“We
must know, we will know.”
-David Hilbert
“If
we wish to foresee the future of mathematics, our proper course is to study the
history and present condition of the science."
-Henri
Poincaré
The inspiring statements above were
just two of the most famous quotations from two of the greatest mathematicians
of the 20th century. The fourth episode of “The Story of Maths” which is
entitled “To Infinity and Beyond”, brought Marcus du Sautoy on the last stage
of his journey on uncovering the history of mathematics, investigating on the
great unsolved problems faced by the great mathematicians on our very own
century, the 20th.
The cornerstone of mathematics
depend on being true and not until there is proof, there would still be doubt
and the problem will remain unsolved that’s why, mathematicians from the past
up to the present century devoted their lives on finding solutions and proofs
on the most complex problems of their time (a love for mathematics that
continued to amaze me up to this very point in time). Thus, the very core of
this episode focused on David Hilbert’s 23 problems most important problems in
mathematics that need to be answered. This comprised the 20th century
mathematics and led to either triumphs or desperation for different
mathematicians who dare tried to break the young German mathematician’s
challenge.
Just like solving a tough maze,
mathematicians struggled on Hilbert’s challenge beginning on Georg Cantor, who
took the challenge on Hilbert’s first problem. He worked on the infinite set of
whole numbers 1, 2, 3 ... ∞ which he compared with the smaller set of numbers
10, 20, 30 ... ∞. Cantor showed that these two infinite sets of numbers
actually had the same size as it was possible to pair each number up; 1 - 10, 2
- 20, 3 - 30 ... etc. If fractions were considered, there were an infinite
number of fractions between any of the two whole numbers, and so, the infinity
of fractions was bigger than the infinity of whole numbers. Yet, Cantor was
still able to pair each such fraction to a whole number 1 - 1/1; 2 - 2/1; 3 -
1/2 ... etc. through to ∞; i.e. and so, the infinities of both fractions and
whole numbers were shown to have the same size.
When the set of all infinite decimal
numbers was considered, Cantor was able to prove that this produced a bigger
infinity. This was because, no matter how one tried to construct such a list,
Cantor was able to provide a new decimal number that was missing from that list
and so, he showed that there were different infinities, some bigger than
others.
However, his problem occurred when
he was unable to solve: Is there an infinity sitting between the smaller
infinity of all the fractions and the larger infinity of the decimals? This
belief made Cantor known for his Continuum Hypothesis, which stated that there
is no such set, as listed in the first problem by Hilbert.
It’s only such a pity knowing that
mathematicians like Cantor, who can be an idol to anyone, would have little
recognition and would suffer from severe manic depression after all his efforts
dedicated to the progress of mathematics.
An accident made this next
mathematician stumble on a theory that had enhanced his reputation. His name was
Henri Poincaré, known for his Chaos Theory that led to a range of “smart
technologies”. This chaotic theory (defined by the word itself) was more on
techniques rather than having an entire solution to a problem, which
mathematicians continued very much different as well as of Poincaré.
But then, besides that, Henri
Poincaré, also worked on the discipline of topology (more concerned with
connections rather than distances like in train stations) and came up with his
conjecture on a topological problem, which was not naming all the possible
shapes for a 3D universe, dealt by Grigori Perelman on a very complex manner,
even mathematicians cannot fully understand. Unfortunately, Perelman refused to
accept questions regarding his solutions, secluding himself and giving up on
his mathematics due to less prizes he received.
An
opposite to Perlman’s view of in life, Marcus du Sautoy cited the life of David
Hilbert himself, a model or epitome of a true mathematician who may not be too
awarded with prizes, yet whole-heartedly gave his life to mathematics. He was a
pleasant and charismatic man who was admired by other mathematicians.
But
as opposed to a traditional mathematician, as we may refer to as nerd or
strange, Hilbert drunk, danced and mingled with his students outside, balancing
social life with his absolute commitment to math. Imagine how cool to have a
professor like that! Indeed, his completely new way of thinking and belief
stimulated the minds of other brilliant mathematicians who made his famous
quotation an inspiration in solving the unsolvable ones. For him, whoever you
are don’t matter for anyone could be an instrument or contributor in
mathematics. Isn't that inspiring for all of us?
Yet, a sickly, strange, Mr. Why as
this mathematician was called during his childhood days, proved this partly
wrong when he had formulated the Incompleteness Theorem based on his study of
Hilbert's second problem: This statement cannot be proved, and discovered the
existence of mathematical statements that were true but were incapable of being
proved. What Kurt Godel discovered surprised him too, a lot. This then created a huge dilemma on the world
of mathematics, and caused series of breakdowns even to him. Love, as always
would save the day, sees the light, and so, a dancer in a local nightclub made him regain his
strength. It’s only the saddest part knowing that Godel decided to remain in
Austria despite the fleeing of his students and co-mathematicians and soon died
with only ten people in his burial.
Some of the students who exiled
Europe and moved to America to the institute which similarly ensemble that of a
European university, that offered greener opportunities to struggling
mathematicians like the best of friends,
George Yuri Rainich, and Albert Einstein.
Exiting Europe for a while and
moving on to the land of brave and the free and of honey and milk, the New
World, Marcus du Sautoy, discussed the biography of the very young American
mathematician, Paul Cohen, who took up the challenge of Cantor's Continuum
Hypothesis. Cohen found 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. Cohen
gained much recognition for this as approved by a famous and reliable
mathematician Godel, and moved on to Hilbert's eighth problem, the Riemann
hypothesis, but this time, failed. Cohen left a mark though being a pleasant
teacher to his students who was so eager to teach what he knew or even not knew
yet, adding to the line of mathematicians to look up to.
An exemption to the rule of
mathematicians as more of males was Julia Robinson, proving that being a woman
is never a hindrance to being great in the field dominated by males. She
pursued her career in mathematics and found the love of her life in the same
field. How romantic that meeting was that evolved on both their passion in
mathematics! It’s amazing how these busy people still found time for love.
Robinson’s Hypothesis stated that to show that there was no such method all you
had 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. Robinson was unable to find
this set as she became inclined to politics and so, this part of the solution
fell to Yuri Matiyasevich who saw how to capture the Fibonacci sequence using
the equations at the heart of Hilbert's tenth.
Lastly, du Sautoy, in a climatic
part in the movie, reflected on the contributions of Alexander Grothendieck,
whose ideas have had a major influence on current mathematical thinking about
the hidden structures behind all mathematics.
In the last part of the film, he
came back to where he began as a mathematician. Being a Filipino, I remembered
our saying that no matter how far we go, we should always go back to our roots
and that’s what he did here, sharing his new found knowledge to the students
like me, and explaining the mystery of the last unsolved problem in mathematics
that became his instant favourite, the Riemann Hypothesis, a conjecture about
the distribution of prime numbers – which are the atoms of the mathematical
universe.
This holy grail of mathematics up to
the present offers a very bright future for the one who can prove it, which
attracts many mathematicians in the whole world. Who knows maybe a Filipino
could be the one? As long as Hilbert’s call of “We must know, we will know”,
will serve as the driving force then nothing would be impossible.
To
sum it all up, it’s the will of finding out answers to unknown solutions that
make mathematics alive. Therefore, this film is an excellent motivator for
anyone who will want to view this one especially to the youth. I hope that more
and more people will voluntarily and unconditionally seek out for new
discoveries in mathematics for the betterment of our world, for mathematicians
are always the most interesting and inspiring creatures (53rd episode
of the radio broadcast Robinson listened to). Yet, good moral values should
always be maintained along with the accomplishments made by any person inclined
with mathematics.
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