mathematics

• 数学

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mathematical

　 　

WordNet

1. a science (or group of related sciences) dealing with the logic of quantity and shape and arrangement (同)math, maths
2. statistically possible though highly improbable; "have a mathematical chance of making the playoffs"
3. beyond question; "a mathematical certainty"
4. characterized by the exactness or precision of mathematics; "mathematical precision"
5. of or pertaining to or of the nature of mathematics; "a mathematical textbook"; "slide rules and other mathematical instruments"; "a mathematical solution to a problem"; "mathematical proof"
6. the branches of mathematics that study and develop the principles of mathematics for their own sake rather than for their immediate usefulness
7. the branches of mathematics that are involved in the study of the physical or biological or sociological world (同)applied math
8. the academic department responsible for teaching and research in mathematics (同)department of mathematics
9. the logical analysis of mathematical reasoning

PrepTutorEJDIC

1. (また《米話》math,《英話》maths)《単数扱い》『数学』 / 《複数扱い》数学的処理,計算
2. 数学の,数学的 / (非常に)正確な

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出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2015/07/26 19:14:15」(JST)

wiki ja

[Wiki ja表示]

数学（すうがく、希: μαθηματικά, 羅: mathematica, 英: mathematics）は、量（数）、構造、空間、変化などを研究する学問である。 数学の正確な範囲と定義について、数学者と哲学者の間で見解の広がりがある。

概要

数学の最も普通の（ありきたりな）定義としては、「数および図形についての学問」というものがある。しかし、19世紀のヨーロッパで集合論が提起されてからは、「数学とは何か」ということが問い直されるようになっており（数学基礎論）、数学の対象、方法、文化史的な価値などについて研究する数理哲学まで生まれている。よって、現代的な意味では数学はもはや「数および図形についての学問」といった単純な定義で済ませておけない状態にある。

上記のありきたりな定義の延長で言うならば、とりあえずは「数学は、量、構造、変化、空間といったものを対象として、いくつかの仮定から始めて、決められた演繹的推論を進めることで得られる体系を研究する学問」などとも言えるかもしれない。 数学とは、狭義には伝統的な数論や幾何学などの分野における研究とその成果の総称として、またそれらの成果を肯定的に内包する公理と推論からなる論理と理論の体系を指して言うものである。また広義には、超数学（メタ数学）などと呼ばれる枠組みに従って、公理と推論規則が定められた体系一般を指す。現代的な数学においては、公理的に定義される抽象的な構造を、数理論理学を共通の枠組みとして用いて探究する。

数学は、西欧の学問分類では一般に「形式科学」に分類され、自然科学とははっきり区別されている。 方法論の如何によらず最終的には、数学としての成果というものは他の自然科学のように実験や観察によるものであってはならない。

数学、特に伝統的な純粋数学では数学研究が自己目的化されており、数学への内的な興味のために研究がなされる。 このような数学ではいかに本質的な概念なり定理なりを得て、体系的な数学を構築するかが重要視されており、数学的対象を記述するのに適した概念や空間を定義したり、数学的事象をうまく表現した定理を得たりすることが数学者の主な仕事である。一方で、美的な理由からそれぞれの分野での研究をしている数学者もいる。彼らは対称性や直観性などその独特の審美眼を以て、数学を芸術に近しいものとみなしているのである。この分野については数学の哲学、数学的な美に詳しい。

伝統的な数学分野で研究される対象は物理現象と深い関わりを持つものが多い。一方、応用分野では数理モデルという形で、例えば計算機や言語などといったものを対象とした研究が行われる。もちろん、数理モデルにおける演繹から得られる成果と実際との間に幾分かのずれを生じることもあるが、そのずれの評価とモデルの実用性・実効性については多くは数学の外の話である。また、数学とパズルの類似性が指摘される事があるが、数学が本質性や体系性を重要視することに照らせば、パズルはむしろ奇をてらい非体系的である。こうした研究姿勢がしばしば様々な数学の諸分野を統一するような概念へと導いたり、他分野の学問の発展に貢献したりすることに繋がる。

なお、数学の語源について日本語「かず」は、説得力のある語源説は示されていない。提唱されたいくつかの説については数を参照。

歴史

詳細は「数学史」を参照

研究

歴史的には、数学の主要な分野は人類が農耕を行うとともに必要となった次の三つの要素から生じたものである。農作物の分配管理や商取引のための計算、農地管理のための測量、そして農作業の時期を知る暦法のための天文現象の周期性の解明。これら三つの必要性は、そのまま数学の大きな三つの区分、構造、空間、変化のそれぞれの研究に大体対応しているといえよう。例えば土木工事などの経験から直角三角形の辺の比は知り得ても、論理的にはこの時点では解明できていない。3 : 4 : 5 は経験的に正しいが、比から導かれる c² = a² + b² （c, b, a は辺の長さ、または比）が普遍的に成立するかは不明である（証明はピタゴラスの定理を参照のこと）。かつて数学が独立した学問でなく、純粋な実用数学であった時代には、あたかも自然科学におけるデータのようにこれらの関係を扱い、例を多数挙げることで正しさを主張するといった手法でもさして問題視されなかった。しかし数は無限に存在するため、たとえコンピュータを使って沢山の数を調べても完全に証明することはできない。よってこのような手法では完全な真偽の判定はできず、数学が一つの学問として研究されるようになって以降は当然ながら別の方法が求められることになり、論理を用いて真偽を判定する「数学的証明」という概念が発達した。そのため現在の数学では証明は非常に重視されている。

現代における純粋数学の研究は主に代数学、幾何学、解析学の三分野に大別される。また、これらの数学を記述するのに必要な道具を与える論理を研究する学問を数学基礎論という。

基礎付け

数学の基礎を明確にすること、あるいは数学そのものを研究することのために、集合論や数理論理学そしてモデル理論は発展してきた。フランスの数学者グループであるニコラ・ブルバキは、集合論による数学の基礎付けを行い、その巨大な体系を『数学原論』として著した。彼らのスタイルはブルバキ主義とよばれ、現代数学の発展に大きな影響をあたえた。個々の対象の持つ性質を中心とする研究方法である集合論とは別の体系として、対象同士の関係性が作るシステムに主眼を置くことにより対象を研究する方法として圏と関手の理論がある。これはシステムという具体性からコンピュータネットワークなどに応用される一方で、極めて高い抽象性を持つ議論を経て極めて具体的な結果を得るようなアブストラクト・ナンセンスなどと呼ばれる形式性も持ち合わせている。

構造

数や関数、図形の中の点などの数学的対象の間に成り立つさまざまな関係を形式化・公理化して調べるという立場がダフィット・ヒルベルトやニコラ・ブルバキによって追求された。数の大小関係や演算、点の近さ遠さなどの関係がそれぞれ順序構造や群の構造、位相構造などの概念として公理化され、その帰結が研究される。特に、様々な代数的構造の性質を研究する抽象代数学は20世紀に大きく発展した。現代数学で取り扱われる構造は上のような基本的な構造にとどまらず、異なった種類の構造を併せて考える位相線型空間や双曲群などさまざまなものがある。

空間

空間の研究は幾何学と共に始まる。初めは、それは身近な三次元におけるユークリッド幾何学や三角法であるが、後にはやはり、一般相対性理論で中心的な役割を演ずる非ユークリッド幾何学に一般化される。長い間未解決だった定規とコンパスによる作図の問題は、最終的にガロア理論によって決着が付いた。現代的な分野である微分幾何学や代数幾何学は幾何学を異なる方向に発展させた：微分幾何学では、座標系や滑らかさ、それに向きの概念が強調されるが、一方で代数幾何学では、代数方程式の解となるような集合を幾何学的な対象とする。集合は数学の基礎を成す重要な概念であるが、幾何学的な側面を強調する場合、集合を空間と言い、その集合の元を点と呼ぶ。群論では対称性という概念を抽象的に研究し、空間と代数構造の研究の間に関連を与える。位相幾何学は連続という概念に着目することで、空間と変化の双方の研究に関係する。

解析

測る量についての変化を理解し、記述することは自然科学の共通の主題であり、微積分学はまさにそのための最も有用な道具として発展してきた。変化する量を記述するのに使われる中心的な道具は関数である。多くの問題は、とても自然に量とその変化の割合との関係になり、そのような問題を解くための手法は微分方程式の分野で研究される。連続的な量を表すのに使われる数が実数であり、実数の性質や実数に値をとる関数の性質の詳しい研究は実解析として知られる。いくつかの理由から、複素数に拡張する方が便利であり、それは複素解析において研究される。関数解析学は関数空間（関数の集合に位相構造を持たせたもの）が興味の中心であり、この分野は量子力学やその他多くの学問の基盤となっている。自然の多くの現象は力学系によって記述され、カオス理論では、多くの系が決定可能であるにもかかわらず予測不可能な現れ方をする、という事実を扱う。

計算機

人類がコンピュータを最初に思いついたとき（それは実際に作られるより遥かに前のことだが）、いくつかの重要な理論的概念は数学者によってかたち作られ、計算可能性理論、計算複雑性理論、情報理論、そしてアルゴリズム情報理論の分野に発展した。これらの問題の内の多くは計算機科学において研究されている。離散数学は計算機科学において有用な数学の分野の総称である。数値解析は、丸め誤差を考慮に入れて、幅広い数学の問題について効率的にコンピュータの上で数値解を求める方法を研究する。また最近では、計算機科学を駆使して自然科学上の問題を解決する計算科学が急速に発展している。

統計

応用数学において、重要な分野に統計学が挙げられる。統計学はランダムな現象の記述や解析や予測を可能にし、全ての科学において、利用されている。統計学は隣接する分野である確率論とは違って実際の統計データを扱うこともあることから、「確率論までは数学だが統計学は違う」という考えを持っている人もいる。

分野

以下の分野や項目の一覧は、数学に対する一つの有機的な見方を反映している。

便宜上の分類

• 代数学
• 幾何学
• 解析学
• 集合論
• 情報科学
• 確率論
• 統計学

量

数--自然数--整数--偶数--奇数--小数--分数--素数--有理数--無理数--実数--複素数--四元数--八元数--十六元数--超実数--順序数--濃度--p進数--巨大数--整数列--数学定数--数の名称--無限

変化

算術--微積分学--ベクトル解析--解析学--微分方程式--力学系--カオス理論--関数一覧

構造

抽象代数学--数論--代数幾何学--群論--モノイド--解析学--位相幾何学--線型代数学--グラフ理論--圏論

空間

解析幾何学--位相幾何学--幾何学--三角法--代数幾何学--微分幾何学--線型代数学--フラクタル幾何--図形--図形の一覧--ベクトル解析

有限数学

組合せ論--素朴集合論--確率論--統計学--計算理論--離散数学--暗号法--暗号理論--グラフ理論--ゲーム理論--数理工学

数理科学

計算科学--数値解析--確率論--逆問題--数理物理学

有名な定理と予想

フェルマーの最終定理--リーマン予想--連続体仮説--P≠NP予想--ゴールドバッハの予想--双子素数の予想--ゲーデルの不完全性定理--ポアンカレ予想--カントールの対角線論法--ピタゴラスの定理--中心極限定理--微積分学の基本定理--代数学の基本定理--四色定理--ツォルンの補題--オイラーの等式--コラッツの予想--合同数の問題--バーチ・スウィンナートン＝ダイアー予想--ヒルベルトの23の問題

基礎と方法

数理哲学--数学的直観主義--数学的構成主義--数学基礎論--集合論--数理論理学--モデル理論--圏と関手の理論--数学的証明--数学記号の表--逆数学

数学の歴史と世界における発展

数学の歴史--ユークリッド原論--和算--インドの数学--中国の数学・中国の剰余定理--アラビア数学--数学年表--数学者--フィールズ賞--アーベル賞--国際数学連合--数学の競技

数学に関する賞

• フィールズ賞（国際数学連合）
• ネヴァンリンナ賞（国際数学連合）
• ガウス賞（国際数学連合）
• アーベル賞（アーベル記念基金）
• 春季賞（日本数学会）
• ヴェブレン賞（アメリカ数学会）
• フランク・ネルソン・コール賞（アメリカ数学会）
• ヨーロッパ数学会賞（ヨーロッパ数学会）
• ウルフ賞数学部門（ウルフ財団）

「ノーベル数学賞」は存在しない。

脚注

参考文献

• 佐藤泰夫・佐藤純『数学とは何だろう—文化としての数学』（森北出版,1998年）

関連項目

• 数学史
• 数秘学
• 日本数学会
• 数学に関する記事の一覧
• 数学上の未解決問題

その他

• 数学に関係する記事の書き方に関してはウィキプロジェクト 数学を参照。
• その他の数学に関係する記事の情報に関してはPortal:数学を参照。

wiki en

[Wiki en表示]

This article is about the study of topics such as quantity and structure. For other uses, see Mathematics (disambiguation).

"Math" redirects here. For other uses, see Math (disambiguation).

Euclid (holding calipers), Greek mathematician, 3rd century BC, as imagined by Raphael in this detail from The School of Athens.

Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics.

Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.

Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. 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 axioms and definitions. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.

Galileo Galilei (1564–1642) said, "The universe cannot be read until we have learned the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth." Carl Friedrich Gauss (1777–1855) referred to mathematics as "the Queen of the Sciences". 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." French mathematician Claire Voisin states "There is creative drive in mathematics, it's all about movement trying to express itself."

Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, finance 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, which has led 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.

History

Evolution

Main article: History of mathematics

The evolution of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, which is shared by many animals, was probably that of numbers: 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.

Greek mathematician Pythagoras (c. 570 – c. 495 BC), commonly credited with discovering the Pythagorean theorem

Mayan numerals

As evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have also recognized how to count abstract quantities, like time – days, seasons, years.

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 earliest uses of mathematics were in trading, land measurement, painting and weaving patterns and the recording of time.

In Babylonian mathematics elementary arithmetic (addition, subtraction, multiplication and division) first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have been many and diverse, with the first known written numerals created by Egyptians in Middle Kingdom texts such as the Rhind Mathematical Papyrus.

Between 600 and 300 BC the Ancient Greeks began a systematic study of mathematics in its own right with Greek mathematics.

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 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 theorems and their proofs."

Etymology

The word mathematics comes from the Greek μάθημα (máthēma), which, in the ancient Greek language, means "that which is learnt", "what one gets to know", hence also "study" and "science", and in modern Greek just "lesson". The word máthēma is derived from μανθάνω (manthano), while the modern Greek equivalent is μαθαίνω (mathaino), both of which mean "to learn". In Greece, the word for "mathematics" came to have the narrower and more technical meaning "mathematical study" even in Classical times. Its adjective is μαθηματικός (mathēmatikós), meaning "related to learning" or "studious", which likewise further came to mean "mathematical". In particular, μαθηματικὴ τέχνη (mathēmatikḗ tékhnē), Latin: ars mathematica, 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 les mathématiques (and the less commonly used singular derivative la mathématique), goes back to the Latin neuter plural mathematica (Cicero), based on the Greek plural τα μαθηματικά (ta mathēmatiká), used by Aristotle (384–322 BC), 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.

Definitions of mathematics

Main article: Definitions of mathematics

Leonardo Fibonacci, the Italian mathematician who established the Hindu–Arabic numeral system to the Western World

Aristotle defined mathematics as "the science of quantity", and this definition prevailed until the 18th century. Starting in the 19th century, when the study of mathematics increased in rigor and began to address abstract topics such as group theory and projective geometry, which have no clear-cut relation to quantity and measurement, mathematicians and philosophers began to propose a variety of new definitions. Some of these definitions emphasize the deductive character of much of mathematics, some emphasize its abstractness, some emphasize certain topics within mathematics. Today, no consensus on the definition of mathematics prevails, even among professionals. There is not even consensus on whether mathematics is an art or a science. A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable. Some just say, "Mathematics is what mathematicians do."

Three leading types of definition of mathematics are called logicist, intuitionist, and formalist, each reflecting a different philosophical school of thought. All have severe problems, none has widespread acceptance, and no reconciliation seems possible.

An early definition of mathematics in terms of logic was Benjamin Peirce's "the science that draws necessary conclusions" (1870). In the Principia Mathematica, Bertrand Russell and Alfred North Whitehead advanced the philosophical program known as logicism, and attempted to prove that all mathematical concepts, statements, and principles can be defined and proven entirely in terms of symbolic logic. A logicist definition of mathematics is Russell's "All Mathematics is Symbolic Logic" (1903).

Intuitionist definitions, developing from the philosophy of mathematician L.E.J. Brouwer, identify mathematics with certain mental phenomena. An example of an intuitionist definition is "Mathematics is the mental activity which consists in carrying out constructs one after the other." A peculiarity of intuitionism is that it rejects some mathematical ideas considered valid according to other definitions. In particular, while other philosophies of mathematics allow objects that can be proven to exist even though they cannot be constructed, intuitionism allows only mathematical objects that one can actually construct.

Formalist definitions identify mathematics with its symbols and the rules for operating on them. Haskell Curry defined mathematics simply as "the science of formal systems". A formal system is a set of symbols, or tokens, and some rules telling how the tokens may be combined into formulas. In formal systems, the word axiom has a special meaning, different from the ordinary meaning of "a self-evident truth". In formal systems, an axiom is a combination of tokens that is included in a given formal system without needing to be derived using the rules of the system.

Mathematics as science

Carl Friedrich Gauss, known as the prince of mathematicians

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; mathematics is in this sense a field of knowledge. The specialization restricting the meaning of "science" to natural science follows the rise of Baconian science, which contrasted "natural science" to scholasticism, the Aristotelean method of inquiring from first principles. 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." More recently, Marcus du Sautoy has called mathematics "the Queen of Science ... the main driving force behind scientific discovery".

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. The theoretical physicist J.M. Ziman proposed that science is public knowledge, and thus includes mathematics. Mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures 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.

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.

Inspiration, pure and applied mathematics, and aesthetics

Main article: Mathematical beauty

Isaac Newton (left) and Gottfried Wilhelm Leibniz (right), developers of infinitesimal calculus

Mathematics arises from many different kinds of problems. At first these were found in commerce, land measurement, architecture and later astronomy; today, 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 relevant only 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 numbers, and in an elegant numerical method that speeds calculation, such as the fast Fourier transform. G.H. Hardy in A Mathematician's Apology 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

Main article: Mathematical notation

Leonhard Euler, who created and popularized much of the mathematical notation used today

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 open and field have been given specialized mathematical meanings. Technical terms such as homeomorphism and integrable 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 "theorems", 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 proofs. Since large computations are hard to verify, such proofs may not be sufficiently rigorous.

Axioms 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

See also: Areas of mathematics and Glossary of areas of mathematics

An abacus, a simple calculating tool used since ancient times

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 structures 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 axiomatic framework, and studying the implications of such a framework. As such, it is home to Gödel's incompleteness theorems which (informally) imply that any effective 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 the rapid advancement 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	Set theory	Category theory	Theory of computation

Pure mathematics

Quantity

The study of quantity starts with numbers, first the familiar natural numbers and integers ("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 numbers ("fractions"). These, in turn, are contained within the real numbers, which are used to represent continuous quantities. Real numbers are generalized to complex numbers. These are the first steps of a hierarchy of numbers that goes on to include quaternions and octonions. Consideration of the natural numbers also leads to the transfinite numbers, which formalize the concept of "infinity". Another area of study is size, which leads to the cardinal numbers and then to another conception of infinity: the aleph numbers, which allow meaningful comparison of the size of infinitely large sets.

Natural numbers	Integers	Rational numbers	Real numbers	Complex numbers

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 integers 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 axioms 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 spaces, 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	Number theory	Group theory	Graph theory	Order theory	Algebra

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 were developed to solve problems in number theory and functional analysis but now are pursued with an eye on applications in optimization and computer science. Within differential geometry are the concepts of fiber bundles and calculus on manifolds, 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 groups 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, and the still unsolved areas of the Hodge 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	Trigonometry	Differential geometry	Topology	Fractal geometry	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 numbers and functions of a real variable is known as real analysis, with complex analysis the equivalent field for the complex numbers. 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 equations. 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.

Calculus	Vector calculus	Differential equations	Dynamical systems	Chaos theory	Complex analysis

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 professional specialty 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, 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 mathematics proposes and studies methods for solving mathematical problems 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 algorithmic matrix and graph theory. Other areas of computational mathematics include computer algebra and symbolic computation.

Mathematical physics	Fluid dynamics	Numerical analysis	Optimization	Probability theory	Statistics	Cryptography
Mathematical finance	Game theory	Mathematical biology	Mathematical chemistry	Mathematical economics	Control theory

Mathematical awards

Arguably the most prestigious award in mathematics is the Fields Medal, established in 1936 and now awarded every four 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 problems, 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. A solution to each of these problems carries a $1 million reward, and only one (the Riemann hypothesis) is duplicated in Hilbert's problems.

See also

Main article: Lists of mathematics topics

• Mathematics and art
• Mathematics education
• Relationship between mathematics and physics
• STEM fields

Notes

References

• Courant, Richard and H. Robbins, What Is Mathematics? : An Elementary Approach to Ideas and Methods, Oxford University Press, USA; 2 edition (July 18, 1996). ISBN 0-19-510519-2.
• Einstein, Albert (1923). Sidelights on Relativity: I. Ether and relativity. II. Geometry and experience (translated by G.B. Jeffery, D.Sc., and W. Perrett, Ph.D). E.P. Dutton & Co., New York. 
• du Sautoy, Marcus, A Brief History of Mathematics, BBC Radio 4 (2010).
• Eves, Howard, An Introduction to the History of Mathematics, Sixth Edition, Saunders, 1990, ISBN 0-03-029558-0.
• Kline, Morris, Mathematical Thought from Ancient to Modern Times, Oxford University Press, USA; Paperback edition (March 1, 1990). ISBN 0-19-506135-7.
• Monastyrsky, Michael (2001). "Some Trends in Modern Mathematics and the Fields Medal" (PDF). Canadian Mathematical Society. Retrieved July 28, 2006. 
• Oxford English Dictionary, second edition, ed. John Simpson and Edmund Weiner, Clarendon Press, 1989, ISBN 0-19-861186-2.
• The Oxford Dictionary of English Etymology, 1983 reprint. ISBN 0-19-861112-9.
• Pappas, Theoni, The Joy Of Mathematics, Wide World Publishing; Revised edition (June 1989). ISBN 0-933174-65-9.
• Peirce, Benjamin (1881). Peirce, Charles Sanders, ed. "Linear associative algebra". American Journal of Mathematics (Corrected, expanded, and annotated revision with an 1875 paper by B. Peirce and annotations by his son, C.S. Peirce, of the 1872 lithograph ed.) (Johns Hopkins University) 4 (1–4): 97–229. doi:10.2307/2369153. JSTOR 2369153. Corrected, expanded, and annotated revision with an 1875 paper by B. Peirce and annotations by his son, C. S. Peirce, of the 1872 lithograph ed. Google Eprint and as an extract, D. Van Nostrand, 1882, Google Eprint. .
• Peterson, Ivars, Mathematical Tourist, New and Updated Snapshots of Modern Mathematics, Owl Books, 2001, ISBN 0-8050-7159-8.
• Popper, Karl R. (1995). "On knowledge". In Search of a Better World: Lectures and Essays from Thirty Years. Routledge. ISBN 0-415-13548-6. 
• Riehm, Carl (August 2002). "The Early History of the Fields Medal" (PDF). Notices of the AMS (AMS) 49 (7): 778–782. 
• Sevryuk, Mikhail B. (January 2006). "Book Reviews" (PDF). Bulletin of the American Mathematical Society 43 (1): 101–109. doi:10.1090/S0273-0979-05-01069-4. Retrieved June 24, 2006. 
• Waltershausen, Wolfgang Sartorius von (1965) [first published 1856]. Gauss zum Gedächtniss. Sändig Reprint Verlag H. R. Wohlwend. ASIN B0000BN5SQ. ISBN 3-253-01702-8. ASIN 3253017028. 

Further reading

• Benson, Donald C., The Moment of Proof: Mathematical Epiphanies, Oxford University Press, USA; New Ed edition (December 14, 2000). ISBN 0-19-513919-4.
• Boyer, Carl B., A History of Mathematics, Wiley; 2nd edition, revised by Uta C. Merzbach, (March 6, 1991). ISBN 0-471-54397-7.—A concise history of mathematics from the Concept of Number to contemporary Mathematics.
• Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Mariner Books; Reprint edition (January 14, 1999). ISBN 0-395-92968-7.
• Gullberg, Jan, 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.
• Jourdain, Philip E. B., The Nature of Mathematics, in The World of Mathematics, James R. Newman, editor, Dover Publications, 2003, ISBN 0-486-43268-8.
• Maier, Annaliese, At the Threshold of Exact Science: Selected Writings of Annaliese Maier on Late Medieval Natural Philosophy, edited by Steven Sargent, Philadelphia: University of Pennsylvania Press, 1982.

External links

At Wikiversity, you can learn
more and teach others about Mathematics at the School of Mathematics.

Library resources about Mathematics

Resources in your library

• Mathematics at Encyclopædia Britannica
• Mathematics on In Our Time at the BBC. (listen now)
• Free Mathematics books Free Mathematics books collection.
• Encyclopaedia of Mathematics 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 physics), 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 archive 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 University
• Open Problem Garden, a wiki 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 TeX 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).
• du Sautoy, Marcus, A Brief History of Mathematics, BBC Radio 4 (2010).
• MathOverflow A Q&A site for research-level mathematics
• Math – Khan Academy
• National Museum of Mathematics, located in New York City

UpToDate Contents

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English Journal

• A mathematical model for Crimean-Congo haemorrhagic fever: tick-borne dynamics with conferred host immunity.

• Switkes J1, Nannyonga B2, Mugisha JY2, Nakakawa J2.
• Journal of biological dynamics.J Biol Dyn.2016 Dec;10(1):59-70. doi: 10.1080/17513758.2015.1102976.
• Crimean-Congo haemorrhagic fever (CCHF) is a highly contagious tick-borne disease that impacts many countries in parts of Africa, Europe, Asia, and the Middle East. Outbreaks are episodic, but deadly. Due to the highly contagious nature of this disease, suspected cases are taken extremely serious,
• PMID 26550705

• Imitation dynamics of vaccine decision-making behaviours based on the game theory.

• Yang J1, Martcheva M2, Chen Y1,3.
• Journal of biological dynamics.J Biol Dyn.2016 Dec;10(1):31-58. doi: 10.1080/17513758.2015.1099749.
• Based on game theory, we propose an age-structured model to investigate the imitation dynamics of vaccine uptake. We first obtain the existence and local stability of equilibria. We show that Hopf bifurcation can occur. We also establish the global stability of the boundary equilibria and persistenc
• PMID 26536171

• A model of the Ebola epidemics in West Africa incorporating age of infection.

• Webb GF1, Browne CJ1.
• Journal of biological dynamics.J Biol Dyn.2016 Dec;10(1):18-30. doi: 10.1080/17513758.2015.1090632.
• A model of an Ebola epidemic is developed with infected individuals structured according to disease age. The transmission of the infection is tracked by disease age through an initial incubation (exposed) phase, followed by an infectious phase with variable transmission infectiousness. The removal o
• PMID 26536076

• A Radial Basis Function (RBF)-Finite Difference (FD) Method for Diffusion and Reaction-Diffusion Equations on Surfaces.

• Shankar V1, Wright GB2, Kirby RM1, Fogelson AL3.
• Journal of scientific computing.J Sci Comput.2016 Jun 1;63(3):745-768.
• In this paper, we present a method based on Radial Basis Function (RBF)-generated Finite Differences (FD) for numerically solving diffusion and reaction-diffusion equations (PDEs) on closed surfaces embedded in ℝ d . Our method uses a method-of-lines formulation, in which surface derivatives that
• PMID 25983388

Japanese Journal

• 適尽方級法の現代数学への応用

• 涌田 和芳,外川 一仁
• 数学史研究 = Journal of history of mathematics
• NAID 40020400132

• 授時暦と関孝和・建部賢弘の招差法対貞享暦と渋川春海の招差法

• 藤井 康生
• 数学史研究 = Journal of history of mathematics
• NAID 40020400097

• 麟徳暦の定朔計算と暦日の問題 : 『日本暦日原点』の問題点を含めて(1)

• 横塚 啓之
• 数学史研究 = Journal of history of mathematics
• NAID 40020400041

• General Mixed Multi-Soliton Solutions to One-Dimensional Multicomponent Yajima–Oikawa System

• Chen Junchao,Chen Yong,Feng Bao-Feng,Maruno Ken-ichi
• Journal of the Physical Society of Japan 84(7), 2015-06-15
• NAID 160000000775

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　　[★]

• adj.

• 数学的な、数理的な

関

mathematical、mathematically