Gödel, Escher, Bach: An Eternal Golden Braid by Douglas R. Hofstadter Read Free Book Online Page A

"holes" in the axiomatic system proposed by Russell and Whitehead, but more generally, that no axiomatic system whatsoever could produce all number-theoretical truths, unless it were an inconsistent system! And finally, the hope of proving the consistency of a system such as that presented in P.M. was shown to be vain: if such a proof could be found using only methods inside P.M., then-and this is one of the most mystifying consequences of Godel's work-P.M. itself would be inconsistent!
    The final irony of it all is that the proof of Gi del's Incompleteness Theorem involved importing the Epimenides paradox right into the heart ofPrincipia Mathematica, a bastion supposedly invulnerable to the attacks of Strange Loops! Although Godel's Strange Loop did not destroy Principia Mathematica, it made it far less interesting to mathematicians, for it showed that Russell and Whitehead's original aims were illusory.

    Babbage, Computers, Artificial Intelligence ...

    When Godel's paper came out, the world was on the brink of developing electronic digital computers. Now the idea of mechanical calculating engines had been around for a while.
    In the seventeenth century, Pascal and Leibniz designed machines to perform fixed operations (addition and multiplication). These machines had no memory, however, and were not, in modern parlance, programmable.
    The first human to conceive of the immense computing potential of machinery was the Londoner Charles Babbage (1792-1871). A character who could almost have stepped out of the pages of the Pickwick Papers,

    Babbage was most famous during his lifetime for his vigorous campaign to rid London of "street nuisances"-organ grinders above all. These pests, loving to get his goat, would come and serenade him at any time of day or night, and he would furiously chase them down the street. Today, we recognize in Babbage a man a hundred years ahead of his time: not only inventor of the basic principles of modern computers, he was also one of the first to battle noise pollution.
    His first machine, the "Difference Engine", could generate mathematical tables of many kinds by the "method of differences". But before any model of the "D.E." had been built, Babbage became obsessed with a much more revolutionary idea: his "Analytical Engine". Rather immodestly, he wrote, "The course through which I arrived at it was the most entangled and perplexed which probably ever occupied the human mind."' Unlike any previously designed machine, the A.E. was to possess both a "store" (memory) and a
    "mill" (calculating and decision-making unit). These units were to be built of thousands of intricate geared cylinders interlocked in incredibly complex ways. Babbage had a vision of numbers swirling in and out of the mill tinder control of a program contained in punched cards-an idea inspired by the jacquard loom, a card-controlled loom that wove amazingly complex patterns. Babbage's brilliant but ill-fated Countess friend, Lady Ada Lovelace (daughter of Lord Byron), poetically commented that "the Analytical Engine weaves algebraic patterns just as the Jacquard-loom weaves flowers and leaves."
    Unfortunately, her use of the present tense was misleading, for no A.E. was ever built, and Babbage died a bitterly disappointed man.
    Lady Lovelace, no less than Babbage, was profoundly aware that with the invention of the Analytical Engine, mankind was flirting with mechanized intelligence-particularly if the Engine were capable of "eating its own tail" (the way Babbage described the Strange Loop created when a machine reaches in and alters its own stored program). In an 1842
    memoir,5 she wrote that the A.E. "might act upon other things besides number". While Babbage dreamt of creating_ a chess or tic-tac-toe automaton, she suggested that his Engine, with pitches and harmonies coded into its spinning cylinders, "might compose elaborate and scientific pieces of music of any degree of complexity or extent." In nearly the same breath,