Hilbert posed the following problem as the 17th in the list of 23 problems in his famous 1900 lecture: Given a multivariate polynomial that takes only non-negative values over the reals, can it be represented as a sum of squares of rational functions? In 1927, E. Artin gave an affirmative answer to this question. His result guaranteed the existence of such a finite representation and raised the following important question: What is the minimum number of rational functions needed to represent any non-negative n-variate, degree d polynomial?
Hilbert was unfailingly optimistic about the future of mathematics, never doubting that his 23 problems would soon be solved. In fact, he went so far as to claim that there are absolutely no unsolvable problems - a famous quote of his (dating from 1930, and also engraved on his tombstone) proclaimed, “We must know! We will know!” - and he was convinced that the whole of mathematics could, and ultimately would, be put on unshakable logical foundations. Another of his rallying cries was “in mathematics there is no ignorabimus”, a reference to the traditional position on the limits of scientific knowledge.
The details of some of these individual problems are highly technical; some are very precise, while some are quite vague and subject to interpretation; several problems have now already been solved, or at least partially solved, while some may be forever unresolvable as stated; some relate to rather abstruse backwaters of mathematical thought, while some deal with more mainstream and well-known issues such as the Riemann hypothesis, the continuum hypothesis, group theory, theories of quadratic forms, real algebraic curves, etc.
Hilbert's problems included the continuum hypothesis, the well ordering of the reals, Goldbach's conjecture, the transcendence of powers of algebraic numbers, the Riemann hypothesis, the extension of Dirichlet's principle and many more. Many of the problems were solved during this century, and each time one of the problems was solved it was a major event for mathematics.
Most of the problems have been partially solved; some have been restated and the new interpretations have been solved; Problem #1 is thought to be solved by some and not by others. Problem #10 is solved, negatively. This means that the collective work of the mathematician working on this problem has proved that it is impossible to derive the process that Hilbert wanted for solving Diophantine equations.
The first half dozen problems pertained to the foundations of mathematics and had been suggested by what he considered the great achievements of the century just past: the discovery of the non-euclidean geometries and the clarification of the concept of the arithmetic continuum, or real number system. These problems showed strongly the influence of the recent work on the foundations of geometry and his enthusiasm for the power of the axiomatic method.
Some of these problems were already long standing and Hilbert himself had made some progress in them, so he knew something about their difficulties. It takes a certain amount of skill and knowledge to be able to ask the the type of questions that people will want to answer. Each problem had to have some kernel that made it important (and difficult) enough to be interesting, and it had to have the potential to lead to some wider results.
At the 1900 International Congress of Mathematicians in Paris, the mathematician David Hilbert presented a list of 23 problems for study in the twentieth century that later guided much of the mathematical research that followed.
In a monumental address, given to the International Congress of Mathematicians in Paris in 1900, David Hilbert, perhaps the most respected mathematician of his time, developed a blueprint for mathematical research in the new century. Jokingly called a natural introduction to thesis writing with examples, this collection of problems has indeed become a guiding inspiration to many mathematicians, and those who succeeded in solving or advancing their solutions form an Honors Class among research mathematicians of this century.
German mathematician David Hilbert was among the most influential mathematicians of the 20th century, conducting landmark research in algebraic number field theory, axiomatics of geometry and mathematics, integral equations, invariant theory, and mathematical physics.