abstractmath.org 2.0
help with abstract math

Produced by Charles Wells     Revised 2016-11-24

Introduction to this website    website TOC    

website index   blog

Mathematical English


Distinctive feature of math English

Assertions and statements



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.

References for 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.

Technical words

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


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



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.


All these sentences are assertions:

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.


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


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. 

See also symbolic assertions.



Creative Commons License

This work is licensed under a Creative Commons Attribution-ShareAlike 2.5 License.