Expressions - Levels of Description - Essential Dictionary II

Mathematical Writing - Vivaldi Franco 2014

Expressions - Levels of Description
Essential Dictionary II

The generic term expression indicates the symbolic encoding of a mathematical object. For instance, the string of symbols ’$$2+3$$’ is a valid expression, and so is ’$$x\mapsto f(x)$$’, while ’$$2+\times 3$$’ is incorrect and does not represent an object.

It would seem that any correct expression should have—in principle, at least—a unique value, representing some agreed ’fully simplified’ form of the expression. For instance, it could be argued that the value of $$13/91$$ is $$1/7$$, and that of $$\sqrt{2187}$$ is $$27\sqrt{3}$$. The simplified value is more concise and informative, and would enable us—among other things—to recognise when two expressions represent the same object.

This is not so simple. For example, the two expressions

$$ \sqrt{3}-\sqrt{2}\qquad \qquad \sqrt{5-2\sqrt{6}} $$

have the same value, yet there is no compelling reason for choosing one over the other. The same object may be viewed from different angles, and our choice of representation will depend on the context.

The following well-known identity makes an even stronger point:

$$ \frac{1-x^n}{1-x}=1+x+x^2+\cdots +x^{n-1}. $$

The right-hand side—the sum of $$n$$ monomials—is the ’fully simplified’ version, while the ’unsimplified’ left-hand side has only four terms. While simplification reveals the true nature of the object—a polynomial as opposed to a rational function—it’s the unsimplified value which provides information about the sum.

Given that agreeing on a unique value of an expression proves difficult, we shift our attention to the type of value (set, number, function, etc.), which assigns the expression to a certain class of objects. This class provides the broadest characterisation of the object in question. Again, we must exercise some judgement. The expressions

$$ 1+1 \qquad \qquad \int \limits _0^\pi \sin (x)\mathrm {d}x $$

have the same value, but their structure is so different that the coincidence of their values seems secondary. Whereas the expression on the left is unquestionably ’a number’, or ’a positive integer’, that on the right is ’a definite integral.’ On the other hand, there may be circumstances in which the reductionist description of the integral as a number is appropriate, for instance when discussing integrability of functions.

Keeping in mind the difficulties mentioned above, we now turn to the description of mathematical expressions.

3.4.1 Levels of Description

We develop the idea of successive refinements in the description of an expression, from the general to the particular. The appropriate level of details to be included will vary, depending on the situation. We treat in parallel verbal and symbolic descriptions, as far as it is reasonable to do so.

Let us consider the definition of a set. The coarsest level of description is

$$ \{\,\ldots \,\} \qquad \qquad {\textit{A set}} $$

where the object’s type is identified by the curly brackets. The use of the indeterminate article—’a’ set rather than ’the’ set—reflects our incomplete knowledge.

The next level in specialisation identifies the ambient set:

$$ \{(x,y)\in \mathbb {Z}^2\,:\,\ldots \,\} \qquad \qquad {\textit{A set of integer pairs}} $$

Now we begin to build the defining properties of our set:

$$ \{(x,y)\in \mathbb {Z}^2\,:\,\mathrm{gcd}(x,y)=1,\,\ldots \,\} \qquad \qquad {\textit{A set of pairs of co-prime integers}} $$

The final step completes the definition:

$$\begin{aligned} \{(x,y)\in \mathbb {Z}^2\,:\,\mathrm{gcd}(x,y)=1,{\quad }2|xy\}&\quad {\textit{The set of pairs of co-prime integers,}}\\&\quad {\textit{with exactly one even component}} \end{aligned}$$

Accordingly, the indefinite article has been replaced by the definite article. Now both words and symbols describe one and the same object, and one should consider the relative merits of the two presentations. The term ’exactly’ is, strictly speaking, redundant, but it helps the reader realise that $$x$$ and $$y$$ cannot both be even. A robotic translation of symbols into words

The set of elements of the cartesian product of the integers with themselves, whose components have greatest common divisor equal to 1, and such that 2 divides the product of the components

lacks the synthesis that comes with understanding.

Expressions may be nested, like boxes within boxes. We begin with the two expressions

$$\begin{aligned}&\bigl (\,\cdots \bigr )^2\qquad \qquad \qquad {\textit{A square}} \\&\sum \cdots \qquad \qquad \qquad {\textit{A sum}} \end{aligned}$$

We only see the outer structure of these objects. We compose them in two different ways:

$$\begin{aligned} \left( \,\sum \cdots \right) ^2&\qquad&{\textit{The square of a sum}} \\ \qquad \qquad \qquad \sum \bigl (\,\cdots \bigr )^2&\qquad&{\textit{A sum of squares}} \end{aligned}$$

The first term in each expression identifies the object’s outer layer. There is still one indefinite article, reflecting a degree of generality. We specialise further:

$$\begin{aligned} \sum _{n=1}^\infty \left( \frac{1}{n}\right) ^2&\qquad \quad {\textit{The sum of the square of the reciprocal of the}}\\&{\textit{natural numbers}} \end{aligned}$$

Words or symbols now define a unique object, with three levels of nesting. By contrast, in the nested expression

$$\begin{aligned}&\qquad \qquad {\textit{The square of the sum of the elements of}}\\ \left( \sum _{n=1}^\infty a_n\right) ^2\ a_n\in \mathbb {Q}&\\&{\textit{a rational sequence}} \end{aligned}$$

the innermost object—a rational sequence—is still generic.

In these examples words and symbols are interchangeable; in actual writing, some concepts are best expressed with words, others with symbols, while most of them require both. For instance, the symbolic expression

$$\begin{aligned} (1-x,2x+x^2,\ldots ,nx^{n-1}+(-x)^n,\ldots ) \end{aligned}$$

(3.20)

defines an infinite sequence succinctly and unambiguously. Using words, we could begin to describe it as follows:

A sequence

An infinite sequence

An infinite sequence of binomials

Increasing further the accuracy of the verbal description is pointless, since the symbolic expression (3.20) is clearly superior in delivering exact information. On the other hand, words can place this expression in a context, which is something symbols cannot do. To illustrate this point, we supplement the description given above with additional information which emphasises a particular property.

An infinite sequence of binomials

with integer coefficients

with unbounded coefficients

with increasing degree

whose leading term alternates in sign.