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In computability theory, the Church–Turing thesis is a thesis about the nature of computable functions. It states that a function on the natural numbers can be calculated by an effective method if and only if it is computable by a Turing machine. The thesis is named after American mathematician Alonzo Church and the British mathematician Alan Turing. Before the precise definition of computable function, mathematicians often used the informal term effectively calculable to describe functions that are computable by paper-and-pencil methods. In the 1930s, several independent attempts were made to formalize the notion of computability:
In mathematics and computer science, the Entscheidungsproblem is a challenge posed by David Hilbert and Wilhelm Ackermann in 1928. It asks for an algorithm that considers an inputted statement and answers "yes" or "no" according to whether it is universally valid, i.e., valid in every structure. Such an algorithm was proven to be impossible by Alonzo Church and Alan Turing in 1936.
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Alonzo Church was an American mathematician, computer scientist, logician, and philosopher who made major contributions to mathematical logic and the foundations of theoretical computer science. He is best known for the lambda calculus, the Church–Turing thesis, proving the unsolvability of the Entscheidungsproblem, the Frege–Church ontology, and the Church–Rosser theorem. Alongside his doctoral student Alan Turing, Church is considered one of the founders of computer science.
Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since expanded to include the study of generalized computability and definability. In these areas, computability theory overlaps with proof theory and effective descriptive set theory.
Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories. Emphasis on metamathematics owes itself to David Hilbert's attempt to secure the foundations of mathematics in the early part of the 20th century. Metamathematics provides "a rigorous mathematical technique for investigating a great variety of foundation problems for mathematics and logic". An important feature of metamathematics is its emphasis on differentiating between reasoning from inside a system and from outside a system. An informal illustration of this is categorizing the proposition "2+2=4" as belonging to mathematics while categorizing the proposition "'2+2=4' is valid" as belonging to metamathematics.
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Heinrich Scholz was a German logician, philosopher, and Protestant theologian. He was a peer of Alan Turing who mentioned Scholz when writing with regard to the reception of "On Computable Numbers, with an Application to the Entscheidungsproblem": "I have had two letters asking for reprints, one from Braithwaite at King's and one from a professor [sic] in Germany... They seemed very much interested in the paper. [...] I was disappointed by its reception here."
References
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- ↑ Turing, A. M. (January 1937). "On computable numbers, with an application to the Entscheidungsproblem". Proceedings of the London Mathematical Society . Series 2. 42: 230–265. doi:10.1112/plms/s2-42.1.230 . Retrieved 2017-12-24.
- ↑ "Rózsa Péter". MacTutor History of Mathematics Archive . School of Mathematics and Statistics, University of St Andrews, Scotland. Retrieved 14 April 2014.
- ↑ Albers, Donald J.; Alexanderson, Gerald L.; Reid, Constance, eds. (1990), "Rozsa Peter 1905–1977", More Mathematical People, Harcourt Brace Jovanovich, p. 149
- ↑ Andrásfai, Béla (1986), "Rózsa (Rosa) Péter", Periodica Polytechnica Electrical Engineering, 30 (2–3): 139–145
- ↑ Lehmann, Inge (1936). "P'". Publications du Bureau Central Séismologique International. A14 (3): S.87–115.
- ↑ Bolt, Bruce A. (1987). "50 years of studies on the inner core". Eos . 68 (6): 73, 80–81. Bibcode:1987EOSTr..68Q..73B. doi:10.1029/eo068i006p00073-01.
- ↑ Bryson, Bill (2004). A Short History of Nearly Everything. Broadway Books. p. 71. ISBN 978-0767908184.
- ↑ Church, Alonzo (1936). "A Note on the Entscheidungsproblem". Journal of Symbolic Logic . 1 (1): 40–41. doi:10.2307/2269326. JSTOR 2269326. S2CID 42323521. Submitted April 15 [sic.]
- ↑ Meijer, C. S. (1936). "Über Whittakersche bzw. Besselsche Funktionen und deren Produkte". Nieuw Archief voor Wiskunde (2) (in German). 18 (4): 10–39. JFM 62.0421.02.
- ↑ Dodson, Peter; et al. "Staurikosaurus". The Age of Dinosaurs. Publications International. p. 45. ISBN 0-7853-0443-6.
- ↑ Selye, Hans (1936). "A Syndrome Produced by Diverse Nocuous Agents". Nature . 138 (3479): 32. Bibcode:1936Natur.138...32S. doi: 10.1038/138032a0 . Archived from the original on 2008-01-07. Retrieved 2020-02-16.
- ↑ Szabo, S.; Yoshida, M.; Filakovszky, J.; Juhasz, G. (2017). ""Stress" is 80 Years Old: From Hans Selye Original Paper in 1936 to Recent Advances in GI Ulceration" (PDF). Current Pharmaceutical Design. 23 (27): 4029–4041. doi:10.2174/1381612823666170622110046. PMID 28641541.
- ↑ "1936". MusicAndHistory. Archived from the original on 2013-06-10. Retrieved 2015-08-16.
- ↑ "Influenza Historic Timeline". CDC. 2019-01-30. Retrieved 2022-03-19.
- ↑ "Egas Moniz – Biography". Nobelprize.org. 9 Jun 2010. Archived from the original on 2010-06-09. Retrieved 11 July 2010.
- ↑ Fanconi, G.; Uehlinger, E.; Knauer, C. (1936). "Das coeliakiesyndrom bei angeborener zysticher pankreasfibromatose und bronchiektasien". Wiener medizinische Wochenschrift. 86: 753–756.
- ↑ Himsworth, H. P. (1936). "Diabetes mellitus: its differentiation into insulin-sensitive and insulin-insensitive types". The Lancet . 227 (5864). London: 127–30. doi:10.1016/S0140-6736(01)36134-2.
- ↑ Hogg, Michael A.; Vaughan, Graham M. (2005). Social Psychology (4th ed.). Harlow: Prentice Hall. ISBN 0273686992.
- ↑ Paddle, Robert (2000). The Last Tasmanian Tiger: the History and Extinction of the Thylacine. Cambridge University Press. ISBN 0-521-53154-3.
- ↑ Watson, D. A. "The Panda Lady: Ruth Harkness (Part 1)". Female explorers. Archived from the original on 15 December 2006. Retrieved 2007-02-01.
- ↑ Corera, Gordon (2021-10-10). "AQ Khan: The most dangerous man in the world?". BBC News . Retrieved 2021-10-10.
- ↑ Dawkins, Richard (2000-03-10). "Obituary". The Independent . London. Archived from the original on 2016-03-04. Retrieved 2021-10-10.