Counting Infinity

              The last episode of the Story of Maths dealt with David Hilbert’s 23 most important problems math have answered which he believed have greatly influenced the mathematics of today. Hilbert’s first problem was the concept of infinity. A man named Georg Cantor was the one who really understand infinity. He also said that there is not only one infinity but a lot of infinities and that these infinities have the same sizes. For example, he paired whole numbers to fractions and concluded they have the same size. Then, he considered the set of all infinite set of decimal numbers in which he proved that this set has a bigger infinity. Yet, Cantor was unable to solve one problem- if there is an existence of infinity in between smaller and larger infinities. He was so bothered with this continuum hypothesis and he believes that this kind of set does not at all exist.

            Another problem in Hilbert’s list was the Poincaré conjecture. Henri Poincaré worked on ‘Bendy Geometry’ wherein he said that a shape can be molded to look like another and vice versa. However, a problem arose when he found out a two-dimensional figure in which he could not solve. Thus, it was called Poincaré conjecture. On the other hand, he had developed mathematical techniques to cosmological studies. In 2002, man named Grigori Perelman found an answer to this conjecture. He stated that a three-dimensional figure could be wrapped up figure with a higher dimension.

            Hilbert, despite being a brilliant mathematician, could not construct a finite set but believes it to exist. His famous line was “We must know, we will know.” However, Kurt Gödel destroyed this belief of Hilbert. As a kid, Gödel was described as ‘Mr. Why’ since he couldn’t stop asking questions. He proposed the incompleteness theorem. You could not say a statement is true but not prove. If a statement is false, it could be proved then it is true and that contradicts your assumption that it is false because it should be true. The problem here is, it should be true but could not be proved.

            During the Nazi’s regime, most of the mathematicians flew to America including Kurt Gödel. Personally, I was shocked to see that Gödel was actually Einstein’s friend more specifically to the fact that Einstein has friends (I always thought he was introvert, proved me wrong). In 1950’s, Paul Cohen worked on mathematical universe. He also tried to solve the Reimann hypothesis, Hilbert’s eighth problem on the list. Hilbert’s tenth problem is if there is a sort of universal method which could tell if any equations have whole number solutions or not. This is when a woman enters a man’s world, Julia Robinson; the first woman elected President of the Mathematical Society (I am so a fan). In 1952, Julia was married to Rafael Robinson and the couple answered the tenth problem on Hilbert’s list, the Robinson hypothesis. This hypothesis stated that in able for you to show there is no method, you choose an equation which solutions is a very specific set of numbers, and that this set should exponentially grow but still be in the premise of the equation. The problem is that Robinson was not able to find this set but fortunately, Yuri Matiyasevich was up for the job. He was able to solve Hilbert’s 10^th problem using the Fibonacci sequence when he was just 22 (I wonder what I would be at my 22^nd year on this planet).

            As the episode approaches its end, it discussed briefly about algebraic geometry. Evariste Galois believed that mathematics should not be a study of shapes and numbers but of structure. He had answered the question whether or not an equation has a solution or not (the Riemann Hypothesis). However, he had a sad fate and ended his life when he was 20. Another mathematician, Andre Weil, was the one who structured algebraic geometry through the works of Galois. He was the leader in creating a fictional mathematician by the name of Nicholas Bourbaki alongside with another mathematician, Alexander Govendick. Bourbaki was made because the above mentioned mathematicians do not desire personal glory. I personally admire these two mathematicians. Then again, this is the last episode of The Story of Maths where Du Sautoy broadened our view and history of mathematics. KUDOS for Marcus du Sautoy.