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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 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.


  1. Goal
  2. Style of exposition
  3. Form of publication
  4. Outline
  5. Topic Posts
  6. References


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:

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:

Form of publication

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


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


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

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


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

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

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.


On line

Books that give intuitive understanding of a math subject

I would appreciate additional references to books like these

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This work is licensed under a Creative Commons Attribution-ShareAlike 2.5 License.