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Alan Turing was the first to make a careful analysis of the potential capabilities of machines, inventing his famous "Turing machines" for the purpos… - Michael J. Beeson
" "Alan Turing was the first to make a careful analysis of the potential capabilities of machines, inventing his famous "Turing machines" for the purpose. He argued that if any machine could perform a computation, then some Turing machine could perform it. The argument focuses on the assertion that any machine's operations could be simulated, one step at a time, by certain simple operations, and that Turing machines were capable of those simple operations. Turing's first fame resulted from applying this analysis to a problem posed earlier by Hilbert, which concerned the possibility of mechanizing mathematics. Turing showed that in a certain sense, it is impossible to mechanize mathematics: We shall never be able to build an "oracle" machine that can correctly answer all mathematical questions presented to it with a "yes" or "no" answer. In another famous paper Turing went on to consider the somewhat different question, "Can machines think?." It is a different question, because perhaps machines can think, but they might not be any better at mathematics than humans are; or perhaps they might be better at mathematics than humans are, but not by thinking, just by brute-force calculation power. These two papers of Turing lie near the roots of the subjects today known as automated deduction and artificial intelligence.
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About Michael J. Beeson
Michael J. Beeson (born August 19, 1945 in Topeka, Kansas) is an American mathematician, and Professor of Mathematics and Computer Science, San Jose State University.
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George Boole took up Leibniz's idea, and wrote a book he called . The laws he formulated are now called ... Boole seems to have had a grandiose vision about the applicability of his algebraic methods to practical problems—his book makes it clear that he hoped these laws would be used to settle practical questions. William Stanley Jevons heard of Boole's work, and undertook to build a machine to make calculations in Boolean algebra. He successfully designed and built... the Logical Piano... the first machine to do mechanical inference.
Gottlob Frege created modern logic including "for all," "there exists," and rules of proof. Leibniz and Boole had dealt only with what we now call "propositional logic" (that is, no "for all" or "there exists"). They also did not concern themselves with rules of proof, since their aim was to reach truth by pure calculation with symbols for the propositions. Frege took the opposite track: instead of trying to reduce logic to calculation, he tried to reduce mathematics to logic, including the concept of number.
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Gottfried Leibniz is famous... for his slogan Calculemus, which means "Let us calculate." He envisioned a formal language to reduce reasoning to calculation, and he said that reasonable men, faced with a difficult question of philosophy or policy, would express the question in a precise language and use rules of calculation to carry out precise reasoning. This is the first reduction of reasoning to calculation ever envisioned. ...he actually designed and built a working calculating machine, the Stepped Reckoner ...inspired by the somewhat earlier work of Pascal, who built a machine that could add and subtract. Leibniz's machine could add, subtract, divide, and multiply, and was apparently the first machine with all four arithmetic capabilities.
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