3.3 Definitions and Naming Conventions

A definition introduces a new term with a fixed meaning. It turns a pattern into a reusable object of thought. Without definitions, mathematics would be a collection of repeated descriptions rather than a system.

A good definition does three things.

For example:

A function f : X → Y is injective if for all x1, x2 ∈ X,
f(x1) = f(x2) implies x1 = x2.

This definition names a property. After it is introduced, we can write “let f be injective” instead of repeating the full condition each time.

Definitions are local contracts. Once a term is defined, every later use should respect the contract. If the definition changes, theorems depending on it may also change.

A mathematical definition usually has this form:

An object X is called [name] if it satisfies [conditions].

or:

We say that X has property P if [conditions].

The conditions should be necessary and sufficient. The reader should be able to determine whether an object satisfies the definition.

Poor:

A good function is one that behaves nicely.

Better:

A function f : X → Y between metric spaces is continuous at x ∈ X
if for every ε > 0 there exists δ > 0 such that
d_X(x,y) < δ implies d_Y(f(x), f(y)) < ε.

The improved definition gives a test. It specifies the objects, the quantifiers, and the relation between input and output.

Definitions should state context before conditions. The phrase “a continuous function” means different things in metric spaces, topological spaces, order theory, and domain theory. A clear definition names the ambient structure.

Mathematics reuses names. This is efficient when contexts are clear and confusing when they are not. The word “normal” has many meanings. A normal subgroup, a normal operator, and a normal topological space are different concepts.

Naming conventions reduce ambiguity. Certain kinds of names signal certain roles.

These patterns are not formal rules, but they help readers infer relationships between concepts.

A definition should be neither too broad nor too narrow. If it is too broad, it captures objects that do not support useful theorems. If it is too narrow, it excludes natural examples.

For example, the definition of a group is broad enough to include integers under addition, invertible matrices, permutations, and symmetries. It is narrow enough to support strong theorems about identity, inverses, subgroups, and homomorphisms.

A useful definition identifies the exact structure needed for later results.

Definitions also control equality. When defining a mathematical object, one should know what counts as the same object.

For example, a graph may be defined as:

G = (V, E)

where V is a set of vertices and E is a set of edges. But two graphs with different vertex names may be structurally the same. This leads to graph isomorphism.

A definition gives raw objects. A notion of equivalence tells us how to classify them.

Good definitions often expose maps. In many fields, the right definition is the one whose structure-preserving maps are natural.

The maps reveal what the definition is designed to preserve.

Definitions may be explicit or implicit.

An explicit definition gives direct conditions.

A group is a set G with a binary operation satisfying associativity,
identity, and inverse laws.

An implicit definition gives axioms that characterize a class of objects.

A field is a structure satisfying the field axioms.

In practice, most mathematical definitions combine both. They describe data and list laws.

A ring consists of a set R with operations + and * such that ...

This “data plus laws” form is common.

Definitions should avoid hidden choices. If an object depends on a choice, the definition should say so.

For example, “choose a basis of V” introduces noncanonical data. Results that depend on the chosen basis may not be intrinsic to V. Results independent of the basis are structural.

A term like “canonical” should be used only when the construction is determined naturally by the given data. If several choices are possible, the construction should be called “chosen,” “fixed,” or “selected,” not canonical.

Definitions also need examples and nonexamples. A definition becomes usable when the reader can distinguish what satisfies it from what fails.

For “group”:

Nonexamples are important because they show which condition matters.

Definitions should be stable under intended operations. If a class of objects is useful, it should often be closed under natural constructions.

For example, subgroups of groups are groups. Products of groups are groups. Isomorphic copies of groups are groups. These closure properties make the definition robust.

A practical checklist for definitions:

Naming should follow the same discipline. A name should be short, stable, and aligned with existing conventions when possible. New names should be introduced only when they reduce repeated explanation or identify a concept that will be used often.

A bad name hides structure. A good name points toward the role of the object.

In mathematical writing, definitions are infrastructure. They are not decoration before theorems. They determine what theorems can mean.

A strong definition minimizes ambiguity, supports examples, exposes structure, and makes later reasoning shorter.