To Infinity and Beyond

4th Installment of The Story of Maths Reaction Paper

“Why is mathematics getting complicated? Are mathematicians not out of ideas yet? Why can’t they just be satisfied for what mathematics is now? What more can mathematics offer?” These questions occasionally pop in my mind whenever I have difficulty in mathematics or when I was just hating on mathematics. But this episode of The Story of Maths showed me the answer to all these questions.

In the 4^th and last installment of The Story of Maths, it started by stating David Hilbert’s 23 most important problems of mathematics to crack. It is said that this would define mathematics in the modern age.  

The first problem was solved by George Cantor. He was the first person to understand the meaning of infinity. Many mathematicians were bothered by the paradox that his works have created but he believed that there are certain things that can be established with mathematical certainty and the absolute infinite which is only in God. He also believes that there is still that final paradox that we are not able to understand that only God does. But, there was one problem that he just could not leave and that is the ‘Continuum Hypothesis’. It asks the question, “Is there an infinity sitting between smaller infinity of the whole numbers and larger infinity of the decimals?” Cantor had his downfall and one mathematician thought that his works was beautiful and that is Henri Poincare. He established the ‘Chaos Theory’ and topological problems which was a powerful new way of looking at shapes. Later, he came up with a topological problem he could not solve. He called it the ‘Poincare Conjecture’ which asks what is the possible shape our universe could be in a 3D perspective. This was solved in 2002 by a Russian mathematician.

David Hilbert believed that mathematics is a universal language and he had no doubt that all his 23 problems would soon be solved and that mathematics would put on unshakable logical foundations. The problems that Hilbert could not prove are also included in his list. The next mathematician mentioned was Kurt Godel. He wanted to solve the second problem in Hilbert’s list. He wanted to find logical foundation for mathematics. He established the ‘Incompleteness Theorem’ which states that a problem is true but cannot be proven. His work led crisis to other mathematician. Due to some circumstances, Godel moved to America and continued his works there.

In the second part of the movie, it moved its plot to America. The first mathematician mentioned was Paul Cohen. He further studied Cantor’s ‘Continuum hypotheses’. No one trusted his opinion but later it was proved that he was correct and that made him rich and famous. Riemann’s hypothesis was introduced next but it was said that it was still unsolved.

Julia Robinson, the only female mathematician mentioned, focused more on Hilbert’s 10^th problem yet it remained unsolved. However, in 1970, Yuri Matiyasevich solved the problem. He captured the Fibonacci sequence of numbers using the equations that was already there in Hilbert’s equation. She later established the ‘Robinson Hypothesis’.

More mathematicians were mentioned in the last part of the movie but I would not focus on them since their topics are generalized. Today, Hilbert’s problems are mostly solved but there was one problem that has yet to be resolved and that is his 8^th problem, the ‘Riemann’s Hypothesis’. This is said to be the “Holy Grail” of mathematics.

I never thought mathematics had all this problems and there are still unsolved ones. Now, I understand why mathematicians are passionate about their work and we should respect them because without them, we would not have understood why things are happening around us.

I liked a quote or section from a book, by Poincare, that the professor read in the middle of the movie and it states that “If we wish to foresee the future of mathematics, our proper course is to study history.” This statement explains the existence of The Story of Maths. This statement could also be seen in a different perspective, for example, in a country. If you want to see the future of a country, we should look at the history and learn from it. From there, we could improve ourselves to make a better country. Similar to mathematics, we can apply this concept. Mathematics will still improve. There are still a lot of unsolved problems in mathematics and like what Poincare said, “There are absolutely no unsolvable problems. We must know, we will know.” Let us all see together the future of mathematics.