For example, in formal languages like mathematics, a ' stipulative ' definition guides a specific discussion.

However, there is no exact, universally agreed, definition of the term " discrete mathematics.

Another definition of the GCD is helpful in advanced mathematics, particularly ring theory.

He presented a paper which posed the question of correctly formed definitions in mathematics, i. e. " how do you define a definition ?".

Many philosophers believe that mathematics is not experimentally falsifiable, and thus not a science according to the definition of Karl Popper.

Category theory, another field within " foundational mathematics ", is rooted on the abstract axiomatization of the definition of a " class of mathematical structures ", referred to as a " category ".

In the broad definition, the parent disciplines of musicology include history ; cultural studies and gender studies ; philosophy, aesthetics and semiotics ; ethnology and cultural anthropology ; archeology and prehistory ; psychology and sociology ; physiology and neuroscience ; acoustics and psychoacoustics ; and computer / information sciences and mathematics.

However this definition does not allow star polytopes with interior structures, and so is restricted to certain areas of mathematics.

In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval.

As discussed below, the definition given above turned out to be inadequate for formal mathematics ; instead, the notion of a " set " is taken as an undefined primitive in axiomatic set theory, and its properties are defined by the Zermelo – Fraenkel axioms.

The thesis states that Turing machines indeed capture the informal notion of effective method in logic and mathematics, and provide a precise definition of an algorithm or ' mechanical procedure '.

In mathematics, the domain of definition or simply the domain of a function is the set of " input " or argument values for which the function is defined.

Many of Plato's dialogues concern the search for a definition of some important concept ( justice, truth, the Good ), and it is likely that Plato was impressed by the importance of definition in mathematics.

While it is often assumed that those with this intelligence naturally excel in mathematics, chess, computer programming and other logical or numerical activities, a more accurate definition places less emphasis on traditional mathematical ability and more on reasoning capabilities, recognizing abstract patterns, scientific thinking and investigation and the ability to perform complex calculations.

A less restrictive definition of linear function, not coinciding with the definition of linear map, is used in elementary mathematics.

The term " dummy variable " is also sometimes used for a bound variable ( more often in general mathematics than in computer science ), but that use creates an ambiguity with the definition of dummy variables in regression analysis.

In mathematics, the inverse of a function is a function that, in some fashion, " undoes " the effect of ( see inverse function for a formal and detailed definition ).

With a definition of effective calculation came the first proofs that there are problems in mathematics that cannot be effectively decided.

" However, another definition, more in keeping with the predominant usage of mathematics, is that mathematical practice is the everyday practice, or use, of math.

" He held that corollarial deduction matches Aristotle's conception of direct demonstration, which Aristotle regarded as the only thoroughly satisfactory demonstration, while theorematic deduction ( A ) is the kind more prized by mathematicians, ( B ) is peculiar to mathematics, and ( C ) involves in its course the introduction of a lemma or at least a definition uncontemplated in the thesis ( the proposition that is to be proved ); in remarkable cases that definition is of an abstraction that " ought to be supported by a proper postulate.

The latter definition is consistent with its uses in higher mathematics such as calculus.

Connes has applied his work in areas of mathematics and theoretical physics, including number theory, differential geometry and particle physics.

One of the most interesting aspects of the axiom of choice is the large number of places in mathematics that it shows up.

In mathematics, the absolute value ( or modulus ) of a real number is the numerical value of without regard to its sign.

In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients ( or equivalently — by clearing denominators — with integer coefficients ).

In mathematics, the unit square is defined to have area one, and the area of any other shape or surface is a dimensionless real number.

In mathematics, the phrase " almost all " has a number of specialised uses.

It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory.

In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field K or the ring R.

Realism in the philosophy of mathematics is the claim that mathematical entities such as number have a mind-independent existence.

Arithmetic or arithmetics ( from the Greek word ἀριθμός, arithmos " number ") is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations.

* Bell number, in mathematics

In mathematics, the Bernoulli numbers B < sub > n </ sub > are a sequence of rational numbers with deep connections to number theory.

With large sets, it becomes necessary to use more sophisticated mathematics to find the number of combinations.

* Catalan number, a concept in mathematics

In mathematics, a countable set is a set with the same cardinality ( number of elements ) as some subset of the set of natural numbers.

Although a " bijection " seems a more advanced concept than a number, the usual development of mathematics in terms of set theory defines functions before numbers, as they are based on much simpler sets.

In 1879, Peirce was appointed Lecturer in logic at the new Johns Hopkins University, which was strong in a number of areas that interested him, such as philosophy ( Royce and Dewey did their PhDs at Hopkins ), psychology ( taught by G. Stanley Hall and studied by Joseph Jastrow, who coauthored a landmark empirical study with Peirce ), and mathematics ( taught by J. J. Sylvester, who came to admire Peirce's work on mathematics and logic ).

The Langlands program is a far-reaching web of these ideas of ' unifying conjectures ' that link different subfields of mathematics, e. g. number theory and representation theory of Lie groups ; some of these conjectures have since been proved.

In mathematics, any number of cases supporting a conjecture, no matter how large, is insufficient for establishing the conjecture's veracity, since a single counterexample would immediately bring down the conjecture.

* Cardinal number, a concept in mathematics

In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties.

In mathematics, the cardinality of a set is a measure of the " number of elements of the set ".

In mathematics, a contraction mapping, or contraction, on a metric space ( M, d ) is a function f from M to itself, with the property that there is some nonnegative real number < math > k < 1 </ math > such that for all x and y in M,