The commutativity of simple operations, such as multiplication and addition of numbers, was for many years implicitly assumed and the property was not named until the 19th century when mathematics started to become formalized.

In higher branches of mathematics, such as analysis and linear algebra the commutativity of well known operations ( such as addition and multiplication on real and complex numbers ) is often used ( or implicitly assumed ) in proofs.

* Lagrange multiplier, a scalar variable used in mathematics to solve an optimisation problem for a given constraint.

Meanwhile, the intuitionistic school had failed to attract adherents among working mathematicians, and floundered due to the difficulties of doing mathematics under the constraint of constructivism.

In mathematics, constraint counting is a crude but often useful way of counting the number of free functions needed to specify a solution to a partial differential equation.

Scientists say that the world and everything in it are based on mathematics.

But because science is based on mathematics doesn't mean that a hot rodder must necessarily be a mathematician.

Pythagoras believed that behind the appearance of things, there was the permanent principle of mathematics, and that the forms were based on a transcendental mathematical relation.

Ethics cannot be based on the authoritative certainty given by mathematics and logic, or prescribed directly from the empirical findings of science.

In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in X

He used his time in Bourg to research mathematics, producing Considérations sur la théorie mathématique de jeu ( 1802 ; “ Considerations on the Mathematical Theory of Games ”), a treatise on mathematical probability that he sent to the Paris Academy of Sciences in 1803.

Two aspects of this attitude deserve to be mentioned: 1 ) he did not only study science from books, as other academics did in his day, but actually observed and experimented with nature ( the rumours starting by those who did not understand this are probably at the source of Albert's supposed connections with alchemy and witchcraft ), 2 ) he took from Aristotle the view that scientific method had to be appropriate to the objects of the scientific discipline at hand ( in discussions with Roger Bacon, who, like many 20th century academics, thought that all science should be based on mathematics ).

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 ).

Though respected for their contributions to various academic disciplines ( respectively mathematics, linguistics, and literature ), the three men became known to the general public only by making often-controversial and disputed pronouncements on politics and public policy that would not be regarded as noteworthy if offered by a medical doctor or skilled tradesman.

" The Four Books on Measurement " were published at Nuremberg in 1525 and was the first book for adults on mathematics in German, as well as being cited later by Galileo and Kepler.

By focusing consciously on an idea, feeling or intention the meditant seeks to arrive at pure thinking, a state exemplified by but not confined to pure mathematics.

On the Infinite was Hilbert ’ s most important paper on the foundations of mathematics, serving as the heart of Hilbert's program to secure the foundation of transfinite numbers by basing them on finite methods.

Abstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena.

In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A.

In mathematics, a binary operation on a set is a calculation involving two elements of the set ( called operands ) and producing another element of the set ( more formally, an operation whose arity is two ).

Following Desargues ' thinking, the sixteen-year-old Pascal produced, as a means of proof, a short treatise on what was called the " Mystic Hexagram ", Essai pour les coniques (" Essay on Conics ") and sent it — his first serious work of mathematics — to Père Mersenne in Paris ; it is known still today as Pascal's theorem.

The first person to describe the mathematics behind Brownian motion was Thorvald N. Thiele in a paper on the method of least squares published in 1880.

Calculus ( Latin, calculus, a small stone used for counting ) is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series.

On November 29, 1921, the trustees declared it to be the express policy of the Institute to pursue scientific research of the greatest importance and at the same time " to continue to conduct thorough courses in engineering and pure science, basing the work of these courses on exceptionally strong instruction in the fundamental sciences of mathematics, physics, and chemistry ; broadening and enriching the curriculum by a liberal amount of instruction in such subjects as English, history, and economics ; and vitalizing all the work of the Institute by the infusion in generous measure of the spirit of research.

* nLab, a wiki project on mathematics, physics and philosophy with emphasis on the n-categorical point of view

In mathematics, specifically in category theory, a preadditive category is a category that is enriched over the monoidal category of abelian groups.

In category theory, a branch of mathematics, an enriched category generalizes the idea of a category by replacing hom-sets with objects from a general monoidal category.

In mathematics, a monoidal category ( or tensor category ) is a category C equipped with a bifunctor

In mathematics, an autonomous category is a monoidal category where dual objects exist.

The most general setting in which these words have meaning is an abstract branch of mathematics called category theory.

Many significant areas of mathematics can be formalised as categories, and the use of category theory allows many intricate and subtle mathematical results in these fields to be stated, and proved, in a much simpler way than without the use of categories.

General category theory, an extension of universal algebra having many new features allowing for semantic flexibility and higher-order logic, came later ; it is now applied throughout mathematics.

These foundational applications of category theory have been worked out in fair detail as a basis for, and justification of, constructive mathematics.

* Timeline of category theory and related mathematics

In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds.

In category theory, a branch of mathematics, a functor is a special type of mapping between categories.

In mathematics, especially in category theory and homotopy theory, a groupoid ( less often Brandt groupoid or virtual group ) generalises the notion of group in several equivalent ways.

In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space.

In certain areas of mathematics, notably category theory, it is valuable to distinguish between equality on the one hand and isomorphism on the other.

Other branches of mathematics, however, such as logic, number theory, category theory or set theory are accepted as part of pure mathematics, though they find application in other sciences ( predominantly computer science and physics ).

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 ".

Because of its applicability in diverse fields of mathematics, mathematicians including Saunders Mac Lane have proposed category theory as a foundational system for mathematics, independent of set theory.

In mathematics, specifically in category theory, the Yoneda lemma is an abstract result on functors of the type morphisms into a fixed object.

In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits.

* Core of a triangulated category in mathematics

In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure ( i. e. the composition of morphisms ) of the categories involved.

In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties.

As an alternative to set theory, others have argued for category theory as a foundation for certain aspects of mathematics.

In category theory, a branch of mathematics, group objects are certain generalizations of groups which are built on more complicated structures than sets.

In mathematics, a category is an algebraic structure that comprises " objects " that are linked by " arrows ".