“We must know, we will know.” – David
Hilbert
In
the fourth and last instalment entitled “To Infinity and Beyond” of the BBC
series, The Story of Maths, we see how mathematics is like a quest. You travel
far and wide in search for something or seeking for something and in this case,
mathematicians do the same thing. Mathematics is all about solving problems.
They pursue the answers to the greatest unsolved problems in mathematics in
order to perform a prescribed feat and to gain acknowledgment. To every
mathematician, there lies joy and excitement whenever they face a challenge in
mathematics that they want to overcome. They work hard to get the fruits of
their labour, the answers to these unsolved mysteries. Marcus du Sautoy
presents us these unsolved problems of the 20th century that make
mathematics fun and lively.
Du
Sautoy’s first stop in this episode is in Paris, France, where he introduces
David Hilbert and his work. Hilbert was a mathematician who set out 23 unsolved
problems that mathematicians should crack. And as he did so, he led mathematics
to the 20th century and his problems became the definition of the
modern age of mathematics.
The
presenter then went to Hanna, East Germany, where Hilbert’s first problem
emerged and where Georg Cantor lived during his adult life. He was the first
person who understood infinity and gave it the precision of mathematics. He
believed there are many and different sorts of infinities like the infinity of
fractions which is greater than the infinity of whole numbers. He gave mathematics
a new way to count and this gave birth to his Continuum Hypothesis.
Du
Sautoy went back to Paris where Henri Poincare became the leading light of this
centre of mathematics. He was the jack of all trades in mathematics. He was
also known for his work and successive approximations of the orbits on the
problem on whether the solar system will continue turning like clockwork or
will it fly off. Though he wasn’t able to solve the problem, his mathematical
technique was too sophisticated that the king still gave him credit. Yet his
mistake on the orbits led to the Chaos Theory.
Du Sautoy then went to the Seven Bridges of Königsberg in Russia, the historically famous problem in
mathematics wherein one must find a route which one only crosses each bridge
once. This mathematical puzzle has baffled many mathematicians and eventually,
Leonard Euler was able to resolve that there was no possible solution where you
can cross every bridge and not cross one of them twice. His resolution then
gave way to a new problem on a new sort of geometry of position, topology.
Du
Sautoy then went to St. Petersburg where he discusses Poincare’s work on the
principle of this “bendy geometry” and how he developed topology into a new way
of looking at shapes. This led to his
Poincare Conjecture, a topological problem which was solved by Grigori Perelman;
he looked at the dynamics of how shapes can flow. Perelman showed everyone that
he solves mathematical problems not for the money but because he gets a kick at
proving theorems.
The
presenter then went to Gottingen, Germany, where David Hilbert’s work on
equations marked him as one of the greats. He was known for his Hilbert space
and Hilbert inequality which mathematicians still use today. He showed how to
divide equations into a finite set creating a more abstract way of looking at
mathematics. He thought that everyone who has a mathematical skill should do
mathematics; it doesn’t matter who does it but their work, the mathematics, should
speak for itself. For Hilbert, mathematics was a universal language that was
able to unlock the truths of the universe.
Du
Sautoy then went to Viena where Kurt Godel started the whole endeavour that
shattered Hilbert’s dream. He puts uncertainty in the very centre of
mathematics. It all began when he wanted to find a logical solution to all of
mathematics but in the end, what he got was what he wasn’t expecting and in
fact he proved exact opposite. He stated that there are some mathematical
statements which are true but can’t be proved. It was during this time that the
Nazis ruled over Austria and Germany and because of this he felt that mathematics
was dying. Hilbert helped his students escape this and as he did, he witnessed
the destruction of one of the greatest mathematical centres of all time.
Mathematicians
and scientists escaped from the Nazi regime and fled to the New World.
Princeton in New Jersey then accommodated some of the greatest mathematicians
and became the next Gottingen. Paul Cohen was one of the many who were
influenced by Godel and he developed a new way of solving problems which was
approved by Godel as well. Then another great mathematician had risen up but
faced many problems along the way because of her gender; this was Sofia
Kovalevskaya who became the first female professor of mathematics at Stockholm
University. Another great female mathematician, Julia Robinson, was the first
woman president of the American Mathematical Society. She wanted to be part of
the world of mathematics so she went to the University of California in Berkeley,
San Francisco where she met a number theorist named Rafael Robinson. They
bonded through their passion for mathematics and later got married. She settled
on solving Hilbert’s tenth problem. With the help of her colleagues, she
developed what became known as the Robinson’s Hypothesis.
Du
Sautoy then went back to St. Petersburg, Russia, where Yuri Matiyasevich was
among the mathematicians presenting their theorems and conjectures. He was
presenting his latest work on the Riemann Hypothesis. Through the influence of
his tutor, he then proceeded to crack Hilbert’s tenth problem and building on
the works of Julia Robinson and her colleagues, Matiyasevich was able to solve
the problem.
The
presenter then went back to Paris, France, where Evariste Galois
refined the language of mathematics and saw mathematics as a study of structure.
His technique of using geometry to solve equations was later picked up by André Weil. He built on Galois ideas and developed algebraic geometry. This
later led to the greatest feat in mathematics, the combination of the number
theorem, algebra, geometry and topology to solve even more equations. Weil was
also the leader of Nicolas Bourbaki who was thought of as a person but was
actually a group of French mathematicians who wrote coherent accounts on the
mathematics of the 20th century. The members of the group valued
their desire of mathematics over personal glory. One of the next great French
mathematicians was Alexandre Grothendieck who proposed that one should
understand the hidden structures of mathematics before understanding
mathematics itself. His vision led other people to view mathematics in colour
than in the usual black and white scenario.
Du Sautoy then went back to England thinking of
David Hilbert and his 23 problems. Among those, the Riemann Hypothesis remains
unsolved and is already acknowledged as the so- called “holy grail of
mathematicians.” As Hilbert’s work started to inspired the younger generations
to follow their dreams and goals in mathematics, Du Sautoy goes back to his
comprehensive school to try to inspire the students there. He believes
mathematicians are pattern searchers. Mathematicians are people who use logic
to understand the patterns and structures all around us. Du Sautoy talks about
how mathematics is now an important aspect in our life as it is connected to
everything we do. The Riemann Hypothesis is now the corner stone of mathematics
and a great prize awaits the person who will be able to crack it. He then takes
us on a recap of everything he has learned on his journey uncovering the story
of mathematics and its evolution.
This series takes us on a journey on the wonders of
mathematics. Through the whole journey we see how mathematicians like
Archimedes, Gauss and others were driven to understand how numbers and space
work together and are correlated. Du Sautoy’s different stops all over the world
showed the evolution of mathematics from simply being a passion to get rid of
doubt to being the key language that will help us understand the universe and
the uncertainties that lie within it. The story of mathematics has taken Du
Sautoy far and wide in tracing its history and the legacy it left behind. The
quest presented by mathematics enchanted and intrigued mathematicians of all
ages. But in the end, it wasn’t for the fame or riches, it was for the glory of
being able to add something or to contribute to the field that they so cherish
and love. These mathematicians became what they are because of their need or
thirst for knowledge and answers to the mysteries of mathematics. I learned
over the course of the series that mathematics is always spoken of by everyday
people in black and white when in fact, it’s more complex than that; it’s more
intriguing than that. I’ve learned of its true beauty and the beauty it brings
to humanity. People create bonds and friendships with people from halfway
around the world through their passion from mathematics. Its precision and
accuracy has lead to the creation of many breathtakingly beautiful works of art
and architecture. I learned of its true power and the gifts that it beholds to
those who take on its challenges. I’ve learned to appreciate mathematics for
what it’s really worth. The quest of mathematics isn’t a walk in the park but
it is fulfilling in all its bounty. And when people have the will or that need
to acquire knowledge, they will get that and much more in the world of
mathematics.
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