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 looks at how mathematics was itself thrown into chaos by the discoveries of Kurt Godel, who showed that the unknowable is an integral part of maths, and Paul Cohen, who established that there were several different sorts of mathematics in which conflicting answers to the same question were possible. He concludes his journey by considering the great unsolved problems of mathematics today, including the Riemann Hypothesis, a conjecture about the distribution of prime numbers. A million dollar prize and a place in the history books await anyone who can prove Riemann's theorem.
When ancient Greece fell into decline, mathematical progress stagnated as Europe entered the Dark Ages, but in the East mathematics reached new heights. Du Sautoy visits China and explores how maths helped build imperial China and was at the heart of such amazing feats of engineering as the Great Wall. In India, he discovers how the symbol for the number zero was invented and Indian mathematicians' understanding of the new concepts of infinity and negative numbers. In the Middle East, he looks at the invention of the new language of algebra and the spread of Eastern knowledge to the West through mathematicians such as Leonardo Fibonacci, creator of the Fibonacci Sequence.
By the 17th century, Europe had taken over from the Middle East as the world's powerhouse of mathematical ideas. Great strides had been made in understanding the geometry of objects fixed in time and space. The race was now on to discover the mathematics to describe objects in motion. In this programme, Marcus du Sautoy explores the work of René Descartes and Pierre Fermat, whose famous Last Theorem would puzzle mathematicians for more than 350 years. He also examines Isaac Newton's development of the calculus, and goes in search of Leonard Euler, the father of topology or 'bendy geometry' and Carl Friedrich Gauss, who, at the age of 24, was responsible for inventing a new way of handling equations: modular arithmetic.
Dawkins opens by talking how organisms "grow up" to understand the universe around them, which requires certain apparatus, such as a brain. But before brains can become large enough to model the universe they must develop from intermediate forms. Dawkins then discusses the digger wasp and the set of experiments conducted by Nikolaas Tinbergen of how the digger wasp models the local geography around its nest. He then talks about the limitations of the digger wasps' brain and concludes that only the human brain is sufficiently developed to model large-scale phenomena about the world. He then shows a MRI scan of a human brain (later revealed to be his own brain) and describes how an image develops from the eye onto the visual cortex. Dawkins discusses how the image on the retina is upside-down and in two dimensions but the overlapping images from each of the eyes are composited to form a three dimensional model in the brain. He shows this by asking the audience to focus on him while holding their hand at eye level which causes them to see two images of their hand; one from each eye. He then describes how using his finger to wriggle his eyeball that the outside world appears to move because he is moving the image on his retina. However this does not happen when he voluntary rolls his eyes from side to side. This is due to the brain using the internal model to compensate for the relative change in position of images on the retina. Dawkins gets someone to wear a virtual reality headset and move around in a 3-D computer generated world and draws an analogy between the model of the universe developed in one's head with the virtual reality universe developed in the computer. He then goes on the show that the brain uses models to describe the universe by looking at how the brain interprets various optical illusions, such as the hollow-face illusion using a rotating hollow mask of Charlie Chaplin, the "impossible" geometry of a Penrose triangle, the shifting interpretations of the Necker cube and the ability of humans to find faces in random shapes.
Dawkins begins by relating the story of asking a little girl "what she thought flowers were 'for'." Her response is anthropocentric, that flowers are there for our benefit. Dawkins points out that many people throughout history have thought that the natural world existed for our benefit, with examples from Genesis and other literature. Author Douglas Adams, who is sitting in the audience, is called to read a relevant passage from his novel The Restaurant at the End of the Universe. Dawkins then asks his audience to put off the idea that the natural world exists for our benefit. He considers the question of flowers seen through the eyes of bees and other pollinators, and performs a series of demonstrations which use ultraviolet light to excite fluorescence in various substances.
Dawkins starts the lecture coming in with a stick insect on his hand. He describes with how much details such a being imitates its environment, its almost like a key that fits a lock. He then shows another insect, namely a Leaf Insect, which basically looks exactly like a dead leaf. He gives some more examples for this amazing imitation of the surrounding, e.g. a Potoo, which looks like a branch of tree and a thorn bug, which gains protection by looking like a rose thorn. He, once again, makes the point that you can compare these beings with a key, which they represent themselves, whereas nature is the lock. Professor Richard Dawkins then explains that a key has to fit a lock exactly, and demonstrates this with a model of a lock. He mentions that a key is something very improbable. However it is hard to measure the probability of such a key, therefore Dawkins takes a bicycle lock for illustration, where you can calculate how likely it is to open the lock, because there is a fixed amount of dials with a fixed amount of positions. In Dawkin's case we have 3 dials, with 6 positions each, so the probability that you open the lock by sheer luck is one to two hundred and sixteen. Dawkins then shows the mechanism of the lock with a big model: Each dial has to be in the correct position in order to open up the lock. The model is then adapted to demonstrate a staged or gradualist solution to finding the right combination to open the lock. The probability of unlocking the combination in three separate phases falls to one in eighteen.
Dawkins' second lecture of the series examines the problem of design. He presents the audience with a number of simple objects, such as rocks and crystals, and notes that these objects have been formed by simple laws of physics and are therefore not designed. He then examines some designed objects - including a microscope, an electronic calculator, a pocket watch, and a clay pot - and notes that none of these objects could have possibly come about by sheer luck. Dawkins then discusses what he calls "designoid objects", which are complex objects that are neither simple, nor designed. Not only are they complex on the outside, they are also complex on the inside - perhaps billions of times more complex than a designed object such as a microscope. Dawkins then shows the audience a number of designed and designoid objects, including the pitcher plant, megalithic mounds built by the compass termite, and pots made by trapdoor spiders, potter wasps, and mason bees. He examines some designoid objects that use camouflage, such as a grasshopper that looks like a stone, a sea horse that looks like sea weed, a leaf insect, a green snake, a stick insect, and a collection of butterflies that look like dead leaves when their wings are closed. Dawkins notes that many animals share similar types of camouflage or protection because of a process called convergent evolution. Examples of such designoid objects include the hedgehog and the spiny anteater (both of which evolved pointed spines along their back) and the marsupial wolf (which looks like a dog but is actually a marsupial). He illustrates the reason why convergent evolution occurs by using two small models of commercial aircraft. The reason they look similar isn't due to industrial espionage, it is due to the fact that they are both built to fly, so they must make use of similar design principles.
In the third and final episode, Dawkins explains why Darwin's theory is one of history's most controversial ideas. Dawkins uses this episode to discuss the opposition that evolution has experienced since it was first discovered. He starts by approaching various anti-evolutionists, ranging from John Mackay from Creation Research, Wendy Wright, President of Concerned Women for America, to English school teacher Nick Cowen. In order to address concerns they bring up, he shows the evidence for evolution, including fossil and DNA evidence.
In the second episode Richard Dawkins deals with some of the philosophical and social ramifications of the theory of evolution. Dawkins starts out in Kenya, speaking with palaeontologist Richard Leakey. He then visits Christ is the Answer Ministries, Kenya's largest Pentecostal church, to interview Bishop Bonifes Adoyo. Adoyo has led the movement to press the National Museums of Kenya to sideline its collection of hominid bones pointing to man's evolution from ape to human.