Successful communication within a scientific subject depends on all participants knowing and following a set of specific conventions. This entails using and extending notation in a standard way, and presenting work in standard order and format.

So every mathematician needs to learn to choose their own symbols. It is an essential part of the wizardry.

### Choosing symbols

Notation takes mathematical objects (their classes, their instances) and gives them names, which can then be usefully manipulated to make meaningful statements about the mathematical objects.

In short: notation is a system of symbolical labels.

If you introduce a function…
$$\mathfrak{X}_{6h_\alpha}^{i\{C\}} \colon \mathbb{R} \to \mathbb{R}$$

(big, scary notation)

…only for the actual definition to be:
$$\mathfrak{X}_{6h_\alpha}^{i\{C\}}(x):=x^2,$$
then you are likely to confuse, distress, and annoy your reader in equal measure.

##### Does it matter?

Philosophically speaking, no, it does not matter, because mathematical truths are independent of labels.

Practically speaking, good notation is crucial for two reasons:

• it allows you to write down a proof that convinces you a theorem is true,
• it allows you to write down a proof that convinces others your theorem is true.

The five key rules that should govern your choice of notation are:

1. Define a symbol before you use it.
2. Do not introduce unnecessary symbols.
3. Use conventional notation.
4. Be wary of unintentionally repeated symbols or symbols that clash.
5. Be consistent.

We discussed the first two points in earlier lectures. Today we discuss the remaining points.

##### Define all the things

You will probably understand what I mean if I write $E = mc^2$.

($E$ is energy, $m$ is mass, $c$ is the speed of light in a vacuum.)

However in general this is not acceptable. Good notation should be unambiguous.

You should always specify exactly what you mean if there is any danger of confusion.

First let us say a few words about how to define terms and objects.

##### Defining terms

Terms that are being defined for the first time should be italicised or bolded. (But not written in ALL CAPS unless you want to fail.)

##### Defining objects

Sometimes it is helpful to give the definition in symbols and in words, providing the reader with two ways of thinking about the new object (even though this creates redundancy). For example:

1. Definition. For any given $\varepsilon> 0$, we define the set $$S_\varepsilon:=\left\{(x,y)\in\mathbb{R}^2 \, \Big| \, \frac{\varepsilon^2}{4}\le x^2+y^2 \le \varepsilon^2, \, x \neq y \right\}.$$ That is, $S_\varepsilon$ is the closed annulus bounded by the circles of radii $\varepsilon$ and $\frac{\varepsilon}{2}$ and not containing any points on the line $x=y$.

### General Conventions in Mathematics

Now we move onto the actual process of choosing which symbols to use in your definitions and notation. Here are a few guidelines, which are valid throughout nearly all mathematical fields. You probably already know most of them intuitively.

• Integers are usually represented by symbols from the middle of the Roman alphabet, such as $i,j,k,m,n$, especially if they are used as subscripts or superscripts.
• Prime numbers are $p$ and $q$.
1. The lowercase $o$ is rarely used because it resembles the digit 0. Also $o$ and $O$ are used for the “little-oh” and “big-oh” notation $o( \log n)$ and $O(\log n)$ etc.
• Real numbers are $x$ and $y$. If the real number represents time, then $t$ is often used. (In general pretty much anything works for a real number, as long as it doesn’t clash with other notation.)
• Complex numbers usually take the letters at the end of the Roman alphabet, either $z$ or $w$ as a first choice, and $z=x + iy$ or  is the usual notation for a number in the complex plane, with $x$ and $y$ the real and imaginary parts. Though in polar coordinates the standard notation is $z=r e^{i\theta}$, where “$r$” stands for radius.
• Vectors are $u$ and $v$ (some weird people like to use $\vec{v}$  or $\mathbf{v}$).
• Matrices are written with capital letters $A,B$ or sometimes $M,N$.
• Functions are typically named $f$, and then $g$ or $h$. However, in some cases the uppercase letters are preferred, e.g. $F(x,y)=(f_1(x,y), f_2(x,y))$. Labels for the standard variables correspond to those of unknowns.
• Functions are typically named $f$, and then $g$ or $h$. However, in some cases the uppercase letters are preferred, e.g. $F(x,y)=(f_1(x,y), f_2(x,y))$. Labels for the standard variables correspond to those of unknowns.

Sometimes functions take Greek letters. Famous examples include:

• the Riemann zeta function $\zeta(z)$,
• the Gamma function $\Gamma(n)$.

These last two functions illustrate our earlier principle quite nicely. Suppose you were unaware (or had forgotten) what sort of function the Riemann zeta function was. The notation $\zeta(z)$ tells you (even if only subconsciously) that this is a function that eats a complex number. Similarly the Gamma function $\Gamma(n)$ eats an integer[1].

• Sets are represented by uppercase letters, with typical generic sets denoted by $A,B,C$ and $X,Y,Z$. Whenever possible, the label for the set reflects the nature or labelling conventions of its elements. So it makes sense to label a sequence of functions as $\mathcal{F}$, a set of points ${p_1, \ldots p_k}$ as $\mathcal{P}$, a set ${a, b, c}$ as $A$.

• Other composite objects, such as groups and spaces, are typically denoted by capital letters that reflect their names. So a group or a graph is $G$ or $\Gamma$; a manifold is $M$, a moduli space is $\mathcal{M}$.

If there is more than one element of the same kind, adjacent letters are used or indices or primes. So we speak of manifolds $M$ and $N$, of groups $G$ and $H$, or spaces $X_1, X_2, X_3$ and $Y, Y'$.

• Similarly, symbols for derived objects are typically chosen to be closely related to the symbols of the objects they are derived from. For example, $\mathbb{Z}^+$ can be used to denote the positive integers[2], and $\mathbb{F}_p$ denotes the prime field obtained from the integers modulo a prime number $p$.

Usual tricks for distinguishing derived objects include (but are not limited to): indices $x_i$, prime $x'$ and double prime $x''$, asterisk $x^*$, overline $\overline{x}$ and underline $\underline{x}$, tilde $\tilde{x}$.

1. As a general rule, don’t combine similar decorations. If $x$ is going to get a tilde, don’t give it a hat as well: $\hat{ \tilde{x}}$ is BAD.
##### Size matters

Certain symbols and letters are usually thought to be “large” or “small”.

Symbol Size
$\varepsilon$ small
$\delta$ small
$n$ medium
$N$ large
$\aleph_0$ very large
$\hbar$ very small

Most of these quantities are often implicitly assumed to be positive.

(Shortest math joke ever: Let $\varepsilon <0$.)

1. Actually the Gamma function extend to a function on the right complex half-plane. But when speaking of this extension, the function is written Γ(z). ↩︎

2. Although I prefer $$\mathbb{N}$$. ↩︎

### Conventions in Specific Fields

In any given field of mathematics or physics there are specific notational conventions, and you should adhere to them.

For example, in symplectic and contact geometry (my field):

Mathematical object Standard symbol
symplectic form $\omega$
contact form $\alpha$
contact distribution $\xi$
Riemannian metric $g$
almost complex structure $J$
points in a cotangent bundle $(q,p)$
pseudoholomorphic map $u$
Hamiltonian function $H$

There is no way to“guess” what the correct conventions are—you learn by osmosis(which means over time, by reading books and papers).

Luckily you only need to know the specific conventions in your field—not all of them!

### Unfortunate Choices

So far we have seen that there are many symbols to choose from but that convention demands you use only certain recognisable combinations.

In a sense, it is a fine line to walk: you must conform to expectations, but you may also have to reach for unused letters in longer papers.

In this section we look at some common errors.

##### Repetition
• At the beginning of Section 1, you define a function $f$.
• At the beginning of Section 2, you define another function $f$, which is alright because it is used to prove a completely disjoint set of theorems than those in Section 1.
• However, in Section 3 you wish to bring the results of the previous two sections together, and you suddenly have to deal with two completely different functions both called $f$.

This is BAD.

Alternatively, you may have called a vector space $V$, but at the same time you are dealing with open sets $U$ and $V$.

Or have a field $F$, and the fibre $F$ of a fibre bundle.

This issue is more common than you think and it is unforgivable. Always relabel your objects!

##### Clashing symbols

The complex number $\sqrt{-1}$ is responsible for many notational crimes. How should you write it?

The standard option is of course $i$, but this can cause sadness if $i$ is used for some other purpose nearby.

1. BAD: Let $i$ and $j$ be positive integers, and consider the complex number $i + ij$.
2. BAD: Let $\{ a_i \mid i \in I\}$ denote a set of complex numbers. Now define $b_i := ia_i$.

There are various ways round this: you could just write $\sqrt{-1}$, or you could use $j$, or use a different font $\mathsf{i}$. The best solution however is to just ensure that your nearby notation doesn’t clash.

Here is another “real life”[1] example of how unintentional notational clashes arise.

• Suppose you in Lecture 1 you work with linear maps from $\mathbb{R}^n$ to $\mathbb{R}^m$.

Why consider maps from $\mathbb{R}^n \to \mathbb{R}^m$ and not the other way round? The reasoning is thus: in elementary linear algebra a general matrix is usually assumed to an $m \times n$ matrix. An $m \times n$ matrix is the same thing as a linear map from $\mathbb{R}^n$ to $\mathbb{R}^m$.

• In a later lecture you define smooth maps between manifolds $\varphi \colon M \to N$.

This is also logical, based on the principal that the first manifold should be $M$ (“M” for “manifold”) and the second manifold should be labelled with the next letter in the alphabet.

• Later still, you realise that because of the way you have set up your notation:

• $M$ has of dimension $n$,
• $N$ is of dimension $m$.

Bummer.

##### Spamming fancy fonts

Beware of using too many different fonts.

Suppose you are running out of names for your sets, you have used up all the symbols, so you reach for another font, and you end up talking about an element in the intersection:
$$a \in \mathcal{A} \cap A \cap \mathscr{A}.$$

Ugh.

In general it is better to decorate the base symbol than use different fonts to label instances of the same kind of object. Using too many special fonts makes it look like you have just discovered how to use $\mathrm{\LaTeX}$ and are “experimenting”[2].

1. Yes, this happened to me when I taught Differential Geometry last year... ↩︎

2. This also reminds me of powerpoint presentations or website from the 90s, which were jam-packed with silly fonts and animations because it was “snazzy”. ↩︎

### Be Consistent

I saved the most important (and most commonly ignored) piece of advice for last:

Do not alter your notation midway through any unit of text that relies on that notation.

This is equivalent to writing a novel and midway through changing the name of the protagonist!