Throughout the year we have seen that mathematics really has become part of our daily lives. From the paying our bills in the bank to driving our automobiles out of town, mathematical theories and principle has been incorporated in everything we do. Mathematics has truly become a part of us. But as we see the era of modernization, we can’t help but wonder of the future that mathematics holds for us. And if we wish to see the future of mathematics we have to look at the history of its past.
And this is what Marcus Du Sautoy has done in the fourth installment of the British television series, The Story of Maths, which outlines the history of mathematics. Entitled “To Infinity and Beyond”, writer and producer Marcus du Sautoy of the University of Oxford documents the 20th century mathematics. This final episode considers the great unsolved problems that confronted many great mathematicians during the 20th century. (Wikipedia, 2013)
Du Sautoy starts his journey in Europe by mentioning David Hilbert and his contribution that posed 23 unsolved problem in mathematics. As he st the agenda for the 20th century mathematics, he implied much importance on these 23 problems, believing that they were of the most immediate importance. Soon Hilbert’s first problem emerged as Georg Cantor was able to understand infinities. Before him no one really understood infinities. Cantor realized that infinities require mathematical precision. He was the one who showed the world that there were different infinities, some bigger than others. And through this Cantor has opened a door and an entirely new mathematics lay before us. (Wikipedia, 2013)
However there was a certain problem that Cantor was not able to solve: “Is there an infinity sitting between the smaller infinity of all the fractions and the larger infinity of the decimals?” And Cantor believed that there is no such set, and this in turn gave birth to what is now known as Continuum Hypothesis. And this would be the first problem listed by Hilbert. (Wikipedia, 2013)
Next Du Sautoy discusses the most famous mathematician of his time, Henri Poincare, and his work on the discipline of 'Bendy geometry'. This states that if two shapes can be molded or morphed to each other's shape then they must have the same topology. Poincare was able to identify all possible two-dimensional topological surfaces; however in 1904 he came up with a topological problem that he could not solve. This problem was also known as the Poincare conjecture that queries on what all the possible shapes for a 3D universe are. (Wikipedia, 2013)
However, this was solved by Grigori Perelman in 2002. Perelman linked the problem to a different area of mathematics. Perelman was able to find all the ways that 3D space could be wrapped up in higher dimensions. (Wikipedia, 2013)
Du Sautoy then goes back to the achievements of David Hilbert. Hilbert was already a star. Aside from his famous 23 unsolved problems, he was also famous for the Hilbert Space, Hilbert Classification and the Hilbert Inequality. And his early work on equations marked him out as a mathematician who was able to think in new ways. Aside from this, Hilbert also showed that while there was infinity of equations, “these equations could be constructed from a finite number of building block like sets”. However Hilbert could not construct that list of sets. “He simply proved that it existed.” But because of this, Hilbert had created a new style of mathematics - a more abstract style of mathematics. (Wikipedia, 2013)
Du Sautoy also mentions that Hilbert believed that mathematics was a “universal language powerful enough to unlock all the truths and solve each of his 23 Problems.” Aside from this, Du Sautoy documents that Hilbert quotes that “we must know, we will know” for he believed that that if his 23 problems were to be solved, mathematics will be placed in an unshakable foundation. (Wikipedia, 2013)
However Kurt Godel had shattered Hilbert’s belief that mathematics was a universal language powerful enough to unlock all the truths and thus solve his 23 problems. Through Godel’s Incompleteness Theorem which was based on Hilbert’s second problem. Godel discovered that if a statement logically cannot be false, then the “existence of mathematical statements that were trues were but incapable of being proved.” (Wikipedia, 2013)
Du Sautoy then moves to America to introduce us to an American mathematician, Paul Cohen who took up the challenge of Cantor’s Continuum Hyphothesis which queries "is there is or isn't there an infinite set of number bigger than the set of whole numbers but smaller than the set of all decimals". Cohen discovered that there existed two equally consistent mathematical worlds. “In one world the Hypothesis was true and there did not exist such a set. Yet there existed a mutually exclusive but equally consistent mathematical proof that Hypothesis was false and there was such a set.” However this was without success. Cohen then worked on Hilbert’s eighth problem, the Riemann Hypothesis, which is also considered as the holy grail of mathematics. (Wikipedia, 2013)
Du Sautoy also tapped on the tenth problem of Hilbert which asks “if there was some universal method that could tell whether any equation had whole number solutions or not”. Many people believed that there was no such method possible, however, the question “how could you prove that no matter how ingenious you are you would never come up with such method?” still lingered. And Julia Robinson created the Robinson Hypothesis to answer this. Robinson Hypothesis states that” to show that there was no such method all you have to do was cook up one equation whose solutions were a very specific set of numbers: The set of numbers needed to grow exponentially yet still be captured by the equations at the heart of Hilbert's problem.” But it was not Robinson who was able to find this set, but rather Yuri Matiyasevich who saw how to capture the Fibonacci sequence through using the equations of Hilbert’s tenth problem. The set of numbers needed to grow exponentially yet still be captured by the equations at the heart of Hilbert's problem.
Then Du Sautoy covers a little on the concept of algebraic geometry. He mentions Evariste Galois who believes that mathematics should not be a study of number of numbers and shape rather it should be the study of structures. He also discovered that the key to tell whether certain equations could have solutions or not lies in the symmetry of certain geometric objects. And this idea inspired Andre Weil to create Algebraic Geometry which is a whole new language that connects number theory, algebra, topology and geometry all together.
Du Sautoy ends his four part British television series Story of Maths through a recap of the history of mathematics in different eras, civilizations, and nations. But as he ends this television series he leaves a mark of the beauty that mathematics hold – a beauty that can only be realized by understanding how mathematics seeks patterns, how mathematics solves the unsolved and how mathematics allows us to understand our world.
Mathematics that seeks patterns. In the first installment of the Story of Maths, we have seen that mathematics makes sense of patterns and this is strengthened by this fourth installment. We can see that mathematicians make sense of patterns because they are pattern searcher. This is what they do. Mathematicians are curious people, they want to see pattern because they want to make sense of those patterns. There is beauty there. In the act of pattern seeking alone lays a beauty already. In turn mathematics has become a tool to search for patterns. Mathematics has somewhat become a mystery seeker. But to unravel the mystery, the mathematicians must seek clues in the form of patterns.
Mathematics that solves the unsolved. Mathematics is about solving problems. These unsolved problems are what bring mathematics to life. This is just like how Hilbert’s 23 problems would provide a strong and unshakable foundation to mathematics. Mathematicians aren’t motivated by money or material gain or even the application of their work, for them it’s in the glory of solving some of the great unsolved problems of the world and in the glory that they have outwitted previous generations of mathematicians. When one person tries to solve a puzzle, that person is motivated by the feeling afterwards once the puzzle is solved. This goes the same for mathematics. The adrenaline rush, the excitement and the glory you get after solving the unsolved is incomparable to any monetary or material gain. It is as if for a second there the world has stopped and has become a better place just because you are able to solve an unsolved problem. Just as Hilbert has once said, “we must know, we will know,” we will solve the unsolved.
Mathematics that allows us to understand. It is impossible to imagine modern life without mathematics. Even before mathematicians were driven to understand how numbers and space works. And not only did mathematics help us understand how numbers and space works, but it too has helped us understand our world. This is because mathematics has become the true language the universe has written; it is the key to understand the world around us. That’s the beauty to mathematics, not only is it a scientific discipline but through it we are able to understand the world, to look at the world and appreciate it for its beauty. If we understand something we are much able to appreciate it. And this is what mathematics is doing, it is helping us appreciate our world.
Through the Story of Maths, we were able to go back and look at the history of mathematics. Were able to travel to different time and nations, we were able to meet new people and we were able to learn mathematical principles, ideas and concepts. But what strikes us is not the mathematical history or concepts that we have learned through that journey but rather what strikes us is that we understand mathematics in a different level now, and by that we see the beauty of mathematics and not just its surface. And by seeing its beauty we are able to value it and appreciate it.
I end with Hilbert’s call, “we must know and we will know”, this does not only apply to the 23 unsolved problems or any mathematical problems, but rather this applies to everything. We must know more about anything first, for only then will we really know what that anything is. We have to understand first in order to fully appreciate it.
Reference: The Story of Maths. (2014, January 21). In Wikipedia. Retrieved February 10, 2014, from http://en.wikipedia.org/wiki/The_Story_of_Maths
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