David Hilbert

• Solving several important problems in the theory of invariants. Hilbert's basis theorem solved the principal problem in the 1800s invariant theory by showing that any form of a given number of variables and of a given degree has a finite, yet complete system of independent rational integral invariants and covariants.
• Unifying the field of algebraic number theory with his 1897 treatise Zahlbericht (literally "report on numbers").
• Providing the first correct and complete axiomatization of Euclidean geometry to replace Euclid's axiomatization of geometry, in his 1899 book Grundlagen der Geometrie ("Foundations of Geometry").
• His suggestion in 1920 that mathematics be formulated on a solid logical foundation (by showing that all of mathematics follows from a system of axioms, and that that axiom system is consistent). Unfortunately, Gödel's Incompleteness Theorem showed that his grand plan was impossible.
• Hilbert's paradox of the Grand Hotel, a musing about strange properties of the infinite.
• Laying the foundations of functional analysis by studying integral equations and Hilbert spaces.
• Putting forth an influential list of 23 unsolved problems in the Paris conference of the International Congress of Mathematicians in 1900.
• Hilbert helped provide the basis for the Theory of Automata which was later built upon by computer scientist Alan Turing.

Further reading

• Jeremy Gray, 2000, The Hilbert Challenge, ISBN 0198506511

External links

• MacTutor Hilbert biography (http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hilbert)