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# Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change. 

## A Mathematician's Apology

A Mathematician's Apology is a 1940 essay by British mathematician G. H. Hardy.

## Abel Prize

The Abel Prize (Abelprisen) is a Norwegian prize awarded annually by the Government of Norway to one or more outstanding mathematicians.

## Abstract algebra

In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures.

## Abstraction

Abstraction in its main sense is a conceptual process where general rules and concepts are derived from the usage and classification of specific examples, literal ("real" or "concrete") signifiers, first principles, or other methods.

## Abstraction (mathematics)

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.

Addition (often signified by the plus symbol "+") is one of the four basic operations of arithmetic; the others are subtraction, multiplication and division.

## Aesthetics

Aesthetics (also spelled esthetics) is a branch of philosophy that explores the nature of art, beauty, and taste, with the creation and appreciation of beauty.

## Albert Einstein

Albert Einstein (14 March 1879 – 18 April 1955) was a German-born theoretical physicist who developed the theory of relativity, one of the two pillars of modern physics (alongside quantum mechanics).

## Aleph number

In mathematics, and in particular set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered.

Alfred North Whitehead (15 February 1861 – 30 December 1947) was an English mathematician and philosopher.

## Algebra

Algebra (from Arabic "al-jabr", literally meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis.

## Algebraic geometry

Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials.

## Algebraic topology

Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces.

## Algorithm

In mathematics and computer science, an algorithm is an unambiguous specification of how to solve a class of problems.

## Analytic geometry

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

## Ancient Egypt

Ancient Egypt was a civilization of ancient Northeastern Africa, concentrated along the lower reaches of the Nile River - geographically Lower Egypt and Upper Egypt, in the place that is now occupied by the countries of Egypt and Sudan.

## Ancient Greece

Ancient Greece was a civilization belonging to a period of Greek history from the Greek Dark Ages of the 13th–9th centuries BC to the end of antiquity (AD 600).

## Ancient Greek

The Ancient Greek language includes the forms of Greek used in ancient Greece and the ancient world from around the 9th century BC to the 6th century AD.

## Anno Domini

The terms anno Domini (AD) and before Christ (BC) are used to label or number years in the Julian and Gregorian calendars.

## Applied mathematics

Applied mathematics is the application of mathematical methods by different fields such as science, engineering, business, computer science, and industry.

## Approximation

An approximation is anything that is similar but not exactly equal to something else.

## Approximation theory

In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby.

## Aristotle

Aristotle (Ἀριστοτέλης Aristotélēs,; 384–322 BC) was an ancient Greek philosopher and scientist born in the city of Stagira, Chalkidiki, in the north of Classical Greece.

## Arithmetic

Arithmetic (from the Greek ἀριθμός arithmos, "number") is a branch of mathematics that consists of the study of numbers, especially the properties of the traditional operations on them—addition, subtraction, multiplication and division.

## Astronomy

Astronomy (from ἀστρονομία) is a natural science that studies celestial objects and phenomena.

## Augustine of Hippo

Saint Augustine of Hippo (13 November 354 – 28 August 430) was a Roman African, early Christian theologian and philosopher from Numidia whose writings influenced the development of Western Christianity and Western philosophy.

## Axiom

An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.

## Axiomatic system

In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems.

## Babylonia

Babylonia was an ancient Akkadian-speaking state and cultural area based in central-southern Mesopotamia (present-day Iraq).

## Babylonian mathematics

Babylonian mathematics (also known as Assyro-Babylonian mathematics) was any mathematics developed or practiced by the people of Mesopotamia, from the days of the early Sumerians to the fall of Babylon in 539 BC.

## Baconian method

The Baconian method is the investigative method developed by Sir Francis Bacon.

## Barbara Oakley

Barbara Ann Oakley (née Grim, November 24, 1955) is a Professor of Engineering at Oakland University.

BBC Radio 4 is a radio station owned and operated by the British Broadcasting Corporation (BBC) that broadcasts a wide variety of spoken-word programmes including news, drama, comedy, science and history.

## Benjamin Peirce

Benjamin Peirce FRSFor HFRSE April 4, 1809 – October 6, 1880) was an American mathematician who taught at Harvard University for approximately 50 years. He made contributions to celestial mechanics, statistics, number theory, algebra, and the philosophy of mathematics.

## Bertrand Russell

Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 &ndash; 2 February 1970) was a British philosopher, logician, mathematician, historian, writer, social critic, political activist, and Nobel laureate.

## Binary relation

In mathematics, a binary relation on a set A is a set of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2.

## Biology

Biology is the natural science that studies life and living organisms, including their physical structure, chemical composition, function, development and evolution.

## Brouwer–Hilbert controversy

In a foundational controversy in twentieth-century mathematics, L. E. J. Brouwer, a supporter of intuitionism, opposed David Hilbert, the founder of formalism.

## Bulletin of the American Mathematical Society

The Bulletin of the American Mathematical Society is a quarterly mathematical journal published by the American Mathematical Society.

## C. R. Rao

Calyampudi Radhakrishna Rao, FRS known as C R Rao (born 10 September 1920) is an Indian-American mathematician and statistician.

## Calculation

A calculation is a deliberate process that transforms one or more inputs into one or more results, with variable change.

## Calculus

Calculus (from Latin calculus, literally 'small pebble', used for counting and calculations, as on an abacus), is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.

## Cardinal number

In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets.

## Carl Benjamin Boyer

Carl Benjamin Boyer (November 3, 1906 – April 26, 1976) was an American historian of sciences, and especially mathematics.

## Carl Friedrich Gauss

Johann Carl Friedrich Gauss (Gauß; Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields, including algebra, analysis, astronomy, differential geometry, electrostatics, geodesy, geophysics, magnetic fields, matrix theory, mechanics, number theory, optics and statistics.

## Category theory

Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms).

## Chaos theory

Chaos theory is a branch of mathematics focusing on the behavior of dynamical systems that are highly sensitive to initial conditions.

## Chemistry

Chemistry is the scientific discipline involved with compounds composed of atoms, i.e. elements, and molecules, i.e. combinations of atoms: their composition, structure, properties, behavior and the changes they undergo during a reaction with other compounds.

## Chern Medal

The Chern Medal is an international award recognizing outstanding lifelong achievement of the highest level in the field of mathematics.

## Cicero

Marcus Tullius Cicero (3 January 106 BC – 7 December 43 BC) was a Roman statesman, orator, lawyer and philosopher, who served as consul in the year 63 BC.

## Combinatorics

Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures.

## Communications on Pure and Applied Mathematics

Communications on Pure and Applied Mathematics is a monthly peer-reviewed scientific journal which is published by John Wiley & Sons on behalf of the Courant Institute of Mathematical Sciences.

## Compass-and-straightedge construction

Compass-and-straightedge construction, also known as ruler-and-compass construction or classical construction, is the construction of lengths, angles, and other geometric figures using only an idealized ruler and compass.

## Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers.

## Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

## Computability theory

Computability theory, also known as recursion theory, is a branch of mathematical logic, of computer science, and of the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees.

## Computational complexity theory

Computational complexity theory is a branch of the theory of computation in theoretical computer science that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other.

## Computational geometry

Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry.

## Computational mathematics

Computational mathematics may refer to two different aspect of the relation between computing and mathematics.

## Computer algebra

In computational mathematics, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions and other mathematical objects.

## Computer science

Computer science deals with the theoretical foundations of information and computation, together with practical techniques for the implementation and application of these foundations.

## Computer-assisted proof

A computer-assisted proof is a mathematical proof that has been at least partially generated by computer.

## Conjecture

In mathematics, a conjecture is a conclusion or proposition based on incomplete information, for which no proof has been found.

## Continuous function

In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.

## Control theory

Control theory in control systems engineering deals with the control of continuously operating dynamical systems in engineered processes and machines.

## Controversy over Cantor's theory

In mathematical logic, the theory of infinite sets was first developed by Georg Cantor.

## Convex geometry

In mathematics, convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space.

## Convex optimization

Convex optimization is a subfield of optimization that studies the problem of minimizing convex functions over convex sets.

## Cost

In production, research, retail, and accounting, a cost is the value of money that has been used up to produce something or deliver a service, and hence is not available for use anymore.

## Counting

Counting is the action of finding the number of elements of a finite set of objects.

## Cryptography

Cryptography or cryptology (from κρυπτός|translit.

## Data compression

In signal processing, data compression, source coding, or bit-rate reduction involves encoding information using fewer bits than the original representation.

## David Hilbert

David Hilbert (23 January 1862 – 14 February 1943) was a German mathematician.

## Decision theory

Decision theory (or the theory of choice) is the study of the reasoning underlying an agent's choices.

## Deductive reasoning

Deductive reasoning, also deductive logic, logical deduction is the process of reasoning from one or more statements (premises) to reach a logically certain conclusion.

## Definition

A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols).

## Definitions of mathematics

Mathematics has no generally accepted definition.

## Design of experiments

The design of experiments (DOE, DOX, or experimental design) is the design of any task that aims to describe or explain the variation of information under conditions that are hypothesized to reflect the variation.

## Deterministic system

In mathematics, computer science and physics, a deterministic system is a system in which no randomness is involved in the development of future states of the system.

## Differential equation

A differential equation is a mathematical equation that relates some function with its derivatives.

## Differential geometry

Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry.

## Differential topology

In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds.

## Diophantus

Diophantus of Alexandria (Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 201 and 215; died around 84 years old, probably sometime between AD 285 and 299) was an Alexandrian Hellenistic mathematician, who was the author of a series of books called Arithmetica, many of which are now lost.

## Discrete geometry

Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects.

## Discretization

In mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts.

## Division (mathematics)

Division is one of the four basic operations of arithmetic, the others being addition, subtraction, and multiplication.

## Dover Publications

Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward Cirker and his wife, Blanche.

## Dynamical system

In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in a geometrical space.

## Elementary arithmetic

Elementary arithmetic is the simplified portion of arithmetic that includes the operations of addition, subtraction, multiplication, and division.

## Encyclopedia of Mathematics

The Encyclopedia of Mathematics (also EOM and formerly Encyclopaedia of Mathematics) is a large reference work in mathematics.

## Entropy (information theory)

Information entropy is the average rate at which information is produced by a stochastic source of data.

## Estimation theory

Estimation theory is a branch of statistics that deals with estimating the values of parameters based on measured empirical data that has a random component.

## Euclid

Euclid (Εὐκλείδης Eukleidēs; fl. 300 BC), sometimes given the name Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the "founder of geometry" or the "father of geometry".

## Euclid's Elements

The Elements (Στοιχεῖα Stoicheia) is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt c. 300 BC.

## Euclidean geometry

Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements.

## Euclidean vector

In mathematics, physics, and engineering, a Euclidean vector (sometimes called a geometric or spatial vector, or—as here—simply a vector) is a geometric object that has magnitude (or length) and direction.

Eugene Paul "E.

## Expected loss

Expected loss is the sum of the values of all possible losses, each multiplied by the probability of that loss occurring.

## Experimental mathematics

Experimental mathematics is an approach to mathematics in which computation is used to investigate mathematical objects and identify properties and patterns.

## Falsifiability

A statement, hypothesis, or theory has falsifiability (or is falsifiable) if it can logically be proven false by contradicting it with a basic statement.

## Fast Fourier transform

A fast Fourier transform (FFT) is an algorithm that samples a signal over a period of time (or space) and divides it into its frequency components.

## Fermat's Last Theorem

In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers,, and satisfy the equation for any integer value of greater than 2.

## Fiber bundle

In mathematics, and particularly topology, a fiber bundle (or, in British English, fibre bundle) is a space that is locally a product space, but globally may have a different topological structure.

## Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.

## Fields Medal

The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years.

## First principle

A first principle is a basic, foundational, self-evident proposition or assumption that cannot be deduced from any other proposition or assumption.

## Fluid dynamics

In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids - liquids and gases.

## Formal system

A formal system is the name of a logic system usually defined in the mathematical way.

## Formalism (philosophy of mathematics)

In foundations of mathematics, philosophy of mathematics, and philosophy of logic, formalism is a theory that holds that statements of mathematics and logic can be considered to be statements about the consequences of certain string manipulation rules.

## Foundations of mathematics

Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathematics.

## Four color theorem

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.

## Fractal

In mathematics, a fractal is an abstract object used to describe and simulate naturally occurring objects.

## Fraction (mathematics)

A fraction (from Latin fractus, "broken") represents a part of a whole or, more generally, any number of equal parts.

## François Viète

François Viète (Franciscus Vieta; 1540 – 23 February 1603), Seigneur de la Bigotière, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to its innovative use of letters as parameters in equations.

## Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

## Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense.

## Fundamental interaction

In physics, the fundamental interactions, also known as fundamental forces, are the interactions that do not appear to be reducible to more basic interactions.

## Fundamental theorem of algebra

The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.

## G. H. Hardy

Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis.

## Galileo Galilei

Galileo Galilei (15 February 1564Drake (1978, p. 1). The date of Galileo's birth is given according to the Julian calendar, which was then in force throughout Christendom. In 1582 it was replaced in Italy and several other Catholic countries with the Gregorian calendar. Unless otherwise indicated, dates in this article are given according to the Gregorian calendar. – 8 January 1642) was an Italian polymath.

## Galois group

In mathematics, more specifically in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension.

## Galois theory

In the field of algebra within mathematics, Galois theory, provides a connection between field theory and group theory.

## Game theory

Game theory is "the study of mathematical models of conflict and cooperation between intelligent rational decision-makers".

## Gödel's incompleteness theorems

Gödel's incompleteness theorems are two theorems of mathematical logic that demonstrate the inherent limitations of every formal axiomatic system containing basic arithmetic.

## General relativity

General relativity (GR, also known as the general theory of relativity or GTR) is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics.

## General topology

In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology.

## Geometry

Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.

## Geometry of numbers

In number theory, the geometry of numbers studies convex bodies and integer vectors in n-dimensional space.

## Giuseppe Peano

Giuseppe Peano (27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist.

## Goldbach's conjecture

Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics.

## Gottfried Wilhelm Leibniz

Gottfried Wilhelm (von) Leibniz (or; Leibnitz; – 14 November 1716) was a German polymath and philosopher who occupies a prominent place in the history of mathematics and the history of philosophy.

## Graph theory

In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.

## Greek mathematics

Greek mathematics refers to mathematics texts and advances written in Greek, developed from the 7th century BC to the 4th century AD around the shores of the Eastern Mediterranean.

## Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.

## Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.

Haskell Brooks Curry (September 12, 1900 – September 1, 1982) was an American mathematician and logician.

## Herbert Robbins

Herbert Ellis Robbins (January 12, 1915 – February 12, 2001) was an American mathematician and statistician.

## Hilbert's problems

Hilbert's problems are twenty-three problems in mathematics published by German mathematician David Hilbert in 1900.

## Hilbert's program

In mathematics, Hilbert's program, formulated by German mathematician David Hilbert in the early part of the 20th century, was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathematics were found to suffer from paradoxes and inconsistencies.

## History of mathematics

The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past.

## Hodge conjecture

In mathematics, the Hodge conjecture is a major unsolved problem in the field of algebraic geometry that relates the algebraic topology of a non-singular complex algebraic variety and the subvarieties of it.

## Homeomorphism

In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function.

## Homotopy

In topology, two continuous functions from one topological space to another are called homotopic (from Greek ὁμός homós "same, similar" and τόπος tópos "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions.

## Hypothesis

A hypothesis (plural hypotheses) is a proposed explanation for a phenomenon.

## If and only if

In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.

## Imre Lakatos

Imre Lakatos (Lakatos Imre; November 9, 1922 – February 2, 1974) was a Hungarian philosopher of mathematics and science, known for his thesis of the fallibility of mathematics and its 'methodology of proofs and refutations' in its pre-axiomatic stages of development, and also for introducing the concept of the 'research programme' in his methodology of scientific research programmes.

## Independence (mathematical logic)

In mathematical logic, independence refers to the unprovability of a sentence from other sentences.

## Infinity

Infinity (symbol) is a concept describing something without any bound or larger than any natural number.

## Information theory

Information theory studies the quantification, storage, and communication of information.

## Integer

An integer (from the Latin ''integer'' meaning "whole")Integer&#x2009;'s first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

## Integral

In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.

## Intuition

Intuition is the ability to acquire knowledge without proof, evidence, or conscious reasoning, or without understanding how the knowledge was acquired.

## Intuitionism

In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality.

## Iris Runge

Iris Anna Runge (1 June 1888 – 27 January 1966) was a German applied mathematician and physicist.

## Isaac Newton

Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, astronomer, theologian, author and physicist (described in his own day as a "natural philosopher") who is widely recognised as one of the most influential scientists of all time, and a key figure in the scientific revolution.

## Islamic Golden Age

The Islamic Golden Age is the era in the history of Islam, traditionally dated from the 8th century to the 14th century, during which much of the historically Islamic world was ruled by various caliphates, and science, economic development and cultural works flourished.

## Jan Gullberg

Jan Gullberg (1936 – 21 May 1998) was a Swedish surgeon and anaesthesiologist, but became known as a writer on popular science and medical topics.

## Karl Popper

Sir Karl Raimund Popper (28 July 1902 – 17 September 1994) was an Austrian-British philosopher and professor.

## Keith Devlin

Keith J. Devlin (born 16 March 1947) is a British mathematician and popular science writer.

## Kepler conjecture

The Kepler conjecture, named after the 17th-century mathematician and astronomer Johannes Kepler, is a mathematical theorem about sphere packing in three-dimensional Euclidean space.

## L. E. J. Brouwer

Luitzen Egbertus Jan Brouwer (27 February 1881 – 2 December 1966), usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician and philosopher, who worked in topology, set theory, measure theory and complex analysis.

## Langlands program

In mathematics, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry.

## Language of mathematics

The language of mathematics is the system used by mathematicians to communicate mathematical ideas among themselves.

## Leonhard Euler

Leonhard Euler (Swiss Standard German:; German Standard German:; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, logician and engineer, who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while also making pioneering contributions to several branches such as topology and analytic number theory.

## Liberal arts education

Liberal arts education (from Latin "free" and "art or principled practice") can claim to be the oldest programme of higher education in Western history.

## Lie group

In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.

## Linear algebra

Linear algebra is the branch of mathematics concerning linear equations such as linear functions such as and their representations through matrices and vector spaces.

## List of mathematical jargon

The language of mathematics has a vast vocabulary of specialist and technical terms.

## Logic

Logic (from the logikḗ), originally meaning "the word" or "what is spoken", but coming to mean "thought" or "reason", is a subject concerned with the most general laws of truth, and is now generally held to consist of the systematic study of the form of valid inference.

## Logicism

Logicism is one of the schools of thought in the philosophy of mathematics, putting forth the theory that mathematics is an extension of logic and therefore some or all mathematics is reducible to logic.

## Loss function

In mathematical optimization, statistics, econometrics, decision theory, machine learning and computational neuroscience, a loss function or cost function is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost" associated with the event.

## Lynn Steen

Lynn Arthur Steen (January 1, 1941 – June 21, 2015) was an American mathematician who was a Professor of Mathematics at St. Olaf College, Northfield, Minnesota in the U.S. He wrote numerous books and articles on the teaching of mathematics.

## MacTutor History of Mathematics archive

The MacTutor History of Mathematics archive is a website maintained by John J. O'Connor and Edmund F. Robertson and hosted by the University of St Andrews in Scotland.

## Manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.

## Marcus du Sautoy

Marcus Peter Francis du Sautoy (born 26 August 1965) is a British mathematician, author, and populariser of science and mathematics.

## Mathematical analysis

Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.

## Mathematical and theoretical biology

Mathematical and theoretical biology is a branch of biology which employs theoretical analysis, mathematical models and abstractions of the living organisms to investigate the principles that govern the structure, development and behavior of the systems, as opposed to experimental biology which deals with the conduction of experiments to prove and validate the scientific theories.

## Mathematical chemistry

Mathematical chemistry is the area of research engaged in novel applications of mathematics to chemistry; it concerns itself principally with the mathematical modeling of chemical phenomena.

## Mathematical economics

Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics.

## Mathematical fallacy

In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept of mathematical fallacy.

## Mathematical finance

Mathematical finance, also known as quantitative finance, is a field of applied mathematics, concerned with mathematical modeling of financial markets.

## Mathematical logic

Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics.

## Mathematical optimization

In mathematics, computer science and operations research, mathematical optimization or mathematical programming, alternatively spelled optimisation, is the selection of a best element (with regard to some criterion) from some set of available alternatives.

## Mathematical physics

Mathematical physics refers to the development of mathematical methods for application to problems in physics.

## Mathematical problem

A mathematical problem is a problem that is amenable to being represented, analyzed, and possibly solved, with the methods of mathematics.

## Mathematical proof

In mathematics, a proof is an inferential argument for a mathematical statement.

## Mathematical Reviews

Mathematical Reviews is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science.

## Mathematical sciences

The mathematical sciences are a group of areas of study that includes, in addition to mathematics, those academic disciplines that are primarily mathematical in nature but may not be universally considered subfields of mathematics proper.

## Mathematical statistics

Mathematical statistics is the application of mathematics to statistics, as opposed to techniques for collecting statistical data.

## Mathematical structure

In mathematics, a structure on a set is an additional mathematical object that, in some manner, attaches (or relates) to that set to endow it with some additional meaning or significance.

## Mathematics and art

Mathematics and art are related in a variety of ways.

## Mathematics education

In contemporary education, mathematics education is the practice of teaching and learning mathematics, along with the associated scholarly research.

## Mathematics Subject Classification

The Mathematics Subject Classification (MSC) is an alphanumerical classification scheme collaboratively produced by staff of, and based on the coverage of, the two major mathematical reviewing databases, Mathematical Reviews and Zentralblatt MATH.

## Measure (mathematics)

In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size.

## Measurement

Measurement is the assignment of a number to a characteristic of an object or event, which can be compared with other objects or events.

## Metaphysics

Metaphysics is a branch of philosophy that explores the nature of being, existence, and reality.

## Metrization theorem

In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space.

## Michiel Hazewinkel

Michiel Hazewinkel (born 22 June 1943) is a Dutch mathematician, and Emeritus Professor of Mathematics at the Centre for Mathematics and Computer and the University of Amsterdam, particularly known for his 1978 book Formal groups and applications and as editor of the Encyclopedia of Mathematics.

## Middle Kingdom of Egypt

The Middle Kingdom of Egypt (also known as The Period of Reunification) is the period in the history of ancient Egypt between circa 2050 BC and 1710 BC, stretching from the reunification of Egypt under the impulse of Mentuhotep II of the Eleventh Dynasty to the end of the Twelfth Dynasty.

## Millennium Prize Problems

The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000.

## Model selection

Model selection is the task of selecting a statistical model from a set of candidate models, given data.

## Model theory

In mathematics, model theory is the study of classes of mathematical structures (e.g. groups, fields, graphs, universes of set theory) from the perspective of mathematical logic.

## Morris Kline

Morris Kline (May 1, 1908 – June 10, 1992) was a Professor of Mathematics, a writer on the history, philosophy, and teaching of mathematics, and also a popularizer of mathematical subjects.

## Morse theory

"Morse function" redirects here.

## Motion (physics)

In physics, motion is a change in position of an object over time.

There is some confusion in the literature on whether al-Khwārizmī's full name is ابو عبد الله محمد بن موسى الخوارزمي or ابو جعفر محمد بن موسی الخوارزمی.

## Multiplication

Multiplication (often denoted by the cross symbol "×", by a point "⋅", by juxtaposition, or, on computers, by an asterisk "∗") is one of the four elementary mathematical operations of arithmetic; with the others being addition, subtraction and division.

## National Museum of Mathematics

The National Museum of Mathematics or MoMath is a museum dedicated to mathematics in Manhattan, New York City.

## Natural number

In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country").

## Natural science

Natural science is a branch of science concerned with the description, prediction, and understanding of natural phenomena, based on empirical evidence from observation and experimentation.

## Non-Euclidean geometry

In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry.

## Number theory

Number theory, or in older usage arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers.

## Numeracy

Numeracy is the ability to reason and to apply simple numerical concepts.

## Numeral system

A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner.

## Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics).

## Numerical linear algebra

Numerical linear algebra is the study of algorithms for performing linear algebra computations, most notably matrix operations, on computers.

## Numerical method

In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems.

## Observational study

In fields such as epidemiology, social sciences, psychology and statistics, an observational study draws inferences from a sample to a population where the independent variable is not under the control of the researcher because of ethical concerns or logistical constraints.

## Octonion

In mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface O or blackboard bold \mathbb O. There are three lower-dimensional normed division algebras over the reals: the real numbers R themselves, the complex numbers C, and the quaternions H. The octonions have eight dimensions; twice the number of dimensions of the quaternions, of which they are an extension.

## Omar Khayyam

Omar Khayyam (عمر خیّام; 18 May 1048 – 4 December 1131) was a Persian mathematician, astronomer, and poet.

## Online Etymology Dictionary

The Online Etymology Dictionary is a free online dictionary written and compiled by Douglas Harper that describes the origins of English-language words.

## Open problem

In science and mathematics, an open problem or an open question is a known problem which can be accurately stated, and which is assumed to have an objective and verifiable solution, but which has not yet been solved (no solution for it is known).

## Open set

In topology, an open set is an abstract concept generalizing the idea of an open interval in the real line.

## Operation (mathematics)

In mathematics, an operation is a calculation from zero or more input values (called operands) to an output value.

## Operations research

Operations research, or operational research in British usage, is a discipline that deals with the application of advanced analytical methods to help make better decisions.

## Order theory

Order theory is a branch of mathematics which investigates the intuitive notion of order using binary relations.

## Organon

The Organon (Greek: Ὄργανον, meaning "instrument, tool, organ") is the standard collection of Aristotle's six works on logic.

## Outline of mathematics

Mathematics is a field of study that investigates topics including number, space, structure, and change.

## Oxford English Dictionary

The Oxford English Dictionary (OED) is the main historical dictionary of the English language, published by the Oxford University Press.

## Oxford University Press

Oxford University Press (OUP) is the largest university press in the world, and the second oldest after Cambridge University Press.

## P versus NP problem

The P versus NP problem is a major unsolved problem in computer science.

## Path integral formulation

The path integral formulation of quantum mechanics is a description of quantum theory that generalizes the action principle of classical mechanics.

## Pattern

A pattern is a discernible regularity in the world or in a manmade design.

## Paul Erdős

Paul Erdős (Erdős Pál; 26 March 1913 – 20 September 1996) was a Hungarian mathematician.

## Philip J. Davis

Philip J. Davis (January 2, 1923 – March 13, 2018) was an American academic applied mathematician.

## Philosophy of mathematics

The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics, and purports to provide a viewpoint of the nature and methodology of mathematics, and to understand the place of mathematics in people's lives.

## Physicist

A physicist is a scientist who has specialized knowledge in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe.

## Physics

Physics (from knowledge of nature, from φύσις phýsis "nature") is the natural science that studies matterAt the start of The Feynman Lectures on Physics, Richard Feynman offers the atomic hypothesis as the single most prolific scientific concept: "If, in some cataclysm, all scientific knowledge were to be destroyed one sentence what statement would contain the most information in the fewest words? I believe it is that all things are made up of atoms – little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another..." and its motion and behavior through space and time and that studies the related entities of energy and force."Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves."Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics. (...) You will come to see physics as a towering achievement of the human intellect in its quest to understand our world and ourselves."Physics is an experimental science. Physicists observe the phenomena of nature and try to find patterns that relate these phenomena.""Physics is the study of your world and the world and universe around you." Physics is one of the oldest academic disciplines and, through its inclusion of astronomy, perhaps the oldest. Over the last two millennia, physics, chemistry, biology, and certain branches of mathematics were a part of natural philosophy, but during the scientific revolution in the 17th century, these natural sciences emerged as unique research endeavors in their own right. Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the fundamental mechanisms studied by other sciences and suggest new avenues of research in academic disciplines such as mathematics and philosophy. Advances in physics often enable advances in new technologies. For example, advances in the understanding of electromagnetism and nuclear physics led directly to the development of new products that have dramatically transformed modern-day society, such as television, computers, domestic appliances, and nuclear weapons; advances in thermodynamics led to the development of industrialization; and advances in mechanics inspired the development of calculus.

## Poincaré conjecture

In mathematics, the Poincaré conjecture is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space.

## Polynomial

In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

## Prehistory

Human prehistory is the period between the use of the first stone tools 3.3 million years ago by hominins and the invention of writing systems.

## Prime number

A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

## Principia Mathematica

The Principia Mathematica (often abbreviated PM) is a three-volume work on the foundations of mathematics written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913.

## Probability theory

Probability theory is the branch of mathematics concerned with probability.

## Projective geometry

Projective geometry is a topic in mathematics.

## Proof theory

Proof theory is a major branchAccording to Wang (1981), pp.

## Proofs and Refutations

Proofs and Refutations is a 1976 book by philosopher Imre Lakatos expounding his view of the progress of mathematics.

## Pure mathematics

Broadly speaking, pure mathematics is mathematics that studies entirely abstract concepts.

## Pythagorean theorem

In mathematics, the Pythagorean theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle.

## Pythagoreanism

Pythagoreanism originated in the 6th century BC, based on the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans, who were considerably influenced by mathematics and mysticism.

## Quantity

Quantity is a property that can exist as a multitude or magnitude.

## Quantum mechanics

Quantum mechanics (QM; also known as quantum physics, quantum theory, the wave mechanical model, or matrix mechanics), including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles.

## Quaternion

In mathematics, the quaternions are a number system that extends the complex numbers.

## Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.

## Real analysis

In mathematics, real analysis is the branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real-valued functions.

## Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

## Recreational mathematics

Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research and application-based professional activity.

## Relationship between mathematics and physics

The relationship between mathematics and physics has been a subject of study of philosophers, mathematicians and physicists since Antiquity, and more recently also by historians and educators.

## Renaissance

The Renaissance is a period in European history, covering the span between the 14th and 17th centuries.

## Reuben Hersh

Reuben Hersh (born 1927) is an American mathematician and academic, best known for his writings on the nature, practice, and social impact of mathematics.

## Rhind Mathematical Papyrus

The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum 10057 and pBM 10058) is one of the best known examples of Egyptian mathematics.

## Richard Courant

Richard Courant (January 8, 1888 – January 27, 1972) was a German American mathematician.

## Richard Feynman

Richard Phillips Feynman (May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics for which he proposed the parton model.

## Riemann hypothesis

In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part.

## Riemann surface

In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold.

## Rigour

Rigour (British English) or rigor (American English; see spelling differences) describes a condition of stiffness or strictness.

## Ring (mathematics)

In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.

## Risk

Risk is the potential of gaining or losing something of value.

## Round-off error

A round-off error, also called rounding error, is the difference between the calculated approximation of a number and its exact mathematical value due to rounding.

## Scholasticism

Scholasticism is a method of critical thought which dominated teaching by the academics ("scholastics", or "schoolmen") of medieval universities in Europe from about 1100 to 1700, and a program of employing that method in articulating and defending dogma in an increasingly pluralistic context.

## Science (journal)

Science, also widely referred to as Science Magazine, is the peer-reviewed academic journal of the American Association for the Advancement of Science (AAAS) and one of the world's top academic journals.

## Science, technology, engineering, and mathematics

Science, Technology, Engineering and Mathematics (STEM), previously Science, Math, Engineering, and Technology (SMET), is a term used to group together these academic disciplines.

## Selection algorithm

In computer science, a selection algorithm is an algorithm for finding the kth smallest number in a list or array; such a number is called the kth order statistic.

## Set (mathematics)

In mathematics, a set is a collection of distinct objects, considered as an object in its own right.

## Set theory

Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects.

## Set-theoretic topology

In mathematics, set-theoretic topology is a subject that combines set theory and general topology.

## Shape

A shape is the form of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture or material composition.

## Sharaf al-Dīn al-Ṭūsī

(c. 1135 – c. 1213) was an Iranian mathematician and astronomer of the Islamic Golden Age (during the Middle Ages).

## Simple random sample

In statistics, a simple random sample is a subset of individuals (a sample) chosen from a larger set (a population).

## Simplicity

Simplicity is the state or quality of being simple.

## Social science

Social science is a major category of academic disciplines, concerned with society and the relationships among individuals within a society.

## Space

Space is the boundless three-dimensional extent in which objects and events have relative position and direction.

Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.

## Statistical hypothesis testing

A statistical hypothesis, sometimes called confirmatory data analysis, is a hypothesis that is testable on the basis of observing a process that is modeled via a set of random variables.

## Statistical inference

Statistical inference is the process of using data analysis to deduce properties of an underlying probability distribution.

## Statistical model

A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of some sample data and similar data from a larger population.

## Statistical theory

The theory of statistics provides a basis for the whole range of techniques, in both study design and data analysis, that are used within applications of statistics.

## Statistics

Statistics is a branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data.

## String theory

In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings.

## Subset

In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. A and B may coincide.

## Subtraction

Subtraction is an arithmetic operation that represents the operation of removing objects from a collection.

## Surveying

Surveying or land surveying is the technique, profession, and science of determining the terrestrial or three-dimensional positions of points and the distances and angles between them.

## Tally stick

A tally stick (or simply tally) was an ancient memory aid device used to record and document numbers, quantities, or even messages.

## Tensor calculus

In mathematics, tensor calculus or tensor analysis is an extension of vector calculus to tensor fields (tensors that may vary over a manifold, e.g. in spacetime).

## TeX

TeX (see below), stylized within the system as TeX, is a typesetting system (or "formatting system") designed and mostly written by Donald Knuth and released in 1978.

## The Mathematical Experience

The Mathematical Experience (1981) is a book by Philip J. Davis and Reuben Hersh that discusses the practice of modern mathematics from a historical and philosophical perspective.

## The Oxford Dictionary of English Etymology

The Oxford Dictionary of English Etymology is an etymological dictionary of the English language, published by Oxford University Press.

## The Unreasonable Effectiveness of Mathematics in the Natural Sciences

"The Unreasonable Effectiveness of Mathematics in the Natural Sciences" is the title of an article published in 1960 by the physicist Eugene Wigner.

## Theorem

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

## Theoretical computer science

Theoretical computer science, or TCS, is a subset of general computer science and mathematics that focuses on more mathematical topics of computing and includes the theory of computation.

## Theoretical physics

Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena.

## Theory of computation

In theoretical computer science and mathematics, the theory of computation is the branch that deals with how efficiently problems can be solved on a model of computation, using an algorithm.

## Timeline of scientific discoveries

The timeline below shows the date of publication of possible major scientific theories and discoveries, along with the discoverer.

## Topological group

In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology.

## Topology

In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.

## Transfinite number

Transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite.

## Trigonometry

Trigonometry (from Greek trigōnon, "triangle" and metron, "measure") is a branch of mathematics that studies relationships involving lengths and angles of triangles.

## Truth

Truth is most often used to mean being in accord with fact or reality, or fidelity to an original or standard.

## Turing machine

A Turing machine is a mathematical model of computation that defines an abstract machine, which manipulates symbols on a strip of tape according to a table of rules.

## Twin prime

A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43).

## Uncertainty

Uncertainty has been called "an unintelligible expression without a straightforward description".

## University of Cambridge

The University of Cambridge (informally Cambridge University)The corporate title of the university is The Chancellor, Masters, and Scholars of the University of Cambridge.

## Uta Merzbach

Uta Caecilia Merzbach (February 9, 1933 – June 27, 2017) was a German-American historian of mathematics who became the first curator of mathematical instruments at the Smithsonian Institution.

## Vector calculus

Vector calculus, or vector analysis, is a branch of mathematics concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3.

## Vector space

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.

W.

## Weaving

Weaving is a method of textile production in which two distinct sets of yarns or threads are interlaced at right angles to form a fabric or cloth.

## Wiki

A wiki is a website on which users collaboratively modify content and structure directly from the web browser.

## Wolf Prize in Mathematics

The Wolf Prize in Mathematics is awarded almost annually by the Wolf Foundation in Israel.

## References

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