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 ".

A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: " Read Euler, read Euler, he is the master of us all.

In mathematics, a lemma ( plural lemmata or lemmas ) from the Greek λῆμμα ( lemma, “ anything which is received, such as a gift, profit, or a bribe ”) is a proven proposition which is used as a stepping stone to a larger result rather than as a statement of interest by itself.

One statement of this philosophy is the thesis that mathematics is not created but discovered.

A lucid statement of this is found in an essay written by the British mathematician G. H. Hardy in defense of pure mathematics.

In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms.

Some, on the other hand, may be called " deep ": their proofs may be long and difficult, involve areas of mathematics superficially distinct from the statement of the theorem itself, or show surprising connections between disparate areas of mathematics.

* Mathematical statement, a statement in logic and mathematics

Although most mathematicians and physicists ( and many philosophers ) would accept the statement " mathematics is a language ", linguists believe that the implications of such a statement must be considered.

In mathematics, Minkowski's theorem is the statement that any convex set in R < sup > n </ sup > which is symmetric with respect to the origin and with volume greater than 2 < sup > n </ sup > d ( L ) contains a non-zero lattice point.

This is the spectral theorem in mathematics, and in a finite state space it is just a statement of the completeness of the eigenvectors of a Hermitian matrix.

In mathematics, the Cauchy integral theorem ( also known as the Cauchy – Goursat theorem ) in complex analysis, named after Augustin-Louis Cauchy, is an important statement about line integrals for holomorphic functions in the complex plane.

In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis.

In mathematics, a proof is a demonstration that if some fundamental statements ( axioms ) are assumed to be true, then some mathematical statement is necessarily true.

In mathematics, an inequation is a statement that an inequality holds between two values.

In mathematics, the proof of a " some " statement may be achieved either by a constructive proof, which exhibits an object satisfying the " some " statement, or by a nonconstructive proof which shows that there must be such an object but without exhibiting one.

Famously, he stated that the point is to know one way or the other what the solution is, and he believed that we always can know this, that in mathematics there is not any " ignorabimus " ( statement that the truth can never be known ).

That is, if a professor of mathematics makes a statement about numbers, it will be assumed to be true in the absence of evidence to the contrary.

Assuming something to be true for all numbers when it has been shown for over 906 million cases would not generally be considered hasty, but in mathematics a statement remains a conjecture until it is shown to be universally true.

* Difference ( mathematics ), a statement about the relative size or order of two objects

In mathematics and logic, a direct proof is a way of showing the truth or falsehood of a given statement by a straightforward combination of established facts, usually existing lemmas and theorems, without making any further assumptions.

The beginning of set theory as a branch of mathematics is often marked by the publication of Cantor's 1874 article, " Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen " (" On a Property of the Collection of All Real Algebraic Numbers ").

* Property ( philosophy ), in philosophy, mathematics, and logic, an abstraction characterizing an object

In mathematics, Property B is a certain set theoretic property.

In mathematics, the Property P conjecture is a statement about 3-manifolds obtained by Dehn surgery on a knot in the 3-sphere.

In mathematics, Property FA is a property of groups first defined by Jean-Pierre Serre.

In mathematics and computational geometry, a Delaunay triangulation for a set P of points in a plane is a triangulation DT ( P ) such that no point in P is inside the circumcircle of any triangle in DT ( P ).

In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set ( P, ≤) is an element of P which is greater than or equal to every element of S. The term lower bound is defined dually as an element of P which is less than or equal to every element of S. A set with an upper bound is said to be bounded from above by that bound, a set with a lower bound is said to be bounded from below by that bound.

Many recognized specialists in the knowledge areas where Korzybski claimed to have anchored general semantics — biology, epistemology, mathematics, neurology, physics, psychiatry, etc .— supported his work in his lifetime, including Cassius J. Keyser, C. B. Bridges, W. E. Ritter, P. W. Bridgman, G. E. Coghill, William Alanson White, Clarence B. Farrar, David Fairchild, and Erich Kähler.

In mathematics, a Diophantine equation is an equation of the form P ( x < sub > 1 </ sub >, ..., x < sub > j </ sub >, y < sub > 1 </ sub >, ..., y < sub > k </ sub >)= 0 ( usually abbreviated P (,)= 0 ) where P (,) is a polynomial with integer coefficients.

Balakirev left the Alexandrovsky Institute in 1853 and entered the University of Kazan as a mathematics student, along with his friend P. D.

There has been considerable formal development on complexity based views like Samuel Buss's Bounded Arithmetic theories which capture mathematics associated with various complexity classes like P and PSPACE.

Ramsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of mathematics that studies the conditions under which order must appear.

* Ehrlich, P. ( 2006 ) The rise of non-Archimedean mathematics and the roots of a misconception.

They include mountain climbers ( Heidi Howkins, class of 1989, the only woman to lead expeditions to both Everest and K-2 ), authors ( such as Harriet Stratemeyer Adams, class of 1914, pen name Carolyn Keene ), astronomers ( including Annie Jump Cannon, class of 1884, who developed the well-known Harvard Classification of stars based upon temperature ), screenwriters, ( including Nora Ephron, class of 1962, famous for such films as When Harry Met Sally and Sleepless in Seattle ), journalists ( Linda Wertheimer, class of 1965, Lynn Sherr, class of 1963, Diane Sawyer, class of 1967, and Cokie Roberts, class of 1964, being a few notable examples ), entrepreneurs ( including Robin Chase, class of 1980, the co-founder of ZipCar ), mathematicians ( Winifred Edgerton Merrill, class of 1883, was the first woman to ever receive a PhD in mathematics ), judges ( including Jane Bolin, class of 1928, the first African-American woman to become a judge, and current federal appeals judges Reena Raggi, Amalya Kearse, and Susan P. Graber ).

In mathematics, the Legendre functions P < sub > λ </ sub >, Q < sub > λ </ sub > and associated Legendre functions P, Q are generalizations of Legendre polynomials to non-integer degree.

In mathematics, the open unit disk ( or disc ) around P ( where P is a given point in the plane ), is the set of points whose distance from P is less than 1:

In mathematics, a predicate is commonly understood to be a boolean-valued function P: X →

* the widely held belief in the banishment of infinitesimals from mathematics until the creation of non-standard analysis by Abraham Robinson in the 1960s, whereas in reality the work on non-Archimedean systems continued unabated, as documented by P. Ehrlich ;

In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product X × G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equipped with

In mathematics, a Ramsey cardinal is a certain kind of large cardinal number introduced by and named after Frank P. Ramsey.