# Mathematics

Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory,[1] algebra,[2] geometry,[1] and analysis,[3][4] respectively. There is no general consensus among mathematicians about a common definition for their academic discipline.

Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature or—in modern mathematics—entities that are stipulated to have certain properties, called axioms. A proof consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, and—in case of abstraction from nature—some basic properties that are considered true starting points of the theory under consideration.[5]

Mathematics is essential in the natural sciences, engineering, medicine, finance, computer science and the social sciences. Although mathematics is extensively used for modeling phenomena, the fundamental truths of mathematics are independent from any scientific experimentation. Some areas of mathematics, such as statistics and game theory, are developed in close correlation with their applications and are often grouped under applied mathematics. Other areas are developed independently from any application (and are therefore called pure mathematics), but often later find practical applications.[6][7] The problem of integer factorization, for example, which goes back to Euclid in 300 BC, had no practical application before its use in the RSA cryptosystem, now widely used for the security of computer networks.

Historically, the concept of a proof and its associated mathematical rigour first appeared in Greek mathematics, most notably in Euclid's Elements.[8] Since its beginning, mathematics was essentially divided into geometry and arithmetic (the manipulation of natural numbers and fractions), until the 16th and 17th centuries, when algebra[a] and infinitesimal calculus were introduced as new areas. Since then, the interaction between mathematical innovations and scientific discoveries has led to a rapid lockstep increase in the development of both.[9] At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the axiomatic method,[10] which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary Mathematics Subject Classification lists more than 60 first-level areas of mathematics.

1. ^ a b "mathematics, n.". Oxford English Dictionary. Oxford University Press. 2012. Archived from the original on November 16, 2019. Retrieved June 16, 2012. The science of space, number, quantity, and arrangement, whose methods involve logical reasoning and usually the use of symbolic notation, and which includes geometry, arithmetic, algebra, and analysis.
2. ^ Kneebone, G. T. (1963). Mathematical Logic and the Foundations of Mathematics: An Introductory Survey. Dover. p. 4. ISBN 978-0-486-41712-7. Retrieved June 20, 2015. Mathematics ... is simply the study of abstract structures, or formal patterns of connectedness.
3. ^ LaTorre, Donald R.; Kenelly, John W.; Biggers, Sherry S.; Carpenter, Laurel R.; Reed, Iris B.; Harris, Cynthia R. (2011). Calculus Concepts: An Informal Approach to the Mathematics of Change. Cengage Learning. p. 2. ISBN 978-1-4390-4957-0. Retrieved June 20, 2015. Calculus is the study of change—how things change, and how quickly they change.
4. ^ Ramana, B. V. (2007). Applied Mathematics. Tata McGraw–Hill Education. p. 2.10. ISBN 978-0-07-066753-2. Retrieved July 30, 2022. The mathematical study of change, motion, growth or decay is calculus.
5. ^ Hipólito, Inês Viegas (2015). "Abstract Cognition and the Nature of Mathematical Proof" (PDF). In Kanzian, Christian; Mitterer, Josef; Neges, Katharina (eds.). Realism – Relativism – Constructivism. 38th International Wittgenstein Symposium, August 9–15, 2015. Kirchberg am Wechsel, Austria. Austrian Ludwig Wittgenstein Society. pp. 132–134. Archived (PDF) from the original on November 7, 2022. Retrieved November 5, 2022. (at ResearchGate Archived November 5, 2022, at the Wayback Machine)
6. ^ Peterson 2001, p. 12.
7. ^ Cite error: The named reference `wigner1960` was invoked but never defined (see the help page).
8. ^ Wise, David. "Eudoxus' Influence on Euclid's Elements with a close look at The Method of Exhaustion". jwilson.coe.uga.edu. Archived from the original on June 1, 2019. Retrieved October 26, 2019.
9. ^ Alexander, Amir (2011). "The Skeleton in the Closet: Should Historians of Science Care about the History of Mathematics?". Isis. 102 (3): 475–480. doi:10.1086/661620. PMID 22073771. S2CID 21629993.
10. ^ Cite error: The named reference `Kleiner_1991` was invoked but never defined (see the help page).

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