Produced by Charles Wells Revised 2016-11-24 Introduction to this website website TOC website index blog
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.
Mathematical English includes:
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 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.
Math English is a kind of technical jargon. All such jargons have examples of ordinary words and technical words used in a special way.
"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".
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.
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:
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.
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