Charles Wells' Research Website at Case Western Reserve University

The Handbook of Mathematical Discourse


Much of the material in the Handbook has been rewritten and included in my website

How to get the Handbook

The Handbook is available in three forms.


\xB7       A paperback from Amazon.\xA0 This does not contain the citations.

\xB7       A free pdf file of the printed text (also no citations).

\xB7       The Handbook in hypertext format, also free. This contains the text and the citations, but not the illustrations. Most of the cross references, including those to the citations, are hyperlinked, but some links don't work. I have lost the original TeX input for both the book and the hypertext, so these problems cannot be corrected.

The citations can be downloaded here.

Purpose of the Handbook

Envision a child at the dinner table being told by a parent, "If you eat all your dinner you can have some ice cream". The child expects that

If she eats all her dinner she will get some ice cream.

If she does not eat all her dinner she will not get any ice cream.

Then she becomes 18 (much faster than the parents imagined) and goes to college, where she takes calculus and learns these things:

If a function is differentiable, it is continuous.
The absolute value function is not differentiable at 0.

She is asked, "Is the absolute value function continuous everywhere?"
Naturally, she says,
"Not at 0!"
being quite proud of herself for putting two facts together that she had just learned. She is quite chagrined to learn that her teacher thinks she has made an elementary logical mistake.

That is because mathematical English is a foreign language.

If you have tried to live in another country, having some knowledge of its language, you can undoubtedly recall instances of bafflement which you may have eventually discovered was due to misunderstanding the meaning of a word or the intent of a certain grammatical construction. Math students in English-speaking countries are faced with the same sort of problem; they find themselves being lectured to in a language (mathematical English) which is so much like English that neither they nor (in many cases) their teacher knows how alien it is.

The Handbook of Mathematical Discourse is a compilation of mathematical usage with a focus on usage that causes problems for students. It also contains words describing behaviors and attitudes that students and instructors might have.

Description of the Handbook


This handbook is an intensive description of many aspects of the vocabulary and forms of the English language used to communicate mathematics. It is designed to be read and consulted by anyone who teaches or writes about mathematics, as a guide to what possible meanings the students or readers will extract (or fail to extract) from what is said or written. Students should also find it useful, especially upper-level undergraduate students and graduate students studying subjects that make substantial use of mathematical reasoning.

This handbook is written from a personal point of view by a mathematician. I have been particularly interested in and observant of the use of language from before the time I knew abstract mathematics existed, and I have taught mathematics for 37 years. During most of that time I kept a file of notes on language usages that students find difficult. Many of those observations may be found in this volume. However, a much larger part of this dictionary is based on the works of others (acknowledged in the individual entries), and the reports of usage are based, incompletely in this early version, citations from the literature.

Someday, I hope, there will be a complete dictionary based on extensive scientific observation of written and spoken mathematical English, created by a collaborative team of mathematicians, linguists and lexicographers. This handbook points the way to such an endeavor. However, its primary reason for being is to provide information about the language to instructors and students that will make it easier for them to explain, learn and use mathematics.

The earliest dictionaries of the English language listed only "difficult"' words. Dictionaries such as Dr. Johnson's that attempted completeness came later. This handbook is more like the earlier dictionaries, with a focus on usages that cause problems for those who are just beginning to learn how to do abstract mathematics.

Point of View

This handbook is grounded in the following beliefs.

The mathematical register

Mathematicians speak and write in a special "register" suited for communicating mathematical arguments. In this book it is called the mathematical register. The mathematical register uses special words as well as ordinary words, phrases and grammatical constructions with special meanings that are different from their meaning in ordinary English.

The standard intepretation

There is a standard interpretation of the mathematical register, in the sense that at least most of the time most mathematicians would agree on the meaning of most statements made in the register. Students have various other interpretations of particular constructions used in the mathematical register, and one of their (nearly always unstated) tasks is to learn how to extract the standard interpretation from what is said and written. One of the tasks of instructors is to teach them how to do that.

Descriptive and Prescriptive

Linguists distinguish between "descriptive" and "prescriptive" treatments of language. A descriptive treatment is intended to describe the language as it is actually used, whereas a prescriptive treatment provides rules for how the author thinks it should be used. This text is mostly descriptive. It is an attempt to describe accurately the language actually used by English-speaking mathematicians in the mathematical register as well as in other aspects of communicating mathematics, rather than some ideal form of the language that they should use. Occasionally I give opinions about usage; they are carefully marked as such.


Entries are supported when possible by "citations", that is, quotations from textbooks and articles about mathematics. This is in accordance with standard dictionary practice. The sources are mostly at the college and early graduate level.

At this writing, many more citations are needed. I encourage readers to send me citations and suggestions of usages that you think should be included in the Handbook. This too would be along the lines of early dictionary practice, particularly that of the Oxford English Dictionary.

Most of the examples are not citations. My intent is that the examples be easy to understand for students beginning to study abstract mathematics, as well as free of extraneous details.

NOTE: The example of the child at the dinner table is from Discrete Mathematics with Applications, 2nd edition, by Susanna Epp.

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