Solving Mathematical Problems: A Personal Perspective

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Before Cantor's study of infinite sets, mathematicians were reluctant to consider actually infinite collections, and considered infinity to be the result of endless enumeration. Cantor's work offended many mathematicians not only by considering actually infinite sets [55] but by showing that this implies different sizes of infinity, per Cantor's diagonal argument. This led to the controversy over Cantor's set theory. [56] 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. The connection between mathematics and material reality has led to philosophical debates since at least the time of Pythagoras. The ancient philosopher Plato argued that abstractions that reflect material reality have themselves a reality that exists outside space and time. As a result, the philosophical view that mathematical objects somehow exist on their own in abstraction is often referred to as Platonism. Independently of their possible philosophical opinions, modern mathematicians may be generally considered as Platonists, since they think of and talk of their objects of study as real objects. [153] Lim, Lisa (December 21, 2018). "Where algebra got its x from, and Xmas its X". South China Morning Post. Archived from the original on August 9, 2022 . Retrieved August 9, 2022.

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O'Connor, J. J.; Robertson, E. F. (September 1998). "Leonhard Euler". MacTutor. School of Mathematics and Statistics, University of St Andrews, Scotland. Archived from the original on November 9, 2022 . Retrieved November 9, 2022. Discrete optimization, including combinatorial optimization, integer programming, constraint programming Wagstaff, Jr., Samuel S. (2021). "History of Integer Factoring" (PDF). In Bos, Joppe W.; Stam, Martijn (eds.). Computational Cryptography, Algorithmic Aspects of Cryptography, A Tribute to AKL. London Mathematical Society Lecture Notes Series 469. Cambridge University Press. pp.41–77. Archived (PDF) from the original on November 20, 2022 . Retrieved November 20, 2022.Affine geometry, the study of properties relative to parallelism and independent from the concept of length. 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. Hamilton, Alan G. (1982). Numbers, Sets and Axioms: The Apparatus of Mathematics. Cambridge University Press. pp.3–4. ISBN 978-0-521-28761-6 . Retrieved November 12, 2022. Game theory (although continuous games are also studied, most common games, such as chess and poker are discrete) 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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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] The most prestigious award in mathematics is the Fields Medal, [196] [197] established in 1936 and awarded every four years (except around World War II) to up to four individuals. [198] [199] It is considered the mathematical equivalent of the Nobel Prize. [199]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] A famous list of 23 open problems, called " Hilbert's problems", was compiled in 1900 by German mathematician David Hilbert. [208] This list has achieved great celebrity among mathematicians, [209] and, as of 2022 [update], at least thirteen of the problems (depending how some are interpreted) have been solved. [208]