¨ University math majors or beginning grad students taking math courses that require working with abstract definitions and understanding and creating proofs.

¨ Teachers of university courses like those just described.

¨ Professionals who need to learn math (in any one of many fields) that is described in terms of mathematical properties with no reference to applications.

¨ Anyone who is curious about advanced math!

What is abstract math?

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

¨ All math involves manipulating symbols (solve a quadratic equation, find a derivative, and so on). Abstract Math provides the conceptual background and theory that justifies these manipulations and explains their real-world applications.

¨ Abstract math requires conceptual reasoning about abstract ideas (as well as manipulating symbols), in particular on understanding and constructing proofs.

¨ Abstract math is mathematics for its own sake. In doing abstract math, you state theorems and prove them mostly in the context of mathematical ideas rather than applications or ideas from other fields.

¨ When you first meet up with abstract math (see appendix) you may find it hard to understand or even bizarre. If you need to know some piece of abstract math you may find the texts in the subject appear to be unmotivated and full of mysterious chains of reasoning. This happens to many people who are quite good at solving trig, derivative and integral problems.

Overview of abstractmath.org

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

The four main parts of the website

This list contains links to the head page of each of the four main parts of abstractmath.org. These head pages explain the ideas of that part in more detail.

The languages of math

Proofs

Understanding math

Doing math

Topic articles

Abstractmath.org has articles on these mathematical topics:

as well as fragmentary discussions of number theory, groups, derivatives and other topics.

¨ These topic articles describe a few of the basic ideas of each topic. They define the ideas precisely and describe how to think about them.

¨ The articles illustrate the techniques and ideas developed in the four main parts of the website.

¨ None of the articles go very far into the subject, but there are links to websites that cover the subjects more thoroughly.

Purple prose

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

Slogans in Purple Prose Displayed Like This

¨ It takes work to understand all the ins and outs of these purple-prose slogans.

¨ Many of them require thinking about things in a way that is very different from the way you think about things in daily life.

¨ Some of them are difficult to believe and put into practice.

Other things on this website

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

Diagnostic examples: These 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.

Gyre&gimble: Most new ideas I have about abstract math and language, some specifically related to abstractmath.org, are posted on this blog.

Astounding Math Stories is a blog containing short articles that are designed to Astound you with the Weird Things that Happen in Math. Sh! They also have an educational purpose.

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

Construction status

This website is under construction. Parts of it are incomplete.

Mistakes

The website contains mistakes.

Please email me if

¨ You find a mistake.

¨ You have a question that is not answered in this site. This means not only questions about math but questions about statements made on the site about language, learning math, etc.

¨ You know of another website, book, or article that is relevant to some topic discussed here. I prefer web links over books, articles and research treatises, but I am not dogmatic about this.

¨ You find broken links or incorrect references.

¨ You disagree with something on the website and want to state another point of view.

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, particularly Math on the Web.

Pennsylvania State math website

Planet Math

Wikipedia

Math Blogs

Gowers’s Weblog

Gyre&gimble

Hilomath

The Math Less Traveled

Mathematics under the Microsope

n-Category Café

News from the World of Maths

New Zealand Math Olympiad blog

Pat’sBlog

The Unapologetic Mathematician

What’s New

Books on beginning abstract math

The books listed below are written 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. The Mathematical Association of America, 2003.

Bloch, Ethan D., Proofs and Fundamentals : A First Course in Abstract Mathematics, Birkhauser, 2000.

Exner, G. R., An Accompaniment to Higher Mathematics. Springer-Verlag, 1996.

Hale, Margie, Essentials of Mathematics: Introduction to Theory, Proof and the Professional Culture. The Mathematical Association of America, 2003.

Solow, Daniel, How to Read and Do Proofs : An Introduction to Mathematical Thought Processes. Wiley, 2003.

Velleman, Daniel J. How to Prove It: A Structured Approach. Cambridge University Press, 1994.

Wells, Charles, Discrete Mathematics Class Notes. Case Western Reserve University, 1999.

Wells, Charles, The Handbook of Mathematical Discourse. Infinity Publishing Company, 2003.

Zakon, Elias, Basic Concepts of Mathematics. Trillia Group, 2003. (This is an e-book based on a book written in 1973.)

My email address

Charles[at]abstractmath.org

Replact “[at]” by the @ sign. I do it this way to stop automated harvesting of email addresses by spammers.

Appendix

University courses containing abstract math

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

¨ Linear Algebra

¨ Abstract Algebra (or Modern Algebra)

¨ Advanced Calculus

¨ Discrete Math

¨ Number Theory

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

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