On a lecture given by a young German
Mathematician, David Hilbert, the 23 most important problems that
mathematicians need to solve were emphasized. The lecture was part of the
International Congress of Mathematicians which happened in the summer of 1900
in Sorbonne, Paris. The “23 problems” became a challenge to every mathematician
on that day. Some tried to crack it; other succeeded while others failed. It
was in Halle, East Germany where the first of Hilbert’s list triumphed. This
town was the place of the great mathematician Georg Cantor. He was the first to
understand the meaning of infinity ad give it mathematical precision. Before
him, no one had understood infinity. According to him, there wasn’t just one
infinity but infinitely many infinities: from the infinity of whole numbers to
the infinity of fractions. Georg Cantor made a clever argument to show how to
construct a new decimal number, this now opens up the idea of infinity. In
spite of this amazing discovery, Cantor suffered from manic depression. There
was no treatment for his illness that time. But he still resumed his investigation
of the infinite until the day he died. Cantor was never worried with the
paradox of the infinite because he believed that the absolute infinite is only
in God. But there was one problem which he really got a hard time to solve, it
is the continuum hypothesis. “Is there an infinity sitting between the smaller infinity
of all the whole numbers and the larger infinity of the decimals? ”
There was a mathematician from
France who once said that Cantor’s new mathematics of infinity was beautiful,
if pathological. He was Henri Poincare. Poincare was good at algebra, geometry,
analysis and everything. He became the leading light of Paris when it became
the center for world mathematics in the 19th century. In 1885, King
Oscar II of Sweden and Norway searched for someone who could establish mathematically
whether the solar system would continue turning like clockwork, or might
suddenly fly apart. The King promised 2500 crowns as a reward. Poincare could
not solve the entire problem but his ideas were refined enough to have the
reward. He developed important mathematical techniques or some sort of great
arsenal techniques to solve the problem. When his paper was now ready for
publication, one of the editors found something unusual. Poincare realized he
committed a mistake. And this now led to what is known as the chaos theory. By
digging deeper into the mathematical rules of chaos, one can explain why a
butterfly’s wings could create tiny changes in the atmosphere that might cause
a tornado or a hurricane on the other side of the world. Poincare’s mistake
enhanced his reputation. He published a lot of his original work throughout his
life. In one of his popular books, he wrote, “If we wish to foresee the future of
mathematics, our proper course is to study the history and present the
condition of the science”. Chaos theory is just one of Poincare’s contributions
to mathematics. The most important lies in The Seven bridges of Konigsberg back
in 1945. At first, it was just an 18th century puzzle. The problem
to solve is to find a route around the city which crosses each of the seven
bridges only once. This was solved by the great mathematician Leonhard Euler
which he claimed in 1735 to be impossible. There could not be a route that did not
cross at least one bridge twice. Through a conceptual leap, Euler was able to
solve the problem. This problem now opens a door to a new sort of problem which
is a problem of topology.
Topology had grown into a powerful new
way of looking at shape, thanks to Henri Poincare. Topology maybe referred to
as a bendy geometry. It is because in topology, two shapes can be the same if
you can bend or transform one into another without cutting it. It was in 1904
when Poincare had a topological problem that he can’t solve. If we are living
in a three-dimensional universe, what are the possible shapes that our universe
can be? This question was later known as the Poincare Conjecture. A Russian
mathematician named Grisha Perelman was able to solve this problem in 2002 by
linking it to a different area of mathematics. Perelman’s proof is very hard to
understand even for mathematicians, also his character. He had turned down all prizes
and offers of professorships from distinguished universities in the west. He
seemed to have given up on mathematics recently and lived peacefully with his
mom. Going back to the one who formulated the list of 23 problems in 1900,
David Hilbert might have commended Perelman for his discovery.
David Hilbert was the most charismatic
mathematician of his age. Hilbert studied number theory and revolutionized the theory
of integral equation. At present, many mathematicians still used the Hilbert
Classification, the Hilbert space, the Hilbert Inequality and several Hilbert
theorems. His early work on equations made him mark as a mathematician thinking
in new ways. Hilbert demonstrated that there are ways to divide infinitely many
equations like a set of building blocks or in a finite set. He could not
construct this finite set actually but he still proved it must exist. In this
way, Hilbert was creating a more abstract approach to mathematics. This
mathematician had indeed a shocking lifestyle. He liked to drink with his students
and dance with women. He also believed that everyone can do mathematics
provided that he/she has the skills. For him, mathematics is a universal language.
He had no doubt that all his 23 problems would be solved. He once said, “We
must know, we will know”. If for Hilbert, all the 23 problems can be proved,
not for this next mathematician. His name is Kurt Godel.
Kurt Godel was an Austrian
mathematician. During his childhood, he used to ask questions which made his
family to call him Mr. Why. Godel improved his skills in mathematics in Vienna
University but most of the time; Godel is in the cafes, in the internet chat
rooms and in places where he can play backgammon and billiards. Godel got himself
to be a member of a highly influential group of philosophers and scientists
called the Vienna Circle. Here, he tried to solve the second problem in Hilbert’s
list but instead, he got something different. It is called the Incompleteness
Theorem. According to him, there will be statements about numbers which are true
but cannot be proved. He began with the statement, “This statement cannot be
proved”. He transformed this statement into a pure statement of arithmetic
which can now be either true or false. If the statement is false, the statement
could be proved, meaning it would be true. But this becomes illogical making the
statement to be true. In simple speaking, this is a mathematical statement
which is true but cannot be proved. Godel’s proof created a crisis in
mathematics and also for him. He got into a series of breakdowns in 1934 and
was contained in a sanatorium. He fell in love with a local night club dancer
and her love kept him alive. When they were walking outside, they were attacked
by Nazi thugs. The Nazis have submitted a law allowing the elimination of any
civil servant who was not Aryan and academics were civil servants.
Mathematicians were much affected; some lose their jobs while some was driven
to suicide or died in concentration camps. But David Hilbert did not lose the
spirit. He made some of his brightest students escape and asked for the
dismissal of his Jewish Colleagues. He did not run away from war for some unclear reasons until he became ill.
Hilbert was left out during the Nazi regime and had witnessed the destruction of
the greatest mathematical centers of all time. He died in 1943 and only ten
people attended his funeral.
Many of the brilliant mathematicians
resumed their lives in America. Two of them were Herman Weyl and John Von
Neumann. The Institute for Advanced Study had been built in Princeton in 1930. The
purpose was to revive the collegiate atmosphere of the old European
universities in rural New Jersey. This institute became the home of these
brilliant European mathematicians including Kurt Godel and his friend Albert
Einstein.
During the 1950s in America,
youngsters were not interested in mathematics except for one who was interested
in cracking the major problems of mathematics. He was Paul Cohen and he wanted
to have a mark in his own field of mathematics. When he read about Cantor’s
Continuum Hypothesis, the 22-year-old Paul Cohen decided that he could do it. Cohen
succeeded to find the truth behind Cantor’s Continuum Hypothesis. But nobody
was sure of his new method and they needed the opinion of the one they trusted
most, Kurt Godel. After Godel gave his approval and said, “Yes, it is correct”,
Cohen’s proof was accepted by everybody. Mathematicians today depends on the
continuum hypothesis wherein one answer can be yes and the other is no. Paul
Cohen was rewarded with fame, riches and a lot of prizes. Up until his death,
he was trying to solve Hilbert’s eighth problem, the Riemann Hypothesis. Cohen
was not the only American mathematician who became great in mathematics in the
1960s, there were many.
Letting go of the male
mathematicians, here come the females. The Russian Sofia Kovalevskaya was the
first female professor of mathematics in Stockholm in 1889. She received very
prestigious French mathematical prize. One of the mathematicians who fled from
the Nazis is Emmy Noether. She was a talented algebraist but unfortunately died
before knowing her potential. And this woman, Julia Robinson, she was the first
woman ever who became president of the American Mathematical Society. When she
was just a child, she and her sister lived with her grandmother because their
mother died when she was just two. She suffered from scarlet fever when she was
seven and spent a year in bed. She was even told that she would not live beyond
40. She really had innate mathematical ability even in a young age but was just
less in encouragement. When she was a teenager, she only knew that there are mathematics
teachers and a radio show made her realize that there were mathematicians. She wanted
to be part of their world. She went to the University of California in Berkeley
and met the number theorist Raphael Robinson. They got married in 1952. Julia
got her PhD and started solving Hilbert’s tenth problem which asked if there
was some universal method that could tell whether any equation had whole number
solutions or not. She and her colleagues developed what is known as the
Robinson Hypothesis. Still, the problem remains unsolved.
Yuri Matiyasevich was the one to
solve the tenth problem by building on the work of Julia and her colleagues. He
was just 22 years old then and he really wanted to thank Julia for what he had
achieved. He sent a mail but got no reply. Back in California, Julia was
hearing rumors that the tenth had been solved. She contacted Yuri and then Yuri
thanked her and gave the much bigger credit to her. Julia and Yuri worked
together in solving other mathematical problems until Julia died in 1985. She was
just 55 then but before she died, she was able to find new ideas to solve
special classes of equations.
Another astonishing story of a
mathematician is in Paris. He was Evariste Galois, the brilliant mathematician
who discovered techniques to tell whether certain equations could have
solutions or not. He was able to do it by using geometry. Galois was killed by a gunshot from a duel. For
mathematics, it was indeed a big loss. Galois’ idea was used by Andre Weil,
another Parisian mathematician in the 1920s. He first developed the algebraic
geometry which is a new language for understanding solutions to equations. Weil’s
theorems connected the number theory, algebra, geometry and topology which
became a great success of modern mathematics. Through Andre Weil, the name
Nicolas Bourbaki was known. Bourbaki is
very popular because of his books in mathematics. But this person really does
not exist. Nicolas Bourbaki was just the nom de plume of a group of
mathematician led by Andre Weil. After the Second World War, the Bourbaki name
was given to the next generation of French mathematicians. Alexandre
Grothendieck was one of them. He was a Structuralist, interested with the
hidden structures underneath all mathematics. Grothendieck discovered a
language that mathematicians keep on using ever since to crack problems in
number theory, geometry and fundamental physics. But Grothendieck chose
politics over mathematics. The threat of nuclear war is more important to him
then mathematics. He was isolated from his old friends and mathematical colleagues,
but his legacy in mathematics never vanished.
Mathematicians are really hard
workers. They are the persons that you can never ever call lazy. Basing on the
four episodes of the Story of Maths, I can call them the “much effort men and
women”. I always say in my three movie reviews, mathematicians lent their lives
for us to experience this great value for mathematics. Mathematics couldn’t be
easy (in just some parts but not totally) without their contributions. We should
be thanking them for the rest of our lives. To be honest, I regard this episode
as the most boring episode of the Story of Maths. But hey, that was just my
first impression. As I’m into the process of understanding the whole story, I’m
learning to love it. For me, this is the most amazing part of the Story of Maths.
I was able to learn much newer information compared to the last three. Well it’s
really obvious because it is the latest episode, but not in that way. It’s just
that these information felt very fresh to me that when I read it, I can’t help
to be in awe. David Hilbert, Georg Cantor, Henri Poincare, Kurt Godel, Paul
Cohen, Julia Robinson and the other mathematicians, they really rocked the
world of mathematics. They proved to us that problems really have solutions and
that all that has the skills can crack a mystery and be famous for it. Do you
know the feeling of being famous? It is when people always remember you for something.
It can be for a place, for a statement, for a thing or for just simply being
who you are. I always wanted to be famous, someday. Who doesn’t want to be
famous? Don’t be a hypocrite. Well, I know that the road to fame is not easy
but at the least, we must try. These famous persons all started from scratch, just
like me and you, just like us. Someday we can be famous in our own fields. But I
don’t expect myself to be famous in the world of mathematics (but we don’t know...).
One of my professors said that in order to achieve something, you need to do
your best always. Do not be lazy and believe that you can do it. Just like
these mathematicians, they tried and tried until they died. (Are you laughing
at my statement?) Yes, it’s quite funny but the idea is there. They certainly
died while giving their very best to solve a problem. So, for us who are still
alive and kicking, be at your very best. Think for the best, speak for the best
and always consult The One who is very best.
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