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Posted 26 July 2008



a, an

The articles in English are the indefinite article "a" (with variant "an") and the definite article "the".  If your native language is not English you may have problems understanding some mathematical discourse because the articles are used in English in some fairly subtle ways.  This is explained in a handout from Rensselaer.   See also Wikipedia.


The word "a" or "an" is the indefinite article, one of two articles in English.

Generic use

In math writing, the indefinite article may be used with the name of a type of math object (producing an indefinite description  a description not meant to refer to an instance of the object that has been previously referred to) to indicate an arbitrary object of that type. Note that plural indefinite descriptions do not use an article.


"Show that an integer that is divisible by four is divisible by two."

¨  Correct interpretation: Show that every integer that is divisible by four is divisible by two.

¨  Incorrect interpretation: Show that some integer that is divisible by four is divisible by two.

So in a sentence like this it the indefinite article has the force of a universal quantifier.   Unfortunately, this is also true of the definite article in some circumstances!   More examples are given in the entry on universal quantifier.

Generic only in subject

An indefinite description has the force of universal quantification when it is the subject of the clause. Consider:

¨  "A number divisible by 4 is even." (Subject of sentence.)

¨  "Show that a number divisible by 4 is even." (Subject of subordinate clause.)

¨  "Problem: Find a number divisible by  4." (Object of verb.) This does not mean find every number divisible by 4; one will do.

Difference with ordinary English

In ordinary English sentences, such as a

"A wolf takes a mate for life."

the meaning is that the assertion is true for a typical individual (typical wolf in this case). In mathematics, however, the assertion is required to be true without exception. 

abstract algebra

See algebra.

abuse of notation

Notation may be called abuse of notation if it involves

¨  suppression of parameters

¨  synecdoche

¨  identifying two structures along an isomorphism between them.

Click on those entries for examples.

        The word “abuse” makes it sound worse than it really is.  Without judicious use of this technique much mathematical writing would be unreadable.


Aleph is the first letter of the Hebrew alphabet, written . It is the only Hebrew letter used widely in mathematics.  Its most common use is to refer to infinite cardinals.


This word has many different meanings in the school system of the USA, and college math majors in particular may be confused by the differences.

High school algebra is primarily algorithmic and concrete in nature.  This is where you learn to solve linear and quadratic equations (MW, Wik) and to apply them using word problems.

College algebra is the name given to a college course, perhaps remedial, covering the material covered in high school algebra.

Linear algebra may be a course in matrix theory (MW, Wik) or a course in linear transformations in a more abstract setting.

A college course for math majors called algebra, abstract algebra, or perhaps modern algebra, is an introduction to groups, rings, fields and perhaps modules (MW, Wik). It is for many students the first course in abstract mathematics and may play the role of a filter course. In some departments, linear algebra plays the role of the first course in abstraction.

Universal algebra (MW, Wik) is a subject math majors don't usually see until graduate school. It is the general theory of structures with n-ary operations subject to equations, and is quite different in character from abstract algebra.


The word algorithm is used in three confusingly similar ways:

algorithm as process

Mathematicians typically use the word “algorithm” for a step by step process for calculating something, as for example the procedure expressed roughly by the description  below.  People who use the word “algorithm” in this way may refer to a program implementing it as the code for the algorithm.

algorithm as program

A program may itself be called “an algorithm”.  In my experience, this is not common usage.

algorithm as mathematical object

In computer science or logic texts, the word “algorithm” may be given a mathematical definition (for example as a Turing machine (MW, Wik)), turning an algorithm into a mathematical object.


You can write a program in Pascal and another one in C to take a list  with at least three entries and swap the second and third entries. There is a sense in which the two programs, although different as programs, implement the “same” algorithm (process): Change  to .   


Newton’s Method gives rise to an algorithm in the sense of a process.


(NM) “Start with a guess x and let   (see colon-equals) repeatedly until either

¨  f(x) gets sufficiently close to 0, in which case x is the answer, or

¨  , or

¨   the process has gone on too long.”

This algorithm is not hard to implement in C, Pascal, Fortran or many other programming languages.  But in all these cases you have to get the syntax exactly right and take care of a lot of details such as assigning labels to certain lines of the program.  Note that (NM) is not any of those programs:  it is the process carried out by those programs.

The different meanings of “algorithm” are discussed with examples in the Wikipedia.  For a detailed development of formal algorithms, see Introduction to Automata Theory, Languages and Computation, by J. E. Hopcroft and J. D. Ullman, Addison-Wesley, 1979.

See also the discussion of functions here.


The symmetry of the square illustrated by the figure below can be described in two different ways.


¨  The corners of the square are relabeled, so that what was labeled A is now labeled D. This is called the alias interpretation of the symmetry.

¨  The square is turned, so that the corner labeled A is now in the upper right instead of the upper left. This is the alibi interpretation of the symmetry.


“Alibi” and “alias” are not mathematical properties of transformations, but ways to think about them. 


See universally true and universally true conditionals.


See universal quantifier.


The word ambient is used to refer to a mathematical object  such as a space that contains a given mathematical object. It is also commonly used to refer to an operation on the ambient space.


“Let A and B be subspaces of a space S and suppose  is an ambient homeomorphism taking A to B.”  The point of this sentence is that A and B are not merely homeomorphic, but they are homeomorphic via an automorphism of the space S.


between assertions

The word “and” between two assertions  P and  Q produces the conjunction of  P and  Q.


The assertion  x is positive and  x is less than 10” is true if both these statements are true: x is positive”, “x is less than 10”.

between verb phrases

The word “and” can also be used between two verb phrases to assert both of them about the same subject.


The assertion “ x is positive and less than 10” means the same thing as “x is positive and  x is less than 10” .

between noun phrases

The word “and” may occur between two noun phrases as well.  In that case the translation from English statement to logical assertion involves subtleties.  This is an example of a translation problem.


¨  “I like red and white wine” means “I like red wine and I like white wine”.  But so does “I like red or white wine!” 

¨  “John and Mary go to school” means the same thing as “John goes to school and Mary goes to school”.

¨  “John and Mary own a car” (probably) does not mean “John owns a car and Mary owns a car”.

¨  Consider also the possible meanings of “John and Mary own cars”.

These examples show that the relationship between sentences containing the English word “and” and their logical equivalent is quite subtle.  It is the main subject of Section 2.4 in

Kamp, H. and U. Reyle (1993), From Discourse to Logic, Parts I and II. Studies in Linguistics and Philosophy. Kluwer Academic Publishers.

Thanks to F. Schweiger for the wine example.


Used to denote the universal quantifier.  Examples are discussed under that heading. See also arbitrary.


Used to emphasize that there is no restriction on the mathematical structure referred to by the noun phrase that follows. You may usually use “any” in this situation instead of "arbitrary".   In most cases the word adds no additional mathematical meaning to the statement.


¨  "The equation  holds in an arbitrary group, but the equation  requires commutativity."   

¨  In a phrase such as "Let S be an arbitrary set" the word arbitrary typically signals an expectation of an upcoming proof by universal generalization.

Consciousness raising about arbitrary

People new to abstract math may have a systematic tendency to underestimate how arbitrary a math object can be.  For example, the set  is a perfectly good set.  It is arbitrary, and, I admit, weird, but it is a set.  Other examples: 

More arbitrary sets.

An arbitrary function.

Another arbitrary function.


This word has three common meanings in mathematical discourse.

¨  The angle a complex number makes with the real axis is called the argument of the number.

¨  The input to a function may be called the argument.

¨  A proof may be called an argument.

This word can cause cognitive dissonance.  In English, “argument” can mean either:

¨  Organized step by step reasoning to support a claim, as in, “The judge’s argument for finding the suspect innocent was based on the fourteenth amendment.”

¨  The verbal expression of a disagreement, as in “George and Martha had an argument about the Venetian blinds.”

The meaning of disagreement is the common one and it may carry a connotation of unpleasantness.  The three meanings in math that are given above have no connotation of unpleasantness.

assume To be written.


An assumption is an assertion that is taken as true in a given block of text that is its scope. "Taken as true" means that any proof in the scope of the assumption may use the assumption to justify a claim without further argument.


¨  "Throughout this chapter, G will denote an arbitrary Abelian group."  In that chapter, a statement such as “The subgroup B of G is normal” can be taken to be true without further justification because every subgroup of an Abelian group is normal.

¨  A statement about a physical situation may be called an assumption.  Such statements are then taken as true for the purposes of constructing a mathematical model.

¨  The hypothesis of a conditional assertion stands as an assumption during the statement of the assertion, and also through the proof if the proof is by modus ponens. 

at least, at most

For real numbers  x and  y, the phrase “x is at least y” means .  The phrase " x is at most y" means .  When I say here that B has at most one element, that means B is either the empty set or a singleton set.