Blackwell Reference Online

Extract

L ogic, philosophy of mathematics, philosophy of science Also called definition by axioms or definition by postulates . In contrast to an explicit definition, which gives the necessary and sufficient conditions for a term to be applied, an implicit definition of a term does not directly state the extension and intension of a term, but defines the term by showing that it satisfies certain axioms, the validity of which is strictly guaranteed. Thus the axioms of a system of geometry implicitly define the primitive geometrical signs that the axioms contain by delimiting the interpretations of the signs that satisfy it. This notion gains its importance in modern mathematics through the work of Hilbert . For he claims that the quest for explicit definitions for many mathematical terms such as “straight line,” “point,” and “plane” is extremely difficult and that we should define such terms implicitly as whatever entities satisfy the formal axioms formulated by means of them. As a result, although non-Euclidean geometry still uses Euclidean terms such as “point,” “place,” and “straight line,” these terms do not mean the same in the two systems, since they are implicitly defined by the postulate set in which they occur. A similar use of implicit definitions in natural science, in which terms are defined through satisfying the theories in which they are embedded, also raises ... log in or subscribe to read full text