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Unformatted text preview: History This is how one pictures the angel of history. His face is turned toward the past. Where we perceive a chain of events, he sees one single catastrophe which keeps piling wreckage and hurls it in front of his feet. The angel would like to stay, awaken the dead, and make whole what has been smashed. But a storm is blowing in from Paradise; it has got caught in his wings with such a violence that the angel can no longer close them. The storm irresistibly propels him into the future to which his back is turned, while the pile of debris before him grows skyward. This storm is what we call progress. (Walter Benjamin, 1940) CS 135 Fall 2009 11: History 1 The Dawn of Computation Babylonian cuneiform circa 2000 B.C. CS 135 Fall 2009 11: History 2 Early Computation “computer” = human being performing computation Euclid’s algorithm circa 300 B.C. Abu Ja’far Muhammad ibn Musa AlKhwarizmi’s books on algebra and arithmetic computation using IndoArabic numerals, circa 800 A.D. Isaac Newton (16431727) CS 135 Fall 2009 11: History 3 Charles Babbage (17911871) Difference Engine (1819) Analytical Engine (1834) Mechanical computation for military applications The specification of computational operations was separated from their execution Babbage’s designs were technically too ambitious CS 135 Fall 2009 11: History 4 Ada Augusta Byron (18151852) Assisted Babbage in explaining and promoting his ideas Wrote articles describing the operation and use of the Analytical Engine The first computer scientist? CS 135 Fall 2009 11: History 5 David Hilbert (18621943) Formalized the axiomatic treatment of Euclidean geometry Hilbert’s 23 problems (ICM, 1900) Problem #2: Is mathematics consistent? CS 135 Fall 2009 11: History 6 The meaning of proof Axiom: ∀ n : n + 0 = n . Math statement: “The square of any even number is even.” Formula: ∀ n ( ∃ k : n = k + k ⇒ ∃ m : m + m = n * n ) Proof: Finite sequence of axioms (basic true statements) and derivations of new true statements (e.g. φ and φ → σ yield σ ). Theorem: A mathematical statement φ together with a proof deriving φ within a given system of axioms and derivation rules. CS 135 Fall 2009 11: History 7 Hilbert’s questions (1920’s) Is mathematics complete? Meaning: for any formula φ , if φ is true, then φ is provable. Is mathematics consistent? Meaning: for any formula φ , there aren’t proofs of both φ and ¬ φ . Is there a procedure to, given a formula φ , produce a proof of φ , or show there isn’t one? Hilbert believed the answers would be “yes”. CS 135 Fall 2009 11: History 8 Kurt G ¨odel (190678) G¨odel’s answers to Hilbert (192930): Any axiom system powerful enough to describe arithmetic on integers is not complete....
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 Fall '07
 VASIGA
 Turing, David Hilbert

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