Writing Definitions - Writing Well

Mathematical Writing - Vivaldi Franco 2014

Writing Definitions
Writing Well

A definition requires a pause, to give the reader time to absorb it. This may be achieved by giving the definition twice, first with words then with symbols (or vice-versa), by using two different formulations, or by supporting the definition with an example. Let us analyse some definitions.

EXAMPLE.

Let $${\fancyscript{P}}$$ be a predicate over the rational numbers, that is, a function of the type

$$\begin{aligned} {\fancyscript{P}}:\mathbb {Q}\rightarrow \{T,F\}. \end{aligned}$$

The definition of the symbol $${\fancyscript{P}}$$ uses some jargon (predicate over a set) so the second part of the sentence recalls its meaning. A previous exposure to this concept is tacitly assumed; for a first encounter we would need a more considerate style, such as in the opening paragraph of Sect. 4.3.

EXAMPLE.

For given $$\varepsilon >0$$, we define the set

$$ S_\varepsilon =\{(x,y)\in \mathbb {R}^2\,:\, |x-y|\leqslant \varepsilon /\sqrt{2}\}, $$

that is, $$S_\varepsilon $$ is a strip of width $$\varepsilon $$ symmetrical with respect to the main diagonal in the cartesian plane.

This time the symbolic (Zermelo) definition appears first, while a verbal explanation clarifies the geometric meaning of $$S_\varepsilon $$, which is not obvious from the formula. The quantity $$\varepsilon $$ appears as a subscript, indicating that it is a parameter. Zermelo definitions should be used sparingly, and should not be used at all if we write for a non-mathematical audience. (Lars Ahlfors in his beautifully written text Complex analysis deliberately avoids them [2].)

EXAMPLE.

For every real number $$\lambda $$, let $$\varPi (\lambda )$$ be the plane in three-dimensional euclidean space orthogonal to the vector $$v(\lambda )=(1,\lambda ,\lambda ^2)$$. Thus $$\varPi (\lambda )$$ consists of the points $$z=(x_1,x_2,x_3)\in \mathbb {R}^3$$ for which the scalar product $$z\cdot v=x_1\,{+}\,x_2\lambda \,{+}\,x_3\lambda ^2$$ is equal to zero.

In this definition the quantity $$\lambda $$ appears as an argument of both $$\varPi $$ and $$v$$, and thus the latter represent functions. The second sentence restates the definition in a form suitable to computation, recalling the connection between orthogonality and scalar product, and introducing further notation.

EXAMPLE.

Let $${\fancyscript{N}}$$ be the set of sequences of natural numbers, such that every natural number is listed infinitely often. For example, the sequence

$$ (1,1,2,1,2,3,1,2,3,4,\ldots ) $$

belongs to $${\fancyscript{N}}$$.

A non-experienced reader will have no idea that such a construction is at all possible, so we give an example.

The definition of a symbol should appear as near as possible to where the symbol is first used; defining a symbol immediately after its first appearance is also acceptable, provided that the definition is given within the same sentence.

EXAMPLE. We give the same definition three times, articulating the changes in emphasis that accompany each version.

Consider the power series

$$ h(x)=\sum _{n=1}^\infty a_n x^n, $$

where the coefficient $$a_n$$ is the square of the n-th triangular number.

The definition of $$a_n$$ immediately follows its first appearance. The displayed formula represents a general power series, and its specific nature is revealed only by reading the entire sentence. For this reason, this format may not be ideal if the formula is to be referred to from elsewhere in the text. This definition puts some burden on readers who are unfamiliar with the term triangular number.

In our second version we change both text and formula.

Let $$t_n$$ be the n-th triangular number. We consider the power series

$$ h(x)=\sum _{n=1}^\infty t^2_n x^n. $$

Now $$t_n$$ is defined before being used, with the symbol $$t$$ chosen so as to remind us of ’triangular’. The formula is clearer. Just glancing at it makes us want to find out what $$t_n$$ is, and to do this one would begin to scan the text preceding, rather than following, the formula.

Our third version combines verbal and symbolic definitions.

We consider the power series $$h(x)$$ whose coefficients are the square of the triangular numbers, namely,

$$ h(x)=\sum _{n=1}^\infty t_n^2 x^n\quad \qquad t_n=\frac{n(n+1)}{2}. $$

The formula is more complex because everything is defined within it. But it is also self-contained, so it’ll be easy to refer to.