David Hilbert

German mathematician

• Born: January 23, 1862
• Birthplace: Königsberg, Prussia (now Kaliningrad, Russia)
• Died: February 14, 1943
• Place of death: Göttingen, Germany

Hilbert occupied a leading position in mathematics at the start of the twentieth century, advocating rigor and precise formalism in mathematics and mathematically rigorous formulations of physics. He is perhaps best known for his list of twenty-three fundamental questions that would guide mathematical work through the twentieth century.

Early Life

David Hilbert was born into a prosperous family of jurists and physicians. His father, Otto, was a district judge. His mother, Maria Therese Erdtman, was the daughter of a merchant and interested in philosophy, astronomy, and mathematics, interests she would share with her young son. At the age of eight, Hilbert was enrolled in a gymnasium (preparatory school), the Freidrichscolleg, which emphasized classical languages. In 1879, he transferred to the Wilhelm Gymnasium, which placed more emphasis on mathematics. Hilbert’s school performance was satisfactory but not exceptional. In 1880 he entered the University of Königsburg as a mathematics student. At Königsburg, the spirit of the great German philosopher Immanuel Kant, who was born in the city and taught at the university, still was evident. Kant’s view, that mathematics was a form of knowledge gained from pure reason alone, would influence Hilbert throughout his career.

The German universities of 1880 were havens of academic freedom, with professors free to lecture on topics of their own choosing and students free to select their own path of study. The universities also allowed for considerable student mobility. Hilbert chose to spend his second semester at the University of Heidelberg, but then returned to Königsburg. In the spring of 1882, he was joined there by the seventeen-year-old mathematical prodigy Hermann Minkowski, also a native of the city, who would become a close friend and collaborator.

Hilbert received his doctor of philosophy degree in 1885, submitting a thesis on algebraic invariants. He submitted a somewhat longer paper on the same subject for his habilitation exam, which made him a Privatdocent who could lecture without a salary to paying students. During the next few years Hilbert would lecture and travel to visit other mathematicians. In 1892 he married Käthe Jerosh. Their only child, Franz, was born a year later. In 1895, Hilbert was called to the University of Göttingen as a professor of mathematics.

Life’s Work

Hilbert is known for important contributions in many areas of mathematics. At the time he received his professorship at Königsburg, the theory of numbers was considered the most important branch of mathematics. In 1893 the Association of German Mathematicians urged Hilbert and Minkowski to prepare a report on recent developments in number theory. The report, published in 1897 and known as the Zahlbericht, received high acclaim for its thoroughness and integrated presentation.

Hilbert then turned his attention to the foundations of geometry. Prior to the nineteenth century, geometry was considered the study of the properties of the physical space in which humans live. According to Kant, geometry could be conducted by reason alone. Geometry allowed for the deduction by logical means of many unclear statements from a small set of apparently self-evident statements systematized by the Greek geometer Euclid. One of these postulates, commonly known as the parallel postulate, asserted that given a straight line and a point not on the line, one could draw one and only one straight line through the point parallel to the line. By Hilbert’s time mathematics was recovering from the finding that alternative postulates allowing no, or many, straight lines to be drawn through the point yielded different, but self-consistent, geometries and that there was no certainty that the space described by Euclidean geometry was in fact the space inhabited by humans.

Hilbert decided that the first step to certain knowledge in geometry was to treat the elementary notions of point, line, and plane as having meaning only in terms of their relation to each other. Hilbert devised twenty axioms, divided into five independent groups. One of the by-products of this work was finding that geometry, in its several forms, could be proved free of contradictions if arithmetic could be proved to be free of contradictions.

Hilbert was invited to make a keynote address at the Second International Congress of Mathematicians in Paris in 1900. He chose to speak on those mathematical problems he thought would be the most important to solve in the new century. In the printed version of his talk he identified twenty-three urgent problems. He concluded his talk by saying, “We must know. We shall know.” Hilbert’s twenty-three item list ranges from the consistency of arithmetic raised to greater importance by his work on the foundations of geometry to the mathematical treatment of the laws of physics.

Physics has long provided a stimulus to the development of mathematics, and Hilbert found himself increasingly drawn to concerns of mathematical physics. He first turned to integral equations, useful in a number of areas of physics. Here he was able to cast the subject of linear integral equations as a case of geometry in an infinite dimensional space, now universally known as Hilbert space .

In 1910, Hilbert became only the second individual to win the Bolyai Prize of the Hungarian Academy of Sciences, an award that consisted of ten thousand gold crowns, solidifying his reputation as one of the two leading mathematicians of his time; the other was the great French mathematicianHenri Poincaré.

By this time, however, Hilbert was devoting most of his attention to the mathematical problems of physics. He even took the unusual step of appointing an assistant to guide him in his study of physics. His last publications, though, from 1934 and 1939, returned to the issue of the consistency of arithmetic.

In 1914, World War I broke out and the German government pressured leading professors to add their signatures to a declaration asserting the justice of the German cause. Hilbert, like Albert Einstein, who was then working in Berlin, refused to sign the document. In refusing to sign, Hilbert retained his good standing in the international mathematics community; the remaining students and faculty at Göttingen, however, shunned him.

Significance

Hilbert was one of the leading figures in German mathematics at a time when the nature of mathematics and its role in physical science was receiving critical reevaluation. His twenty-three problems for future mathematicians would yield a number of surprising solutions. One problem prompted British mathematician Alan Mathison Turing to demonstrate the possibility of a universal automaton for solving problems of symbol manipulation, solutions that became the conceptual ancestor of the programmable digital computer. Perhaps most surprising was the attempt to demonstrate the consistency of arithmetic. Austrian logician Kurt Gödel demonstrated that any set of axioms consistent with ordinary multiplication would allow the formulation of assertions that could not be proved or disproved within the system.

Hilbert’s work on the differential equations of physics challenged physicists to function at a new level of rigor. Physicists now routinely make use of the theory of Hilbert spaces, in which functions are represented in a geometrical space of an infinite number of dimensions. Over his long career Hilbert served as mentor to many distinguished mathematicians and physicists, including Nobel laureate Max Born and mathematical physicist Emma Noether, who would determine the fundamental connection between symmetry and conservation laws. Her work guided much elementary particle theory into the twenty-first century.

The University at Göttingen would come to play a pivotal role in the development of quantum physics, attracting students from the United States as well as Europe. After 1933, as the Nazi Party came to power, Hilbert and his colleagues did what they could to help former students and professors find academic positions outside Germany.

Bibliography

Hilbert, David. Foundations of Geometry. LaSalle, Ill.: Open Court, 2006. An English translation of the book in which Hilbert formulates the postulates of geometry in logically independent form. This work has been reprinted many times in English and German.

Klein, Morris. Mathematics: The Loss of Certainty. New York: Oxford University Press, 1980. Klein details the loss of confidence in mathematical intuition occasioned by the discovery of non-Euclidean geometries and the paradoxes of set theory. Hilbert’s many contributions in this area are highlighted.

Reid, Constance. Hilbert. New York: Springer, 1996. A detailed biography by a nonmathematician. Includes many photographs of key mathematicians and the Göttingen locale in Hilbert’s time.

Yandell, Ben H. The Honors Class: Hilbert’s Problems and Their Solvers. Natick, Mass.: A. K. Peters, 2002. Following a brief biography of Hilbert, Yandell describes the twenty-three fundamental problems advanced by Hilbert at the 1900 international mathematics conference. Details the progress on each problem and includes biographical details on those who solved them.