# Don’t Use Reuse Mathematical Symbols for Different Meanings

One of the surest sources of confusion in mathematical writing is to use one expression to mean multiple things. Authors often make this mistake with functions. To illustrate, consider a function $f$ from $\mathbb{R}$ to $\mathbb{R}$. The expressions “$f$” and “$f(x)$” have different meanings. The statement “$f(x)$ is monotonically decreasing” is meaningless because “$f(x)$” is a constant real number. Instead, write “$f$ is monotonically decreasing” or “$x \mapsto f(x)$ is monotonically decreasing”. The latter form is useful if you want to talk about the composition of functions without introducing a new symbol: “$x \mapsto f(1 + x^3)$ is monotonically decreasing”.

### When is it OK to reuse a symbol?

A complete ban on reusing symbols in a mathematical text would cause you to quickly run out of symbols, especially in long texts. Similar to variables in a programming language, symbols have a particular scope.

The tightest scope is what I will call a *bound symbol*.
A bound symbol is defined and used only in a single expression.
The symbol $x$ is a bound variable in the following expressions: “$\forall x\ P(x)$”, “$\exists x\ P(x)$”, and “$x \mapsto f(x)$.”
It is OK to immediately use the same bound symbol in multiple expressions, such as, “Consider the functions $x \mapsto f(x)$ and $x \mapsto g(x)$.”
To aid the reader, however, each time a bound symbol is reused in a section of text, it should have the same type of meaning.

Symbols introduced in a proof should be restricted to that proof (many readers skip proofs on their first read of a text), so we can treat a proof as a symbol scope. If we introduce “$\varepsilon > 0$ and $\delta = 2\varepsilon$” in the proof of Theorem 1, then it is OK to define “$\delta \in (0, 1)$ and $\varepsilon = \sqrt{\delta}$” in the proof of Theorem 2.

Symbols introduced in definitions and theorems are not scoped in the same way.