![]() Naive set theory: The empty set is a primitive notion.A set is a plurality thought of as a unit." As evidence, she quotes Felix Hausdorff: "A set is formed by the grouping together of single objects into a whole. As Mary Tiles writes: 'definition' of 'set' is less a definition than an attempt at explication of something which is being given the status of a primitive, undefined, term. Set theory: The concept of the set is an example of a primitive notion.The necessity for primitive notions is illustrated in several axiomatic foundations in mathematics: This is not a superficial problem but lies at the root of all knowledge it is necessary to begin somewhere, and to make progress one must clearly state those elements and relations which are undefined and those properties which are taken for granted. ![]() To a non-mathematician it often comes as a surprise that it is impossible to define explicitly all the terms which are used. The sentence which determines the meaning of a term in this way is called a DEFINITION.Īn inevitable regress to primitive notions in the theory of knowledge was explained by Gilbert de B. At the same time we adopt the principle: not to employ any of the other expressions of the discipline under consideration, unless its meaning has first been determined with the help of primitive terms and of such expressions of the discipline whose meanings have been explained previously. Details Īlfred Tarski explained the role of primitive notions as follows: When we set out to construct a given discipline, we distinguish, first of all, a certain small group of expressions of this discipline that seem to us to be immediately understandable the expressions in this group we call PRIMITIVE TERMS or UNDEFINED TERMS, and we employ them without explaining their meanings. Instead of attempting to define them, their interplay is ruled (in Hilbert's axiom system) by axioms like "For every two points there exists a line that contains them both". Formal theories cannot dispense with primitive notions, under pain of infinite regress (per the regress problem).įor example, in contemporary geometry, point, line, and contains are some primitive notions. Some authors refer to the latter as "defining" primitive notions by one or more axioms, but this can be misleading. In an axiomatic theory, relations between primitive notions are restricted by axioms. ![]() ![]() It is often motivated informally, usually by an appeal to intuition and everyday experience. In mathematics, logic, philosophy, and formal systems, a primitive notion is a concept that is not defined in terms of previously-defined concepts. Concept that is not defined in terms of previously defined concepts ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |