The "Hilbert" Games

     For the last installment of the series The Story of Maths, Marcus du Sautoy anticipates some of the great unsolved problems that oppose mathematics and discovers the enigma of the life of mathematicians who tried to crack them.

     During the 20^th century of mathematics, German mathematician David Hilbert boldly set out the 23 most important problems for mathematicians to crack. These Hilbert problems would define the mathematics of the modern age.

     The first Hilbert problem took place at Halle in East Germany, where Georg Cantor became the first person to really understand the meaning of infinity and gave it mathematical precision. He concluded that the infinity of fractions is much bigger than the infinity of whole numbers. Due to this intellectual breakthrough, Cantor became one of the most celebrated mathematicians during his time. But as he successfully understood infinity, there is one problem in which he failed to solve that bothered almost his life—the continuum hypothesis.

     On the other side, a French mathematician spoke up for him arguing that Cantor’s new mathematics of infinity was “beautiful, if pathological”. This is no other than Henri Poincare. During 1885, King Oscar II of Sweden and Norway offered a prize of 2,500 crowns for anyone who could establish mathematically once and for all whether the solar system would continue turning like clockwork, or might suddenly fly apart. This problem stumped even the great Newton. However, Poincare simplified the problem by making the successive approximations to the orbits which he believed wouldn't affect the final outcome significantly. But as soon as the King’s scientific writer was about to publish his work, the editor found some errors. Poincare noticed his mistake and asked for the paper to be not published. Although he made such mistake, he was still known as a mathematician who developed arsenal techniques to solve problems.

     Poincare also knew all the possible two-dimensional topological surfaces and in 1904, came up with a topological problem he just couldn't solve. This is called the Poincare conjecture. But in 2007, a Russian mathematician came to halt as he finally solved the problem that bothered Poincare, his name was Grigori Perelman. Though his solution was hard to understand even for mathematicians, it was still accepted and recognized. Today, Perelman lived his life shutting his doors to fame and money but continues to live by his passion—mathematics.

     In Gottingen, Germany; we've met the person that started it all off, David Hilbert. Hilbert became by far the most charismatic mathematician of his age. He showed that though there are infinitely many equations, there are ways to divide them up so that they are built out just a finite set, like a set of building blocks. Though he couldn't construct this finite set, he just proved it must exist. What’s special to Hilbert was his creation of new style of mathematics into a more abstract approach to the subject even though he couldn't construct it explicitly. He just stated “We must know, we will know” to problems who've seem to be bugging many mathematicians. Truly, he was a legend.

     But that doesn't stop there, in fact there is this Austrian mathematician who shattered Hilbert’s beliefs and attempted to solve the second problem. Kurt Godel not only couldn't provide the guarantee Hilbert wanted, but instead proved the opposite. And that he came up to his own Incompleteness Theorem, which he proved that within any logical system for mathematics there will be statements about numbers which are true but which you cannot prove. On the contrary, if the statement is false that means the statement could be proved, which may be true, and that is the contradiction.

     As the mathematical centre in Europe shattered, the flame shifted to the New World. Where mathematicians like Paul Cohen and Julia Robinson grapple some of the major problems in mathematics.

        Paul Cohen solved Cantor’s continuum hypothesis by proving that there wasn't a set between the whole numbers and the infinite decimals, but there was an equally consistent mathematics where the continuum hypothesis could also be assumed to be false. In 2007, he died even though he was trying to prove Riemann’s hypothesis. But like other mathematicians in the past, he also failed.

     Julia Robinson on the other side cracked Hilbert’s 10^th problem.  Instead of proving, she constructed Robinson hypothesis and on the bright side, started the trail of its success. In St. Petersburg, January 1970, Yuri Matiyasevich had finally solved the tenth problem through Robinson’s first mark. He saw how to capture the famous Fibonacci sequence of number using the equations that were at the heart of Hilbert’s problem.

     Evariste Galois, a French mathematician, believed that mathematics shouldn't be just a study of numbers and shape, but a study of structure. With that he discovered new techniques to be able to tell whether certain equations could have solutions or not. His idea of using geometry to analyse equations was picked up by another Parisian mathematician Andre Weil in the 1920s. Galois had shown how new mathematical structures can be used to reveal the secrets behind equations, while Weil’s work led him to theorems that connected number theory, algebra, geometry and topology.

     Andre Weil also led a group of French mathematicians by the name of Nicolas Bourbaki. The project aims to override personal glory. Meanwhile, a Bourbaki member Alexander Grothendiek produced a new powerful language to see structures in a new way. Indeed he is the dominant figure of the 20^th century.

Before the curtains closed, Marcus concludes in his journey that there are still great mathematical mysteries that remain unsolved—including the Riemann hypothesis. Today, there is now $1 million prize and a place in the history books for anyone who can prove Riemann’s theory.

     It is true that mathematics pervades every aspect of our lives. But we must not forget the people behind this history written journey for they serve the vital importance in the complexity of what we now know mathematics. As what the folk always say, when you lost something there will a new one which will come along the way. Truly, that’s why mathematics has evolved; a cycle that involved the birth and death of brilliant minds that contributed so much for the love of mathematics. It may seem bizarre how math keeps the fire burning in the souls of many people. How it made someone triumphant and at the same time miserable. But what keeps it rolling, is the language it speaks. The same language it heeds millennium ago to the modern age today. Indeed, mathematics is the true language the universe is written in, the key to understanding the world around us.