From the ancient cultures of Egypt, Mesopotamia and Greece
– cultures that created the basic language of numbers and calculation, to the
East – where mathematics reached dynamic heights and to the 17th
Century in Europe where great strides have been made to understand geometry, we
finally come down to the finale of Marc de Sautoy’s investigation into the
history of maths - the 20th century. In this 4th and
final episode of BBC’S Story of Maths tells us about some of the great unsolved
problems that confronted mathematician in the 20th century.
David Hilbert, a
young German mathematician, set out what he assumed the 23 most imperative
problems for mathematicians to crack. These problems defined the mathematics of
the modern age. Georg Cantor, a great mathematician from Halle, in East
Germany, was the first person who fully understood the gist of infinity and
gives it mathematical precision. Before Cantor, no one had really understood
infinity. It was complex and crafty that
didn’t seem to go anywhere. But Cantor showed that infinity can be impeccably
understandable. He also considered the
set of all infinite decimal numbers and produced a clever argument to show how
to construct a new decimal number that was missing from your list then the idea
of infinity opens up. But Cantor suffered from manic depression. He found
solace in the university’s sanitarium which gave him the mental strength to
resume his exploration of the infinite. But there was one problem that Cantor
couldn’t leave, a problem he wrestled with for the rest of his life. It became
the known as the continuum hypothesis.
Henri Poincare, according to Bertrand Russell a respected
mathematician in France, considered him “the greatest man France had produced”..
He also simplified the problem whether the solar system would continue turning
like clockwork or might suddenly fly apart by making successive approximation
to the orbits which he believed wouldn’t affect the final outcome
significantly. But then Poincare realized that he made a mistake. Even a small
change in the initial conditions could end up producing vastly different orbits
and these orbits that he discovered led to what we now know as The Chaos
Theory. Poincare continued to produce a wide range of original work throughout
his life and also wrote popular books admiring the importance of mathematics.
The Seven Bridges of Konigsberg known today as Kaliningrad,
started as an 18th century puzzle. The problem was, “Is there a
route around the city which crosses each of these seven bridges only once?” The
problem was solved by the great mathematician, Leonhard Euler. Euler solved the
problem by making a conceptual leap. He realized that the distance between the
bridges doesn’t matter but it is how the bridges are connected together. This
is a problem of a new sort of geometry of position – a problem of topology.
Although topology had its foundation in the bridges of Konigsberg, it was in
the hands of Poincare that the subject evolved into a powerful new way. Some
people refer to topology as bendy geometry because two shapes are the same if
you can bend or morph onto into another without cutting it. In 1904, Poincare
came up with a new topological problem that he just couldn’t solve which became
known as the Poincare Conjecture. Grisha Perelman, a Russian mathematician,
finally solved the problem in the year 2002 in St. Petersburg. His proof was
very difficult to understand, even for mathematicians. Perelman solved the
problem by linking it to a completely different area of mathematics. He looked
in the dynamics of the way things can flow over the shape.
Back to David Hilbert, he didn’t offered no prize or reward
beyond those mathematicians who was able to solve any of his 23 problems. He
was by far the most charming mathematician of his age. He studied the number
theory and revolutionized the theory of integral equation. Up to now, his works
and his name is attached to many mathematical terms like the Hilbert Space, the
Hilbert Classification, the Hilbert Inequality and several Hilbert theorems. Mathematics
for Hilbert was a universal language and believed that this language was
powerful enough to unlock all the truths of mathematics.
Next to Hilbert was Kurt Godel, an Austrian mathematician
who destroyed Hilbert’s belief. Godel lived in Vienna in the 1920s and 1930s
and studied mathematics at Vienna University. He was the one who derived the
Incompleteness Theorem by which he proved that within any logical system for
mathematics there will be statements about numbers which are true but which you
cannot prove. Godel’s proof led to a crisis in mathematics which eventually led
to him on a series of breakdowns and spent time in a sanitarium. But thanks to
a woman named Adele Nimbursky he was saved and was back on his own feet. During
the new German empire in the late 1930s, the Nazis passed a law allowing the
removal of any civil servants who wasn’t Aryan and Mathematicians suffered the
most. Some lost their jobs and the others were driven to suicide or died in
concentration camps. The center for world mathematics for 500 years was over
and it is time for the mathematical baton to be handed to the New World.
The Institute for Advance Study had been set up in
Princeton which aims to reproduce the collegiate atmosphere of the old European
universities in rural New Jersey. Fleeing European mathematicians like Hermann
Weyl and John Von Neumann made the Institute quickly became the perfect to
create another eminent mathematician. Godel, who also fled to America from the Nazis,
was also there along with his closest friend Albert Einstein. Einstein was full
of mirth while Godel on the other hand became increasingly lugubrious and
pessimistic. He spent less and less time with his fellow mathematicians and
preferred to come to the beach instead but still his influence on American
mathematics paradoxically was growing stronger and stronger.
Paul Cohen, a young mathematician from New Jersey who was
winning mathematical competitions and prized decided to solve Cantor’s
Continuum Hypothesis, Hilbert’s first problem way back in 1900. The problem was
whether there is or there isn’t an infinite set of number bigger than the set
of all whole numbers but smaller that the set of all decimals. According to
Cohen, there was one mathematics where the continuum hypothesis could be
assumed to be true. Cohen’s solution was daring and was so new that made nobody
felt absolutely sure. Cohen then visited Godel at the Institute who gave him
his stamp of approval which made everything change. Today, mathematicians insert
a statement that says whether the result depends on the continuum hypothesis. Cohen
has really rocked the mathematical universe which gave him fame and riches but
still he wasn’t able to the Riemann Hypothesis.
In the story of mathematics, nearly all the truly great
mathematicians have been men but there were some women who made their way to
the top. Female mathematicians like the Russian Sofia Kovalevskaya who became
the first female professor of mathematics in Stockholm in 1889, Emmy Noether a
talented and algebraist and lastly, Julia Robinson, the first woman ever to be
elected president of the American Mathematical Society.
Julia Robinson was born in St. Louis in 1919 and grew up as
a shy sickly girl but right from the very start she had an inherent
mathematical ability. Despite showing an early brilliance, she had to
continually fight at school and college simply to be allowed to keep doing
maths. She was finally able to express her true self at the University of
California and was also found the other part of her life in the person of
Raphael Robinson, a number theorist whom she married in 1952. Julia got her PhD
and worked on with Hilbert’s tenth problem which she became obsessively with.
Hilbert’s tenth problem asked if there was some universal method that could
tell whether any equation had whole number solutions or not. With the help of
her colleagues, Julia was able to develop the Robinson Hypothesis that argued that
in order to show that no such method existed; all you have to do was to cook up
one equation whose solutions were a very specific set of number. Robinson tried
her best but still, she wasn’t able to find the set. For the tenth problem to
be finally solved, they needed some fresh inspiration.
Yuri Matiyasevich from St. Petersburg in Russia, found how
to capture the famous Fibonacci sequence of number in January 1970 using the
equation that were at the heart of Hilbert’s problem. Building on the work of
Julia and her colleagues, he was able to solve the tenth problem and he was
only 22 years at that time. Julia heard about rumors that the problem was
finally solved and contacted Yuri herself. Yuri thanked her and reassured her
that the credit is as much hers as it is his. After that incident, Julia and
Yuri collaborated with each other and worked on several other mathematical problems
until Julia died in 1985 at the age of just 55 years old.
Evariste Galois, a young republican revolutionary who also
had a passion in mathematics discovered new techniques to be able to tell
whether certain equations could have solutions or not. The symmetry of certain geometric
objects seemed to be the key. Galois died at the very young age due to a saga
of unrequited love which turned into duel that killed him. However, his idea of
using geometry to analyze equations would be picked up by Andre Weil, a
Parisian mathematician. Weil built on the ideas of Galois and first developed
algebraic geometry a whole new language for understanding solutions to
equations. His work led him to theorems that connected to number theory,
algebra, geometry and topology and are one of the greatest achievements of
modern mathematics. Andre Weil also who wrote a coherent account of the
mathematics of the 20th century then led a group of French
mathematician who assumed the name of Nicolas Bourbaki. The Bourbaki baton was
handed down to the next generation of French mathematicians and their most
brilliant member was Alexandre Grothendieck.
Alexandre Grothendieck is a Structuralist. He was
interested in the hidden structures underneath all mathematics. But in the late
1960s, Grotendieck turned his back on mathematics and pursued politics. In the
end, Grotendieck became recluse and lost contact with his old friends and
mathematical colleagues. Still, his achievements weren’t forgotten and stands
alongside Canto, Godel and Hillbert.
Though most of Hilbert’s 23 problems
were solved, the Riemann Hypothesis, the eight on Hilbert’s list still remained
the holy grail of mathematics. Hilbert’s lecture inspired a generation to pursue
their mathematician dreams.
A mathematician is pattern searcher.
They try and understand the patterns and the structure and the logic to explain
the way the world around us works. They aren’t motivated by money and material
gain or even by practical applications of their work. The glory of solving one of the great unsolved
problems that have outwitted previous generations of mathematician.
Mathematics really is the Empress of
the Sciences. Without her, there would be neither physics, nor chemistry and so
on. Any field of study depends on Mathematics. She is dominant and has an
unceasing power. Imperious and unyielding, mathematics brooks no dissent and
tolerates no error. In an age of uncertainty, mathematics is the only
discipline that generates knowledge that’s immutably, incontestably, and
eternally true. Mathematics had surely made its way through different
generations enriching its identity that made her the most prominent phenomenon known
to mankind.
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