Deciding the Undecidable: Wrestling with Hilbert’s Problems

In the year 1900, the German mathematician David Hilbert gave a dramatic address in Paris, at the meeting of the 2nd International Congress of Mathematicians—an address which was to have lasting fame and importance. Hilbert was at that point a rapidly rising star, if not superstar, in mathematics, and before long he was to be ranked with Henri Poincaré as one of the two greatest and most influential mathematicians of the era. Like Poincaré, Hilbert worked in an exceptional variety of areas. He had already made fundamental contributions to algebra, number theory, geometry and analysis. After 1900 he would expand his researches further in analysis, then move on to mathematical physics and finally turn to mathematical logic. In his work, Hilbert demonstrated an unusual combination of direct intuition and concern for absolute rigor. With exceptional technical power at his command, he would tackle outstanding problems, usually with great originality of approach. The title of Hilbert’s lecture in Paris was simply, “Mathematical Problems”. In it he emphasized the importance of taking on challenging problems for maintaining the progress and vitality of mathematics. And with this, he expressed a remarkable conviction in the solvability of all mathematical problems, which he even called an axiom. To quote from his lecture: