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Mathematics is often defined as the study of certain subjects, such as quantity, structure, space, and change. Others define mathematics as that body of knowledge discovered by deductive reasoning, starting from axioms and definitions, as in Euclid's Elements (circa 300 BC).

Many cultures have added to our mathematical knowledge. Particularly notable are the Greek speaking Hellenic culture 4th Century BC, the civilizations of Egypt, Arabia, India, China, and Japan, and the culture of Renaissance Europe. The positional base ten number system, invented in India, made mathematics much more accessible. New discoveries in science and technology require new mathematics, and more new mathematics has been published in the last century than in all the time that had gone before.

The word mathematics is also used to refer to the knowledge gained by people by doing mathematics, also known as the body of mathematical knowledge. This latter meaning of mathematics includes the mathematics used to do calculations or mathematical modeling and is an indispensable tool in the natural sciences, engineering, economics, and medicine.

The word "mathematics" comes from the Greek μάθημα (máthema) meaning "science, knowledge, or learning" and μαθηματικός (mathematikós) meaning "fond of learning". It is often abbreviated maths in Commonwealth English and math in American English.

History

Main article: History of mathematics

The evolution of mathematics can be seen to be an ever increasing series of abstractions. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought. In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. From counting, naturally followed arithmetic (e.g. addition, subtraction, multiplication and division).

However, mathematics undoubtedly could not have developed out of simple counting and arithmetic without writing and a way of writing numbers. Perhaps prehistoric peoples first expressed quantity by drawing lines in the ground or scratching wood. People of the Inca empire, which lacked any other writing system, represented and stored numerical data using a complex system of knots and rope called khipu.

Historically, the major disciplines within mathematics 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 studies of structure, space and change.

Inspiration, aesthetics and pure and applied mathematics

Mathematics arises wherever there are difficult problems that merit careful mental investigation. At first these were found in commerce, land measurement and later astronomy. Nowadays, mathematics derives much inspiration from the natural sciences and it is not uncommon for new mathematics to be pioneered by physicists, although it may need to be recast into more rigorous language. Some notable examples of this happening are Newton inventing calculus and Feynman inventing his Feynman path integral, but it also happens with results from string theory. The mathematics arising from this immediately has relevance for the subject which inspired it and can be applied to solve problems in that subject. Mathematics which can be so used is called applied mathematics as opposed to pure mathematics. In this way applied mathematics is an indispensable tool. With the increase in our mathematical knowledge, mathematics itself has become a source of inspiration. Mathematics is inspiring to mathematicians because it has some intrinsic aesthetics or inner beauty, which is hard to explain. Mathematicians value especially simplicity and generality and when these seemingly incompatible properties combine in a piece of mathematics, as in a unifying generalization for several subfields, or in a helpful tool for common calculations, often that piece of mathematics is called beautiful. Since the result of mathematics inspired by mathematics is often pure mathematics and thus has no applications outside of mathematics yet, the only value it has is in its aesthetics. Surprisingly often, it has happened that pure mathematics, which was considered only of interest to mathematicians, has become applied mathematics because of some new insight, as if it anticipated later needs.

Notation, language and rigor

Main article: Mathematical notation

Mathematicians strive to be as clear as possible in the things they say and especially in the things they write, something which mathematicians refer to as rigor. To accomplish rigor, mathematicians have extended natural language with precisely defined vocabulary and grammar for referring to mathematical objects and stating certain common relations, with accompanying notation. Some of the terms they use also have a meaning outside of mathematics, such as ring, group and category, but some are specific to mathematics, such as homotopy and Hilbert space.

Even so, in the past it sometimes happened that something which had supposedly been proved turned out to be false. This was possible because mathematics was done using natural language. To prevent this from happening, mathematicians wanted their theorems to follow mechanically from a few simple incontrovertible truths and for this they invented axioms and axiomatic reasoning.

An axiom is just a string of symbols which have an intrinsic meaning because of all derivable formulas. 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 (strong enough) axiom system has undecidable formulas so a final axiomatization of mathematics is unavailable. Nonetheless mathematics is often imagined to be nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory. But for most of mathematics this complete rigor is far too cumbersome and mathematical language and notation are supposed to suffice.

Is mathematics a science?

Carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. The mathematician-physicist Leon M. Lederman has quipped "The physicists defer only to mathematicians, and the mathematicians defer only to God (though you may be hard pressed to find a mathematician that modest)."

If one considers science to be strictly about the physical world, then mathematics itself is not a science. That is, mathematical knowledge exists separate from the physical world. Mathematics shares much in common with the sciences. Experimentation plays a large role in the formulation of reasonable conjectures, and therefore is not by any means excluded from use by research mathematicians. However, theorems are only accepted if rigorous proofs have been found for them.

Overview of fields of mathematics

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 (i.e. algebra, geometry and analysis). In addition to these three main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic and other simpler systems (foundations) and to the empirical systems of the various sciences (applied mathematics).

The study of structure starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are characterized 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 generalize 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 mathematical logic and then set theory 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.

Major themes in mathematics

An alphabetical and subclassified list of mathematical topics is available. The following list of themes and links gives just one possible view. For a fuller treatment, see Areas of mathematics or the list of lists of mathematical topics.

Quantity

This starts from explicit measurements of sizes of numbers or sets, or ways to find such measurements.

Natural numbers Integers Rational numbers Real numbers Complex numbers
Number – Natural number – Integers – Rational numbers – Real numbers – Complex numbers – Hypercomplex numbers – Quaternions – Octonions – Sedenions – Hyperreal numbers – Surreal numbers – Ordinal numbers – Cardinal numbers – p-adic numbers – Integer sequences – Mathematical constants – Number names – Infinity – Base

Change

Ways to express and handle change in mathematical functions, and changes between numbers.
Arithmetic Calculus Vector calculus Analysis
Differential equations Dynamical systems Chaos theory
Arithmetic – Calculus – Vector calculus – Analysis – Differential equations – Dynamical systems – Chaos theory – List of functions

Structure

Pinning down ideas of size, symmetry, and mathematical structure.
File:Rubik float.png
Abstract algebra Number theory Group theory
File:Lattice of the divisibility of 60.png
Topology Category theory Order theory
Abstract algebra – Number theory – Algebraic geometry – Group theory – Monoids – Analysis – Topology – Linear algebra – Graph theory – Universal algebra – Category theory – Order theory – Measure theory

Spatial relations

A more visual approach to mathematics.
File:Pythagorean.png File:Taylorsine.png File:OsculatingCircle.png
Topology Geometry Trigonometry Differential geometry Fractal geometry
Topology – Geometry – Trigonometry – Algebraic geometry – Differential geometry – Differential topology – Algebraic topology – Linear algebra – Fractal geometry

Discrete mathematics

Discrete mathematics involves techniques that apply to objects that can only take on specific, separated values.


 File:Venn A intersect B.png File:Fsm moore model door control.jpg File:Caesar3.png
Combinatorics Naive set theory Theory of computation Cryptography Graph theory
Combinatorics – Naive set theory – Theory of computation– Cryptography – Graph theory

Applied mathematics

Applied mathematics uses the full knowledge of mathematics to solve real-world problems.
Mathematical physics – Mechanics – Fluid mechanics – Numerical analysis – Optimization – Probability – Statistics – Financial mathematics – Game theory – Mathematical biology – Cryptography – Information theory

Famous theorems and conjectures

These theorems have interested mathematicians and non-mathematicians alike.
Pythagorean theorem – Fermat's last theorem – Goldbach's conjecture – Twin Prime Conjecture – Gödel's incompleteness theorems – Poincaré conjecture – Cantor's diagonal argument – Four color theorem – Zorn's lemma – Euler's identity – Church-Turing thesis – Collatz conjecture

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 hypothesis – Continuum hypothesis – P=NP – Pythagorean theorem – Central limit theorem – Fundamental theorem of calculus – Fundamental theorem of algebra – Fundamental theorem of arithmetic – Fundamental theorem of projective geometry – classification theorems of surfaces – Gauss-Bonnet theorem

Foundations and methods

Approaches to understanding the nature of mathematics also influence the way mathematicians study their subject.
Philosophy of mathematics – Mathematical intuitionism – Mathematical constructivism – Foundations of mathematics – Set theory – Symbolic logic – Model theory – Category theory – Logic – Reverse Mathematics – Table of mathematical symbols

History and the world of mathematicians

See also list of mathematics history topics

History of mathematics – Timeline of mathematics – Mathematicians – Fields medal – Abel Prize – Millennium Prize Problems (Clay Math Prize) – International Mathematical Union – Mathematics competitions – Lateral thinking – Mathematical abilities and gender issues

Mathematics and other fields

Mathematics and architecture – Mathematics and education – Mathematics of musical scales


Common misconceptions

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:

The case of Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work. And like astronomy, mathematics owes much to amateur contributors such as Fermat and Mersenne.

Mathematics is not accountancy. Although arithmetic computation is crucial to accountants, their main concern is to verify 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.

Mathematics does not produce absolute facts about the physical world. In the case of geometry, for example, it is not relevant to mathematics to know whether points and lines exist in any physical sense, as geometry starts from axioms and postulates about abstract entities called "points" and "lines" that we feed into the system. While these axioms are derived from our perceptions and experience, they are not dependent on them. And yet, mathematics is extremely useful for solving real-world problems. It is this fact that led Eugene Wigner to write an essay on The Unreasonable Effectiveness of Mathematics in the Natural Sciences.

Mathematics is not about theorem proving. To say that it was is akin to saying that literature is about the construction of grammatically correct sentences. Indeed, theorems may be encapsulated in formal languages, and computers may then be used to prove them, by means of automated theorem provers. However, such techniques, applied in general, will not generate mathematics any more than the proverbial thousand monkeys, seated at typewriters, might accidentally produce a play worthy of Shakespeare. A computer cannot tell apart an interesting theorem from a boring one.

See also

Bibliography

  • Benson, Donald C., The Moment Of Proof: Mathematical Epiphanies (1999).
  • 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).
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