[graycat]

Ah, in math writing, it's easy enough to say more and be not boring and at times be at least interesting, inviting, even exciting!

Let's have some examples!!

(1) Dimension.

So, suppose we are in the first class in linear algebra:

"Maybe you have heard that the real line has 1 dimension, is 1 dimensional, the plane is 2 dimensional, and the space we live in is 3 dimensional. Well, that's all true enough, but in linear algebra we do better and have more: For one, we get to say clearly what is meant by dimension, that, in particular, why the line, plane, and space are 1, 2, 3 dimensional. For much more, for any positive integer n we have n-dimensional space.

Next, in linear algebra n-dimensional space is a relatively easy generalization of what we already know well in dimensions 1, 2, 3.

Why might we care? For example, we know well what distance is in dimensions 1, 2, 3, and distance in n dimensions is a straight forward generalization. In dimensions 2 and 3, we understand angle, and also that carries over to n dimensions. For more, with computing it is common to have a list of, say, 15 numbers. Well, for just one benefit, with linear algebra we get to regard that list as a point in n = 15 dimensional space, and doing so lets us do some powerful things with representing and approximating that list."

So, we get some sense of previews of coming attractions and some invitation to higher dimensions.

(2) Optimization.

"There is a subject, with a lot of development just after WWII, called linear programming (LP). The programming is in the English sense of operational planning as in war logistics and planning as was crucial in WWII. The linear is the same as in linear algebra.

The main goal, point of LP is to find how to exploit the freedom we have in doing the operations, the work to be done, to get the work done as fast or cheaply as possible, that is, to find an optimal way to do the work.

So, the subject LP is part of optimization. There have been some Nobel prizes from applications of LP and other math of optimization to economics. There have been applications of LP to feed mixing, oil refinery operation, management of large projects, and parts of transportation."

(3) The Simplex Algorithm.

"Maybe in high school algebra you saw the topic of systems of linear equations. Well, it is fair to say that the standard way to solve such a system is Gauss elimination due to C. F. Gauss.

The idea is simple: Multiplying one of the equations by some non-zero number and adding the resulting equation to another of the equations does not change the set of solutions. So, doing that in a slightly clever way results in the system of equations with a lot of zeros, about half all zeros, so that the set of solutions is obvious just by inspection.

Then for linear programming, in practice the main solution technique is the simplex algorithm, and it is just but done with optimization in mind."

(4) Completeness.

A rational number can be written as p/q for integers p and q. We will see, easily, that the rational numbers are not up to carrying the load, are not up to doing the work we need done. So we need a more powerful system of numbers -- we need the real numbers.

Here is a really simple place the rational numbers fail to do what we want: At times we consider square roots. E.g., the square root of 9 is 3. Well, what is the square root of 2? Suppose that square root were a rational number, i.e., so that

(p/q)^2 = 2

Then we have

p^2 = 2q^2

so that the left side has an even number of factors of 2 while the right side has an odd number. Tilt. Bummer! That can't be. That's a contradiction.

So, there is no rational number that is the square root of 2. So, for something really simple, just finding a square root, the rational numbers fail us, can't carry the load or do the work.

The real numbers will let us find the square root of 2 and much more. With the real numbers we get what we call completeness. A joke, basically correct, is that calculus is the elementary consequences of the completeness property of the real numbers. Then we generalize: Banach space is a complete normed linear space. Hilbert space is a complete inner product space. The Fourier transform works because of completeness. So, we move on and see how the real numbers are complete ...."

[Upitor]

These are fine intros, but then you have to actually dive into these topics. Sometimes there is no real interesting way to explain these topics. Fx, iirc the construction of the real numbers is rather tedious. But I agree, that more effort could be done to motivate many of these topics (at least that was my experience studying math)

[graycat]

"Sometimes"? Yup!

"Tedious"? Yup! Can use Dedekind cuts or maybe something called the normal completion, and especially the first is darned tedious!

There is a good math writer G. F. Simmons, of Introduction to Topology and Modern Analysis, who stated that the two pillars of analysis were linearity and continuity -- nice remark. He also stated that really to understand, have to chew on all the arguments, etc. or some such.

Then I decided to study the proofs really carefully, so to "chew", and in hopes of finding techniques I could use elsewhere. When I mentioned that study technique, objective to my department Chair his remark was "There is no time." -- he also had a point.

Commonly there is an intuitive explanation of what is going on and some views that can provide motivation to study the stuff at all.

There are a lot of books and papers. As a student, I saw a lot of the books, got copies of some of them, put them on my TODO reading list, etc. Eventually, after falling far enough behind on the list, I wondered just where all those books were coming from? It dawned on me, profs need to publish so they do. They are also supposed to have grad students and do. Then the grad students take the advanced course by their major prof and end up with a big pile of notes. Then the grad student, as an assistant prof, wants to publish so cleans up the pile of notes and contacts the usual publishers to publish a book. The top university libraries are essentially required to buy the books, so they get published and bought. And, then, often, there the books sit, gathering dust. I won't say that writing those books was a total waste, and I won't say that students should spend more time reading those books. Or, the books are there on the shelves. They are not really difficult to find. The books have work that was done. Maybe the work is useful now; maybe someday it will be useful; whatever, the work is done, the results found, and there in case they do become useful.

In the meanwhile, back to the mainline of math education, research, applications, usually there can be some helpful intuitive explanations and motivating example applications!

Apparently some authors just give up and assume that their books will mostly just gather dust. But once I wrote Paul Halmos, likely my favorite author, and got back a nice letter from him with "It warms the heart of an author actually to be read, and clearly understood, by ordinary humans." -- at the time I had no academic affiliation and was just reading his book on my own. So, Halmos was surprised that an ordinary human would be reading and understanding his book.

Ah, in what I wrote, I left out that also in linear algebra in n dimensional space, the Pythagorean theorem still holds, that is, an n dimensional version holds!