- Students taking math courses that require working with abstract definitions and understanding and creating proofs.
- Teachers of university courses like those just described.
- Professionals (in any one of many fields) who need to learn math that is described in terms of mathematical properties with little or 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”.
- Abstract Math provides the conceptual background and theory that justifies 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.
- Many of the references in Other Sources for math discuss various aspects of abstract 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:
Other articles
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.
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.
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 |
- I put the slogans in purple prose because they are aimed at correcting deep misunderstandings that some readers will have about certain topics. Take purple prose statements seriously.
- It takes work to understand all the ins and outs of 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.
Links
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 |
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).
Mathematica
Many of the abmath articles are illustrated by diagrams created in Mathematica.
- The programming for these illustrations are in Mathematica notebooks, all of which have a file extension ".nb".
- The illustrations are all .gif, .jpeg or .png files.
- There are also links to interactive documents with extension ".cdf".
All these files are in the directory Mathematica, many of them in subdirectories.
How to use the Mathematica files
- If you own or have access to Mathematica 10.4 or later, you can load an .nb or a .cdf file into Mathematica and run the interactive commands (the ones with sliders), turn any 3D diagram to get better views, and change and run any command in any way you want. Like everything in abstractmath.org, the files are licensed under a Creative Commons Attribution-ShareAlike 2.5 License.
- If you own CDF Player (which is free), you can load an .nb or .cdf file into the player and run the interactive commands and turn 3D diagrams around to get different views. But you can't change the document or compile any command in it.
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.
MathOverflow A question and answer site for professional mathematicians
MathStackExchange A question and answer site for people studying math at any level.
Math Blogs
The unapologetic mathematician
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:
- Abstract Algebra (or Modern Algebra)
- Advanced Calculus
- Discrete Math
- Group Theory
- Linear Algebra
- Number Theory
All of them may involve abstract definitions and require doing proofs.
Acknowledgments
I am grateful to Case Western Reserve University for providing software and library privileges.
