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Produced by Charles Wells     Revised 2015-12-03
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THE LANGUAGES OF MATH

In science one tries to tell people, in such a way as to be understood by every­one, something that no one ever knew before. But in poetry, it's the exact opposite. --Paul Dirac

Mathematics in the English-speaking world is communicated using two languages:

• Mathematical English is a special form of English.
• It uses ordinary words with special meanings.
• Some of its structural words (“if”, “or”) have different meanings from those of ordinary English.
• It is both written and spoken.
• Other languages also have special mathematical forms.
• The symbolic language of math is a distinct, special-purpose language.
• It has its own symbols and rules that are rather different from spoken languages.
• It is not a dialect of English.
• It is largely a written language.
• Simple expressions can be pronounced, but complicated expressions may only be pointed to or referred to.
• It is used by all mathematicians, not just those who write math in English.

Math in writing and in lectures involve both mathematical English and the symbolic language embedded in each other and referring back and forth to each other.

Contents

The languages of math are covered in three chapters, each with several parts. Some things are not covered; see Notes.

• Mathematical English
• Assertions
• Names
• Glossary
• The symbolic language of math
• Symbolic expressions
• Grammar of the symbolic language
• Alphabets
• Delimiters
• Other symbols
• More about math languages
• Context
• Embedding
• Displayed symbolic expressions
• Parenthetic assertions
• Preconditions and postconditions
• Incomplete notation
• Synecdoche
• Overloaded notation
• Redundancy
• Parameters
• Intent of assertions
• Variations in meaning
• Conventions
• Defaults
• Scope

Links

• Communicating mathematics: useful ideas from computer science, by Charles Wells.
• Earliest Known Uses of Some of the Words of mathe­matics, by Jeff Miller
• Gyre&Gimble, a blog about math and language by Charles Wells
• The Handbook of Mathematical Discourse, by Charles Wells
• The history of mathematical symbols, by Douglas Weaver
• The Language of mathe­matics, by Mohan Ganesalingam. Not available on the internet.
• The language and grammar of mathe­matics, by Timothy Gowers
• On the communication of mathematical reasoning, by Atish Bagchi and Charles Wells.
• On-line etymological dictionary
• Varieties of mathematical prose, by Atish Bagchi and Charles Wells

Note

Math communication also uses pictures, graphs and diagrams, which are discussed briefly in the chapters Images and Metaphors and Images and Metaphors of Functions.

Abstractmath.org does not cover the history and etymology of mathematical notation.

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