**Mathematics**is the study of quantity

, space

, structure

, and change

. Mathematician

s seek out patterns and formulate new conjecture

s. Mathematicians resolve the truth or falsity of conjectures by mathematical proof

s, which are arguments sufficient to convince other mathematicians of their validity. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. However, mathematical proofs are less formal and painstaking than proofs in mathematical logic

.

Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere.

If in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics.

Great fleas have little fleas upon their backs to bite 'em,And little fleas have lesser fleas, and so ad infinitum,And the great fleas themselves, in turn, have greater fleas to go on,While these again have greater still, and greater still, and so on.

Numbers exist only in our minds. There is no physical entity that is number 1. If there were, 1 would be in a place of honor in some great museum of science, and past it would file a steady stream of mathematicians gazing at 1 in wonder and awe.

Euler calculated the force of the wheels necessary to raise the water in a reservoir … My mill was carried out geometrically and could not raise a drop of water fifty yards from the reservoir. Vanity of vanities! Vanity of geometry!

**Mathematics**is the study of quantity

, space

, structure

, and change

. Mathematician

s seek out patterns and formulate new conjecture

s. Mathematicians resolve the truth or falsity of conjectures by mathematical proof

s, which are arguments sufficient to convince other mathematicians of their validity. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. However, mathematical proofs are less formal and painstaking than proofs in mathematical logic

. Since the pioneering work of Giuseppe Peano

(1858-1932), David Hilbert

(1862-1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth

by rigorous deduction

from appropriately chosen axiom

s and definition

s. When those mathematical structures are good models of real phenomena, then mathematical reasoning often provides insight or predictions.

Through the use of abstraction

and logic

al reasoning, mathematics developed from counting

, calculation

, measurement

, and the systematic study of the shape

s and motions

of physical objects. Practical mathematics has been a human activity for as far back as written records

exist. Rigorous arguments

first appeared in Greek mathematics

, most notably in Euclid's

*ElementsEuclid's ElementsEuclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...*

. Mathematics continued to develop, for example in China in 300 BC, in India in AD 100, and in the Muslim world

in AD 800, until the Renaissance

, when mathematical innovations interacting with new scientific discoveries

led to a rapid increase in the rate of mathematical discovery that continues to the present day.

The mathematician Benjamin Peirce

(1809-1880) called mathematics "the science that draws necessary conclusions". David Hilbert said of mathematics: "We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise." Albert Einstein

(1879-1955) stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality".

Mathematics is used throughout the world as an essential tool in many fields, including natural science

, engineering

, medicine

, and the social sciences

. Applied mathematics

, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as statistics

and game theory

. Mathematicians also engage in pure mathematics

, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered.

## Etymology

The word "mathematics" comes from the Greek μάθημα (*máthēma*), which means in ancient Greek

*what one learns*,

*what one gets to know*, hence also

*study*and

*science*, and in modern Greek just

*lesson*.

The word

*máthēma*comes from μανθάνω (

*manthano*) in ancient Greek and from μαθαίνω (

*mathaino*) in modern Greek, both of which mean

*to learn*.

The word "mathematics" in Greek came to have the narrower and more technical meaning "mathematical study", even in Classical times. Its adjective is (

*mathēmatikós*), meaning

*related to learning*or

*studious*, which likewise further came to mean

*mathematical*. In particular, (

*mathēmatikḗ tékhnē*), , meant

*the mathematical art*. In Latin, and in English until around 1700, the term "mathematics" more commonly meant "astrology" (or sometimes "astronomy") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This has resulted in several mistranslations: a particularly notorious one is Saint Augustine's warning that Christians should beware of "mathematici" meaning astrologers, which is sometimes mistranslated as a condemnation of mathematicians.

The apparent plural form in English, like the French plural form (and the less commonly used singular derivative ), goes back to the Latin neuter plural (Cicero

), based on the Greek plural , used by Aristotle

(384-322BC), and meaning roughly "all things mathematical"; although it is plausible that English borrowed only the adjective

*mathematic(al)*and formed the noun

*mathematics*anew, after the pattern of physics

and metaphysics

, which were inherited from the Greek. In English, the noun

*mathematics*takes singular verb forms. It is often shortened to

*maths*or, in English-speaking North America,

*math*.

## History

The evolution of mathematics might be seen as an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction, which is shared by many animals, was probably that of number

s: the realization that a collection of two apples and a collection of two oranges (for example) have something in common, namely quantity of their members.

In addition to recognizing how to count

*physical*objects, prehistoric

peoples also recognized how to count

*abstract*quantities, like time – days, season

s, years. Elementary arithmetic

(addition

, subtraction

, multiplication

and division

) naturally followed.

Since numeracy pre-dated writing

, further steps were needed for recording numbers such as tallies or the knotted strings called quipu

used by the Inca to store numerical data. Numeral system

s have been many and diverse, with the first known written numerals created by Egyptians

in Middle Kingdom

texts such as the Rhind Mathematical Papyrus

.

The earliest uses of mathematics were in trading

, land measurement

, painting

and weaving

patterns and the recording of time. More complex mathematics did not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra and geometry for taxation and other financial calculations, for building and construction, and for astronomy

. The systematic study of mathematics in its own right began with the Ancient Greeks between 600 and 300 BC.

Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science

, to the benefit of both. Mathematical discoveries continue to be made today. According to Mikhail B. Sevryuk, in the January 2006 issue of the

*Bulletin of the American Mathematical SocietyBulletin of the American Mathematical SocietyThe Bulletin of the American Mathematical Society is a quarterly mathematical journal published by the American Mathematical Society...*

, "The number of papers and books included in the Mathematical Reviews

database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorem

s and their proofs

."

## Inspiration, pure and applied mathematics, and aesthetics

Mathematics arises from many different kinds of problems. At first these were found in commerce, land measurement

, architecture

and later astronomy

; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. For example, the physicist

Richard Feynman

invented the path integral formulation

of quantum mechanics

using a combination of mathematical reasoning and physical insight, and today's string theory

, a still-developing scientific theory which attempts to unify the four fundamental forces of nature

, continues to inspire new mathematics. Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. A distinction is often made between pure mathematics

and applied mathematics

. However pure mathematics topics often turn out to have applications, e.g. number theory

in cryptography

. This remarkable fact that even the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics

".

As in most areas of study, the explosion of knowledge in the scientific age has led to specialization: there are now hundreds of specialized areas in mathematics and the latest Mathematics Subject Classification

runs to 46 pages. Several areas of applied mathematics have merged with related traditions outside of mathematics and become disciplines in their own right, including statistics

, operations research

, and computer science

.

For those who are mathematically inclined, there is often a definite aesthetic aspect to much of mathematics. Many mathematicians talk about the

*elegance*of mathematics, its intrinsic aesthetics

and inner beauty

. Simplicity

and generality are valued. There is beauty in a simple and elegant proof, such as Euclid

's proof that there are infinitely many prime number

s, and in an elegant numerical method that speeds calculation, such as the fast Fourier transform

. G. H. Hardy

in

*A Mathematician's ApologyA Mathematician's ApologyA Mathematician's Apology is a 1940 essay by British mathematician G. H. Hardy. It concerns the aesthetics of mathematics with some personal content, and gives the layman an insight into the mind of a working mathematician.-Summary:...*

expressed the belief that these aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. He identified criteria such as significance, unexpectedness, inevitability, and economy as factors that contribute to a mathematical aesthetic. Mathematicians often strive to find proofs that are particularly elegant, proofs from "The Book" of God according to Paul Erdős

. The popularity of recreational mathematics

is another sign of the pleasure many find in solving mathematical questions.

## Notation, language, and rigor

Most of the mathematical notation in use today was not invented until the 16th century. Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery. Euler(1707–1783) was responsible for many of the notations in use today. Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. It is extremely compressed: a few symbols contain a great deal of information. Like musical notation

, modern mathematical notation has a strict syntax (which to a limited extent varies from author to author and from discipline to discipline) and encodes information that would be difficult to write in any other way.

Mathematical language

can be difficult to understand for beginners. Words such as

*or*and

*only*have more precise meanings than in everyday speech. Moreover, words such as

*openOpen setThe concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...*

and

*fieldField (mathematics)In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...*

have been given specialized mathematical meanings. Technical terms such as

*homeomorphismHomeomorphismIn the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...*

and

*integrableIntegralIntegration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...*

have precise meanings in mathematics. Additionally, shorthand phrases such as "iff" for "if and only if" belong to mathematical jargon

. There is a reason for special notation and technical vocabulary: mathematics requires more precision than everyday speech. Mathematicians refer to this precision of language and logic as "rigor".

Mathematical proof

is fundamentally a matter of rigor. Mathematicians want their theorems to follow from axioms by means of systematic reasoning. This is to avoid mistaken "theorem

s", based on fallible intuitions, of which many instances have occurred in the history of the subject. The level of rigor expected in mathematics has varied over time: the Greeks expected detailed arguments, but at the time of Isaac Newton

the methods employed were less rigorous. Problems inherent in the definitions used by Newton would lead to a resurgence of careful analysis and formal proof in the 19th century. Misunderstanding the rigor is a cause for some of the common misconceptions of mathematics. Today, mathematicians continue to argue among themselves about computer-assisted proof

s. Since large computations are hard to verify, such proofs may not be sufficiently rigorous.

Axiom

s in traditional thought were "self-evident truths", but that conception is problematic. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system

. It was the goal of Hilbert's program

to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (sufficiently powerful) axiomatic system has undecidable

formulas; and so a final axiomatization of mathematics is impossible. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory

in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.

## Fields of mathematics

Mathematics can, broadly speaking, be subdivided into the study of quantity, structure, space, and change (i.e. arithmetic, algebra

, geometry

, and analysis

). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic

, to set theory

(foundations

), to the empirical mathematics of the various sciences (applied mathematics

), and more recently to the rigorous study of uncertainty

.

### Foundations and philosophy

In order to clarify the foundations of mathematics, the fields of mathematical logic

and set theory

were developed. Mathematical logic includes the mathematical study of logic

and the applications of formal logic to other areas of mathematics; set theory is the branch of mathematics that studies sets or collections of objects. Category theory

, which deals in an abstract way with mathematical structure

s and relationships between them, is still in development. The phrase "crisis of foundations" describes the search for a rigorous foundation for mathematics that took place from approximately 1900 to 1930. Some disagreement about the foundations of mathematics continues to the present day. The crisis of foundations was stimulated by a number of controversies at the time, including the controversy over Cantor's set theory

and the Brouwer-Hilbert controversy.

Mathematical logic is concerned with setting mathematics within a rigorous axiom

atic framework, and studying the implications of such a framework. As such, it is home to Gödel's incompleteness theorems

which (informally) imply that any formal system

that contains basic arithmetic, if

*sound*(meaning that all theorems that can be proven are true), is necessarily

*incomplete*(meaning that there are true theorems which cannot be proved

*in that system*). Whatever finite collection of number-theoretical axioms is taken as a foundation, Gödel showed how to construct a formal statement that is a true number-theoretical fact, but which does not follow from those axioms. Therefore no formal system is a complete axiomatization of full number theory. Modern logic is divided into recursion theory

, model theory

, and proof theory

, and is closely linked to theoretical

computer science

, as well as to Category Theory

.

Theoretical computer science

includes computability theory, computational complexity theory

, and information theory

. Computability theory examines the limitations of various theoretical models of the computer, including the most well known model – the Turing machine

. Complexity theory is the study of tractability by computer; some problems, although theoretically solvable by computer, are so expensive in terms of time or space that solving them is likely to remain practically unfeasible, even with rapid advance of computer hardware. A famous problem is the "P=NP?" problem, one of the Millennium Prize Problems

. Finally, information theory is concerned with the amount of data that can be stored on a given medium, and hence deals with concepts such as compression

and entropy.

Mathematical logic Mathematical logic Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics... |
Set theory Set theory Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics... |
Category theory Category theory 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 , where these collections satisfy certain basic conditions... |
Theory of computation Theory of computation In theoretical computer science, the theory of computation is the branch that deals with whether and how efficiently problems can be solved on a model of computation, using an algorithm... |

#### Quantity

The study of quantity starts with numbers, first the familiar natural number

s and integer

s ("whole numbers") and arithmetical operations on them, which are characterized in arithmetic

. The deeper properties of integers are studied in number theory

, from which come such popular results as Fermat's Last Theorem

. The twin prime

conjecture and Goldbach's conjecture

are two unsolved problems in number theory.

As the number system is further developed, the integers are recognized as a subset

of the rational number

s ("fractions

"). These, in turn, are contained within the real number

s, which are used to represent continuous

quantities. Real numbers are generalized to complex number

s. These are the first steps of a hierarchy of numbers that goes on to include quarternions and octonion

s. Consideration of the natural numbers also leads to the transfinite number

s, which formalize the concept of "infinity

". Another area of study is size, which leads to the cardinal number

s and then to another conception of infinity: the aleph number

s, which allow meaningful comparison of the size of infinitely large sets.

Natural number Natural number In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively... s |
Integer Integer The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively... s |
Rational number Rational number In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number... s |
Real number Real number In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π... s |
Complex number Complex number A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part... s |

#### Structure

Many mathematical objects, such as sets of numbers and functions, exhibit internal structure as a consequence of operations

or relations

that are defined on the set. Mathematics then studies properties of those sets that can be expressed in terms of that structure; for instance number theory

studies properties of the set of integer

s that can be expressed in terms of arithmetic

operations. Moreover, it frequently happens that different such structured sets (or structures

) exhibit similar properties, which makes it possible, by a further step of abstraction

, to state axiom

s for a class of structures, and then study at once the whole class of structures satisfying these axioms. Thus one can study groups

, rings

, fields

and other abstract systems; together such studies (for structures defined by algebraic operations) constitute the domain of abstract algebra

. By its great generality, abstract algebra can often be applied to seemingly unrelated problems; for instance a number of ancient problems concerning compass and straightedge constructions were finally solved using Galois theory

, which involves field theory and group theory. Another example of an algebraic theory is linear algebra

, which is the general study of vector space

s, whose elements called vectors have both quantity and direction, and can be used to model (relations between) points in space. This is one example of the phenomenon that the originally unrelated areas of geometry

and algebra

have very strong interactions in modern mathematics. Combinatorics

studies ways of enumerating the number of objects that fit a given structure.

Combinatorics Combinatorics Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,... |
Number theory Number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well... |
Group theory Group theory In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and... |
Graph theory Graph theory In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of... |
Order theory Order theory Order theory is a branch of mathematics which investigates our intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and gives some basic definitions... |

#### Space

The study of space originates with geometry– in particular, Euclidean geometry

. Trigonometry

is the branch of mathematics that deals with relationships between the sides and the angles of triangles and with the trigonometric functions; it combines space and numbers, and encompasses the well-known Pythagorean theorem

. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries

(which play a central role in general relativity

) and topology

. Quantity and space both play a role in analytic geometry

, differential geometry, and algebraic geometry

. Convex

and discrete geometry

was developed to solve problems in number theory

and functional analysis

but now is pursued with an eye on applications in optimization and computer science

. Within differential geometry are the concepts of fiber bundles and calculus on manifold

s, in particular, vector and tensor calculus. Within algebraic geometry is the description of geometric objects as solution sets of polynomial

equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie group

s are used to study space, structure, and change. Topology

in all its many ramifications may have been the greatest growth area in 20th century mathematics; it includes point-set topology, set-theoretic topology

, algebraic topology

and differential topology

. In particular, instances of modern day topology are metrizability theory, axiomatic set theory, homotopy theory, and Morse theory

. Topology also includes the now solved Poincaré conjecture

. Other results in geometry and topology, including the four color theorem

and Kepler conjecture

, have been proved only with the help of computers.

Geometry Geometry Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers .... |
Trigonometry Trigonometry Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves... |
Differential geometry | Topology Topology Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing... |
Fractal geometry Fractal A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity... |
Measure theory |

#### Change

Understanding and describing change is a common theme in the natural sciences, and calculus

was developed as a powerful tool to investigate it. Functions

arise here, as a central concept describing a changing quantity. The rigorous study of real number

s and functions of a real variable is known as real analysis

, with complex analysis

the equivalent field for the complex number

s. Functional analysis

focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics

. Many problems lead naturally to relationships between a quantity and its rate of change, and these are studied as differential equation

s. Many phenomena in nature can be described by dynamical system

s; chaos theory

makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic

behavior.

Calculus Calculus Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem... |
Vector calculus | Differential equation Differential equation A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders... s |
Dynamical system Dynamical system A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a... s |
Chaos theory Chaos theory Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the... |
Complex analysis 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. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,... |

### Applied mathematics

Applied mathematics concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge. The term "applied mathematics" also describes the professionalspecialty in which mathematicians work on practical problems; as a profession focused on practical problems,

*applied mathematics*focuses on the

*formulation, study, and use of mathematical models*in science

, engineering

, and other areas of mathematical practice.

In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics, where mathematics is developed primarily for its own sake. Thus, the activity of applied mathematics is vitally connected with research in pure mathematics

.

#### Statistics and other decision sciences

Applied mathematics has significant overlap with the discipline of statistics, whose theory is formulated mathematically, especially with probability theory

. Statisticians (working as part of a research project) "create data that makes sense" with random sampling and with randomized experiments

; the design of a statistical sample or experiment specifies the analysis of the data (before the data be available). When reconsidering data from experiments and samples or when analyzing data from observational studies

, statisticians "make sense of the data" using the art of modelling

and the theory of inference

– with model selection

and estimation

; the estimated models and consequential predictions should be tested

on new data.

Statistical theory

studies decision problems such as minimizing the risk

(expected loss) of a statistical action, such as using a procedure in, for example, parameter estimation, hypothesis testing, and selecting the best

. In these traditional areas of mathematical statistics

, a statistical-decision problem is formulated by minimizing an objective function, like expected loss or cost

, under specific constraints: For example, a designing a survey often involves minimizing the cost of estimating a population mean with a given level of confidence. Because of its use of optimization, the mathematical theory of statistics shares concerns with other decision sciences, such as operations research

, control theory

, and mathematical economics

.

#### Computational mathematics

Computational mathematicsproposes and studies methods for solving mathematical problem

s that are typically too large for human numerical capacity. Numerical analysis

studies methods for problems in analysis using functional analysis

and approximation theory

; numerical analysis includes the study of approximation

and discretization

broadly with special concern for rounding errors. Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithm

ic matrix

and graph theory

. Other areas of computational mathematics include computer algebra and symbolic computation

.

Mathematical physics Mathematical physics Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and... |
Fluid dynamics Fluid dynamics In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics... |
Numerical analysis Numerical analysis Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis .... |
Optimization | Probability theory Probability theory Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single... |
Statistics Statistics Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments.... |
Cryptography Cryptography Cryptography is the practice and study of techniques for secure communication in the presence of third parties... |

Mathematical finance Mathematical finance Mathematical finance is a field of applied mathematics, concerned with financial markets. The subject has a close relationship with the discipline of financial economics, which is concerned with much of the underlying theory. Generally, mathematical finance will derive and extend the mathematical... |
Game theory Game theory Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others... |
Mathematical biology Mathematical biology Mathematical and theoretical biology is an interdisciplinary scientific research field with a range of applications in biology, medicine and biotechnology... |
Mathematical chemistry 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 Mathematical economics is the application of mathematical methods to represent economic theories and analyze problems posed in economics. It allows formulation and derivation of key relationships in a theory with clarity, generality, rigor, and simplicity... |
Control theory Control theory Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference... |

## Mathematics as profession

Arguably the most prestigious award in mathematics is the , established in 1936 and now awarded every 4 years. The Fields Medal is often considered a mathematical equivalent to the Nobel Prize.

The Wolf Prize in Mathematics

, instituted in 1978, recognizes lifetime achievement, and another major international award, the Abel Prize

, was introduced in 2003. The Chern Medal was introduced in 2010 to recognize lifetime achievement. These accolades are awarded in recognition of a particular body of work, which may be innovational, or provide a solution to an outstanding problem in an established field.

A famous list of 23 open problem

s, called "Hilbert's problems

", was compiled in 1900 by German mathematician David Hilbert

. This list achieved great celebrity among mathematicians, and at least nine of the problems have now been solved. A new list of seven important problems, titled the "Millennium Prize Problems

", was published in 2000. Solution of each of these problems carries a $1 million reward, and only one (the Riemann hypothesis

) is duplicated in Hilbert's problems.

## Mathematics as science

Carl Friedrich Gauss referred to mathematics as "the Queen of the Sciences". In the original Latin*Regina Scientiarum*, as well as in German

*Königin der Wissenschaften*, the word corresponding to

*science*means a "field of knowledge", and this was the original meaning of "science" in English, also. Of course, mathematics is in this sense a field of knowledge. The specialization restricting the meaning of "science" to

*natural scienceNatural scienceThe natural sciences are branches of science that seek to elucidate the rules that govern the natural world by using empirical and scientific methods...*

follows the rise of Baconian science

, which contrasted "natural science" to scholasticism

, the Aristotelean method

of inquiring from first principles

. Of course, the role of empirical experimentation and observation is negligible in mathematics, compared to natural sciences such as psychology

, biology

, or physics

. Albert Einstein

stated that

*"as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.*"

Many philosophers believe that mathematics is not experimentally falsifiable

, and thus not a science according to the definition of Karl Popper

. However, in the 1930s Gödel's incompleteness theorems

convinced many mathematicians that mathematics cannot be reduced to logic alone, and Karl Popper concluded that "most mathematical theories are, like those of physics

and biology

, hypothetico

-deductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently." Other thinkers, notably Imre Lakatos

, have applied a version of falsificationism to mathematics itself.

An alternative view is that certain scientific fields (such as theoretical physics

) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is

*public knowledge*and thus includes mathematics. In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences

of assumptions. Intuition

and experiment

ation also play a role in the formulation of conjecture

s in both mathematics and the (other) sciences. Experimental mathematics

continues to grow in importance within mathematics, and computation and simulation are playing an increasing role in both the sciences and mathematics, weakening the objection that mathematics does not use the scientific method

.

The opinions of mathematicians on this matter are varied. Many mathematicians feel that to call their area a science is to downplay the importance of its aesthetic side, and its history in the traditional seven liberal arts

; others feel that to ignore its connection to the sciences is to turn a blind eye to the fact that the interface between mathematics and its applications in science and engineering

has driven much development in mathematics. One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematics is

*created*(as in art) or

*discovered*(as in science). It is common to see universities

divided into sections that include a division of

*Science and Mathematics*, indicating that the fields are seen as being allied but that they do not coincide. In practice, mathematicians are typically grouped with scientists at the gross level but separated at finer levels. This is one of many issues considered in the philosophy of mathematics

.

## See also

- Definitions of mathematicsDefinitions of mathematicsMathematics has no generally accepted definition. Different schools of thought, particularly in philosophy, have put forth radically different definitions...
- DyscalculiaDyscalculiaDyscalculia is a specific learning disability involving innate difficulty in learning or comprehending simple arithmetic. It is akin to dyslexia and includes difficulty in understanding numbers, learning how to manipulate numbers, learning maths facts, and a number of other related symptoms...
- IatromathematiciansIatromathematiciansIatromathematicians was a school of physicians in 17th century Italy who tried to apply laws of mathematics and mechanics in order to understand functioning of the human body...
- List of mathematics competitions
- Mathematical anxietyMathematical anxietyMathematical anxiety is anxiety about one's ability to do mathematics independent of skill.-Math anxiety:Math anxiety is a phenomenon that is often considered when examining students’ problems in mathematics. Mark H. Ashcraft, Ph.D. defines math anxiety as “a feeling of tension, apprehension, or...
- Mathematical gameMathematical gameA mathematical game is a multiplayer game whose rules, strategies, and outcomes can be studied and explained by mathematics. Examples of such games are Tic-tac-toe and Dots and Boxes, to name a couple. On the surface, a game need not seem mathematical or complicated to still be a mathematical game...
- Mathematical modelMathematical modelA mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...
- Mathematics and artMathematics and artMathematics and art have a long historical relationship. The ancient Egyptians and ancient Greeks knew about the golden ratio, regarded as an aesthetically pleasing ratio, and incorporated it into the design of monuments including the Great Pyramid, the Parthenon, the Colosseum...
- Mathematics educationMathematics educationIn contemporary education, mathematics education is the practice of teaching and learning mathematics, along with the associated scholarly research....
- PseudomathematicsPseudomathematicsPseudomathematics is a form of mathematics-like activity that does not work within the framework, definitions, rules, or rigor of formal mathematical models...

## Further reading

- Benson, Donald C.,
*The Moment of Proof: Mathematical Epiphanies*, Oxford University PressOxford University PressOxford University Press is the largest university press in the world. It is a department of the University of Oxford and is governed by a group of 15 academics appointed by the Vice-Chancellor known as the Delegates of the Press. They are headed by the Secretary to the Delegates, who serves as...

, USA; New Ed edition (December 14, 2000). ISBN 0-19-513919-4. - Boyer, Carl B.,
*A History of Mathematics*, Wiley; 2 edition (March 6, 1991). ISBN 0-471-54397-7. — A concise history of mathematics from the Concept of Number to contemporary Mathematics. - Davis, Philip J.Philip J. DavisPhilip J. Davis is an American applied mathematician.Davis was born in Lawrence, Massachusetts. He is known for his work in numerical analysis and approximation theory, as well as his investigations in the history and philosophy of mathematics...

and Hersh, ReubenReuben HershReuben Hersh is an American mathematician and academic, best known for his writings on the nature, practice, and social impact of mathematics. This work challenges and complements mainstream philosophy of mathematics.After receiving a B.A...

,*The Mathematical Experience*. Mariner Books; Reprint edition (January 14, 1999). ISBN 0-395-92968-7.The Mathematical ExperienceThe Mathematical Experience is a 1981 book by Philip J. Davis and Reuben Hersh that discusses the practice of modern mathematics from a historical and philosophical perspective... - Gullberg, JanJan GullbergJan Gullberg, was a Swedish writer on popular Science and Medical topics. He is the author of Mathematics: From the Birth of Numbers, published by W. W. Norton in 1997 . The book was positively reviewed in Scientific American, but more reservedly in New Scientist.-References:...

,*Mathematics — From the Birth of Numbers*. W. W. Norton & Company; 1st edition (October 1997). ISBN 0-393-04002-X. - Hazewinkel, Michiel (ed.),
*Encyclopaedia of Mathematics*. Kluwer Academic Publishers 2000. — A translated and expanded version of a Soviet mathematics encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM, and online.Encyclopaedia of MathematicsThe Encyclopaedia of Mathematics is a large reference work in mathematics. It is available in book form and on CD-ROM.... - Jourdain, Philip E. B.,
*The Nature of Mathematics*, in*The World of Mathematics*, James R. Newman, editor, Dover PublicationsDover PublicationsDover Publications is an American book publisher founded in 1941 by Hayward Cirker and his wife, Blanche. It publishes primarily reissues, books no longer published by their original publishers. These are often, but not always, books in the public domain. The original published editions may be...

, 2003, ISBN 0-486-43268-8.

## External links

- Free Mathematics books Free Mathematics books collection.
- Encyclopaedia of MathematicsEncyclopaedia of MathematicsThe Encyclopaedia of Mathematics is a large reference work in mathematics. It is available in book form and on CD-ROM....

online encyclopaedia from Springer, Graduate-level reference work with over 8,000 entries, illuminating nearly 50,000 notions in mathematics. - HyperMath site at Georgia State University
- FreeScience Library The mathematics section of FreeScience library
- Rusin, Dave:
*The Mathematical Atlas*. A guided tour through the various branches of modern mathematics. (Can also be found at NIU.edu.) - Polyanin, Andrei:
*EqWorld: The World of Mathematical Equations*. An online resource focusing on algebraic, ordinary differential, partial differential (mathematical physicsMathematical physicsMathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...

), integral, and other mathematical equations. - Cain, George: Online Mathematics Textbooks available free online.
- Tricki, Wiki-style site that is intended to develop into a large store of useful mathematical problem-solving techniques.
- Mathematical Structures, list information about classes of mathematical structures.
- Mathematician Biographies. The MacTutor History of Mathematics archiveMacTutor History of Mathematics archiveThe 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...

Extensive history and quotes from all famous mathematicians. -
*Metamath*. A site and a language, that formalize mathematics from its foundations. - Nrich, a prize-winning site for students from age five from Cambridge UniversityUniversity of CambridgeThe University of Cambridge is a public research university located in Cambridge, United Kingdom. It is the second-oldest university in both the United Kingdom and the English-speaking world , and the seventh-oldest globally...
- Open Problem Garden, a wikiWikiA wiki is a website that allows the creation and editing of any number of interlinked web pages via a web browser using a simplified markup language or a WYSIWYG text editor. Wikis are typically powered by wiki software and are often used collaboratively by multiple users. Examples include...

of open problems in mathematics -
*Planet Math*. An online mathematics encyclopedia under construction, focusing on modern mathematics. Uses the Attribution-ShareAlike license, allowing article exchange with Wikipedia. Uses TeXTeXTeX is a typesetting system designed and mostly written by Donald Knuth and released in 1978. Within the typesetting system, its name is formatted as ....

markup. - Some mathematics applets, at MIT
- Weisstein, Eric et al.:
*MathWorld: World of Mathematics*. An online encyclopedia of mathematics. - Patrick Jones' Video Tutorials on Mathematics
- Citizendium: Theory (mathematics).