I am revising the (large) section of abstractmath.org that concerns the languages of math. Below is most of the the introduction to that section, which contains in particular detailed links to its contents. All of these are now available, but only a few of them have been revised. They are the ones that say “Abstractmath 2.0” in the header.

Introduction

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 unlike those that spoken languages have.
• 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. They are embedded in each other and refer 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
• Variables
• Substitution
• Symbols
• 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 to other sites

• 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 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

Notes

Math communication also uses pictures, graphs and diagrams, which abstractmath.org doesn’t discuss. Also not covered is the history and etymology of mathematical notation.

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