abstractmath.org 2.0
help with abstract math

Revised 2015-05-10      website TOC      website index   blog


CONTENTS

# INTRODUCTION

## Who is this website for?

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

• 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.

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

• Abstract Math provides the conceptual background and theory that justfies the way math is used in 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, 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 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

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

Abmath also has articles on certain mathematical 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.
• 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 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.

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.

## 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

Mathworld

Planet Math

+plus magazine

Wikipedia

### Math Blogs

Gowers’s Weblog

Gyre&gimble

Numberwarrior

E. Kowalski’s blog

Pat’s Blog

### 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.

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

### Articles and blog posts about abstract math

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

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

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

### University courses containing abstract math

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

• Linear Algebra
• Abstract Algebra (or Modern Algebra)