The Handbook of Mathematical Discourse

Abstractmath.org: a website for newcomers to abstract math based in part on the Handbook.

The Handbook is based on citations from the literature which are available as a PDF file by clicking here. To find a citation, type control-shift n and then the number of the citation.

The Handbook of Mathematical Discourse is a compilation of mathematical usage with a focus on the words and phrases that cause problems for students at the postcalculus level, when they are beginning to study abstract mathematics. It also contains words describing behaviors and attitudes that students and instructors might have. The focus is on American usage/

Its point of view is that 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.
It uses familiar grammatical constructions with different meanings. For example, the logic underlying “If you do the laundry then I will go to the store” is different from the logic of “If a prime is bigger than 2 then it is odd.”

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.

Description of the Handbook

Overview

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 32 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.

Citations

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. The citations are available here as a PDF file. To find a citation, type control-shift n and then the number of the citation.

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.

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