Perisho, Margaret W. (Spring 1965). "The Etymology of Mathematical Terms". Pi Mu Epsilon Journal. 4 (2): 62–66. JSTOR 24338341. The validity of a mathematical theorem relies only on the rigor of its proof, which could theoretically be done automatically by a computer program. This does not mean that there is no place for creativity in a mathematical work. On the contrary, many important mathematical results (theorems) are solutions of problems that other mathematicians failed to solve, and the invention of a way for solving them may be a fundamental way of the solving process. [178] [179] An extreme example is Apery's theorem: Roger Apery provided only the ideas for a proof, and the formal proof was given only several months later by three other mathematicians. [180] Stewart, Ian (2018). "Mathematics, Maps, and Models". In Wuppuluri, Shyam; Doria, Francisco Antonio (eds.). The Map and the Territory: Exploring the Foundations of Science, Thought and Reality. The Frontiers Collection. Springer. pp.345–356. doi: 10.1007/978-3-319-72478-2_18. ISBN 978-3-319-72478-2 . Retrieved November 17, 2022. Some renowned mathematicians have also been considered to be renowned astrologists; for example, Ptolemy, Arab astronomers, Regiomantus, Cardano, Kepler, or John Dee. In the Middle Ages, astrology was considered a science that included mathematics. In his encyclopedia, Theodor Zwinger wrote that astrology was a mathematical science that studied the "active movement of bodies as they act on other bodies". He reserved to mathematics the need to "calculate with probability the influences [of stars]" to foresee their "conjunctions and oppositions". [151] Bishop, Alan (1991). "Environmental activities and mathematical culture". Mathematical Enculturation: A Cultural Perspective on Mathematics Education. Norwell, Massachusetts: Kluwer Academic Publishers. pp.20–59. ISBN 978-0-7923-1270-3 . Retrieved April 5, 2020.

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Harper, Douglas. "mathematical (adj.)". Online Etymology Dictionary. Archived from the original on November 26, 2022 . Retrieved November 26, 2022.

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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] Marker, Dave (July 1996). "Model theory and exponentiation". Notices of the American Mathematical Society. 43 (7): 753–759. Archived from the original on March 13, 2014 . Retrieved November 19, 2022.

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For considering as reliable a large computation occurring in a proof, one generally requires two computations using independent software 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°. The two subjects of mathematical logic and set theory have belonged to mathematics since the end of the 19th century. [52] [53] Before this period, sets were not considered to be mathematical objects, and logic, although used for mathematical proofs, belonged to philosophy and was not specifically studied by mathematicians. [54] Oaks, J. A. (2018). "François Viète's revolution in algebra" (PDF). Archive for History of Exact Sciences. 72 (3): 245–302. doi: 10.1007/s00407-018-0208-0. S2CID 125704699. Archived (PDF) from the original on November 8, 2022 . Retrieved November 8, 2022.

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Weil, André (2007). Number Theory, An Approach Through History From Hammurapi to Legendre. Birkhäuser Boston. pp.1–3. ISBN 978-0-8176-4571-7 . Retrieved March 19, 2023. Moschovakis, Joan (September 4, 2018). "Intuitionistic Logic". Stanford Encyclopedia of Philosophy. Archived from the original on December 16, 2022 . Retrieved November 12, 2022. Downey, Rod (2014). Turing's Legacy: Developments from Turing's Ideas in Logic. Issue 42 of Lecture Notes in Logic. Cambridge University Press. pp.260–261. ISBN 978-1-107-04348-0 . Retrieved November 10, 2022.

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Mathematicians can find an aesthetic value to mathematics. Like beauty, it is hard to define, it is commonly related to elegance, which involves qualities like simplicity, symmetry, completeness, and generality. G. H. Hardy in A Mathematician's Apology expressed the belief that the aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. He also identified other criteria such as significance, unexpectedness, and inevitability, which contribute to mathematical aesthetic. [182] Paul Erdős expressed this sentiment more ironically by speaking of "The Book", a supposed divine collection of the most beautiful proofs. The 1998 book Proofs from THE BOOK, inspired by Erdős, is a collection of particularly succinct and revelatory mathematical arguments. Some examples of particularly elegant results included are Euclid's proof that there are infinitely many prime numbers and the fast Fourier transform for harmonic analysis. [183] 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.Asper, Markus (2009). "The two cultures of mathematics in ancient Greece". In Robson, Eleanor; Stedall, Jacqueline (eds.). The Oxford Handbook of the History of Mathematics. Oxford Handbooks in Mathematics. OUP Oxford. pp.107–132. ISBN 978-0-19-921312-2 . Retrieved November 18, 2022.