# ABSTRACTING ALGEBRA

Charles Wells

Last revised 2015 1130

This article describes a plan for a book, tentatively titled Abstracting algebra (AbAl), which takes you through algebra in its successive levels of abstraction. I expect to write pieces of the book as AbAl topic posts, and to reuse some stuff that I have written on this blog and in abstractmath.org. No guarantees on if or when the book will be published.

I would also like to bundle up parts of abstractmath and turn them into book(s), using the same methodology.

## Goal

The goal of the book is to convey an understanding of how mathematicians think about numbers and algebra, with particular emphasis on the use of abstraction. In fact, the intention is for the reader to develop an understanding of abstraction in the context of abstract algebra.

The book is intended for people who are interested in math and want to gain an intuitive understanding of some aspect of math. This includes math students, STEM people and math fans:

• People who work in a field that uses math and want to know more about their tools.
• People who are taking math courses -- high school or university -- and are actually interested in what they are learning .
• People who are interested in math as a hobby -- the kind of people who read Martin Gardner's column in Scientific American.

Would a text-to-speech reader know how to pronounce the word "read" in the previous sentence?

Important: The book may be helpful to math majors, but it is not designed to teach the subjects discussed in the depth that math majors need to learn a subject. Math majors do need intuitive understanding, but they also need to learn how to read formally written math, understand and create proofs, and to use what they have learned in applications.

I would appreciate links to other books and web sites that give non-mathematicians an intuitive understanding of some part of math. There are some good ones (see References) but not enough.

## Style of exposition

The book will be in the style of my posts on G&G:

• Strong emphasis on understanding concepts, and understanding how they evolved.
• Informal writing. In particular, free use of italics, boldface and colors. (See my post The revolution in technical exposition II.)
• Lots of graphs and pictures, in particular interactive graphs.
• Thick descriptions of concepts.
• Using metaphors and many different representations to emphasize the many ways you can think about any important part of math. This overlaps with thick description, which also includes context and application.
• Oversimplified descriptions (with warnings). See Example C4 in Representations of mathematical objects.
• Extensive discussion of naive perceptions of abstract objects.
• In particular, two different descriptions of a kind of math object can be perceived as completely different things, even though mathematicians may "know" they are different aspects of the same kind of object. Reductivist mathematicians ("the number $2$ is the set $\{\{0\}\}$") will not like this book.
• Using formal definitions and proofs only when I think they communicate something important. The bare definition of a concept is nearly useless in understanding it, and some proofs illuminate while other proofs obfuscate. The reader is not being prepared to do proofs in abstract algebra, but rather to have a rich mental picture of the concepts I discuss.
• Using bullet lists in the way that the Mathematica Help topics are written (see Plot for an example). These topics have one or two levels of hierarchy that include bullet lists for subsidiary topics (instead of more hierarchy) in which the successive statements are not necessarily parallel and occur in a flattened list. This makes it easy to find short pieces of information -- provided the bullet list is not to long. This list is an example.
• Links to places on the internet that give more insight or information about a topic. References to books will be necessary, too, but young people rarely follow up on them.

## Form of publication

Abstracting algebra will be published on e-readers that must have these capabilities:

• Use TeX to generate the math and have the math show up seamlessly on the e-reader. I use MathType in Microsoft Word for abstractmath and MathJax for this blog. The html generated by MT and Word renders differently on different browsers and loads slowly. MathJax requires a connection to the internet.
• Include graphics generated by xypic and by Mathematica. I currently display these using jpeg but it is not very sharp. I am investigating direct rendering of PDF files. I have just discovered the Stackoverflow discussion of how to put PDF files into html and will experiment with these methods on G&G posts.
• Incorporate interactive demonstrations using Mathematica CDF files. I don't know if this is possible on any e-reader now.
• Make side comments in text boxes inset into the main body or in the margin. I used this a lot in the printed version of the Handbook (done in LaTeX) and somewhat less in abstractmath (using Word textboxes). But comments in small print, as in the remark above about text-to-speech, can serve as a pretty good substitute.

## Outline

The topics in the outline will be linked to the topic posts as they appear.

### Numbers

The chapter on numbers would describe some of the aspects of numbers that are relevant for algebra and abstract algebra.

• One thing is to describe our naive intuitions about number. The intuitions that many abstract algebra ideas are based on are different from our naive intuitions and produce a gulf of understanding between mathematicians and students.
• For example, most people naturally think of addition as an operation on lists of numbers, not as a binary operation. Just as in a monad.
• A deeper example, and prehaps the biggest gap in students' understanding, is the mathematicians' understanding of the difference between a number as an abstract object and a representation of the number.

I am not clear just what topics the numbers section should include. It should not balloon into a book about numbers. But at least:

• Base notation, which introduces representations.
• Algorithms, for example for addition and multiplication. These algorithms are likely the first thing that comes to mind when most people think of numbers.
• Product of primes representation, for integers, which shows that different representations are good for different things.
• Commutativity, associativity, inverses etc should be discussed in terms of examples of specific numbers. No variables. (I am not sure how far to go with this.)

### Algebra

By "algebra" I mean high school and college algebra.

• The use of of variables. There are different kinds of variables and many different ways of thinking about variables. See Susanna Epp's article and the abmath article.
• Free variables and bound variables cause familiar problems in understanding algebra.
• Another sort of variable is a parameter. People who want to reduce everything to logic mutter that they are merely free variables. Category theorists mutter that they are completely explained by cartesian closedness. In fact, they are psychologically complex and result in difficult (for beginners) mathematical objects such as families of functions.
• Polynomials and other functions. Well, they are also expressions. A big difficulty hides there.
• Equations. There are many kinds of equation, such as equations with a finite number of solutions, constraints, identities, the equals sign used after "Let...". You may mutter that equations with a finite number of solutions are constraints, but they are quite different psychologically, which this book is especially concerned about.
• Rules of transformation of equations based on properties of operations. This is what people think of when they hear about "algebra".
• Algebra is usually presented only in the usual algebraic notation. It should be presented in many ways, including tree form, which carries a much lower abstraction load. See Visible algebra II and Making visible the abstraction in algebraic notation for some examples of what I mean.

### Abstract algebra

Abstract algebra in academia is really two or three subjects. One is called abstract algebra and is concerned mostly with groups, rings and fields. These are three important branches of math that fit nicely together into one course: In other words, "abstract algebra" is not really a single field of mathematics. But all three result from taking the examples of symmetries (for groups), number systems (for rings and fields) and linear algebra (for rings).

"Rings" is not the right word; the subject I am thinking of includes modules and algebras over rings and fields, too. The previous paragraph is an example of an oversimplified description that (I think) nevertheless gives a clue about the big picture.

These topics arose by strong abstraction, by which I mean taking properties of some examples and turning them into axioms imposed on a mathematical structure with an underlying set whose elements are structureless "points". I probably should introduce the subject with some simpler examples of axiomatically defined structures. One of my favorites is equivalence relations = partitions, where two different structures turn out to be strongly equivalent. But that might be a distraction...

I have a weak understanding of what I would mean by "weak abstraction" -- it would probably include, for example, turning constants into parameters (from the plane and 3-space to $n$-dimensional spaces), weakening an axiom on a structure without changing the structural data (fields to near fields), or using a more general structure with essentially similar axioms (numerical arithmetic into linear algebra).

The other subject that belongs in this section is universal algebra, the general study of operations with axioms imposed on them. This is not so often taught in colleges and universities, but I certainly need to explain the idea of universal algebra and in particular how properties of operations affect the theorems that are true about them, because that turns into a big part of the last section of the book.

### Abstract abstract algebra

This phrase refers to several second order strong abstractions of abstract algebra.
Monads These give a very different perspective to universal algebra and works in any category. It is important to give a detailed example of a strong monad, used in computing science, but I will have to learn more about them. (Strong monads are related to monads but they are not a special kind of monad.)
Lawvere theories Equivalent to monads in terms of the algebras they generate but a very different point of view. Gets away from presentations and their often misleading peculiarities.
Sketches Another second order strong abstraction of abstract algebra that covers all of universal algebra and more. Multisorted and generalized to all categories. Their corresponding theories eliminate presentations. See the entries on sketches in the references.
Forms A strong abstraction of sketches, so this is a triple abstraction of abstract algebra. Dare I add a section called "Abstract abstract abstract algebra"? I invented this idea in 1990 (Wells). Eric Palmgren and Steve Vickers independently worked out an equivalent methodology (P&V) using finite-limits logic. The paper by Bagchi & Wells give more details of my original conception, based on graphs and diagrams in categories. My unpublished intuitive introduction Forms can be the start of this part of the book.

## Topic Posts

AbAl topic posts will be posts on Gyre&Gimble and each post will be listed in the abstracting algebra category listed under Subjects in the right column of the blog. This subject category will not appear in the listing until I publish my first topic post, which will be Real Soon Now.

Word Press is designed to use the word "categories". AbAl will say a lot about mathematical categories. I think I know how to get into the inner workings of Word Press to change their word "categories" to some other word, such as "topics". Maybe some day I will do it.

Each AbAL topic post concerns a particular subject that that book will cover. The post is not even a first draft of the AbAl section on that subject; instead it is a partly filled-in outline. Each topic post will consist of

• Parts of the text that I have already written.
• An outline of what other ideas it will go over.
• Meta-comments in smaller print on how I present the material. Some of them will probably not appear in the published book.
• References to other topics, mainly in Wikipedia or abmath. Many of these will be changed to references to other sections of the book when they are written.
• Descriptions in smaller print of other diagrams and demos that are planned.

I am doing it this way because the topic posts will contain many pictures and diagrams, including demos (interactive diagrams) based on Mathematica which take me a long time to create.