Finally, we are at the last instalment of The Story of Maths
but actually this is where math really went nuts. Most of the mathematician had
a bad life, but that didn’t keep them from being great, instead they became one
of the people who made the world what is it now, easier and secure and
pleasurable. We owe mathematics a huge amount of recognition for what great
things that math contributed to our world.
The
first mathematician featured in this episode is Georg Kantor. He was the first
one to really understand the concept of infinity by comparing numbers 1, 2, 3
and so on to 10, 20, 30 … that it has the same size. So what about the
fractions in between the numbers? He created an infinite grid which the first
round contain the whole numbers fractions with one on the bottom in the second
row. 10 harms fractions with on the bottom and so on. Every fraction appears
somewhere in this grid for example 4/5. So 4th column 5th
row is the location where 4/5 is located. He then snaked the grid and turned it
into a straight line which every fraction has a corresponding whole number. So
the fraction can go to infinity as well as the whole numbers and this created a
new idea, a new hypothesis which is the continuum hypothesis. The next person
featured is Henri Poincare. He solved the problem in the solar system which
involved the revolving of the sun earth and moon together and how to solve each
missing variables like exact coordinates, velocity, and direction. He sorted
these out and simplified them by making approximations to the orbits. He was
the guy that likes techniques in solving such complicated math expressions. But
one person found out that the simplifications of Poincare are wrong because a
small difference in values could create a new orbit. This ‘mistake’ of Poincare
was an indirect discovery of a theory now known as the Chaos Theory. He was
also an author writing about the importance of math but the work that made him
to the famous status was topology, enabling us to have a new way of looking at
shape. Topology gave Poincare a question he cannot seem to answer, If you've
got a flat two-dimensional universe then Poincare worked out all the possible
shapes he could wrap that universe up into. It could be a ball or a bagel with
one hole, two holes or more holes in. But we live in a three-dimensional
universe so what are the possible shapes that our universe can be? That question
became known as the Poincare Conjecture. This was solved by Grisha Perelman by
linking it to as different mathematical area and his proof was very hard to
understand even for mathematicians.
David Hilbert was a mathematician who studied
the number theory and revolutionized the theory of integral equation but the
work that made him stand out is his work on equations showed that although
there are infinitely many equations, there are ways to divide them up so that
they are built out of just a finite set, like a set of building blocks. The
interesting part of his proof is that he didn’t construct it but he just proved
that they should exist. It is just like telling that there is a way from Los
Angles to Bay Area but I just don’t know how to get there. He said that there
are no equations that can’t be solved and we should and will know. You know
when you are full of hope and your day seems to be good then just one person
shatters all those? That’s what Kurt Godel did to Mr. Hilbert, and he also put
uncertainty to mathematics. He was a person of overflowing curiosity when
growing up. Godel proved that within any logical system for mathematics there
will be statements about numbers which are true but which you cannot prove
known as the Incompleteness theorem. He started with a statement, “This
statement cannot be proved”. If the statement is false, that means the
statement could be proved, which means it would be true, and that's a
contradiction. So that means, the statement must be true. In other words, here
is a mathematical statement that is true but can't be proved.
When David Hilbert died, the
mathematical prowess of Europe died too and the baton was passed to the
Americans. It was in Princeton, New Jersey that they tried to rebuild the
environment of mathematics that was lost. Kurt Godel was one of the European
exiles from the Nazi empire and he was neighbours with the great Albert
Einstein. The sanguine type personality of Einstein made Godel a much more
quiet and pessimistic person that led to his lunacy. The next mathematician
that evolved during the hedonistic and fast food life of the Americans was Paul
Cohen. He was brilliant winning awards and prizes but he wanted to make a mark
in mathematics and what field should he leave a mark. He then began to work
with Cantor’s problems where he can’t seem to solve but Cohen found a proof
that did prove the Continuum hypothesis. Many doubted the solution of Cohen
because it is new but there is one man that everyone looked up to for the
solution’s approval, Kurt Godel. He then went to Princeton so his work could be
checked by Godel and Godel did give his stamp of approval giving Paul Cohen
more recognition and prizes. Then after gaining confidence from solving Hilbert’s
1st problem, he attempted to solve the most important problem in
Cantor’s problem, #8 the Riemann Hypothesis. When he died, he was still trying
to solve it but he just couldn’t just like other mathematicians before him.
That doesn’t mean that he was forgotten, in fact he was a successful product of
the American dream because from being an exile he became a top professor.
Then the next thing just caught my
attention. A female mathematician was featured and this is interesting because
there are just 3 mathematicians that have been really known. The female
American mathematicians is the first ever woman to be the president of the
American Mathematical Society, she is Julia Robinson. She had an obsession to
solve Hilbert’s 10th problem and she said that she just wouldn’t want
to die without knowing the answer. With the help of her colleagues, she then
developed the Robinson Hypothesis She had inspiration from Yuri Matiyasevich
from St. Petersburg, Russia. Yuri found the last important piece to solving
Hilbert’s 10th problem. He saw how to capture the Fibonacci sequence
in the problem using the equations at the heart of Hilbert’s equations. The
next mathematician featured was Evariste Galois, a republican revolutionary in
France. He believed that mathematics should not be the study of shape and
number but of structure. Galois had developed a technique where certain equations
could have solutions or not, and shown how new mathematical structures can be
used to reveal the solutions to the equation and this is because of his idea to
analyse these equations. This idea made light to Andre Weil, a prisoner of a
war but he became a soldier. In his jail time, he first developed algebraic
geometry and built on the ideas of Galois, a whole new way of understanding
solutions to equations. This led to the connections to theorems connected to
geometry, topology, and Algebra, all this led to one of the greatest
achievement of modern mathematics. The next man is Alexander Grothendieck who
is a pretty interesting person because he is a structuralist, interested in
hidden structures underneath
mathematics. I like him because we can understand the most complicated things
when we go back to the basic then from there we could work our way up. He discovered
a new way to look at patterns in mathematics which are being used until today
to solve problems and equations. But sadly, he turned his back on mathematics
when he entered politics. One thing that still remains in doubt today is the
Riemann Hypothesis.
Maybe one of you reading this article may have
that world changing idea, not just math. Sometimes math may intimidate most of
us, but it is the people who step out and get that intimidation out of their
head who can truly get them to that changing idea. People wanted to do simple
things and in order to do simple things, some should do the hard things first.
This is what all the mathematicians showed us, erase that uncertainty and doubt
so we could have that feeling that we can do things very easy. Math isn’t all
about hard equations to solve, it is how we use the basic things to solve the
hard ones, like what Grothendieck did, he understood complicated things easily because
he used the general and basic terms first before all the hard ones so math is
possible for all of us. I hope this will
serve as an inspiration to all of you out there, Peace!
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