German mathematician, philosopher, and physicist whose work in all three disciplines was brilliantly innovative and fundamental to further development. An excellent teacher, unsurpassed in lucidity of exposition, his influence on 20th-century mathematics has been enormous. He is particularly remembered for his research on the theory of algebraic invariants, on the theory of algebraic numbers, on the formulation of abstract axiomatic principles in geometry, on analysis and topology (in which he derived what is now called Hilbert's theory of spaces), on theoretical physics, and finally on the philosophical foundations of mathematics.
Hilbert was born on 23 January 1862 in Königsberg, then in German Prussia, now Kaliningrad in Russia. He grew up and was educated there, attending Königsberg University – an ancient and venerable seat of learning – 1880–85, when he received his PhD. He then studied further in Leipzig and in Paris before returning to Königsberg University to become an unsalaried lecturer. Six years later he was appointed professor, and three years later still (in 1895) he was offered the highly prestigious post of professor of mathematics at Göttingen University. He accepted, and held the position until he retired in 1930. Although he developed pernicious anaemia in 1925, he recovered, and died in Göttingen on 14 February 1943.
Hilbert's first period of research, 1885–92), was on algebraic invariants; he tried to find a connection between invariants and fields of algebraic functions and algebraic varieties. Representing the rational function in terms of a square, he eventually arrived at what is known as Hilbert's irreducibility theorem, which states that, in general, irreducibility is preserved if, in a polynomial of several variables with integral coefficients, some of the variables are replaced by integers. He later investigated ninth-degree equations, solving them by using algebraic functions of only four variables. In this way, Hilbert had by the end of the period not only solved all the known central problems of this branch of mathematics, he had in his methodology introduced sweeping developments and new areas for research (particularly in algebraic topology, which he himself returned to later).
In 1897, with some help from his colleague and friend Hermann Minkowski, Hilbert produced Der Zahlbericht, in which he gathered together all the relevant knowledge of algebraic number theory, reorganized it, and laid the basis for the developing class-field theory. He abandoned this work, however, when there was still much to be done.
Two years later, having moved to another area of study, Hilbert published his classic work, Grundlagen der Geometrie. In it, he gave a full account of the development of geometry in the 19th century, and although on this occasion his innovations were (for him) relatively few, his use of geometry, and of algebra within geometry, to devise systems incorporating abstract yet rigorously axiomatic principles, was important both to the further development of the subject and to Hilbert's own later work in logic and consistency proofs. In the related field of topology, he referred back to his previous work on invariants in order to derive his theory of spaces in an infinite number of dimensions.
In 1900, attending the International Congress of Mathematicians in Paris, Hilbert set the congress a total of 23 hitherto unsolved problems. Many have since been solved – but solved or not, the problems stimulated considerable scientific debate, research, and fruitful development.
In a study of mathematical analysis some years afterwards, Hilbert used a new approach in tackling Dirichlet's problem (see Lejeune Dirichlet), and made other contributions to the calculus of variations. In 1909 he provided proof of Waring's hypothesis (of a century earlier) of the representation of integers as the sums of powers. From that time forward, Hilbert worked on problems of physics, such as the kinetic theory of gases, and the theory of relativity – problems, he said, too difficult to be confined to physics and physicists. His deep research led finally to his critical work on the foundations of mathematical logic, in which his contribution to proof theory was extremely important by itself.
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