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Mathematics, often abbreviated maths in British English and math in American English, is the science of abstraction. The first abstraction was probably that of numbers, whole positive numbers. The realization that two apples and two pears do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought. Following this the interactions of numbers, for example addition, were abstracted.

Nowadays mathematics is the investigation of axiomatically defined abstract structures using mathematical notation and symbolic logic. It is also commonly defined as the study of patterns of structure, change, and space; even more informally, one might say it is the study of "figures and numbers". Mathematics is widely used for the development and communication of ideas, and particularly quantitative relationships in scientific observation, reasoned analysis and prediction.

Our knowledge in many fields of mathematics is constantly growing, through research and application. Mathematics is usually regarded as an important tool for science, even though the development of mathematics is not necessarily done with science in mind (See pure mathematics and applied mathematics.).

The specific structures that are investigated by mathematicians sometimes do have their origin in natural and social sciences, including particularly physics and economics. Some contemporary mathematics also has its origins in computer science and communication theory.


The nature and scope of mathematics

The word "mathematics" comes from the Greek μάθημα (máthema) which means "science, knowledge, or learning"; μαθηματικός (mathematikós) means "fond of learning".

In the formalist view, widely accepted as a description by professionals in the field, the definition used is the one given at the beginning of this article, except that it also requires the use of mathematical notation, a tautologous usage. Mathematics might accordingly be seen as an extension of spoken and written natural languages, with an extremely precisely defined vocabulary and grammar, for the purpose of describing and exploring physical and conceptual relationships. While this is not obviously inaccurate, there are other views, and some are described in the article on the philosophy of mathematics.

Mathematics itself is usually considered absolute, without any reference. Mathematicians define and investigate some structures for reasons purely internal to mathematics; they may provide, for instance, a unifying generalization for several subfields, or a helpful tool for common calculations. Purely speculative extensions can be exercises in abstraction, apparently for its own sake. Surprisingly often, they turn out to anticipate later needs. What is and is not sterile amongst abstract developments is one topic where mathematicians often refer to ideas of good taste , as well as applicability and the subject's own tradition.

Despite an enormous expansion since the year 1800, and the appearance of numerous specializations within it, mathematics is still normally considered to be a single, distinctive field of research. Its teaching also forms a recognised academic field. The rudiments, starting with arithmetic and moving on to basic application of mathematics disciplines including algebra, geometry, trigonometry, statistics, and calculus, constitute a core school subject in primary and secondary education: also referred to as mathematics, even though it is a very restricted view of the subject taken as a whole. Certain mathematical subfields are commonly studied by students in a wide variety of other academic fields, for example at post-secondary/tertiary institutions; the teaching of these is sometimes known as service teaching .

In the end, many mathematicians work for purely aesthetic reasons, viewing mathematics more as an art ('pure mathematics') rather than for its practical application ('applied mathematics'); this is the same kind of motivation as poets and philosophers may experience, and no more explicable. Albert Einstein referred to the subject as the Queen of the Sciences in his book Ideas and Opinions (a phrase first used by Carl Friedrich Gauss).

Overview of fields of mathematics

See the article on the history of mathematics for past development

The major disciplines within mathematics first arose out of the need to do calculations in commerce, to measure land and to predict astronomical events. These three needs can be roughly related to the broad subdivision of mathematics into the study of structure, space and change.

The study of structure starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are recorded in elementary algebra. The deeper properties of whole numbers are studied in number theory. The investigation of methods to solve equations leads to the field of abstract algebra, which, among other things, studies rings and fields, structures that generalize the properties possessed by everyday numbers. Long standing questions about ruler-and-compass constructions were finally settled by Galois theory. The physically important concept of vectors, generalized to vector spaces and studied in linear algebra, belongs to the two branches of structure and space.

The study of space originates with geometry, first the Euclidean geometry and trigonometry of familiar three-dimensional space (also applying to both more and fewer dimensions), later also generalized to non-Euclidean geometries which play a central role in general relativity. The modern fields of differential geometry and algebraic geometry generalize geometry in different directions: differential geometry emphasizes the concepts of functions, fiber bundles, derivatives, smoothness, and direction, while in algebraic geometry geometrical objects are described as solution sets of polynomial equations. Group theory investigates the concept of symmetry abstractly; topology, the greatest growth area in the twentieth century, has a focus on the concept of continuity. Both the group theory of Lie groups and topology reveal the intimate connections of space, structure and change.

Understanding and describing change in measurable quantities is the common theme of the natural sciences, and calculus was developed as a most useful tool for that. The central concept used to describe a changing variable is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods to solve these are studied in the field of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. For several reasons, it is convenient to generalise to the complex numbers which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions, laying the groundwork for quantum mechanics among many other things. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior.

In order to clarify the foundations of mathematics, the fields first of set theory and then mathematical logic were developed. Mathematical logic, which divides into recursion theory, model theory and proof theory, is now closely linked to computer science. When electronic computers were first conceived, several essential theoretical concepts were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, and information theory. Many of those topics are now investigated in theoretical computer science. Discrete mathematics is the common name for the fields of mathematics most generally useful in computer science.

An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis and prediction of phenomena where chance plays a part. It is used in all sciences. Numerical analysis investigates methods for efficiently solving a broad range of mathematical problems numerically on computers, beyond human capacities, and taking rounding errors and other sources of error into account to obtain credible answers.

Topics in mathematics

An alphabetical and subclassified list of mathematical topics is available. The following list of subfields and topics reflects one organizational view of mathematics. For a fuller treatment, see Areas of mathematics


In general, these topics and ideas present explicit measurements of sizes of numbers or sets, or ways to find such measurements.

NumberNatural numberIntegersRational numbersReal numbersComplex numbersHypercomplex numbersQuaternionsOctonionsSedenionsHyperreal numbersSurreal numbersOrdinal numbersCardinal numbersp-adic numbersInteger sequencesMathematical constantsNumber namesInfinityBase


These topics give ways to measure change in mathematical functions, and changes between numbers.

ArithmeticCalculusVector calculusAnalysisDifferential equationsDynamical systems and chaos theoryList of functions


These branches of mathematics measure size and symmetry of numbers, and various constructs.

Abstract algebraNumber theoryAlgebraic geometryGroup theoryMonoidsAnalysisTopologyLinear algebraGraph theoryUniversal algebraCategory theoryOrder theory

Spatial relations

These topics tend to quantify a more visual approach to mathematics than others.

TopologyGeometryTrigonometryAlgebraic geometryDifferential geometryDifferential topologyAlgebraic topologyLinear algebraFractal geometry

Discrete mathematics

Topics in discrete mathematics deal with branches of mathematics with objects that can only take on specific, separated values.

CombinatoricsNaive set theoryProbabilityTheory of computationFinite mathematicsCryptographyGraph theory

Applied mathematics

Fields in applied mathematics use knowledge of mathematics to solve real world problems.

MechanicsNumerical analysisOptimizationProbabilityStatisticsFinancial mathematicsGame theoryMathematical biologyCryptographyInformation theoryFluid dynamics

Famous theorems and conjectures

These theorems have interested mathematicians and non-mathematicians alike.

Pythagorean theoremFermat's last theoremGoldbach's conjectureTwin Prime ConjectureGödel's incompleteness theoremsPoincaré conjectureCantor's diagonal argumentFour color theoremZorn's lemmaEuler's identityScholz ConjectureChurch-Turing thesis

Important theorems and conjectures

See list of theorems, list of conjectures for more

These are theorems and conjectures that have changed the face of mathematics throughout history.

Riemann hypothesisContinuum hypothesisP=NPPythagorean theoremCentral limit theoremFundamental theorem of calculusFundamental theorem of algebraFundamental theorem of arithmeticFundamental theorem of projective geometryclassification theorems of surfacesGauss-Bonnet theorem

Foundations and methods

Such topics are approaches to mathematics, and influence the way mathematicians study their subject.

Philosophy of mathematicsMathematical intuitionismMathematical constructivismFoundations of mathematicsSet theorySymbolic logicModel theoryCategory theoryLogicReverse MathematicsTable of mathematical symbols

History and the world of mathematicians

See also list of mathematics history topics

History of mathematicsTimeline of mathematicsMathematiciansFields medalAbel PrizeMillennium Prize Problems (Clay Math Prize)International Mathematical UnionMathematics competitionsLateral thinkingMathematical abilities and gender issues

Mathematics and other fields

Mathematics and architectureMathematics and educationMathematics of musical scales

Mathematical coincidences

List of mathematical coincidences

Mathematical tools



Common misconceptions

Although Einstein called it "the Queen of the Sciences", by one not-unusual definition, mathematics itself is not a science, because it is not empirical. That is, what counts as mathematical knowledge is determined by proof rather than experiment. Mathematics is therefore not simply an aspect of physics, the science closest to it (historically speaking), as physics is an empirical science. On the other hand, experiment plays a large role in the formulation of reasonable conjectures, and therefore is not by any means excluded from use by research mathematicians.

Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems.

Pseudomathematics is a form of mathematics-like activity undertaken outside academia: and occasionally by mathematicians themselves. It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally-accepted mathematics is similar to that between pseudoscience and real science. The misconceptions involved are normally based on:

  • misunderstanding of the implications of mathematical rigour;
  • attempts to get round the usual criteria for publication in a learned journal after peer review, with assumptions of bias;
  • lack of familiarity with, and therefore underestimation of, the existing literature.

The case of Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work.

Mathematics is not accountancy. Although arithmetic computation is crucial to accountants, their mainly concern is to verifying that computations are correct through a system of doublechecks. Advances in abstract mathematics are mostly irrelevant to the efficiency of concrete bookkeeping, but the use of computers clearly does matter.

Mathematics is not numerology. Numerology uses modular arithmetic to reduce names and dates down to numbers, but assigns emotions or traits to these numbers intuitively or on the basis of traditions.


  • Courant, R. and H. Robbins, What Is Mathematics? (1941);
  • Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Birkhäuser, Boston, Mass., 1980. A gentle introduction to the world of mathematics.
  • Boyer, Carl B., History of Mathematics, Wiley, 2nd edition 1998 available, 1st edition 1968 . A concise history of mathematics from the Concept of Number to contemporary Mathematics.
  • Gullberg, Jan, Mathematics--From the Birth of Numbers. W.W. Norton, 1996. An encyclopedic overview of mathematics presented in clear, simple language.
  • Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics. Kluwer Academic Publishers 2000. A translated and expanded version of a Soviet math encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM.
  • Kline, M., Mathematical Thought from Ancient to Modern Times (1973).
  • Pappas, Theoni, The Joy Of Mathematics (1989).

External links

Last updated: 10-10-2005 07:55:46
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