Last edited 1/17/2008 11:12
AM
HOME
PAGE
Contents
The
four main parts of the website
Books
on beginning abstract 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
¨ College math majors or beginning grad students taking math courses that require working with abstract definitions and understanding and creating proofs.
¨ Teachers of college 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?
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¨ 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 is based on conceptual reasoning (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 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.
Topic articles
Abstractmath.org has articles on these 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.
¨ 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.
Background and Attitude
The article on attitude shows some of the thinking behind this website.
Diagnostic examples
The diagnostic 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.
Blog
Most new ideas I have about abstract math and language, some specifically related to abstractmath.org, are posted on my blog gyre&gimble.
Astounding Math Stories
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.
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.
Construction status
This website is under construction. Parts of it are incomplete.
Comments
Comments written like this mention gaps, omissions and parts that need rewriting.
Mistakes
The website also contains mistakes.
Please email me or post to the discussion group 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.
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Problems with Firefox
When you view abstractmath.org via Firefox two things go wrong:
¨ Sometimes the columnar Google ads push the text down below them. I have a fix for this but it involves cutting and pasting in each file individually, then regenerating the file via MathType and reposting it. I have already done most of them.
¨ Links inside text boxes don't work. They work if I convert them to frames, and I have started doing that.
Other sources
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
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
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
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.
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.