Definition is the description of the meaning of a word, phrase, or concept. There are various kinds of definition, including truth-functional, recursive (or inductive), lexical, stipulative, ostensive, definition by abstraction, and definition by genus and difference. The truth-functional definitions and definitions by abstraction are the most commonly used in logic and mathematics.
A lexical definition is the report of a meaning a word already has. This is mainly what a dictionary definition is. A stipulative definition is one which assigns a new meaning to a word. Stipulative definitions are often used by theoreticians attempting to use a word in a way that will capture an idea in a way not thought previously. An ostensive definition is an example of the word to be defined. Pointing to an apple would be an ostensive definition of the word, "apple." The definiendum (that which is defined in a definition) in a definition by abstraction is a class term, and it is defined by the properties a thing must have in order to be a member of that class. For example, whatever properties are necessary for calling something a human constitute a definition by abstraction. A definition by genus and difference, like the definition by abstraction, is determined by classes. The genus is the larger class, and the species is the smaller class. For example, human being is a species of the genus animal. What differentiates humans from other species in the genus is a peculiar form of rationality. The genus is the class, and the 'difference' is the property or properties that distinguish a species of a genus. So, human being is defined by its genus, animal and its difference, rationality, from other species.
The definition of a logical operator is called a truth-functional definition, or simply a truth-function. This definition is determined by the components of the statement which are connected by the operator. For example, the definition for the conjunction or "and" operator is determined by the truth of the statement's components so that the conjunction is true if and only if both sides of the conjunct are true. If one side of the conjunct is false, the whole statement is false. Likewise, if both sides are false, the whole statement is false. The statement, "Tom and Jane are in class," is true if and only if Tom is in class and Jane is in class. The definition of conjunction applies to any statement employing a conjunction, as illustrated by the Tom and Jane example.
A recursive, or inductive definition applies a definition repeatedly to various instances, like a formula. For example, the series of natural numbers can be defined recursively. 1 is the first in the series, then 2, 3...n, n+1, and so on, where n is some successive number in the series, and the series is defined by applying the formula n+1 to any number in order to generate the next in the series.
Definition is important to mathematics because it provides clarity to concepts that are meant to have universal application. Analytic geometry, for example, first presented by Descartes in the 17th century, is defined by the relationship of points on an axis. The "Cartesian coordinates," as they are now known, show us that all quadratic equations become geometrical figures when graphed as connected points on an axis of intersecting lines.
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