# Mathematics

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

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") includes the study of such topics as quantity,[1] structure,[2] space,[1] and change.[3][4][5]

Mathematicians seek and use patterns[6][7] to formulate new conjectures; they 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 from 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.[8]

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."[9] Carl Friedrich Gauss (1777–1855) referred to mathematics as "the Queen of the Sciences".[10] Benjamin Peirce (1809–1880) called mathematics "the science that draws necessary conclusions".[11] 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."[12] 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."[13]

Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics, or mathematics for its own sake, without having any application in mind. Practical applications for what began as pure mathematics are often discovered.[14][15]

## History

The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.
Archimedes used the method of exhaustion to approximate the value of pi.
The numerals used in the Bakhshali manuscript, dated between the 2nd century BCE and the 2nd century CE.

The history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, which is shared by many animals,[16] 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.

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.[17]

Evidence for more complex mathematics does 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.[18] The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry. It is in Babylonian mathematics that elementary arithmetic (addition, subtraction, multiplication and division) first appear in the archaeological record. The Babylonians also possessed a place-value system, and used a sexagesimal numeral system, still in use today for measuring angles and time.[19]

Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics.[20] Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom, theorem, and proof. His textbook Elements is widely considered the most successful and influential textbook of all time.[21] The greatest mathematician of antiquity is often held to be Archimedes (c. 287–212 BC) of Syracuse.[22] He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.[23] Other notable achievements of Greek mathematics are conic sections (Apollonius of Perga, 3rd century BC),[24] trigonometry (Hipparchus of Nicaea (2nd century BC),[25] and the beginnings of algebra (Diophantus, 3rd century AD).[26]

The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, and an early form of infinite series.

A page from al-Khwārizmī's Algebra

During the Golden Age of Islam, especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics. The most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī.

During the early modern period, mathematics began to develop at an accelerating pace in Western Europe. The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries. Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, analysis, differential geometry, matrix theory,number theory, and statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system that is consistent will contain unprovable propositions.

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."[27]

### Etymology

The word mathematics comes from Ancient Greek μάθημα (máthēma), meaning "that which is learnt",[28] "what one gets to know", hence also "study" and "science". The word for "mathematics" came to have the narrower and more technical meaning "mathematical study" even in Classical times.[29] 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".

Similarly, one of the two main schools of thought in Pythagoreanism was known as the mathēmatikoi (μαθηματικοί)—which at the time meant "teachers" rather than "mathematicians" in the modern sense.

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. For example, Saint Augustine's warning that Christians should beware of mathematici, meaning astrologers, is sometimes mistranslated as a condemnation of mathematicians.[30]

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 τὰ 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 Greek.[31] In English, the noun mathematics takes a singular verb. It is often shortened to maths or, in North America, math.[32]

Other Languages
Afrikaans: Wiskunde
Alemannisch: Mathematik
Ænglisc: Rīmcræft
العربية: رياضيات
aragonés: Matematicas
armãneashti: Mathematică
অসমীয়া: গণিত
asturianu: Matemátiques
Avañe'ẽ: Papapykuaa
Aymar aru: Jakhu
azərbaycanca: Riyaziyyat
تۆرکجه: ریاضیات
বাংলা: গণিত
Bahasa Banjar: Matamatika
Bân-lâm-gú: Sò͘-ha̍k
Basa Banyumasan: Matematika
башҡортса: Математика
беларуская: Матэматыка
беларуская (тарашкевіца)‎: Матэматыка
भोजपुरी: गणित
Bislama: Matematikis
български: Математика
Boarisch: Mathematik
བོད་ཡིག: རྩིས་རིག
bosanski: Matematika
brezhoneg: Matematik
català: Matemàtiques
Чӑвашла: Математика
Cebuano: Matematika
čeština: Matematika
Chamoru: Matematika
chiShona: Masvomhu
corsu: Matematica
Cymraeg: Mathemateg
dansk: Matematik
Deutsch: Mathematik
ދިވެހިބަސް: ރިޔާޟިއްޔާތު
dolnoserbski: Matematika
Ελληνικά: Μαθηματικά
emiliàn e rumagnòl: Matemâtica
эрзянь: Математика
español: Matemáticas
Esperanto: Matematiko
estremeñu: Matemáticas
euskara: Matematika
فارسی: ریاضیات
Fiji Hindi: Mathematics
føroyskt: Støddfrøði
français: Mathématiques
Frysk: Wiskunde
furlan: Matematiche
Gaeilge: Matamaitic
Gàidhlig: Matamataig
galego: Matemáticas
ГӀалгӀай: Математика

ગુજરાતી: ગણિત

хальмг: Математика
한국어: 수학
Hausa: Lissafi
Hawaiʻi: Makemakika
հայերեն: Մաթեմատիկա
हिन्दी: गणित
hrvatski: Matematika
Ilokano: Matematika
বিষ্ণুপ্রিয়া মণিপুরী: গণিত
Bahasa Indonesia: Matematika
interlingua: Mathematica
Interlingue: Matematica
isiXhosa: I-Mathematics
isiZulu: Imathemathiki
íslenska: Stærðfræði
italiano: Matematica
עברית: מתמטיקה
Basa Jawa: Matématika
kalaallisut: Matematikki
ಕನ್ನಡ: ಗಣಿತ
къарачай-малкъар: Математика
ქართული: მათემატიკა
kaszëbsczi: Matematika
қазақша: Математика
Kiswahili: Hisabati
Kreyòl ayisyen: Matematik
kurdî: Matematîk
Кыргызча: Математика
Latina: Mathematica
latviešu: Matemātika
Lëtzebuergesch: Mathematik
lietuvių: Matematika
Ligure: Matematica
Limburgs: Mathematiek
Lingua Franca Nova: Matematica
Livvinkarjala: Matematiekku
la .lojban.: cmaci
Luganda: Ekibalangulo
lumbaart: Matemàtega
magyar: Matematika
मैथिली: गणित
македонски: Математика
Malagasy: Fanisana
മലയാളം: ഗണിതം
Malti: Matematika
मराठी: गणित
მარგალური: მათემატიკა
مصرى: رياضيات
مازِرونی: ریاضی
Bahasa Melayu: Matematik
Mìng-dĕ̤ng-ngṳ̄: Só-hŏk
Mirandés: Matemática
монгол: Математик
မြန်မာဘာသာ: သင်္ချာ
Nederlands: Wiskunde
Nedersaksies: Wiskunde
नेपाली: गणित
नेपाल भाषा: ल्याःज्या

Nordfriisk: Matematiik
Norfuk / Pitkern: Maethamatiks
norsk: Matematikk
norsk nynorsk: Matematikk
Nouormand: Caltchul
Novial: Matematike
occitan: Matematicas
олык марий: Математике
ଓଡ଼ିଆ: ଗଣିତ
Oromoo: Herrega
oʻzbekcha/ўзбекча: Matematika
ਪੰਜਾਬੀ: ਗਣਿਤ
पालि: गणितं
Pangasinan: Matematiks
پنجابی: میتھمیٹکس
Patois: Matimatix
ភាសាខ្មែរ: គណិតវិទ្យា
Picard: Matématikes
Piemontèis: Matemàtica
Tok Pisin: Ol matematik
Plattdüütsch: Mathematik
polski: Matematyka
português: Matemática
Qaraqalpaqsha: Matematika
qırımtatarca: Riyaziyat
română: Matematică
Runa Simi: Yupay yachay
русиньскый: Математіка
русский: Математика
саха тыла: Математика
ᱥᱟᱱᱛᱟᱲᱤ: ᱮᱞᱠᱷᱟ
Gagana Samoa: Matematika
संस्कृतम्: गणितम्
Seeltersk: Mathematik
Setswana: Mathematics
shqip: Matematika
sicilianu: Matimàtica
සිංහල: ගණිතය
Simple English: Mathematics
سنڌي: رياضي
SiSwati: Tekubala
slovenčina: Matematika
slovenščina: Matematika
ślůnski: Matymatyka
Soomaaliga: Xisaab
کوردی: ماتماتیک
Sranantongo: Sabi fu Teri
српски / srpski: Математика
srpskohrvatski / српскохрватски: Matematika
Basa Sunda: Matematika
svenska: Matematik
Tagalog: Matematika
தமிழ்: கணிதம்
Taqbaylit: Tusnakt
татарча/tatarça: Математика
తెలుగు: గణితము
тоҷикӣ: Риёзиёт
ತುಳು: ಗಣಿತ
Türkçe: Matematik
Türkmençe: Matematika
Thuɔŋjäŋ: Akuënkäŋ
ᨅᨔ ᨕᨘᨁᨗ: ᨆᨈᨛᨆᨈᨗᨀ
українська: Математика
اردو: ریاضی
Vahcuengh: Soqyoz
vèneto: Matemàtega
vepsän kel’: Matematik
Tiếng Việt: Toán học
Volapük: Matemat

West-Vlams: Wiskunde
Winaray: Matematika
Wolof: Xayma

ייִדיש: מאטעמאטיק
Yorùbá: Mathimátíkì

Zazaki: Matematik
Zeêuws: Wiskunde
žemaitėška: Matematėka