The Origins of Cyberspace collection described as lots 1-255 will first be offered as a single lot, subject to a reserve price. If this price is not reached, the collection will be immediately offered as individual lots as described in the catalogue as lots 1-255.
HILBERT, David. "Probleme der Grundlegung der Mathematik." In Atti del Congresso Internazionale dei Matematici, Bologna 3-10 settembre 1928 (VI), I, 135-41. Bologna: Nicola, Zanichelli, [1929-32].
Details
HILBERT, David. "Probleme der Grundlegung der Mathematik." In Atti del Congresso Internazionale dei Matematici, Bologna 3-10 settembre 1928 (VI), I, 135-41. Bologna: Nicola, Zanichelli, [1929-32].
6 volumes, 4o. Original blue cloth.
In 1928 the first International Congress of Mathematicians since 1912 was held in Italy. Hilbert, the leader of the German delegation, spoke on the fundamentals of mathematics, a topic to which he had devoted a great deal of thought and effort over the years. Unlike several of his younger colleagues, he was convinced that it was possible to find an incontrovertible demonstration that mathematics formed a complete and consistent whole. Returning to the second of the twenty-three problems posed in his 1900 paper--can the axioms of arithmetic be proved consistent--Hilbert added two more questions, bringing the number to three. First, was mathematics complete, in the technical sense that every statement (such as "every integer is the sum of four squares") could be either proved, or disproved. Second, was mathematics consistent, in the sense that the statement "2 + 2 = 5" could never be arrived at by a sequence of valid steps of proof. And thirdly, was mathematics decidable? By this he meant, did there exist a definite method which could, in principle, be applied to any assertion, and which was guaranteed to produce a correct decision as to whether that assertion was true (Hodges, Turing: The Enigma, 1983, 91).
Three years later, the Czech mathematician Kurt Gödel published "Über formal unentscheidbare Sdtze der Principia mathematica und verwandter Systeme I" (Monatshefte für Mathematik und Physik 38 [1931]: 173-98), in which he answered the first two of Hilbert's questions in the negative: mathematics was neither complete nor consistent. Hilbert's third question, known as the Entscheidungsproblem, was addressed independently in 1936 by Church (see lot 86), Post (see lot 86), and Turing (see lot 196), each of whom presented proofs that mathematics was also not decidable; Turing's proof involved the creation of his hypothetical "universal computing machine" (Turing machine). OOC 321.
6 volumes, 4
In 1928 the first International Congress of Mathematicians since 1912 was held in Italy. Hilbert, the leader of the German delegation, spoke on the fundamentals of mathematics, a topic to which he had devoted a great deal of thought and effort over the years. Unlike several of his younger colleagues, he was convinced that it was possible to find an incontrovertible demonstration that mathematics formed a complete and consistent whole. Returning to the second of the twenty-three problems posed in his 1900 paper--can the axioms of arithmetic be proved consistent--Hilbert added two more questions, bringing the number to three. First, was mathematics complete, in the technical sense that every statement (such as "every integer is the sum of four squares") could be either proved, or disproved. Second, was mathematics consistent, in the sense that the statement "2 + 2 = 5" could never be arrived at by a sequence of valid steps of proof. And thirdly, was mathematics decidable? By this he meant, did there exist a definite method which could, in principle, be applied to any assertion, and which was guaranteed to produce a correct decision as to whether that assertion was true (Hodges, Turing: The Enigma, 1983, 91).
Three years later, the Czech mathematician Kurt Gödel published "Über formal unentscheidbare Sdtze der Principia mathematica und verwandter Systeme I" (Monatshefte für Mathematik und Physik 38 [1931]: 173-98), in which he answered the first two of Hilbert's questions in the negative: mathematics was neither complete nor consistent. Hilbert's third question, known as the Entscheidungsproblem, was addressed independently in 1936 by Church (see lot 86), Post (see lot 86), and Turing (see lot 196), each of whom presented proofs that mathematics was also not decidable; Turing's proof involved the creation of his hypothetical "universal computing machine" (Turing machine). OOC 321.
Further details
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