abstractmath.org 2.0
help with abstract math

Revised 2017-04-24     website TOC      website index   blog	



Who is this website for?

What is abstract math?


Reading this website

Other sources for math

Who is this website for?

Abstractmath.org is designed for people who are beginning the study of some part of abstract math. This includes:

What is abstract math?

Abstract math is my name for what is often called “higher math” or “pure math”.

Overview of the site

This website is a multiple-entry site with many cross-links. This overview will give you a start on finding out what is on it.

The four main parts of the website

These head pages explain the ideas of each part in more detail:

The languages of math

Mathematical reasoning.

Understanding math

Doing math

Other articles

Abmath also has articles on certain mathematical topics:




Other things on this website

Background and Attitude: This article describes some of the thinking behind this website.

Diagnostic examples: These examples illustrate some of the many kinds of difficulty people meet with when studying and doing abstract math. Each example gives links to the relevant sections of the website or elsewhere.

Gyre&Gimble: A blog that discusses new ideas I have about abstract math and language, some specifically related to abstractmath.org.

Discrete Mathematics Class Notes: An introduction to abstract math for computing science students based on some of the ideas of abstractmath.org.

Mathematica notebooks and CDF files: These are the sources for most of the diagrams, including interactive diagrams, in abmath and in the Gyre&Gimble posts. There is more information about these files below.

Expository articles.

All the abstractmath.org articles and all the other documents mentioned in this section are licensed under a Creative Commons Attribution-ShareAlike 2.5 License.

Reading this website

Purple prose

Many of the important ideas about mathematics in this site are summarized in

Slogans in Purple Prose


This website is aimed at people with widely varying knowledge of parts of math. When some sort of math object or method is mentioned, it is usually linked to a definition or explanation elsewhere in abstractmath.org or in Wikipedia.

This website is intended to be read
on the screen, not on paper.
If you are not sure you understand a word
or phrase underlined in blue
I encourage you to click on it.

This is particularly important if the word is an ordinary English word rather than a technical word such as "homomorphism", because mathematicians routinely give ordinary words technical meanings which may be quite different from their ordinary meanings. Examples: "definition" and "group".

Images and metaphors

In many places in abmath you will find a section or an article with the phrase "images and metaphors" in its title. An article title "Images and metaphors for [topic]" will tell you about how to think intuitively about [topic]. Gaining intuition about a topic in math is as important as learning how to understand and come up with proofs.

Intuition is both vitally necessary and dangerous, since it can mislead you. The article baldly titled Images and Metaphors explains this aspect of doing math in general, and it is worth looking at to get an overview of what is going on when you read (for example) Functions: Images Metaphors and Representations (big article) or Images and Metaphors for Rational Numbers (little bitty section).


Many of the abmath articles are illustrated by diagrams created in Mathematica.

All these files are in the directory Mathematica, many of them in subdirectories.

How to use the Mathematica files

Other sources for math

Math sites

I provide links to other treatments of a specific topic at the point where they are discussed. These general links are particularly useful for learning about various aspects of math:

Mathematical Association of America

American Mathematical Society.

Math Forum


MathOverflow A question and answer site for professional mathematicians

MathStackExchange A question and answer site for people studying math at any level.

Planet Math

+plus magazine


Wolfram Alpha

Math Blogs

Division by zero

Explaining Mathematics

Flying colours

God plays dice

Gowers’s Weblog


Math Counterexamples

Math with Bad Drawing



$n$-Category Cafe

Pat’s Blog

Plus Magazine Blog

Roots of unity

Tanya Kovanova's Math Blog

The unapologetic mathematician

Visual Insight

Books on beginning abstract math

The books listed below are suitable for people beginning abstract math.  Except for the Handbook, they emphasize different aspects of abstract math from what this website emphasizes. 

Ash, Robert, A Primer of Abstract Mathematics.

Gowers, Timothy, Mathematics: a very short introduction.

Hale, Margie, Essentials of Mathematics: Introduction to Theory, Proof and the Professional Culture.

Solow, Daniel, How to Read and Do Proofs : An Introduction to Mathematical Thought Processes. 

Wells, Charles, The Handbook of Mathematical Discourse.

Articles and blog posts about abstract math

My blog Gyre&Gimble contains many articles on the difficulties students have with beginning abstract math. You can find them by selecting the category "student difficulties". (Finding the button for selecting categories may require widening the window.)

Some other articles:

These articles are all available for free. I have avoided referring to articles behind paywalls in abstractmath.org.

Aubrey, Mathieu, Metaphors in mathematics.

Bagchi, Atish and Charles Wells, Varieties of Mathematical Prose (1997).

Bagchi, Atish and Charles Wells, Communicating Logical Reasoning PRIMUS (1998).

Edwards, Barbara and Michael B. Ward, Surprises from mathematics education research: student (mis)use of mathematical definitions.

Epp, Susanna, A unified framework for proof and disproof.

Epp, Susanna, Discrete mathematics for computer science.

Epp, Susanna, Proof issues with existential quantification.

Epp, Susanna, Variables in mathematics education.

Maurer, Stephen, Advice for undergraduate on special aspects of writing mathematics.

Quora, Are there any mathematics studies higher than calculus? If so, which ones?.

Quora: How can "abstract thinking" be best defined or exemplified?

Strogatz, Steven The elements of math.

Tao, Terry, There is more to mathematics than rigour and proofs.

Wells, Charles, Communicating mathematics: useful ideas from computer science. American Mathematical Monthly 102, 1995.

Note: There are many other articles and blog posts about math at the abstract level that need to be added to this section.

University courses containing abstract math

Courses that math majors must take typically include some of these:

All of them may involve abstract definitions and require doing proofs.


I am grateful to Case Western Reserve University for providing software and library privileges.

Creative Commons License

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