Student Reports: A Rewarding Undertaking - Undergraduate Report

Hilbert's Tenth Problem

David Hilbert was born on January 23rd, 1862 in Königsberg, Prussia. He worked in many areas of mathematics including geometry, algebra, number theory, and mathematical physics to just name a few. In 1899, Hilbert greatest work of the time was his book Grundlagen der Geometrie (Foundations of Geometry), which put Euclidian geometry in a formal axiomatic setting and removed the flaws from Euclidean Geometry. It was of Hilbert's belief, or inspiration, that all of mathematics could be placed in a formal axiomatic setting and thus all mathematical theorems could be proven. He began what is known as the "formalist school" of mathematics.

In 1900, he was selected as the leading mathematician of the time to present the keynote speech at the 2nd International Congress of Mathematics in Paris. In his introduction of his lecture he states:

“Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries? What particular goals will there be toward which the leading mathematical spirits of coming generations will strive? What new methods and new facts in the wide and rich field of mathematical thought will the new centuries disclose?”

Hilbert went on to state the nature of mathematical problems, in that they should be difficult, but not completely inaccessible, and what mathematicians should do about them, then he went on to state 23 problems that he felt were relevant to the foundation and future research of mathematics as seen at the turn of the century.

“The great importance of definite problems for the progress of mathematical science in general... is undeniable. ... [for] as long as a branch of knowledge supplies a surplus of such problems, it maintains its vitality.... every mathematician certainly shares.. the conviction that every mathematical problem is necessarily capable of strict resolution ... we hear within ourselves the constant cry: There is the problem, seek the solution. You can find it through pure thought...”

The 23 Paris problems became the backbone for mathematical research in the 20th century. Many of the problems were standing open problems of the time. Many of the problems have been proven. Several are still open problems.

Without doubt, Hilbert was the dominant influence on 20'' century mathematics. Such a grand body of work and new areas of mathematics have evolved from attacking his challenges and visions. His greatness can be summed up by his first student Otto Blumenthal:

“In the analysis of mathematical talent one has to differentiate between the ability to create new concepts that generate new types of thought structures and the gift for sensing deeper connections and underlying unity. In Hilbert’s case, his greatness lies in an immensely powerful insight that penetrates into the depths of a question. All of his works contain examples from far flung fields in which only he was able to discern an interrelatedness and connection with the problem at hand.  From these, the synthesis, his work of art, was ultimately created.  Insofar as the creation of new ideas is concerned, I would place Minkowski higher, and of the classical great ones, Gauss, Galois, and Riemann. But when it comes to penetrating insight, only a few of the very greatest were the equal of Hilbert.” - Otto Blumenthal, Hilbert's first student.