One of the important problems for logicians in the 1930s was David Hilbert's Entscheidungsproblem, which asked whether there was a mechanical procedure for separating mathematical truths from mathematical falsehoods.
Defenders of classical mathematics, such as David Hilbert, have argued that it is easier to work in, and is most fruitful; although they acknowledge non-classical mathematics has at times led to fruitful results that classical mathematics could not (or could not so easily) attain, they argue that on the whole, it is the other way round.
The Entscheidungsproblem, proposed by David Hilbert, asked whether there is an effective procedure to determine which mathematical statements (coded as natural numbers) are true.
"The foundations of mathematics," with comment by Weyl and Appendix by Bernays, 464–89.
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At the University of Göttingen, Hilbert was surrounded by a social circle of some of the most important mathematicians of the 20th century, such as Emmy Noether and Alonzo Church.
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One who had to leave Germany, Paul Bernays, had collaborated with Hilbert in mathematical logic, and co-authored with him the important book Grundlagen der Mathematik (which eventually appeared in two volumes, in 1934 and 1939).
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Hilbert, the first of two children of Otto and Maria Therese (Erdtmann) Hilbert, was born in the Province of Prussia - either in Königsberg (according to Hilbert's own statement) or in Wehlau (known since 1946 as Znamensk) near Königsberg where his father worked at the time of his birth.
In the words of David Hilbert (referring to geometry), "it does not matter if we call the things chairs, tables and beer mugs or points, lines and planes."
In mathematics, the Hilbert cube, named after David Hilbert, is a topological space that provides an instructive example of some ideas in topology.
The Hilbert spectrum (sometimes referred to as the Hilbert amplitude spectrum), named after David Hilbert, is a statistical tool that can help in distinguishing among a mixture of moving signals.
The Hilbert transform is named after David Hilbert, who first introduced the operator in order to solve a special case of the Riemann–Hilbert problem for holomorphic functions.
In mathematical logic, the Hilbert–Bernays provability conditions, named after David Hilbert and Paul Bernays, are a set of requirements for formalized provability predicates in formal theories of arithmetic (Smith 2007:224).
In 1900, David Hilbert posed the problem of proving their consistency using only finitistic methods as the second of his twenty-three problems.
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Towards the beginning of the twentieth century, results similar to that of Arzelà and Ascoli began to accumulate in the area of integral equations, as investigated by David Hilbert and Erhard Schmidt.
Probably Gauss first realized this, and used it to prove the impossibility of some constructions; only much later did Hilbert find a complete set of axioms for geometry.
1917 — Albert Einstein and David Hilbert had dawn-to-dusk discussions of physics; and they continued their debate in writing, although Felix Klein records that they talked past each other, as happens not infrequently between simultaneously producing mathematicians.
In 1928, Ackermann helped David Hilbert turn his 1917 – 22 lectures on introductory mathematical logic into a text, Principles of Mathematical Logic.