David Hilbert

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David Hilbert


German Mathematician

Perhaps the most famous event from the long and fruitful career of David Hilbert was his 1900 address to the International Congress of Mathematicians (ICM) in Paris. A new century was dawning, and Hilbert, at 38 already well known, proposed a set of 23 problems. These problems, he suggested, would keep mathematicians occupied throughout the coming century and, indeed, some did. But long before the century had reached its midway point, Hilbert's brilliant career had peaked and entered a headlong dive—another casualty of the political nightmare unfolding in Hitler's Germany.

Hilbert was born on January 23, 1862, in Königsberg, Prussia (now Kaliningrad in Russia). His father, Otto, was a lawyer, and his parents were protective of their social standing. This fact would become clear years later, when they expressed their disapproval of Hilbert's friendship with fellow mathematician Hermann Minkowski (1864-1909), who not only came from the lower classes but was a Jew.

Hilbert entered the University of Königsberg in 1880 and earned his doctorate in 1885. Following his graduation, he went to work at the university, becoming a professor in 1893. Two years later, he took a chair at the University of Göttingen, where he would spend his entire career. In 1892 he married Kathe Jerosch and they had one child, a son named Franz, who appears to have had a mental disorder.

Hilbert's first important work was on the theory of invariants, involving the expression of an entity that remains the same throughout a variety of transformations. Classifying invariants through calculation had proven a herculean task, but Hilbert cut through the haze by showing that such calculations were unnecessary. These revelations proved so unsettling to the mathematical community that invariant theory itself went out of vogue for a time, but, when it was later reopened for study, mathematicians discovered that Hilbert's observations were accurate.

At this early stage in his career, Hilbert struck up a friendship with Minkowski, and together they labored over a summary of the state of number theory, commissioned in 1893 by the German Mathematical Association. Minkowski eventually abandoned the enterprise, but Hilbert's 1897 "Number Report" helped formulate the terms by which the theory of numbers would be discussed throughout the twentieth century.

Hilbert turned from this task to a reassessment of Euclidean principles, a project that had become necessary in the face of non-Euclidean geometry—not to mention the fact that mathematicians had become increasingly aware of the many untested assumptions at the heart of Euclidean theory. Hilbert proposed that in order to prevent untested assumptions from entering into geometry it was necessary to focus on the form of an axiom. This led him into debate with Gottlob Frege (1848-1925), one of the founders of mathematical logic, and the ensuing controversy only served to promote Hilbert's name further.

At the high point of his career, Hilbert addressed the ICM. Among the most enduring of the problems he presented was the first, involving the question of how many real numbers there were compared to the number of whole numbers, and the seventh, on the irrationality of certain real numbers. The first was not solved until 1963 and the seventh until 1934, when Aleksandr Gelfond (1906-1968) presented his theorem concerning transcendental numbers. All in all, Hilbert's problems created a veritable industry, with numerous books and conferences devoted to individual questions.

In the years that followed, Hilbert contributed to the study of mathematical analysis. During this period, he became something of a celebrity, noted for his Panama hat and his beard. In the increasingly tense environment of Germany during World War I and afterward, he repeatedly showed himself to be a mathematician and a human being first and a German second. (He defended Emmy Noether [1882-1935], a Jewish mathematician, against whom the German mathematical establishment discriminated.)

Hilbert later devoted himself to theoretical physics and formalism, neither of which proved to be pursuits of enduring impact. Though formalism at one time was an influential theory of mathematics, attacks from many sides—most notably from the incompleteness theorem of Kurt Gödel (1906-1978)—rendered it obsolete. Hilbert began to find himself increasingly isolated in the 1930s as more Jewish members of German academia fled the country. He refused to go at his age, and he died in the middle of World War II on February 14, 1943, in Göttingen. Only about a dozen people attended his funeral.


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