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Mathematics 𓆩⚪𓆪｜Definition｜1st｜20251119205401-00-⌔

Mathematics - Wikipedia

Mathematics

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Mathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities. It uses logical reasoning and proof to study and establish their properties, often expressed as theorems, formulas, and equations. Mathematics is used to model and solve problems in science, engineering, technology, economics, and everyday life.

There are many areas of mathematics, including number theory (the study of integers and their properties), algebra (the study of operations and the structures they form), geometry (the study of shapes and spaces that contain them), analysis (the study of approximating continuous changes), and set theory (presently used as a foundation for all mathematics).

Mathematics involves the description and manipulation of abstract objects that are either abstractions from nature or purely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to prove the properties of objects through proofs, which consist of a succession of applications of deductive rules to already established results. These results, called theorems, 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.1

Mathematics is essential in the natural sciences, engineering, medicine, finance, computer science, and the social sciences. Although mathematics is widely used to model empirical phenomena, its results are established by deductive proof rather than by experiment. The relationship between mathematical truth, logic, and reality is a subject of philosophical debate. Some areas of mathematics, such as 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 but often find practical applications later.23

Mathematical written records first appeared in Ancient Egypt and Mesopotamia, but the concept of proof and its associated mathematical rigor began in Ancient Greek mathematics, exemplified in Euclid’s Elements.4 Mathematics was primarily divided into geometry and arithmetic until the 16th and 17th centuries, when algebra5 and infinitesimal calculus evolved into new fields. Since then, the interaction between mathematical innovations and scientific discoveries has led to a correlated increase in the development of both.6 At the end of the 19th century, the foundational crisis of mathematics led to the systematic use of the axiomatic method,7 which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary Mathematics Subject Classification lists more than sixty first-level areas of mathematics.89

Printed 2026-06-28.

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Footnotes

  1. Hipólito, Inês Viegas (Aug 9–15, 2015). “Abstract Cognition and the Nature of Mathematical Proof”. In Kanzian, Christian; Mitterer, Josef; Neges, Katharina (eds.). Realismus – Relativismus – Konstruktivismus: Beiträge des 38. Internationalen Wittgenstein Symposiums [Realism – Relativism – Constructivism: Contributions of the 38th International Wittgenstein Symposium] (PDF) (in German and English). Vol. 23. Kirchberg am Wechsel, Austria: Austrian Ludwig Wittgenstein Society. pp. 132–134. ISSN 1022-3398. OCLC 236026294. Archived (PDF) from the original on 2022-11-07. Retrieved 2024-01-17. (at ResearchGate Archived November 5, 2022, at the Wayback Machine) ↩

  2. Peterson 1988, p. 12. ↩

  3. Wigner, Eugene (1960). “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”. Communications on Pure and Applied Mathematics. 13 (1): 1–14. Bibcode:1960CPAM…13…1W. doi:10.1002/cpa.3160130102. S2CID 6112252. Archived from the original on February 28, 2011. ↩

  4. Wise, David. “Eudoxus’ Influence on Euclid’s Elements with a close look at The Method of Exhaustion”. The University of Georgia. Archived from the original on 2019-06-01. Retrieved 2024-01-18. ↩

  5. Here, algebra is taken in its modern sense, which is, roughly speaking, the art of manipulating formulas. ↩

  6. Alexander, Amir (Sep 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. MR 2884913. PMID 22073771. ↩

  7. Kleiner, Israel (Dec 1991). “Rigor and Proof in Mathematics: A Historical Perspective”. Mathematics Magazine. 64 (5). Taylor & Francis, Ltd.: 291–314. doi:10.1080/0025570X.1991.11977625. eISSN 1930-0980. ISSN 0025-570X. JSTOR 2690647. LCCN 47003192. MR 1141557. OCLC 1756877. ↩

  8. “MSC2020-Mathematics Subject Classification System” (PDF). zbMath. Associate Editors of Mathematical Reviews and zbMATH. Archived (PDF) from the original on 2024-01-02. Retrieved 2025-11-13. ↩

  9. Dunne, Edward; Hulek, Klaus (Mar 2020). “Mathematics Subject Classification 2020” (PDF). Notices of the American Mathematical Society. 67 (3): 410–411. doi:10.1090/noti2052. eISSN 1088-9477. ISSN 0002-9920. LCCN sf77000404. OCLC 1480366. Archived (PDF) from the original on 2021-08-03. Retrieved 2024-02-03. The new MSC contains 63 two-digit classifications, 529 three-digit classifications, and 6,006 five-digit classifications. ↩

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