Choosing Words - Writing Well

Mathematical Writing - Vivaldi Franco 2014

Choosing Words
Writing Well

Franco Vivaldi1

(1)

School of Mathematical Sciences, Queen Mary, University of London, London, UK

Franco Vivaldi

Email: f.vivaldi@qmul.ac.uk

In this chapter we consider some techniques for writing mathematics. We deal with small-scale features: choosing an appropriate terminology and notation, writing clear formulae, mixing words and symbols, writing definitions, introducing a new concept. These are pre-requisites for the more substantial task of structuring and delivering a mathematical argument, to be tackled in later chapters.

A universal problem besets any form of specialised writing: how much explanation should we provide? Ideally, we ought to explain the meaning of every word that we use. But this is impossible: we would need to explain the words used in the explanation, and so on. Instead, we should only explain a word or symbol if our explanation will make it clearer than it was before. Accordingly, we shall call a word or symbol primitive 1 if it’s suitable to use without an explanation of its meaning.

An ordinary English word like ’thousand’ is obviously primitive, but for more specialised words we must consider the context. When we communicate to the general public, the term multiplication can safely be regarded as primitive. Likewise, there should be no need to explain to a mathematician what an eigenvalue is, while a number theorist will be familiar with conductor. Then there are extremes of specialisation: only a handful of people on this planet will know the meaning of Hsia kernel. Finally, terms such as exceptional set mean different things in different contexts. Understanding what constitutes an appropriate set of primitives is an essential pre-requisite for the communication of complex knowledge. Getting it right is tricky.

The writing developed in this chapter is addressed to a mathematically mature audience. Occasionally, we will consider writing for the general public.

6.1 Choosing Words

Precision must be the primary concern of anyone who writes mathematics. We list some common mistakes and inaccuracies that result from a poor choice of words.

·  BAD: the equation $$x-3\leqslant 0$$

·  GOOD: the inequality $$x-3\leqslant 0$$

·  BAD: the equation $$x^2-1=(x-1)(x+1)$$

·  GOOD: the identity $$x^2-1=(x-1)(x+1)$$

·  BAD: the identity $$x=\sqrt{x^2}$$

·  GOOD: the equation $$x=\sqrt{x^2}$$

·  BAD: the solution of $$x^k=x$$

·  GOOD: a solution of $$x^k=x$$

·  BAD: the minima of a quadratic function

·  GOOD: the minimum of a quadratic function

·  BAD: the function $$\sin (x)$$

·  GOOD: the sine function

·  BAD: the function $$f(A)$$ of the set $$A$$

·  GOOD: the image of the set $$A$$ under $$f$$

·  BAD: the interval $$[1,\infty )$$

·  GOOD: the ray $$[1,\infty )$$ (the infinite interval $$[1,\infty )$$)

·  BAD: the set $$\mathbb {Z}$$ minus $$k\mathbb {Z}$$

·  GOOD: the set difference of $$\mathbb {Z}$$ and $$k\mathbb {Z}$$

·  BAD: the area of the unit circle

·  GOOD: the area of the unit disc

·  BAD: the coordinates of a complex number

·  GOOD: the real and imaginary parts of a complex number

·  BAD: the absolute value is positive

·  GOOD: the absolute value is non-negative

·  BAD: the function crosses the vertical axis at a positive point

·  GOOD: the graph of the function intersects the ordinate axis at a positive point.

Once our writing is accurate, we consider refining the choice of words to differentiate meaning, improve legibility, or avoid repetition.

A set is mostly called a set, although collection or family are useful alternatives (as in the title of Sect. 2.3.1). A set becomes a space if it has an added structure, like a metric space (a set with a distance) or a vector space (a set with an addition and a scalar multiplication).

The word element expresses a special kind of subsidiary relationship. If $$A$$ is a set and $$a\in A$$, then we say that $$a$$ is an element of $$A$$. The term member is a variant, and so is point, which is the default choice for geometrical sets. So if $$A\subset \mathbb {R}^n$$, then $$x$$ is a point of $$A$$. For sequences, we may replace ’element’ by term. This is necessary if the elements of a sequence are added (multiplied) together, in which case the operands are the terms of the sum (product), not the elements. However, if $$V=(v_k)$$ is a vector, then $$v_k$$ is a component of $$V$$, not a term (even though a vector is just a finite sequence). But if $$M=(m_{i,j})$$ is a matrix, then $$m_{i,j}$$ is an element or an entry of $$M$$, not a component (even though a matrix is just a sequence of vectors).

The word variable is used in connection with functions and equations. In the former case it has the same meaning as argument, and it refers to the function’s input data. In the latter case it means unknown—a quantity whose value is to be found. Polynomials and rational functions may represent functions or algebraic objects; for the latter, the term indeterminate is preferable to variable.

The term parameter is used to identify a variable which is assigned a value that remains fixed in the subsequent discussion. For example, the indefinite integral of a function is a function which depends on a parameter, the integration constant:

$$ \int g(x)dx=f(x)+c. $$

The two symbols $$x$$ and $$c$$ play very different roles here, so we have a one-parameter family of functions, rather than a function of two variables.

A function is not always called a function. This may be true for real functions, but in more general settings the terms map, mapping, and transformation are at least as common. So a complex function $$\mathbb {C}\rightarrow \mathbb {C}$$ may be called a mapping, and a function between euclidean spaces $$\mathbb {R}^n\rightarrow \mathbb {R}^m$$ is a mapping or a transformation. Some functions are called operators. These include basic functions, like addition of numbers or intersection of sets, represented by specialised symbols ($$+,\cap $$) and syntax ($$x+y$$ rather than $$+(x,y)$$). The arithmetical operators, the set operators, the relational operators, and the logical operators are functions of this type. The attribute binary means that there are two operands, while unary is used for a single operand, like changing the sign of a number. The term operator is also used to characterise functions acting on functions which produce other functions as a result. The differentiation operator is a familiar example. A real-valued function acting on functions is called a functional. So definite integration is a functional.

In logic, a function is called a predicate, a boolean function, or a characteristic function. These terms represent the same thing, but they are not completely interchangeable. ’Characteristic function’ should be used if there is explicit reference to the associated set, and ’predicate’ (or ’boolean function’) otherwise. If the arguments of a function are also boolean, then there is a strong case for using ’boolean function’. Some examples:

The negation of a predicate is a predicate.

Let $$\chi $$ be the characteristic function of the prime numbers.

Let us consider the boolean function $$(x,y)\mapsto \lnot x \wedge y$$.

The predicate ’$$n\mapsto (3\mid n(n+1))$$’ is the characteristic function of a proper subset of the integers.

A relational operator is a boolean function of two variables.

In mathematics it is acceptable to change the meaning of established words. The following re-definitions of universal geometrical terms were taken from mathematics research literature:

By a triangle we mean a metric space of cardinality three.

By a segment we mean a maximal subpath of P that contains only light or only heavy edges.

By a circle we mean an affinoid isomorphic to $$\max {{\mathbf {C}}_p}(T,T-1)$$.

Re-defining such common words requires some self-confidence, but in an appropriate context this provocative device may be quite effective. Clearly the new meaning is meant to remain confined to the document in which it’s introduced.