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MATHEMATICAL ENGLISH    Incomplete

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.

Introduction

Distinctive features of mathematical English include

¨ Ordinary words used in a technicall sense, for example, "function", "include", "integral", and "group".  

¨ Technical words special to the subject, such as "topology", "polynomial", and "homeomorphism".

¨ 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.

No standards body

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. 

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.

Etymology

I say nothing about the etymology of math words here.  A good source is Schwartzman, S., The Words of Mathematics. American Mathematical Society, 1996.

Assertions and statements

Statements

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

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  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).

Examples

¨ “The integer n is even”.  This is true if n = 3, but is false if n = 4. 

¨ “The set S has three elements.”  This is true if S = {1, 4, 6}  but false if S = {2, 4, 6, 8}. 

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

¨ The truth set of the assertion “  ” is the set {2, 2}.

¨ The truth set of “The integer n is even” is the set of all even integers.

Terminology

I use assertion here for the type of sentence described here, but that is not standard terminology.   In mathematical logic, assertions are called predicates or formulas.  I don’t use those words because they can cause semantic contamination.

Assertions in math English play the same roles as symbolic assertions in the symbolic language. 

Intent of assertions

English sentences can state facts, ask question, give commands, and other things.  The intent  of an English sentence is often obvious, but sometimes it can be unexpectedly different from what is apparent in the sentence.  For example, the statement “Could you turn the TV down?” is apparently a question expecting a yes or no answer, but in fact it may be a request.  (See Wikipedia about this phenomenon.)  Such things are normally understood by people who know each other, but people for whom English is a foreign language or who have a different culture have difficulties with them.

There are some problems of this sort in math English, too.  An assertion can have the intent of being a claim, a definition, or a constraint.  Sentences in math English, as in ordinary English, can also be questions or commands.    

Most of the time the intent of an assertion in math English is obvious.  But there are conventions and special formats that newcomers to abstract math may not recognize, so they misunderstand the point of the assertion.   This section takes a brief look at some of the problems.  There is a discussion like this about the intent of symbolic assertions here.

This section is not based on lexicographical research and does not appear in the Handbook.

Terminology

The way I am using the words “assertion”, “claim”, “constraint” is not standard usage in math, logic or linguistics. 

Claims

In most circumstances, you would expect that if a lecturer or author makes a math assertion they are claiming that it is a true statement, and you would be right.   

Examples

a)      “The 240th digit of  after the decimal point is 4.”

b)      “An integer divisible by 4 must be even.”

c)      “If a function is differentiable, it is continuous.”

Remarks

¨ Note that you don’t have to know whether these statements are true or not to recognize them as claims.  An incorrect claim is still a claim. 

¨ The assertion in (a) is a statement, in this case a false one.  If it claimed the googolth digit was 4 you would never be able to tell whether it is true or not, but it still would be an assertion.

¨ The assertions in (b) and (c) both use the standard math convention that an indefinite noun phrase (such as “a widget”) in the subject of a sentence is universally quantified (see also here).  In other words, “An integer divisible by 4 must be even” means “Any integer divisible by 4 must be even.”  This is a statement, too, and it is true.  (Similar remarks for (c).) 

See here for claims in the symbolic language.

Definitions

Definitions are discussed primarily in the chapter on definitions. 

Definitions are not the same thing as claims. 

Example

If you see the definition

“An integer is even if it is divisible by 2”

then you know that the claim

“An integer is even if and only if it is divisible by 2”

is true. The definition makes the claim true. 

Newcomers may not understand that the intent of an assertion is that it is a definition. 

This can be the author’s fault; some texts are very sloppy about this.  You can also write down a statement that a lecturer made and fail to write down that they said it was a definition, causing you no end of confusion later when you review your notes.

Examples

¨ Suppose that the concept of “even integer” was new to you and the book said, “A number is even if it is divisible by 4.”  Later the book refers to 6 as even and you pull your hair out wondering why.   The statement is a correct claim but an incorrect definition.  A GOOD writer would write something like “Recall that a number is even if it is divisible by 2, so that in particular it is even if it is divisible by 4.”

¨ Suppose you had heard the word “even” but took overly sketchy notes.  For example, perhaps the lecturer said, “By definition, an integer is even if it is divisible by 2.” and you wrote in your notes “An integer is even if it is divisible by 2.”  Later you get all panicky wondering How Did She Know That?? 

¨ The confusion in the previous example can also occur if a book says “An integer is even if it is divisible by 2” and you don’t know about the convention that when an author puts a word or phrase in boldface or italics it may mean they are defining it.

Constraints

Here are two assertions that contain variables. 

¨  “n is even”. 

¨  “  ”.

¨ Each of them may be true or false depending on what value is used for the variable. 

Such an assertion is a constraint or a condition if it is intended that the statement will hold in that part of the text (the scope of the constraint).  The part of the text in which it holds is usually the immediate vicinity unless the authors explicitly says it will hold in a larger part of the text such as “this chapter” or “in the rest of the book”.

Examples

¨  Sometimes the wording makes it clear that the phrase is a constraint, as in “Suppose n is even.” 

¨  It may appear in parentheses as a condition on an equation, for example

“ .                                                          (  )”

which means that if the constraint “x > 1” holds, then “  ” holds.  In this statement “  ” is not a constraint, but a claim.  A constraint written this way generally holds only for the displayed equation in which it occurs.

¨ A condition for which you are told to find the solution(s) is a constraint.  For example

“Solve the equation .”

need more examples and more detail

Where is a constraint in effect?   To do

Questions   To do

Commands   To do

Names

Glossary

Appendix

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.

Ordinary words

¨ “Strike” in baseball is an ordinary word used in a technical sense that directly contradicts its use in ordinary English (nothing is hit!).  

¨ “Grand Slam” is a technical phrase in both baseball and bridge.  

¨ Particle physicists use ordinary words such as “flavor” and “charm” 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” and “photon”.