Complicated Math that Made Life Easier

Finally, we are at the last instalment of The Story of Maths but actually this is where math really went nuts. Most of the mathematician had a bad life, but that didn’t keep them from being great, instead they became one of the people who made the world what is it now, easier and secure and pleasurable. We owe mathematics a huge amount of recognition for what great things that math contributed to our world.

                The first mathematician featured in this episode is Georg Kantor. He was the first one to really understand the concept of infinity by comparing numbers 1, 2, 3 and so on to 10, 20, 30 … that it has the same size. So what about the fractions in between the numbers? He created an infinite grid which the first round contain the whole numbers fractions with one on the bottom in the second row. 10 harms fractions with on the bottom and so on. Every fraction appears somewhere in this grid for example 4/5. So 4^th column 5^th row is the location where 4/5 is located. He then snaked the grid and turned it into a straight line which every fraction has a corresponding whole number. So the fraction can go to infinity as well as the whole numbers and this created a new idea, a new hypothesis which is the continuum hypothesis. The next person featured is Henri Poincare. He solved the problem in the solar system which involved the revolving of the sun earth and moon together and how to solve each missing variables like exact coordinates, velocity, and direction. He sorted these out and simplified them by making approximations to the orbits. He was the guy that likes techniques in solving such complicated math expressions. But one person found out that the simplifications of Poincare are wrong because a small difference in values could create a new orbit. This ‘mistake’ of Poincare was an indirect discovery of a theory now known as the Chaos Theory. He was also an author writing about the importance of math but the work that made him to the famous status was topology, enabling us to have a new way of looking at shape. Topology gave Poincare a question he cannot seem to answer, If you've got a flat two-dimensional universe then Poincare worked out all the possible shapes he could wrap that universe up into. It could be a ball or a bagel with one hole, two holes or more holes in. But we live in a three-dimensional universe so what are the possible shapes that our universe can be? That question became known as the Poincare Conjecture. This was solved by Grisha Perelman by linking it to as different mathematical area and his proof was very hard to understand even for mathematicians.

 David Hilbert was a mathematician who studied the number theory and revolutionized the theory of integral equation but the work that made him stand out is his work on equations showed that although there are infinitely many equations, there are ways to divide them up so that they are built out of just a finite set, like a set of building blocks. The interesting part of his proof is that he didn’t construct it but he just proved that they should exist. It is just like telling that there is a way from Los Angles to Bay Area but I just don’t know how to get there. He said that there are no equations that can’t be solved and we should and will know. You know when you are full of hope and your day seems to be good then just one person shatters all those? That’s what Kurt Godel did to Mr. Hilbert, and he also put uncertainty to mathematics. He was a person of overflowing curiosity when growing up. Godel proved that within any logical system for mathematics there will be statements about numbers which are true but which you cannot prove known as the Incompleteness theorem. He started with a statement, “This statement cannot be proved”. If the statement is false, that means the statement could be proved, which means it would be true, and that's a contradiction. So that means, the statement must be true. In other words, here is a mathematical statement that is true but can't be proved.

When David Hilbert died, the mathematical prowess of Europe died too and the baton was passed to the Americans. It was in Princeton, New Jersey that they tried to rebuild the environment of mathematics that was lost. Kurt Godel was one of the European exiles from the Nazi empire and he was neighbours with the great Albert Einstein. The sanguine type personality of Einstein made Godel a much more quiet and pessimistic person that led to his lunacy. The next mathematician that evolved during the hedonistic and fast food life of the Americans was Paul Cohen. He was brilliant winning awards and prizes but he wanted to make a mark in mathematics and what field should he leave a mark. He then began to work with Cantor’s problems where he can’t seem to solve but Cohen found a proof that did prove the Continuum hypothesis. Many doubted the solution of Cohen because it is new but there is one man that everyone looked up to for the solution’s approval, Kurt Godel. He then went to Princeton so his work could be checked by Godel and Godel did give his stamp of approval giving Paul Cohen more recognition and prizes. Then after gaining confidence from solving Hilbert’s 1^st problem, he attempted to solve the most important problem in Cantor’s problem, #8 the Riemann Hypothesis. When he died, he was still trying to solve it but he just couldn’t just like other mathematicians before him. That doesn’t mean that he was forgotten, in fact he was a successful product of the American dream because from being an exile he became a top professor.

Then the next thing just caught my attention. A female mathematician was featured and this is interesting because there are just 3 mathematicians that have been really known. The female American mathematicians is the first ever woman to be the president of the American Mathematical Society, she is Julia Robinson. She had an obsession to solve Hilbert’s 10^th problem and she said that she just wouldn’t want to die without knowing the answer. With the help of her colleagues, she then developed the Robinson Hypothesis She had inspiration from Yuri Matiyasevich from St. Petersburg, Russia. Yuri found the last important piece to solving Hilbert’s 10^th problem. He saw how to capture the Fibonacci sequence in the problem using the equations at the heart of Hilbert’s equations. The next mathematician featured was Evariste Galois, a republican revolutionary in France. He believed that mathematics should not be the study of shape and number but of structure. Galois had developed a technique where certain equations could have solutions or not, and shown how new mathematical structures can be used to reveal the solutions to the equation and this is because of his idea to analyse these equations. This idea made light to Andre Weil, a prisoner of a war but he became a soldier. In his jail time, he first developed algebraic geometry and built on the ideas of Galois, a whole new way of understanding solutions to equations. This led to the connections to theorems connected to geometry, topology, and Algebra, all this led to one of the greatest achievement of modern mathematics. The next man is Alexander Grothendieck who is a pretty interesting person because he is a structuralist, interested in hidden structures underneath mathematics. I like him because we can understand the most complicated things when we go back to the basic then from there we could work our way up. He discovered a new way to look at patterns in mathematics which are being used until today to solve problems and equations. But sadly, he turned his back on mathematics when he entered politics. One thing that still remains in doubt today is the Riemann Hypothesis.

 Maybe one of you reading this article may have that world changing idea, not just math. Sometimes math may intimidate most of us, but it is the people who step out and get that intimidation out of their head who can truly get them to that changing idea. People wanted to do simple things and in order to do simple things, some should do the hard things first. This is what all the mathematicians showed us, erase that uncertainty and doubt so we could have that feeling that we can do things very easy. Math isn’t all about hard equations to solve, it is how we use the basic things to solve the hard ones, like what Grothendieck did, he understood complicated things easily because he used the general and basic terms first before all the hard ones so math is possible for all of us.  I hope this will serve as an inspiration to all of you out there, Peace!