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Produced by Charles Wells     Revised 2016-11-24

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# Mathematical English

CONTENTS

Mathematical English is a special form of the English language used for making formal mathematical statements, specifically to communicate definitions, theorems, proofs and examples. Many ordinary English words are used in math English with different meanings. In some ways, math English is a foreign language.

## Distinctive features of math English

### Vocabulary and structure

Mathematical English includes:

1. Ordinary words used in a technical sense, for example, "function", "include", "integral", and "group".
2. Technical words special to the subject, such as "topology", "polynomial", and "homeomorphism".
3. Words and phrases used to communicate the logic of an argument that are similar to those in ordinary English but often with differences in meaning.

All technical jargons have examples of (a) and (b) (see note).  Mathematical English is the only technical jargon that I know of that has examples of (c).  Some of the words and phrases mentioned in (c) are a major stumbling block for people new to abstract math.  "If…then" is one of the worst.  These words and phrases are discussed in the Chapter on Mathematical Reasoning.

### Mathematical register

Mathematical English is an example of a techni­cal regis­ter or jargon.  Math texts also may include dis­cus­sions of history, intui­tive des­crip­tions of phe­nom­ena and appli­cations, and so on, that are in a general academic regis­­ter rather than the math­ematical register.

### Other technical jargons

Math English is a kind of technical jargon.  All such jargons have examples of ordinary words and technical words used in a special way.

• "Strike" in baseball is an ordinary word used in a technical sense that directly contradicts its use in ordinary English (a strike often occurs when the batter doesn’t  hit the ball.).
• "Grand Slam" is a technical phrase in both baseball and bridge.
• Particle physicists use ordinary words such as "flavor", "charm" and "color" in special senses.

#### Technical words

• Computer people use words such as "byte" and "wysiwyg" that do not exist outside their jargon.
• Particle physicists have invented words such as "electron", "photon" and "quark".
• "Quark" is in fact an uncommon English word with two different meanings aside from the meaning in physics ("caw" like a crow and a kind of cottage cheese), but I wouldn’t call it "ordinary".

### No standards

There is no national or international body setting standards for math terminology, unlike for example the one for anatomy.   There is a good reason for this:  research in abstract math often leads to new ways of understanding some type of math object that calls for new terminology.

The lack of standards is discussed at greater length in Definitions.

It is also true that some mathematicians abuse their freedom, using definitions of words and phrases that are different from the customary ones for no good reason, and often without even pointing out that their definitions are different.  This is discussed briefly in the Handbook, page 204.

## Assertions and statements

### Statements

Math English, just like everyday English, is used for making statements. Every statement is either true or false.  (See terminology).

#### Examples

• "$2$ is an even integer".  This is a true statement.
• "Every set has at least three elements."  This is a false statement, but it is still a statement.
• "The googolth digit of $\pi$ is $7$."  This statement is either true or false, but I don’t know which.  Maybe no one will ever know.  But it is still a statement.

### Assertions

Mathematical English also has sentences that are like statements, but may contain variables and may be true or false depending on the values chosen for the variables.  In abstractmath.org these are called assertions. In particular, any statement is regarded as an assertion with no variables.  (See boundary values of definitions).

In some articles in abstractmath.org I use the word "statement" when I should say "assertion". Eventually this inconsistency will be rectified.

#### Examples

All these sentences are assertions:

• "The integer $n$ is even".  This is true if $n= 4$, but is false if $n = 5$.
• "The set $S$ has three elements."  This is true if $S= \{1, 4, 6\}$  but false if $S=\{2, 4, 6, 8\}$.
• "$x^2-4=(x-4)(x+4)$". This is true for every real number $x$.
• "$x+1\lt x$". This is not true for every real number $x$.

### Truth set

The truth set of an assertion is the set of all objects that make the assertion true when substituted for the variable(s) in the assertion.

#### Examples

In the following examples, $x$ is a real variable and $n$ is an integer variable.

• The truth set of the assertion "${{x}^{2}}-4=0$" is the set $\{2, –2\}$.
• The truth set of "The integer $n$ is even" is the set of all even integers.
• The truth set of the assertion "$x^2-4=(x-4)(x+4)$" is $\mathbb{R}$ (the set of real numbers).
• The truth set of "$x+1\lt x$" is the empty set.

### Terminology

My use of the words statement and assertion
is not standard terminology.

In mathematical logic, statements may be called propositions or sentences and assertions may be called predicates or formulas.   I don’t use those words because they can cause semantic contamination.

The words "statement" and "assertion" also have connotations in English that are not relevant here.

• In the abstractmath usage, a statement is simply a sentence that is true or false; it doesn’t have to be a witness’s report, for example.
• An assertion is a sentence that becomes true or false when you substitute values for the variables.  It doesn’t have to be emphatic.