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 thought and experience."

Fundamentally, Hilbert wished to provide greater precision to the reasoning and deduction of mathematics, developing basic axioms on which all other axioms could be based. He contended that all mathematical problems could be correctly solved by applying logical reasoning to a finite number of processes. In short, he demanded rigor in reasoning, with this rigor based on exactly formulated statements. Hilbert hoped that by making the logic of mathematics itself more exacting, fewer complications and paradoxes would be encountered in the study of mathematics. He also noted that forcing the rigor of mathematical proofs to a well-understood and developed chain of axioms, the simplest (and thus more understandable) proofs would be developed. Hilbert particularly opposed the idea that the axioms of geometry, through which greater understanding of the axioms of arithmetic have been achieved, are not disposed to such rigorous proofs, disputing that rigorous proof is confined to concepts of analysis, or perhaps arithmetic alone.

While such fundamental axioms were possible in some fields of mathematicsÐ'--particularly arithmetic--more abstract fields of mathematics did not lend themselves easily to this approach. As a result, Hilbert's objectives to derive all areas of mathematics from a few fundamental principles are not likely to be realized. Formalism, it seems, has limits when apply to abstraction. Important strides, however, have been made on the governing axioms in many areas of mathematics, leading to the efficiency and simplicity for which Hilbert strived.

Briefly stated, Hilbert's 23 problems are as follows:

Problems related to the foundation of mathematical science

Ð'* The cardinal number of the continuum (based on Cantor's continuum hypothesis)

Ð'* The consistency of the axioms of arithmetic

Ð'* The equality equality of volumes of two tetrahedra with equal heights and bases

Ð'* The systematic review and development of the axioms of geometry

Ð'* The establishment of axioms for group theory independent of the assumption