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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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1886

1900

David Hilbert (1862 - 1943)


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1886

1900

David Hilbert (1862 - 1943)


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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

    • .....


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Jacques Herbrand (1908 - 1931)

Alonzo Church (1903 - 1995)



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Alan Turing (1912 - 1954)

Emil Post (1897 - 1954)


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Alan Turing (1912 - 1954)

Emil Post (1897 - 1954)


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Alan Turing (1912 - 1954)

Emil Post (1897 - 1954)



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Definition:

e.g., perfect squares are diophantine

Davis Normal Form (1949): each r.e. set admits definition in terms of a

Diophantine equation


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What set associated with this family of equations?

Davis "daring" conjecture (1953):

The recursively enumerable sets are precisely the diophantine sets.







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New (2000) lists:

Clay Mathematics Institute: 7 prize problems ($1 million each)

P = NP ?

Riemann hypothesis

Poincare conjecture: 3-sphere simply connected?


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  • 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?


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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

  • .....


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  • 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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