In the fourth episode, Professor Marcus du Sautoy concludes his investigation into the history of mathematics with a look at some of the great unsolved problems that confronted mathematicians in the 20th century. After exploring Georg Cantor's work on infinity and Henri Poincare's work on chaos theory, he sees how mathematics was itself thrown into chaos by the discoveries of Kurt Godel and Paul Cohen, before completing his journey by considering some unsolved problems of maths today, including the Riemann Hypothesis.

By our third year, most of us will have learned to count. Once we know how, it seems as if there would be nothing to stop us counting forever. But, while infinity might seem like an perfectly innocent idea, keep counting and you enter a paradoxical world where nothing is as it seems. Mathematicians have discovered there are infinitely many infinities, each one infinitely bigger than the last. And if the universe goes on forever, the consequences are even more bizarre. In an infinite universe, there are infinitely many copies of the Earth and infinitely many copies of you. Older than time, bigger than the universe and stranger than fiction. This is the story of infinity.

Mathematician Dr Hannah Fry explores the mystery of maths. It underpins so much of our modern world that it's hard to imagine life without its technological advances, but where exactly does maths come from? Is it invented like a language or is it something discovered and part of the fabric of the universe? It's a question that some of the most eminent mathematical minds have been wrestling with. To investigate this question, Hannah goes head first down the fastest zip wire in the world to learn more about Newton's law of gravity, she paraglides to understand where the theory of maths and its practice application collide, and she travels to infinity and beyond to discover that some infinities are bigger than others.
In this episode, Hannah goes back to the time of the ancient Greeks to find out why they were so fascinated by the connection between beautiful music and maths. The patterns our ancestors found in music are all around us, from the way a sunflower stores its seeds to the number of petals in a flower. Even the shapes of some of the smallest structures in nature, such as viruses, seem to follow the rules of maths. All strong evidence for maths being discovered. But there are those who claim maths is all in our heads and something we invented. To find out if this is true, Hannah has her brain scanned. It turns out there is a place in all our brains where we do maths, but that doesn't prove its invented.
Experiments with infants, who have never had a maths lesson in their lives, suggests we all come hardwired to do maths. Far from being a creation of the human mind, this is evidence for maths being something we discover. Then along comes the invention of zero to help make counting more convenient and the creation of imaginary numbers, and the balance is tilted in the direction of maths being something we invented. The question of whether maths is invented or discovered just got a whole lot more difficult to answer

The Pythagorian philosopher Plato hinted enigmatically that there was a golden key that unified all of the mysteries of the universe. It is this golden key that we will return to time and again throughout our exploration. The golden key is the intelligence of the logos, the source of the primordial om. One could say that it is the mind of God. With our limited senses we are observing only the outer manifestation of the hidden mechanics of self similarity. The source of this divine symmetry is the greatest mystery of our existence.

Writing itself is 5,000 years old, and for most of that time words were written by hand using a variety of tools. The Romans were able to run an empire thanks to documents written on papyrus. Scroll books could be made quite cheaply and, as a result, ancient Rome had a thriving written culture. With the fall of the Roman Empire, papyrus became more difficult to obtain. Europeans were forced to turn to a much more expensive surface on which to write: Parchment. Medieval handwritten books could cost as much as a house, they also represent a limitation on literacy and scholarship.
No such limitations were felt in China, where paper had been invented in the second century. Paper was the foundation of Chinese culture and power, and for centuries how to make it was kept secret. When the secret was out, paper mills soon sprang up across central Asia. The result was an intellectual flourishing known as the Islamic Golden Age. Muslim scholars made discoveries in biology, geology, astronomy and mathematics. By contrast, Europe was an intellectual backwater.
That changed with Gutenberg’s development of movable type printing. The letters of the Latin alphabet have very simple block-like shapes, which made it relatively simple to turn them into type pieces. When printers tried to use movable type to print Arabic texts, they found themselves hampered by the cursive nature of Arabic writing. The success of movable type printing in Europe led to a thousand-fold increase in the availability of information, which produced an explosion of ideas that led directly to the European Scientific Revolution and the Industrial Revolution that followed.

Chaos theory has a bad name, conjuring up images of unpredictable weather, economic crashes and science gone wrong. But there is a fascinating and hidden side to Chaos, one that scientists are only now beginning to understand. It turns out that chaos theory answers a question that mankind has asked for millennia - how did we get here? In this documentary, Professor Jim Al-Khalili sets out to uncover one of the great mysteries of science - how does a universe that starts off as dust end up with intelligent life? How does order emerge from disorder? It's a mindbending, counterintuitive and for many people a deeply troubling idea. But Professor Al-Khalili reveals the science behind much of beauty and structure in the natural world and discovers that far from it being magic or an act of God, it is in fact an intrinsic part of the laws of physics. Amazingly, it turns out that the mathematics of chaos can explain how and why the universe creates exquisite order and pattern. And the best thing is that one doesn't need to be a scientist to understand it. The natural world is full of awe-inspiring examples of the way nature transforms simplicity into complexity. From trees to clouds to humans - after watching this film you'll never be able to look at the world in the same way again.