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A Brief History of the Computability Problem

A Brief History of the Computability Problem. Gottfried Wilhelm Leibniz (1646 - 1716). Leibniz's computer. Georg Cantor (1845 - 1918). 1886. 1900. David Hilbert (1862 - 1943). 1886. 1900. David Hilbert (1862 - 1943). Hilbert's List (1900). Foundations (general)

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A Brief History of the Computability Problem

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  1. A Brief History of the Computability Problem

  2. Gottfried Wilhelm Leibniz (1646 - 1716)

  3. Leibniz's computer

  4. Georg Cantor (1845 - 1918)

  5. 1886 1900 David Hilbert (1862 - 1943)

  6. 1886 1900 David Hilbert (1862 - 1943)

  7. Hilbert's List (1900) • Foundations (general) • cardinal number of the continuum • compatibility of arithmetic axioms • Foundations (specific areas) • tetrahedra with equal bases and altitudes (geometry) • straight line as shortest distance between points (alternative geometries) • Lie group without differentiability (analysis) • axiomatization of Physics • Number theory • irrationality and transcendence of certain numbers • prime numbers • most general law of reciprocity in number field • algorithm for solvability of a diophantine equation • quadratic form with algebraic coefficients • extend Kronecker's Abelian field result to algebraic realms • Algebra and Geometry • ..... • Analysis • .....

  8. Jacques Herbrand (1908 - 1931) Alonzo Church (1903 - 1995)

  9. Kurt Godel (1906 - 1978)

  10. Alan Turing (1912 - 1954) Emil Post (1897 - 1954)

  11. Alan Turing (1912 - 1954) Emil Post (1897 - 1954)

  12. Alan Turing (1912 - 1954) Emil Post (1897 - 1954)

  13. Martin Davis (1928 - ), Julia Robinson (1919 - 1985), Yuri Matiyasevich (1947 - )

  14. Definition: e.g., perfect squares are diophantine Davis Normal Form (1949): each r.e. set admits definition in terms of a Diophantine equation

  15. What set associated with this family of equations? Davis "daring" conjecture (1953): The recursively enumerable sets are precisely the diophantine sets.

  16. Alfred Tarski (1902 - 1983)

  17. Problem J.R.

  18. Hilary Putnam (1926 - )

  19. Hilary Putnam (1926 - )

  20. Yuri Matiyasevich (1947 - )

  21. New (2000) lists: Clay Mathematics Institute: 7 prize problems ($1 million each) P = NP ? Riemann hypothesis Poincare conjecture: 3-sphere simply connected?

  22. Landau's problems (1912): • Goldbach conjecture integer > 4 is sum of three primes even integer > 3 is sum of two primes • Twin prime conjecture • A prime between every pair of adjacent squares?

  23. Steven Smale (1930 - ) • Smale's list (2000): • Goldbach conjecture • Riemann hypothesis • Poincare conjecture • Twin prime conjecture • P = NP ? • Theoretical limits of intelligence, human and artificial • .....

  24. References: • The Honors Class: Hilbert's Problems and Their Solvers by Benjamin Yandell (2002) • The Unknowable by Gregory Chaitin (1999) • Conversations with a Mathematician by Gregory Chaitin (1999) • The Universal Computer: The Road from Leibniz to Turing by Martin Davis (2000)

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