The interactive example in this post require installing Wolfram CDF player, which is free and works on most desktop computers using Firefox, Safari and Internet Explorer, but not Chrome. The source code is the Mathematica Notebook algebra2.nb, which is available for free use under a Creative Commons Attribution-ShareAlike 2.5 License. The notebook can be read by CDF Player if you cannot make the embedded versions in this post work.
Notes on viewing
Explaining math
I am in the process of writing an explanation of monads for people with not much math background. In that article, I began to explain my ideas about exposition for readers at that level and after I had written several paragraphs decided I needed a separate article about exposition. This is that article. It is mostly about language.
Who is it written for?
Interested laypeople
There are many recent books explaining some aspect of math for people who are not happy with high school algebra; some of them are listed in the references. They must be smart readers who know how to concentrate, but for whom algebra and logic and definition-theorem-proof do not communicate. They could be called interested laypeople, but that is a lousy name and I would appreciate suggestions for a better name.
Math newbies
My post on monads is aimed at people who have some math, and who are interested in "understanding" some aspect of "higher math"; not understanding in the sense of being able to prove things about monads, but merely how to think about them. I will call them math newbies. Of course, I am including math majors, but I want to make it available to other people who are willing to tackle mathematical explanations and who are interested in knowing more about advanced stuff.
These "other people" may include people (students and practitioners) in other science & technology areas as well as liberal-artsy people. There are such people, I have met them. I recall one theologian who asked me about what was the big deal about ruler-and-compass construction and who seemed to feel enlightened when I told him that those constructions preserve exactly the ideal nature of geometric objects. (I later found out he was a famous theologian I had never heard of, just like Ngô Bảo Châu is a famous mathematician nonmathematicians have never heard of.)
Algebra and other foreign languages
If you are aiming at interested laypeople you absolutely must avoid algebra. It is a foreign language that simply does not communicate to most of the educated people in the world. Learning a foreign language is difficult.
So how do you avoid algebra? Well, you have to be clever and insightful. The book by Matthew Watkins (below) has absolutely wonderful tricks for doing that, and I think anyone interested in math exposition ought to read it. He uses metaphors, pictures and saying the same thing in different words. When you finish reading his book, you won't know how to prove statements related to the prime number theorem (unless you already knew how) but you have a good chance of understanding the statement of some theorem in that subject. See my review of the book for more details.
If your article is for math newbies, you don't have to avoid algebra completely. But remember they are newbies and not as fluent as you are — they do things analogous to "Throw Mama from the train a kiss" and "I can haz cheeseburger?". But if you are trying to give them some way of thinking about a concept, you need many other things (metaphors, illustrative applications, diagrams…) You don't need the definition-theorem-proof style too common in "exposition". (You do need that for math majors who want to become professional mathematicians.)
Unfamiliar notation
In writing expositions for interested laypeople or math newbies, you should not introduce an unfamiliar notation system (which is like a minilanguage). I expect to write the monad article without commutative diagrams. Now, commutative diagrams are a wonderful invention, the best way of writing about categories, and they are widely used by other than category theorists. But to explain monads to a newbie by introducing and then using commutative diagrams is like incorporating a short grammar of Spanish which you will then use in an explanation of Sancho Panza's relationship with Don Quixote.
The abstractmath article on and, or and not does not use any of the several symbolic notations for logic that are in use. The explanations simply use "and", "or" and "not". I did introduce the notation, but didn't use it in the explanations. When I rewrite the article I expect to put the notation at the end of the article instead of in the middle. I expect to rewrite the other articles on mathematical reasoning to follow that practice, too.
Technical terminology
This is about the technical terminology used in math. Technical terminology belongs to the math dialect (or register) of English, which is not a foreign language in the same sense as algebra. One big problem is changing the meaning of ordinary English words to a technical meaning. This requires a definition, and definitions are not something most people take seriously until they have been thoroughly brainwashed into using mathematical methodology. Math majors have to be brainwashed in this way, but if you are writing for laypeople or newbies you cannot use the technology of formal definition.
Groups, simple groups
"You say the Monster Group is SIMPLE??? You must be a GENIUS!" So Mark Ronan in his book (below) referred to simple groups as atoms. Marcus du Sautoy calls them building blocks. The mathematical meaning of "simple group" is not a transparent consequence of the meanings of "simple" and "group". Du Sautoy usually writes "group of symmetries" instead of just "group", which gives you an image of what he is talking about without having to go into the abstract definition of group. So in that usage, "group" just means "collection", which is what some students continue to think well after you give the definition.
A better, but ugly, name for "group" might be "symmetroid". It sounds technical, but that might be an advantage, not a disadvantage. "Group" obviously means any collection, as I've known since childhood. "Symmetroid" I've never heard of so maybe I'd better find out what it means.
In beginning abstract math courses my students fervently (but subconsciously) believe that they can figure out what a word means by what it means already, never mind the "definition" which causes their eyes to glaze over. You have to be really persuasive to change their minds.
Prime factorization
Matthew Watkins referred to the prime factorization of an integer as a cluster. I am not sure why Watkins doesn't like "prime factorization", which usually refers to an expression such as $p^{n_1}_1p^{n_2}_2\ldots p^{n_k}_k$. This (as he says) has a spurious ordering that makes you have to worry about what the uniqueness of factorization means. The prime factorization is really a multiset of primes, where the order does not matter.
Watkins illustrates a cluster of primes as a bunch of pingpong balls stuck together with glue, so the prime factorization of 90 would be four smushed together balls marked 2, 3, 3 and 5. Below is another way of illustrating the prime factorization of 90. Yes, the random movement programming could be improved, but Mathematica seduces you into infinite playing around and I want to finish this post. (Actually, I am beginning to think I like smushed pingpong balls better. Even better would be a smushed pingpong picture that I could click on and look at it from different angles.)
Metaphors, pictures, graphs, animation
Any exposition of math should use metaphors, pictures and graphs, especially manipulable pictures (like the one above) and graphs. This applies to expositions for math majors as well as laypeople and newbies. Calculus and other texts nowadays have begun doing this, more with pictures than with metaphors.
I was turned on to these ideas as far back as 1967 (date not certain) when I found an early version of David Mumford's "Red Book", which I think evolved into the book The Red Book of Varieties and Schemes. The early version was a revelation to me both about schemes and about exposition. I have lost the early book and only looked at the published version briefly when it appeared (1999). I remember (not necessarily correctly) that he illustrated the spectrum as a graph whose coordinates were primes, and generic points were smudges. Writing this post has motivated me to go to the University of Minnesota math library and look at the published version again.
References
Expositions for educated non-mathematicians
Previous posts in G&G
Relevant abmath articles
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