Subwiki:Definition

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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.

In this article, we use the technical word definiendum for the term being defined, though it may also be used more loosely for the concept, idea, or notion behind the term.

Goals of the definition section

Potential audience for a definition section

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

  • People encountering the definiendum for the first time.
  • People who've seen the definiendum 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 presenting intensional definitions

Most definitions in mathematics are intensional definitions. A mathematical object is defined by specifying necessary and sufficient conditions for what it means to be such an object. Intensional definitions are also found for some simple terms outside mathematics.

Intensional definitions are in sharp contrast with extensional definitions, that define a concept through a range of representative examples. They also differ, for instance, from sense definitions, that define an object in terms of how it appears to the senses.

The main feature of definitions in mathematics, and of intensional definitions in general, is that the definition is de facto always true, and a complete description. Thus, it is not usually necessary to justify this definition against prior notions of what the term or concept means. This sharply contrasts with definitions involving terms that are already in daily use and carry strong and diverse connotations in people's minds: terms like society, environment, terrorism, and patriotism.

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 idea. 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 understand the definition gets a better understanding of what the term means.

The challenge of symbols and expressions

For new definitions, specially complicated ones, notation is often necessary, or at any rate, helpful. Thus, we often start definitions like: A ... is a set with ....

Symbols and expressions can be useful to set up a definition, particularly when the number of variables involved, or the nature of operations, is too large for the nouns and pronouns and constructs of natural language.

However, they also have a crucial disadvantage: they may not carry conceptual clarity. In many situations, the mind relates more clearly to natural language than to variables and formulas (we've learned natural language since preschool, but algebra is usually introduced in middle school). Also, symbols are often at odds with the use of further ingredients or concepts.

The subject wikis meet this challenge through a somewhat unusual method: most definitions carry with them two sections, the symbol-free definition, which seeks to use natural language only without any variables or symbols, and the definition with symbols, which expresses everything in symbols (so whatever objects are referred to are given symbols).

The challenge of stickiness

Definitions can often be dull and dry, and the tremendous conceptual compactification they represent may not be obvious to people just starting. A definition in isolation, thus, may not be too sticky.

The subject wikis try to remedy this in a number of ways. The most commonly implemented way is the use of Quick Phrases. On some definition pages, a Quick Phrases box appears right on top of the definition section, with phrases that could be used to compactly remember the definition, or the essential meaning of the term. Accuracy is sometimes sacrificed for memorability or analogical usefulness in the quick phrases.

The challenge of misinterpretation and confusion

Definitions, read in isolation, are prone to being poorly understood, misinterpreted, and wrongly applied. All these cannot be remedied simply by a better definition. Other techniques, including the use of examples, variations, analogies, formalisms for definitions, and further exploration and analysis of the concept, are necessary to ensure a proper and useful understanding of the term being defined.

Nonetheless, certain things are done on the subject wikis to reduce the chances of confusion:

  • Redundancy and multiple perspectives: The same definition is written in many different ways, both with and without symbols, as well as with quick phrases
  • Links to definition understanding pages, funky definitions pages (pages with preposterous, weird, and sometimes circular definitions for the term)
  • Links to definition equivalence pages for proofs of the equivalence of definitions

The challenge of motivation

Probably the biggest challenge, that definitions on subject wikis fail to satisfy, is the challenge of motivation. Why is this definition important? Why were things defined this way, rather than that?

Conventional wisdom suggests that every definition should be preceded by examples. This is often excellent pedagogy; however, in a structure of this kind, there is no natural notion of order, and illustrative examples that build up to the definition may take up too much space on the definition page, and may not be very useful to people who want a quick review of the definition. On the other hand, the linked structure of the wiki makes it possible to do the examples elsewhere, but link to them, allowing people to zoom in.

Some of the special features in the definition section to cater to the challenge of motivation are:

  • Links to definition understanding pages. These pages often explore the definition and various aspects of the definition in detail.
  • The definition section is preceded by a History section that may give information, links from the historical perspective. The definition section is succeeded by an Examples section that explores examples of various kinds (learn more at Subwiki:Examples).

Genus-differentia definitions

Many definitions are of the genus-differentia kind. These define new terms by providing differentiating criteria within an existing genus. For instance:

  • The definition of prime number starts with the genus (generic term) of natural numbers and provides the differentiating criterion (differentiae specificae) of being prime.
  • The definition of skyscraper starts with the genus of buildings and provides the differentiating criterion of being tall.
  • The definition of e-book starts with the genus of book and provides the differentiating criterion of being in an electronic format.

Genus-differentia definitions are of different kinds, and each operates somewhat differently:

  1. Property definitions: These are most common in precise fields like mathematics where practically all definitions are intensional. Given a collection of objects, or a context space, a property on that context space is something that everything in the context space either satisfies or does not satisfy. Thus, a property picks out a subcollection of the context space comprising those things that do satisfy the property.
  2. Modifier definitions: These are found both in precise fields like mathematics and in other fields. Here, the differentia may not strictly lie within the genus, but rather, may modify it somewhat. For instance, ecoterrorism is obtained by modifying terrorism, but may not itself be a form of terrorism.
  3. Relationships of physical containment: Here, the definiendum is physically contained inside the general term. For instance, the general term might be the city of London while the definiendum might be Bloomsbury, a district of London).
  4. Taxonomical classification: For instance, the Linnaean taxonomy for living beings.

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:

  1. More effective for reverse search i.e. locating the term from the definition.
  2. Might be easier to read if you already know the definition and are trying to recall it (because of less demand on working memory)
  3. Easier to keep in mind, memorize, and act upon in more diverse situations where the notation and symbols differ
  4. 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 of a group is termed permutable if for every subgroup , .

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:

  1. It is the kernel of a homomorphism from the group.
  2. 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).
  3. 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).
  4. Its left cosets are the same as its right cosets (that is, it commutes with every element of the group)
  5. It is a union of conjugacy classes.

The definition with symbols of normal subgroup:

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

  1. There is a homomorphism from to another group such that the kernel of is precisely .
  2. For all in , . More explicitly, for all , we have .
  3. For all in , .
  4. For all in , .
  5. 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.