In mathematics, a well-order relation ( or well-ordering ) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering.

In mathematics, the well-ordering theorem states that every set can be well-ordered.

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.

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

The anthropic principle is often criticized for lacking falsifiability and therefore critics of the anthropic principle may point out that the anthropic principle is a non-scientific concept, even though the weak anthropic principle, " conditions that are observed in the universe must allow the observer to exist ", is " easy " to support in mathematics and philosophy, i. e. it is a tautology or truism

The originality of Descartes ' thinking, therefore, is not so much in expressing the cogito — a feat accomplished by other predecessors, as we shall see — but on using the cogito as demonstrating the most fundamental epistemological principle, that science and mathematics are justified by relying on clarity, distinctiveness, and self-evidence.

In mathematics, the Hausdorff maximal principle is an alternate and earlier formulation of Zorn's lemma proved by Felix Hausdorff in 1914 ( Moore 1982: 168 ).

* Large deviation principle, the rate function in mathematics

The ubiquity of groups in numerous areas within and outside mathematics makes them a central organizing principle of contemporary mathematics.

In mathematics, the principle that says that if the number of players is one more than the number of chairs, then one player is left standing, is the pigeonhole principle.

More generally, non-standard analysis is any form of mathematics that relies on non-standard models and the transfer principle.

Deeper theorems from real analysis were not included, but by the end of the third volume it was clear to experts that a large amount of known mathematics could in principle be developed in the adopted formalism.

The Church – Turing thesis states that this is a law of mathematics — that a universal Turing machine can, in principle, perform any calculation that any other programmable computer can.

In mathematics, the Banach fixed-point theorem ( also known as the contraction mapping theorem or contraction mapping principle ) is an important tool in the theory of metric spaces ; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, and provides a constructive method to find those fixed points.

This means that man becomes immortal only if and to the extent that he acquires knowledge of what he can in principle know, e. g. mathematics and the natural sciences.

According to Lewis ( 1918 ), the " principle of these diagrams is that classes set ( mathematics ) | set s be represented by regions in such relation to one another that all the possible logical relations of these classes can be indicated in the same diagram.

Both Spinoza and Leibniz asserted that, in principle, all knowledge, including scientific knowledge, could be gained through the use of reason alone, though they both observed that this was not possible in practice for human beings except in specific areas such as mathematics.

Although the concepts of a law or principle in nature is borderline to philosophy, and presents the depth to which mathematics can describe nature, scientific laws are considered from a scientific perspective and follow the scientific method ; they " serve their purpose " rather than " questioning reality " ( philosophical ) or " statements of logical absolution " ( mathematical ).

Calculus, though allowed in solutions, is never required, as there is a principle at play that anyone with a basic understanding of mathematics should understand the problems, even if the solutions require a great deal more knowledge.

In mathematics, a nonlinear system is one that does not satisfy the superposition principle, or one whose output is not directly proportional to its input ; a linear system fulfills these conditions.

* The superposition principle in physics, mathematics, and engineering, describes the overlapping of waves.

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.

In mathematics, the Borsuk – Ulam theorem, named after Stanisław Ulam and Karol Borsuk, states that every continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point.

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.

In mathematics, the four color theorem, or the four color map theorem states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color.

In mathematics, specifically commutative algebra, Hilbert's basis theorem states that every ideal in the ring of multivariate polynomials over a Noetherian ring is finitely generated.

Much constructive mathematics uses intuitionistic logic, which is essentially classical logic without the law of the excluded middle which states that for any proposition, either that proposition is true, or its negation is.

The thought experiment illustrates quantum mechanics and the mathematics necessary to describe quantum states.

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 '.

Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order ( number of elements ) of every subgroup H of G divides the order of G. The theorem is named after Joseph Lagrange.

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.

During the 290s BC, Hellenistic civilization begins its emergence throughout the successor states of the former Argead Macedonian Empire of Alexander the Great resulting in the diffusion of Greek culture throughout the Ancient world and advances in Science, mathematics, philosophy and etc.

In mathematics, Tait's conjecture states that " Every 3-connected planar cubic graph has a Hamiltonian cycle ( along the edges ) through all its vertices ".

In mathematics, the convolution theorem states that under suitable

In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side ( and, if the setting is a Euclidean space, then the inequality is strict if the triangle is non-degenerate ).< ref name = Khamsi >

In mathematics, de Moivre's formula ( a. k. a. De Moivre's theorem and De Moivre's identity ), named after Abraham de Moivre, states that for any complex number ( and, in particular, for any real number ) x and integer n it holds that

The Sumerians were incredibly advanced: as well as inventing writing, they also invented early forms of mathematics, early wheeled vehicles, astronomy, astrology and the calendar and they created the first city states / nations such as Uruk, Ur, Lagash, Isin, Umma, Eridu, Nippur and Larsa.

In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact.

In mathematics, and more specifically in homological algebra, the splitting lemma states that in any abelian category, the following statements for short exact sequence are equivalent.

In mathematics, the parity of an object states whether it is even or odd.

In mathematics the modularity theorem ( formerly called the Taniyama – Shimura – Weil conjecture and several related names ) states that elliptic curves over the field of rational numbers are related to modular forms.

In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space.

In engineering, mathematics and the physical and biological sciences, common terms for the points around which the system gravitates include: attractors, stable states, eigenstates / eigenfunctions, equilibrium points, and setpoints.