Infinity to the 23rd Power

1900’s, Summer, at Sorbonne Paris where one of the greatest International Congress Mathematician was held. The lecture of a young German mathematician named David Hilbert set out his ‘23 most important problems for mathematicians to crack’ that gave mathematics definition of the modern age.

The first problem in Hilbert’s list emerged in Halle, Germany. Georg Cantor is the man that first understood the concept of a tricky and slippery concept of infinity. Despite of his Manic Depression Illness, he continued to strive and follow his passion. What Cantor did, is that he paired up all whole numbers with infinite of fractions and found out that fractions have the same sort of infinity as the whole numbers. Furthermore, the decimal form of the fractions has much bigger infinity than whole numbers such; Cantor built in an argument on how to construct next new decimal number from a missing decimal number from an infinite form of decimal number. However, Cantor struggled most of his life about ‘an infinity existing between the smaller infinity of all the whole numbers and the larger infinity of the decimals’ or the ‘Continuum Hypothesis’.

Henri Poincare a French mathematician who believed that Cantor’s math is “beautiful IF PATHOLOGICAL”.  He became the leading light of some French Mathematician and he was good in everything. Suddenly, when he was getting on a bus and he got a flash of Inspiration that lead to a success. In 1885, King Oscar II of Sweden and Norway offered 2500 crowns for anyone who will establish a mathematical system on predicting whether the solar system could continue turning or might suddenly fall apart. Poincare then develops arsenal techniques that lead him and his simplification enough to win the prize without using formulas. But when his idea is about to publish, he realized a mistake of his simplification. He realized that a small change of the initial condition could disrupt the orbit. Fortunately, at the end, the mistake is what matters most; the discovery about ‘Chaos Theory’, that explains on how a butterfly’s wing can change the atmosphere that might create a tornado or hurricane, is the mistake that Poincare done.

The Seven Bridges of Konigsberg became an instrument for the development of geometry of position where what matters is not about distances but connections between many objects. This development became a very useful for us in our everyday life like in station. We normally forget about the distances of each station but we are conscious about the connection between stations. But how it did develop? It was first became a puzzle on how can we get inside a town using the seven bridges of Konigbeg but passed them only at once. And it was solved by a great mathematician Leonhard Euler by using a conceptual leap of topology. Meanwhile, Poincare developed a new topological problem that he couldn’t solve and this is called Poincare Conjecture, it is about knowing the possible shape of our three dimensional universe.

Hilbert love creating new mathematical things. Revising mathematical theory and developing them. Hilbert Space, the Hilbert Classification, Hilbert Inequality and several Hilbert theorems are some of the terms because of his flexible approach of Mathematics. But an Austrian Mathematician will shatter Hilbert’s belief. He is Kurt Godel. He studied at Vienna University and spent his life in the cafe where all of his idea emerged that revolutionised mathematics. By solving the second Hilbert’s Problem, he came up to an opposite solution. He proved that within any logical system for mathematics there will be statements about numbers which are true but which you cannot prove.

In the time of Adolf Hitler, all mathematicians and scientist pulled out from the regime of Nazi party except Hilbert. He saw the fall of the centre of mathematics (Europe) until he died.

The Institute for Advanced Study had been set up in Princeton in 1930. The idea was to reproduce the collegiate atmosphere of the old European universities in rural New Jersey. Many of the brightest European mathematicians were fleeing the Nazis for America.

At very early age, Paul Cohen was winning mathematical competitions. Until he was challenged and found his passion about mathematics when he read the Cantor’s Continuum Hypothesis and began solving Hilbert’s first problem. Cohen’s proof has a daring solution that is so new that nobody seemed to be sure, until it was discovered and that gave him riches, fame and galore. He settled down for a moment, and then began solving the eight Hilbert problems, the Riemann Hypothesis. But like the other mathematicians Cohen has been defeated.

Julia Robinson, a female mathematician who was been elected as a first female president on the American Mathematical Society. Julia got her PhD and got married indeed; she got an obsession about the 10^th Hilbert Problem. The tenth problem says that if there is any universal equation that could tell whether any equation had whole number solutions or not. And she came up with Robinson’s Hypothesis but still failed.

Yuri Matiyasevich, who has been told during his school year to get on the tenth Hilbert problem which Julia Robinson failed- the Riemann Hypothesis. In January 1970 he found out how to capture the famous Fibonacci numbers by using Hibert’s equation and he solved the 10^th when he was just 22 years old.

During the reign of Charles X, a man named Evariste Galois was killed by a gunshot when he was 22. But he leaved a legacy, a legacy that Andre Weil used to discover and developed Algebraic Geometry. And without Andre Weil, Nicolas Bourbaki would never be heard because it really never exists. It is a code name for French mathematicians. And one of the brilliant member is Alexandre Grothendieck, he is interested with the hidden structures of mathematics; he is a structuralist. He made a new powerful language to see structures in a new way. But suddenly, the weather changes and he escaped from Paris and became one of the recluses. But still he is one that lines up of great mathematician.

Most of the Hilbert’s 23 problems had been solved and found out applications behind on it.  But still the 8^th problem remains unsolved- Riemann Hypothesis, the conjecture about the distribution of prime. And whoever proved this corner-stone of math will never be forgotten, he/she will be positioned ahead of any other mathematicians.

Mathematics is a search for pattern. And the more abstract and difficult the mathematics is the more application to the real world can be deduced from it. And because of the unsolved problems, mathematics will remain a living subject that will continue inspire mathematicians and students to discover and understand things; things that we must know and thus we will know.