Hilbert’s Problems

Hilbert's Problems

In an address to the International Council of Mathematicians in 1900, David Hilbert (1863-1942), a professor of mathematics at the University of Goettingen, outlined 23 significant problems in mathematics for the community to research in the new century. The problems cross many areas of mathematics, including set theory, arithmetic, geometry, group theory, variable calculus, algebra, and others. Some problems were relatively straightforward and were thus quickly solved, but others were expansive and may never be completely resolved. Some mathematicians consider Hilbert's address as one of the most influential ever made in contemporary study of the field.

Beyond stating the problems Hilbert felt important for the mathematics community to address in the coming years, Hilbert expressed his address in the context of his particular philosophy about the field. Hilbert believed that the development of mathematics stemmed from two sources--practicality and reason--with each intertwined since ancient times. The idea of a straight line as the shortest distance between two points, for example, arose from practicality--taking a straight path or using a straight-edge measurement is often the most efficient course of action. Other developments in mathematics, however, are based on reason, either inductive or deductive. Such inductive or deductive reason may be based on observations of the surrounding environment or simply the logical consideration of an active mind. Each gain of new knowledge, however, grows with others, and it becomes difficult, if not impossible, to retrace absolutely the steps of discovery. Hilbert referred to this phenomena as the "ever-recurring interplay between