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In mathematics, the term undefined refers to a value, function, or other expression that cannot be assigned a meaning within a specific formal system. [ 1 ] Attempting to assign or use an undefined value within a particular formal system, may produce contradictory or meaningless results within that system.
Although applicable to any area of mathematics, geometry is the branch of elementary mathematics in which this method has most extensively been successfully applied. [1] There are several components of an axiomatic system. [2] Primitives (undefined terms) are the most basic ideas. Typically they include objects and relationships.
The necessity for primitive notions is illustrated in several axiomatic foundations in mathematics: Set theory : The concept of the set is an example of a primitive notion. As Mary Tiles writes: [ 6 ] [The] 'definition' of 'set' is less a definition than an attempt at explication of something which is being given the status of a primitive ...
A lesson learned by mathematics in the last 150 years is that it is useful to strip the meaning away from the mathematical assertions (axioms, postulates, propositions, theorems) and definitions. One must concede the need for primitive notions, or undefined terms or concepts, in any study. Such abstraction or formalization makes mathematical ...
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry, Elements.Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions from these.
This is an accepted version of this page This is the latest accepted revision, reviewed on 9 January 2025. Look up undefined in Wiktionary, the free dictionary. Undefined may refer to: Mathematics Undefined (mathematics), with several related meanings Indeterminate form, in calculus Computing Undefined behavior, computer code whose behavior is not specified under certain conditions Undefined ...
Mathematics in the twentieth century evolved into a network of axiomatic formal systems. This was, in considerable part, influenced by the example Hilbert set in the Grundlagen .
Besides the volume, a measure generalizes the notions of area, length, mass (or charge) distribution, and also probability distribution, according to Andrey Kolmogorov's approach to probability theory. A "geometric body" of classical mathematics is much more regular than just a set of points. The boundary of the body is of zero volume.