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a b Ferreirós, J. (2007). "Ό Θεὸς Άριθμητίζει: The Rise of Pure Mathematics as Arithmetic with Gauss". In Goldstein, Catherine; Schappacher, Norbert; Schwermer, Joachim (eds.). The Shaping of Arithmetic after C.F. Gauss's Disquisitiones Arithmeticae. Springer Science & Business Media. pp.235–268. ISBN 978-3-540-34720-0.

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Harper, Douglas. "mathematical (adj.)". Online Etymology Dictionary. Archived from the original on November 26, 2022 . Retrieved November 26, 2022. Medicine uses statistical hypothesis testing, run on data from clinical trials, to determine whether a new treatment works. [ citation needed] Ore, Øystein (1988). Number Theory and Its History. Courier Corporation. pp.19–24. ISBN 978-0-486-65620-5 . Retrieved November 14, 2022. In the 19th century, the internal development of geometry (pure mathematics) led to definition and study of non-Euclidean geometries, spaces of dimension higher than three and manifolds. At this time, these concepts seemed totally disconnected from the physical reality, but at the beginning of the 20th century, Albert Einstein developed the theory of relativity that uses fundamentally these concepts. In particular, spacetime of special relativity is a non-Euclidean space of dimension four, and spacetime of general relativity is a (curved) manifold of dimension four. [126] [127]

Game theory (although continuous games are also studied, most common games, such as chess and poker are discrete) Integration, measure theory and potential theory, all strongly related with probability theory on a continuum; Main article: Geometry On the surface of a sphere, Euclidean geometry only applies as a local approximation. For larger scales the sum of the angles of a triangle is not equal to 180°. Takase, M. (2014). "Pure Mathematics and Applied Mathematics are Inseparably Intertwined: Observation of the Early Analysis of the Infinity". A Mathematical Approach to Research Problems of Science and Technology. Mathematics for Industry. Vol.5. Tokyo: Springer. pp.393–399. do

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Main articles: Calculus and Mathematical analysis A Cauchy sequence consists of elements such that all subsequent terms of a term become arbitrarily close to each other as the sequence progresses (from left to right). A notable example is the prime factorization of natural numbers that was discovered more than 2,000 years before its common use for secure internet communications through the RSA cryptosystem. [124] A second historical example is the theory of ellipses. They were studied by the ancient Greek mathematicians as conic sections (that is, intersections of cones with planes). It is almost 2,000 years later that Johannes Kepler discovered that the trajectories of the planets are ellipses. [125] Restivo, S. (December 2013). Mathematics in Society and History. Springer Netherlands. pp.14–15. ISBN 978-94-011-2944-2 . Retrieved March 19, 2023. Corry, Leo (December 6, 2012). Modern Algebra and the Rise of Mathematical Structures. Birkhäuser Basel. pp.247–252. ISBN 978-3-0348-7917-0 . Retrieved March 19, 2023. The aftermath of World War II led to a surge in the development of applied mathematics in the US and elsewhere. [114] [115] Many of the theories developed for applications were found interesting from the point of view of pure mathematics, and many results of pure mathematics were shown to have applications outside mathematics; in turn, the study of these applications may give new insights on the "pure theory". [116] [117]Hilbert, David (1962). The Foundations of Geometry. Open Court Publishing Company. p.1 . Retrieved March 22, 2023. Notes that sound well together to a Western ear are sounds whose fundamental frequencies of vibration are in simple ratios. For example, an octave doubles the frequency and a perfect fifth multiplies it by 3 2 {\displaystyle {\frac {3}{2}}} . [186] [187] [ bettersourceneeded] Letourneau, Mary; Wright Sharp, Jennifer (2017). "AMS style guide" (PDF). American Mathematical Society. p.75. Archived (PDF) from the original on December 8, 2022 . Retrieved November 16, 2022.

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Krömer, Ralph (2007). Tool and Object: A History and Philosophy of Category Theory. Science Networks. Historical Studies. Vol.32. Springer Science & Business Media. pp.xxi–xxv, 1–91. ISBN 978-3-7643-7524-9 . Retrieved November 25, 2022. Archibald, Raymond Clare (January 1949). "History of Mathematics After the Sixteenth Century". The American Mathematical Monthly. Part 2: Outline of the History of Mathematics. 56 (1): 35–56. doi: 10.2307/2304570. JSTOR 2304570. A fundamental innovation was the ancient Greeks' introduction of the concept of proofs, which require that every assertion must be proved. For example, it is not sufficient to verify by measurement that, say, two lengths are equal; their equality must be proven via reasoning from previously accepted results ( theorems) and a few basic statements. The basic statements are not subject to proof because they are self-evident ( postulates), or are part of the definition of the subject of study ( axioms). This principle, foundational for all mathematics, was first elaborated for geometry, and was systematized by Euclid around 300 BC in his book Elements. [32] [33]This became the foundational crisis of mathematics. [57] It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a formalized set theory. Roughly speaking, each mathematical object is defined by the set of all similar objects and the properties that these objects must have. [23] For example, in Peano arithmetic, the natural numbers are defined by "zero is a number", "each number has a unique successor", "each number but zero has a unique predecessor", and some rules of reasoning. [58] This mathematical abstraction from reality is embodied in the modern philosophy of formalism, as founded by David Hilbert around 1910. [59] Stolz, Michael (2002). "The History Of Applied Mathematics And The History Of Society". Synthese. 133: 43–57. doi: 10.1023/A:1020823608217. S2CID 34271623 . Retrieved November 20, 2022. Sipser, Michael (July 1992). The history and status of the P versus NP question. STOC '92: Proceedings of the twenty-fourth annual ACM symposium on Theory of Computing. pp.603–618. doi: 10.1145/129712.129771. Halpern, Joseph; Harper, Robert; Immerman, Neil; Kolaitis, Phokion; Vardi, Moshe; Vianu, Victor (2001). "On the Unusual Effectiveness of Logic in Computer Science" (PDF). Archived (PDF) from the original on March 3, 2021 . Retrieved January 15, 2021.