebra. more or less the ballpark)

ok, now, back to the “V over k” phraseology thingy.

First, what does it mean? It means this:

when mathematicians says “blab blab is a algebraic structure V over k”. It means,

it's a set G… (actually let's mirror our explanation of group above here.) Let's denote G's elements to be g1, g2, g3, etc. Now, here's the interesting thing. The elements of G, those g1, g2, g3, are not simple atomic things. They are themselves algebraic structures. That is, for example, the element of G may be a paired number, written as {x,y}, where x and y are real numbers, where you can speak of x+y or x*y. (remember that x and y are component of the element of the set G). So, overal all, we want to talk about the function with element of G, those g1, g2, g3, but, we also want to talk about operations that occure between the component of a element of G, such as whether {3*x,y} is a element of G.

Basically, the gist is that, when we define some algebraic structure, often, the elements of the set is a composite entity, that itself is a algebraic structure, and it is often some ordered n-tuple of numbers, and they constitute a field or ring, with their own number of binary operations. (often the standard addition and multiplication)

So, thus the phrase “something over something”, where the first “something” is a algebraic structure the mathematician talking about, the main subject, but the second “something” is is a structure formed by the component of a element of the structure, usually one of field or ring.

do you remember your linear algebra?

here's Wikipedia first paragraph about Vector Space:

A vector space is a mathematical structure formed by a collection of elements called vectors, which may be added together and multiplied (“scaled”) by numbers, called scalars in this context. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. T

now, see, the formal definition section, quote:

A vector space over a field F is a set V together with two operations that satisfy the eight axioms listed below.

you see? this “some over some” is actually notionally fairly complex thing. You have a structure within a structure. Now, this also gets more interesting.

you (or I) begin to wonder, hold on a sec, now i study algebra for its elegant simplicity, why now we often have this structure within structure shit? Why is a algebraic structure almost always some over some field/ring? I don't want the field/ring thing. I want a simple structure and study just that. Or, fine, if you have structure within structure, but i don't want anything to do with the “over” stuff. I want it to be explained/exposited as structure within structure. I don't want to hear about the inner structure itself like a side-effect.

anyhow, now and then i've thought about how to rephrase this. But haven't been powerful enough to do. One thing to exam is, when “Some over x” is said, to what extend we NEED to mention the x? Sure, the x part gives context, but asides from giving context, is it necessary in the study/definition the “Some” part?

linear algebra is one of the math i love the most and understand the most. But one day, when i tried to understand it abstractly, as in algebraic structure, it's beautiful, but i realized it's rather extremely complex. You see, a vector space, isn't just about a set, with some binary function defined for the set. It involves a extraneous thing, called “scalar”, that interacts with the element of the set via the substructure's functions. With this in mind, suddenly it becomes a complex ugly thing. (and, the proper abstract study of this structure, algebraic structure, i've later learned, is called a “module”: Module (mathematics) )

also, you know, one way to shape my perspective in a normally understoodable way is to think of calculational math (movement/approach) or computer math system approach. There, you won't have this “over” thing, period. It is very, very, annoying.

ok, so it's “over”, as in “over my head”. Is there a “under”?

#math #linguistics #algebra