In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that " the product of a collection of non-empty sets is non-empty ".

The axiom of choice is avoided in some varieties of constructive mathematics, although there are varieties of constructive mathematics in which the axiom of choice is embraced.

Despite these facts, most mathematicians accept the axiom of choice as a valid principle for proving new results in mathematics.

The axiom of choice has also been thoroughly studied in the context of constructive mathematics, where non-classical logic is employed.

The status of the axiom of choice varies between different varieties of constructive mathematics.

Although the axiom of countable choice in particular is commonly used in constructive mathematics, its use has also been questioned.

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

The axiom of regularity is arguably the least useful ingredient of Zermelo – Fraenkel set theory, since virtually all results in the branches of mathematics based on set theory hold even in the absence of regularity ( see chapter 3 of ).

It is ' naive ' in that the language and notations are those of ordinary informal mathematics, and in that it doesn't deal with consistency or completeness of the axiom system.

Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory.

The status of the axiom of choice in constructive mathematics is complicated by the different approaches of different constructivist programs.

The field of mathematics known as proof theory studies formal axiom systems and the proofs that can be performed within them.

They also had to make several compromises in order to develop so much of mathematics, such as an " axiom of reducibility ".

Many formalists would say that in practice, the axiom systems to be studied will be suggested by the demands of science or other areas of mathematics.

Hilbert's goals of creating a system of mathematics that is both complete and consistent were dealt a fatal blow by the second of Gödel's incompleteness theorems, which states that sufficiently expressive consistent axiom systems can never prove their own consistency.

Other formalists, such as Rudolf Carnap, Alfred Tarski and Haskell Curry, considered mathematics to be the investigation of formal axiom systems.

In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo – Fraenkel set theory.

The axiom of extensionality is generally uncontroversial in set-theoretical foundations of mathematics, and it or an equivalent appears in just about any alternative axiomatisation of set theory.

Linear equations occur with great regularity in applied mathematics.

Arbuthnot returned to mathematics in 1710 with An argument for Divine Providence, taken from the constant regularity observed in the births of both sexes ( linked below ) in the Royal Society's Philosophical Transactions, where he analysed birth data and demonstrated that males were born at a greater rate than females.

There are an extremely large number of unrelated notions of " regularity " in mathematics.

In mathematics, the Szemerédi regularity lemma states that every large enough graph can be divided into subsets of about the same size so that the edges between different subsets behave almost randomly.

In mathematics, the Karush – Kuhn – Tucker ( KKT ) conditions ( also known as the Kuhn – Tucker conditions ) are first order necessary conditions for a solution in nonlinear programming to be optimal, provided that some regularity conditions are satisfied.

Blind students also complete mathematical assignments using a braille-writer and Nemeth code ( a type of braille code for mathematics ) but large multiplication and long division problems can be long and difficult.

It is also commonly used in mathematics in algebraic solutions representing quantities such as angles.

The term may be also used loosely or metaphorically to denote highly skilled people in any non -" art " activities, as well — law, medicine, mechanics, or mathematics, for example.

He also applied mathematics in generalizing physical laws from these experimental results.

Despite being quite religious, he was also interested in mathematics and science, and sometimes is claimed to have contradicted the teachings of the Church in favour of scientific theories.

He was educated at the Collège des Quatre-Nations ( also known as Collège Mazarin ) from 1754 to 1761, studying chemistry, botany, astronomy, and mathematics.

His criticisms of the scientific community, and especially of several mathematics circles, are also contained in a letter, written in 1988, in which he states the reasons for his refusal of the Crafoord Prize.

He is also noted for his mastery of abstract approaches to mathematics and his perfectionism in matters of formulation and presentation.

It can also be used in topics as diverse as mathematics, gastronomy, fashion and website design.

In classical mathematics, analytic geometry, also known as coordinate geometry, or Cartesian geometry, is the study of geometry using a coordinate system and the principles of algebra and analysis.

He won a scholarship to the University and majored in mathematics, and also studied astronomy, physics and chemistry.

His father, Étienne Pascal ( 1588 – 1651 ), who also had an interest in science and mathematics, was a local judge and member of the " Noblesse de Robe ".

In chemistry, physics, and mathematics, the Boltzmann distribution ( also called the Gibbs Distribution ) is a certain distribution function or probability measure for the distribution of the states of a system.

Bioinformatics also deals with algorithms, databases and information systems, web technologies, artificial intelligence and soft computing, information and computation theory, structural biology, software engineering, data mining, image processing, modeling and simulation, discrete mathematics, control and system theory, circuit theory, and statistics.

It can also be used to denote abstract vectors and linear functionals in mathematics.

In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space.

The term can also be applied to some degree to functions in mathematics, referring to the anatomy of curves.

Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, and combinatorics also has many applications in optimization, computer science, ergodic theory and statistical physics.

It has also given rise to a new theory of the philosophy of mathematics, and many theories of artificial intelligence, persuasion and coercion.

Most undergraduate programs emphasize mathematics and physics as well as chemistry, partly because chemistry is also known as " the central science ", thus chemists ought to have a well-rounded knowledge about science.

Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows ( also called morphisms, although this term also has a specific, non category-theoretical meaning ), where these collections satisfy some basic conditions.