Uncertainty

“Mathematics is about solving problems and it’s the great unsolved problems that make math really alive”- this was the opening sentence in the final instalment of the story of maths entitled To Infinity and Beyond. This first sentence had truly struck me. It made my thoughts wander the depths of space. I began to question myself; can mathematics still solve even the hardest of the hardest questions that a person genius enough can think of? What if all those great problems are already solved, what will happen to mathematics? Or is it even possible that all problems will be solved?

David Hilbert, a young German mathematician, boldly set out what he believed were the 23 most important problems for mathematicians to crack. The first problem on Hilbert’s list emerged from Halle, East Germany. It was where the great mathematician Georg Cantor spent all his adult life and where he became the first person to really understand the meaning of infinity and give it mathematical precision. He showed that infinity could be perfectly understandable. There wasn’t just one infinity, but infinitely many infinities. He suffered from manic depression and some says that it was because he dealt with some mathematical abstraction which is difficult to deal with. There was one problem he tried to unravel and it became known as the continuum hypothesis. Henri Poincare spent most of his life in Paris. He was hood at everything, his work would lead to all kinds of applications. In 1885, King Oscar II of Sweden and Norway offered a prize of 2,500 crowns for anyone who could establish mathematically whether the solar system would continue turning like clockwork, or might suddenly fly apart. Poincare simplified the problem, his ideas were sophisticated enough to win him the prize. When Poincare’s paper was being prepared for publication, one of the editors found a problem. Poincare, realised he’d made a mistake. Contrary to what he had originally thought, even a small change in the initial conditions could end up producing vastly different orbits, and this led to the chaos theory. Leonhard Euler, solved the problem to find a route around the city which crosses each of the seven bridges only once, by making a conceptual leap. This was a problem of topology, which other people refer to bendy geometry. In 1904 he came up with a topological problem he just couldn’t solve. What led to it  was known as the Poincare Conjecture. It was finally solved in 2002 by a Russian mathematician, Grisha Perelman. Perelman solved the problem by linking it to a completely different area of mathematics. Somebody criticized that for some time whatever’s been discussed it may seemed to be theology and not mathematics but what they’re missing is that Hilbert was creating a new style of mathematics, a more abstract approach to the subject. Hilbert believed that mathematics was a universal language, that was powerful enough to unlock all the truths of mathematics. He declared that there are no absolutely unsolvable problems, “We must now, we will know”. Kurt Godel was responsible for putting an uncertainty in the heart of mathematics. Godel proved the opposite of what Hilbert wanted, which is called the Incompleteness Theorem. He proved that within any logical system for mathematics there will be statements about numbers which are true but which you cannot prove. This led to crisis even for him. David Hilbert died in 1943. Paul Cohen proved that the continuum hypothesis could be assumed to be false. Sofia Kovalevskaya, the first female professor of mathematics in Stockholm and won a French mathematical prize. Emmy Noether, a talented algebraist who died before she fully realised her potential. Julia Robinson, the first woman to be elected president of the American Mathematical Society. Hilbert’s tenth problem became her lifetime work, she tends to think of it all the time. Soon, with the help of her colleagues, she developed the Robinson hypothesis. Matiyasevich soon solved the tenth. Galois had discovered new techniques to be able to tell whether a certain equations could have solutions or not. . . and many mathematicians made their contributions and the like.

I am wondering if what sort of problems does Hilbert’s problems seem like. That many mathematicians have made it the lifetime work just to be solve. I mean, was it really that difficult or what? But, I better not meddle with those kind of stuff, I already know for myself that I wasn’t born to be on those kind of things, but still, I am open for such. Still, we don’t know what the future is to bring us, what we could do is to wait and be prepared whenever, wherever. It is still not clear to me, whether all problems can be solved or not. “We must know, we will know” this quote had quite inspired me, though almost everything seems to be blurry to me but still I got this quote to guide me whatever the circumstances maybe. Our world, are so vast and full of mysteries and problems waiting to be figured out and unravelled and we are certain that mathematics will took a huge part in unravelling and solving such uncertainties.