Charles Wells' Research Website at Case Western
Reserve University

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

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.

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.**

- It uses familiar words with
different meanings. Sometimes the meanings are only a
*little*different and sometimes they are*very*different. - As illustrated, it uses familiar grammatical constructions with different meanings.

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.

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.

This handbook is grounded in the following beliefs.

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.

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.

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.

Charles Wells' Website

Email Charles Wells at charles@abstractmath.org