FAME and FOREVER

            On a lecture given by a young German Mathematician, David Hilbert, the 23 most important problems that mathematicians need to solve were emphasized. The lecture was part of the International Congress of Mathematicians which happened in the summer of 1900 in Sorbonne, Paris. The “23 problems” became a challenge to every mathematician on that day. Some tried to crack it; other succeeded while others failed. It was in Halle, East Germany where the first of Hilbert’s list triumphed. This town was the place of the great mathematician Georg Cantor. He was the first to understand the meaning of infinity ad give it mathematical precision. Before him, no one had understood infinity. According to him, there wasn’t just one infinity but infinitely many infinities: from the infinity of whole numbers to the infinity of fractions. Georg Cantor made a clever argument to show how to construct a new decimal number, this now opens up the idea of infinity. In spite of this amazing discovery, Cantor suffered from manic depression. There was no treatment for his illness that time. But he still resumed his investigation of the infinite until the day he died. Cantor was never worried with the paradox of the infinite because he believed that the absolute infinite is only in God. But there was one problem which he really got a hard time to solve, it is the continuum hypothesis. “Is there an infinity sitting between the smaller infinity of all the whole numbers and the larger infinity of the decimals?  ”

            There was a mathematician from France who once said that Cantor’s new mathematics of infinity was beautiful, if pathological. He was Henri Poincare. Poincare was good at algebra, geometry, analysis and everything. He became the leading light of Paris when it became the center for world mathematics in the 19^th century. In 1885, King Oscar II of Sweden and Norway searched for someone who could establish mathematically whether the solar system would continue turning like clockwork, or might suddenly fly apart. The King promised 2500 crowns as a reward. Poincare could not solve the entire problem but his ideas were refined enough to have the reward. He developed important mathematical techniques or some sort of great arsenal techniques to solve the problem. When his paper was now ready for publication, one of the editors found something unusual. Poincare realized he committed a mistake. And this now led to what is known as the chaos theory. By digging deeper into the mathematical rules of chaos, one can explain why a butterfly’s wings could create tiny changes in the atmosphere that might cause a tornado or a hurricane on the other side of the world. Poincare’s mistake enhanced his reputation. He published a lot of his original work throughout his life. In one of his popular books, he wrote,  “If we wish to foresee the future of mathematics, our proper course is to study the history and present the condition of the science”. Chaos theory is just one of Poincare’s contributions to mathematics. The most important lies in The Seven bridges of Konigsberg back in 1945. At first, it was just an 18^th century puzzle. The problem to solve is to find a route around the city which crosses each of the seven bridges only once. This was solved by the great mathematician Leonhard Euler which he claimed in 1735 to be impossible. There could not be a route that did not cross at least one bridge twice. Through a conceptual leap, Euler was able to solve the problem. This problem now opens a door to a new sort of problem which is a problem of topology.

            Topology had grown into a powerful new way of looking at shape, thanks to Henri Poincare. Topology maybe referred to as a bendy geometry. It is because in topology, two shapes can be the same if you can bend or transform one into another without cutting it. It was in 1904 when Poincare had a topological problem that he can’t solve. If we are living in a three-dimensional universe, what are the possible shapes that our universe can be? This question was later known as the Poincare Conjecture. A Russian mathematician named Grisha Perelman was able to solve this problem in 2002 by linking it to a different area of mathematics. Perelman’s proof is very hard to understand even for mathematicians, also his character. He had turned down all prizes and offers of professorships from distinguished universities in the west. He seemed to have given up on mathematics recently and lived peacefully with his mom. Going back to the one who formulated the list of 23 problems in 1900, David Hilbert might have commended Perelman for his discovery.    

            David Hilbert was the most charismatic mathematician of his age. Hilbert studied number theory and revolutionized the theory of integral equation. At present, many mathematicians still used the Hilbert Classification, the Hilbert space, the Hilbert Inequality and several Hilbert theorems. His early work on equations made him mark as a mathematician thinking in new ways. Hilbert demonstrated that there are ways to divide infinitely many equations like a set of building blocks or in a finite set. He could not construct this finite set actually but he still proved it must exist. In this way, Hilbert was creating a more abstract approach to mathematics. This mathematician had indeed a shocking lifestyle. He liked to drink with his students and dance with women. He also believed that everyone can do mathematics provided that he/she has the skills. For him, mathematics is a universal language. He had no doubt that all his 23 problems would be solved. He once said, “We must know, we will know”. If for Hilbert, all the 23 problems can be proved, not for this next mathematician. His name is Kurt Godel.

            Kurt Godel was an Austrian mathematician. During his childhood, he used to ask questions which made his family to call him Mr. Why. Godel improved his skills in mathematics in Vienna University but most of the time; Godel is in the cafes, in the internet chat rooms and in places where he can play backgammon and billiards. Godel got himself to be a member of a highly influential group of philosophers and scientists called the Vienna Circle. Here, he tried to solve the second problem in Hilbert’s list but instead, he got something different. It is called the Incompleteness Theorem. According to him, there will be statements about numbers which are true but cannot be proved. He began with the statement, “This statement cannot be proved”. He transformed this statement into a pure statement of arithmetic which can now be either true or false. If the statement is false, the statement could be proved, meaning it would be true. But this becomes illogical making the statement to be true. In simple speaking, this is a mathematical statement which is true but cannot be proved. Godel’s proof created a crisis in mathematics and also for him. He got into a series of breakdowns in 1934 and was contained in a sanatorium. He fell in love with a local night club dancer and her love kept him alive. When they were walking outside, they were attacked by Nazi thugs. The Nazis have submitted a law allowing the elimination of any civil servant who was not Aryan and academics were civil servants. Mathematicians were much affected; some lose their jobs while some was driven to suicide or died in concentration camps. But David Hilbert did not lose the spirit. He made some of his brightest students escape and asked for the dismissal of his Jewish Colleagues. He did not run away from war  for some unclear reasons until he became ill. Hilbert was left out during the Nazi regime and had witnessed the destruction of the greatest mathematical centers of all time. He died in 1943 and only ten people attended his funeral.

            Many of the brilliant mathematicians resumed their lives in America. Two of them were Herman Weyl and John Von Neumann. The Institute for Advanced Study had been built in Princeton in 1930. The purpose was to revive the collegiate atmosphere of the old European universities in rural New Jersey. This institute became the home of these brilliant European mathematicians including Kurt Godel and his friend Albert Einstein.

            During the 1950s in America, youngsters were not interested in mathematics except for one who was interested in cracking the major problems of mathematics. He was Paul Cohen and he wanted to have a mark in his own field of mathematics. When he read about Cantor’s Continuum Hypothesis, the 22-year-old Paul Cohen decided that he could do it. Cohen succeeded to find the truth behind Cantor’s Continuum Hypothesis. But nobody was sure of his new method and they needed the opinion of the one they trusted most, Kurt Godel. After Godel gave his approval and said, “Yes, it is correct”, Cohen’s proof was accepted by everybody. Mathematicians today depends on the continuum hypothesis wherein one answer can be yes and the other is no. Paul Cohen was rewarded with fame, riches and a lot of prizes. Up until his death, he was trying to solve Hilbert’s eighth problem, the Riemann Hypothesis. Cohen was not the only American mathematician who became great in mathematics in the 1960s, there were many.

            Letting go of the male mathematicians, here come the females. The Russian Sofia Kovalevskaya was the first female professor of mathematics in Stockholm in 1889. She received very prestigious French mathematical prize. One of the mathematicians who fled from the Nazis is Emmy Noether. She was a talented algebraist but unfortunately died before knowing her potential. And this woman, Julia Robinson, she was the first woman ever who became president of the American Mathematical Society. When she was just a child, she and her sister lived with her grandmother because their mother died when she was just two. She suffered from scarlet fever when she was seven and spent a year in bed. She was even told that she would not live beyond 40. She really had innate mathematical ability even in a young age but was just less in encouragement. When she was a teenager, she only knew that there are mathematics teachers and a radio show made her realize that there were mathematicians. She wanted to be part of their world. She went to the University of California in Berkeley and met the number theorist Raphael Robinson. They got married in 1952. Julia got her PhD and started solving Hilbert’s tenth problem which asked if there was some universal method that could tell whether any equation had whole number solutions or not. She and her colleagues developed what is known as the Robinson Hypothesis. Still, the problem remains unsolved.

            Yuri Matiyasevich was the one to solve the tenth problem by building on the work of Julia and her colleagues. He was just 22 years old then and he really wanted to thank Julia for what he had achieved. He sent a mail but got no reply. Back in California, Julia was hearing rumors that the tenth had been solved. She contacted Yuri and then Yuri thanked her and gave the much bigger credit to her. Julia and Yuri worked together in solving other mathematical problems until Julia died in 1985. She was just 55 then but before she died, she was able to find new ideas to solve special classes of equations.

            Another astonishing story of a mathematician is in Paris. He was Evariste Galois, the brilliant mathematician who discovered techniques to tell whether certain equations could have solutions or not. He was able to do it by using geometry.  Galois was killed by a gunshot from a duel. For mathematics, it was indeed a big loss. Galois’ idea was used by Andre Weil, another Parisian mathematician in the 1920s. He first developed the algebraic geometry which is a new language for understanding solutions to equations. Weil’s theorems connected the number theory, algebra, geometry and topology which became a great success of modern mathematics. Through Andre Weil, the name Nicolas Bourbaki was known.  Bourbaki is very popular because of his books in mathematics. But this person really does not exist. Nicolas Bourbaki was just the nom de plume of a group of mathematician led by Andre Weil. After the Second World War, the Bourbaki name was given to the next generation of French mathematicians. Alexandre Grothendieck was one of them. He was a Structuralist, interested with the hidden structures underneath all mathematics. Grothendieck discovered a language that mathematicians keep on using ever since to crack problems in number theory, geometry and fundamental physics. But Grothendieck chose politics over mathematics. The threat of nuclear war is more important to him then mathematics. He was isolated from his old friends and mathematical colleagues, but his legacy in mathematics never vanished.

            Mathematicians are really hard workers. They are the persons that you can never ever call lazy. Basing on the four episodes of the Story of Maths, I can call them the “much effort men and women”. I always say in my three movie reviews, mathematicians lent their lives for us to experience this great value for mathematics. Mathematics couldn’t be easy (in just some parts but not totally) without their contributions. We should be thanking them for the rest of our lives. To be honest, I regard this episode as the most boring episode of the Story of Maths. But hey, that was just my first impression. As I’m into the process of understanding the whole story, I’m learning to love it. For me, this is the most amazing part of the Story of Maths. I was able to learn much newer information compared to the last three. Well it’s really obvious because it is the latest episode, but not in that way. It’s just that these information felt very fresh to me that when I read it, I can’t help to be in awe. David Hilbert, Georg Cantor, Henri Poincare, Kurt Godel, Paul Cohen, Julia Robinson and the other mathematicians, they really rocked the world of mathematics. They proved to us that problems really have solutions and that all that has the skills can crack a mystery and be famous for it. Do you know the feeling of being famous? It is when people always remember you for something. It can be for a place, for a statement, for a thing or for just simply being who you are. I always wanted to be famous, someday. Who doesn’t want to be famous? Don’t be a hypocrite. Well, I know that the road to fame is not easy but at the least, we must try. These famous persons all started from scratch, just like me and you, just like us. Someday we can be famous in our own fields. But I don’t expect myself to be famous in the world of mathematics (but we don’t know...). One of my professors said that in order to achieve something, you need to do your best always. Do not be lazy and believe that you can do it. Just like these mathematicians, they tried and tried until they died. (Are you laughing at my statement?) Yes, it’s quite funny but the idea is there. They certainly died while giving their very best to solve a problem. So, for us who are still alive and kicking, be at your very best. Think for the best, speak for the best and always consult The One who is very best.