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A Handbook of Mathematical Discourse

Charles Wells
Infinity Publishing
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Annie Selden
, on

What sort of book is this? It is a dictionary of sorts of all those words and conventions you had questions about as an undergraduate or graduate student but were afraid to ask, for fear of sounding dumb. Nobody, especially not your professors, bothered to explain these words, because they knew them so well and used them so automatically that it never occurred to them that you might not know to use them.

For example, a student might be confused by the many different ways mathematicians use let. This book explains, with illustrative examples, that let can mean assume or suppose, that it can be used

  1. to introduce a new symbol when considering successive cases (Let \(n>0\).... Now let \(n<0\)),
  2. to specify a condition (Let \(n\) be an integer. Then \(n\) is…),
  3. to introduce an arbitrary object when proving a for all statement (Let \(g\in G\); we need to prove that…), or
  4. to define a concept (Let an integer be even if it is divisible by 2),

as well as several other meanings. That students are not clear about the use of words like let can be seen from Steve Maurer’s PRIMUS article, “Advice for undergraduates on special aspects of writing mathematics” (Vol. 1, pp. 9–28, 1991).

A student might want to know what a bound variable is — not many transition-to-proof course textbooks cover that very well, if at all. There is a definition here, and it comes with a picture. Whether or not you like the somewhat quirky line drawings, however, depends on your sense of humor: next to the entry for bound variable, one finds an \(X\) with lots of rope around its middle. If you know already know the meaning of bound variable, you may be amused by this play on words. However, if you are a student trying to understand its meaning, I doubt it would help.

You can browse the book like a coffee table book (though its size is much smaller at 8 by 8 inches) or like a dictionary, which it resembles. Give it to your favorite math major or beginning graduate student to help enculturate him/her into mathematicians’ sometimes unusual usage of terms and phrases. You might also consider using it as a prize for a math contest or as an addition to your departmental math library.

Some might consider the paperback version a bit pricey at $24.95. It can be obtained from Infinity Publishing of Haverford, PA, a print-on-demand publisher of camera-ready author-submitted works. However, you can also download it (Version 0.95, March 8, 2003) for free at The advantage of the online version is that words in blue are clickable. One the other hand, none of the quirky line drawings are there.

Annie Selden is Adjunct Professor of Mathematics at New Mexico State University and Professor Emerita of Mathematics from Tennessee Technological University. In 2002, she was recipient of the Association for Women in Mathematics 12th Annual Louise Hay Award for Contributions to Mathematics Education. She remains active in mathematics education research and curriculum development.

The table of contents is not available.

Comments's picture

I read this one from cover to cover, and wrote lots of comments in the margin. It is a lot of fun to read even if you disagree with some of Wells's recommendations. I think at times he is too focused on formal logic. He also misses some of the main mathematical words: there are no entries for "straightforward", "deep", or "pure thought".

I suspect instructors will profit more from the book than students will. At the very least, I hope they will learn that they do have to explain some of the standard terms we use.