Subwiki:Definition

Each subject wiki has certain kinds of articles called definition articles or terminology articles. Each such article is about a term being defined, and the essential component of any such article is the definition of that term. The definition forms a separate section of the article, and this section is designed keeping a number of objectives in mind.

Goals of the definition section

Potential audience for a definition section

Some people know the term and are looking for its definition. This could include:

People encountering the term for the first time
People who've seen the term before, and know what it means, but want a precise definition
People who already know one definition, but want to know/read multiple definitions

On the other hand, there could also be people who know the definition already, and are looking for the precise term having that definition.

What the definition should provide

A definition should be clear, succinct, actionable, and interesting.

It must clearly answer the question: what is this
If the definition provides a differentiation or augmentation of an existing concept (for instance, a particular breed of dog, or a particular property of groups, or a particular type of number) then it should give a clear differentiating criterion.
It should give an idea of what related terms and facts are and how one can explore the notion better
It should be amenable to reverse search

Challenges in creating a definition section

The challenge of ingredients and dependencies

To define one term often requires other terms as ingredients. For instance:

The definition of a prime number relies on the definitions of natural number and divisor.
The typical definition of a finitely generated group relies on the definitions of group, generating set of a group, and, of course, the notion of finiteness.
The typical definition of Hausdorff space relies on the definition of topological space.

To cope with this challenge, we need to decide what ingredients are taken for granted (i.e., their definitions are not included) and what ingredients are to be defined along with the main term we're defining.

The way chosen in subject wikis is to have a certain primitive definition -- a definition that has as few dependencies as is possible while maintaining the conceptual integrity. Then, we create further definitions that are shorter and simpler, but rely on concepts beyond just the primitives.

The challenge of multiple definitions

Important terms can be defined from multiple perspectives. In fact, the equivalence of different definitions often gives an insight into why the concept is important.

Multiple definitions are both a boon and a challenge. Conventional wisdom suggests that giving a barrage of definitions can create a cognitive overload that makes it hard for people to grasp a new ideas. On the other hand, theories of learning, association and memory also suggest that the more hooks we have on an idea, the easier it is to understand.

In addition to genuinely different definitions, there are also equivalent definitions obtained by a slight rewording and change in the defining ingredients. For instance, as discussed in the previous subsection, a primitive definition may be made more compact and conceptually elegant by introducing further defining ingredients. In this case, both the primitive and the compacted definition are stated as equivalent definitions.

This again has advantages for definition cognition and definition understanding: in the effort to try to reconcile various rewordings of the definition, somebody trying to learn and understnad the definition gets a better understanding of what the term means.

Formats of definition

Quick phrases

Quick phrases are to be seen at the top of definition sections in some articles. These are phrases which one can use to remember the term being defined in a very easy way. For instance, a quick phrase for a prime number is number with no nontrivial factorization.

For quick phrases, conceptual ease matters more than precise meaning, so some of the quick phrases may not make precise mathematical sense.

Symbol-free definition

A symbol-free definition is a definition that avoids that use of algebraic symbols and notation, relying instead on constructs of natural language such as pronouns and prepositions. For instance, here's a symbol-free definition of permutable subgroup:

A subgroup of a group is termed permutable if its product with every subgroup of the group is a subgroup, or equivalently, if it commutes with every subgroup.

Here, symbols for the group and the two subgroups ar enot introduced. Rather, constructs of natural language, in particular, the pronoun it, are used.

Sometimes 1-2 symbols may be introduced in a symbol-free definition to avoid ambiguous pronouns.

Some advantages of symbol-free definitions:

More effective for reverse search i.e. locating the term from the definition.
Might be easier to read if you already know the definition and are trying to recall it (because of less demand on working memory)
Easier to keep in mind, memorize, and act upon in more diverse situations where the notation and symbols differ
More effective linking with other concepts and ideas

Definition with symbols

This gives the definition with explicit introduction of symbols for every quantified object. For instance the definition with symbols of the property of being a permutable subgroup is:

A subgroup $H$ of a group $G$ is termed permutable if for every subgroup $K$ , $HK=KH$ .

The definition with symbols has a number of advantages, for instance:

It may be better for first-time reading
It may be more actionable
It may require less knowledge of the subject or of terminology in the subject

Multiple definitions

Equivalent definitions as a list

Often, the same term may have multiple equivalent definitions. In this case, all the equivalent forms are stated as bullet points. The same format is repeated in the symbol-free definition and in the definition with symbols.

For instance, the symbol-free definition of normal subgroup: A subgroup of a group is said to be normal if it satisfies the following equivalent conditions:

It is the kernel of a homomorphism from the group.
It is invariant under all inner automorphisms. Thus, normality is the invariance property with respect to the property of an automorphism being inner. This definition also motivates the term invariant subgroup for normal subgroup (which was used earlier).
It equals each of its conjugates in the whole group. This definition also motivates the term self-conjugate subgroup for normal subgroup (which was used earlier).
Its left cosets are the same as its right cosets (that is, it commutes with every element of the group)
It is a union of conjugacy classes.

The definition with symbols of normal subgroup:

A subgroup $N$ of a group $G$ is said to be normal in $G$ (in symbols, $N\triangleleft G$ or $G\triangleright N$ Template:Subgroup-notation-page) if the following equivalent conditions hold:

There is a homomorphism $\phi$ from $G$ to another group such that the kernel of $\phi$ is precisely $N$ .
For all $g$ in $G$ , $gNg^{-1}\subseteq N$ . More explicitly, for all $g\in G,h\in N$ , we have $ghg^{-1}\in N$ .
For all $g$ in $G$ , $gNg^{-1}=N$ .
For all $g$ in $G$ , $gN=Ng$ .
$N$ is a union of conjugacy classes

We try to follow these conventions:

When multiple definitions are given, the same order is maintained for the definitions in the symbol-free format and the definition with symbols format.
The multiple definitions should preferably be numbered, making it easier to refer to the numbering
The definitions are ordered in a way that makes the equivalence between them as self-evident as possible.

Separate definitions as separate subsections

Sometimes, if the equivalent definitions are long and intricate or each requires their own machinery, they may be developed in separate subsections. The title of each subsection gives some title to the definition.

For instance, the two definitions of group are labeled as the textbook definition and the universal algebraic definition.

Equivalence of multiple definitions

Usually, there is a separate section within the definition part explaining the equivalence of definitions, or giving a link to a page that gives a full proof of this equivalence.

Defining ingredients

Linking style

Whenever a term is being defined using other terms, put links to those terms. For instance, when defining a group property, begin with: A group is said to be ... if ...

Typically, the symbol-free definition will not attempt to define other terms referenced in the definition. The definition with symbols, because it gives a more explicit description, may also give a clearer idea of other terms. For instance, in the symbol-free definition of normality, the second point states that a subgroup is normal if it is invariant under all inner automorphisms. In the definition with symbols, the explicit form of inner automorphisms is given.

Locating defining ingredients

The wiki markup contains information about what terms are used as ingredients in defining a particular term (using semantic MediaWiki). At the bottom of the article, a box called Facts gives a list of defining ingredients. Clicking on the magnifying glass icon next to any of these ingredients gives a list of all the other terms that use the same defining ingredient.

Links to survey articles clarifying the definition

For definitions of important terms, the definition section may begin with a box containing links to survey articles clarifying the definition.

Other comments

The definition section may also contain a few other comments, in so far as these comments play a direct role in clarifying the definition. Examples are typically reserved for another section, and under special circumstances, there is a separate section titled Importance.