Module 11

Module 11 - History This is how one pictures the angel of...

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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 2008 History 1 The Dawn of Computation Babylonian cuneiform circa 2000 B.C. CS 135 Fall 2008 History 2 Early Computation computer = human being performing computation Euclids algorithm circa 300 B.C. Abu Jafar Muhammad ibn Musa Al-Khwarizmis books on algebra and arithmetic computation using Indo-Arabic numerals, circa 800 A.D. Isaac Newton (1643-1727) CS 135 Fall 2008 History 3 Charles Babbage (1791-1871) Difference Engine (1819) Analytical Engine (1834) Mechanical computation for military applications The specification of computational operations was separated from their execution Babbages designs were technically too ambitious CS 135 Fall 2008 History 4 Ada Augusta Byron (1815-1852) 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 2008 History 5 David Hilbert (1862-1943) Formalized the axiomatic treatment of Euclidean geometry Hilberts 23 problems (ICM, 1900) Problem #2: Is mathematics consistent? CS 135 Fall 2008 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. and yield ). Theorem: A mathematical statement together with a proof deriving within a given system of axioms and derivation rules. CS 135 Fall 2008 History 7 Hilberts questions (1920s) Is mathematics complete? Meaning: for any formula , if is true, then is provable. Is mathematics consistent? Meaning: for any formula , there arent proofs of both and . Is there a procedure to, given a formula , produce a proof of , or show there isnt one? Hilbert believed the answers would be yes. CS 135 Fall 2008 History 8 Kurt G odel (1906-78) Godels answers to Hilbert (1929-30): Any axiom system powerful enough to describe arithmetic on integers is not complete....
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This note was uploaded on 10/21/2010 for the course CS 135 taught by Professor Vasiga during the Fall '07 term at Waterloo.

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Module 11 - History This is how one pictures the angel of...

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