Melvyn Bragg and guests discuss negative numbers, a history of mystery and suspicion. In 1759 the British mathematician Francis Maseres wrote that negative numbers "darken the very whole doctrines of the equations and make dark of the things which are in their nature excessively obvious and simple". Because of their dark and mysterious nature, Maseres concluded that negative numbers did not exist, as did his contemporary, William Friend. However, other mathematicians were braver. They took a leap into the unknown and decided that negative numbers could be used during calculations, as long as they had disappeared upon reaching the solution. The history of negative numbers is one of stops and starts. The trailblazers were the Chinese who by 100 BC were able to solve simultaneous equations involving negative numbers. The Ancient Greeks rejected negative numbers as absurd, by 600 AD, the Indians had written the rules for the multiplication of negative numbers and 400 years later, Arabic mathematicians realised the importance of negative debt. But it wasn't until the Renaissance that European mathematicians finally began to accept and use these perplexing numbers. Why were negative numbers considered with such suspicion? Why were they such an abstract concept? And how did they finally get accepted? With Ian Stewart , Professor of Mathematics at the University of Warwick; Colva Roney-Dougal , Lecturer in Pure Mathematics at the University of St Andrews; Raymond Flood , Lecturer in Computing Studies and Mathematics at Kellogg College, Oxford.

Melvyn Bragg and guests discuss the story of human evolution, which stretches back over six million years. It is not the story of one species but of several diverse species, some of whom walked the Earth at the same time. From the earliest hominids to the early Homo sapiens, there was nothing inevitable about the course of human evolution. But what conditions created the opportunity for diverse human species to thrive? What environmental factors led to the survival of one human species, but contributed to the extinction of so many others? What can the fossil record and the science of genetics tell us about our ancestors? How does the brain make modern man so unique in the natural world? With Steve Jones, Professor of Genetics in the Galton Laboratory at University College London; Fred Spoor, Professor of Evolutionary Anatomy at University College London; Margaret Clegg, Honorary Research Fellow in the Department of Biological Anthropology at University College London.

Melvyn Bragg and guests discuss relativism, a philosophy of shifting sands. "Today, a particularly insidious obstacle to the task of educating is the massive presence in our society and culture of that relativism which, recognizing nothing as definitive, leaves as the ultimate criterion only the self with its desires. And under the semblance of freedom it becomes a prison for each one, for it separates people from one another, locking each person into his or her own 'ego'." Pope Benedict XVI, in a speech given in June 2005, showed that the issue of relativism is as contentious today as it was in Ancient Greece, when Plato took on the relativist stance of Protagoras and the sophists. Relativism is a school of philosophical thought which holds to the idea that there are no absolute truths. Instead, truth is situated within different frameworks of understanding that are governed by our history, culture and critical perspective. Why has relativism so radically divided scholars and moral custodians over the centuries? How have its supporters answered to criticisms that it is inherently unethical? And if there are universal standards such as human rights, how do relativists defend culturally specific practices such as honour killings or female infanticide? With Barry Smith, Senior Lecturer in Philosophy at Birkbeck College, University of London; Jonathan Rée, freelance philosopher who holds visiting professorships at the Royal College of Art and Roehampton University; Kathleen Lennon, Senior Lecturer in Philosophy at the University of Hull.

Melvyn Bragg and guests discuss prime numbers: 2, 3, 5, 7, 11, 13, 17 … This sequence of numbers goes on literally forever. Recently, a team of researchers in Missouri successfully calculated the highest prime number - it has 9.1 million digits. For nearly two and a half thousand years, since Euclid first described the prime numbers in his book Elements, mathematicians have struggled to write a rule to predict what comes next in the sequence. The Swiss mathematician Leonhard Euler feared that it is "a mystery into which the human mind will never penetrate." But others have been more hopeful... In the middle of the nineteenth century, the German mathematician Bernhard Riemann discovered a connection between prime numbers and a complex mathematical function called the 'zeta function'. Ever since, mathematicians have laboured to prove the existence of this connection and reveal the rules behind the elusive sequence. What exactly are prime numbers and what secrets might they unlock about our understanding of atoms? What are the rules that may govern the prime sequence? And is it possible that the person who proves Riemann's Hypothesis may bring about the collapse of the world financial system? With Marcus du Sautoy, Professor of Mathematics and Fellow of Wadham College at the University of Oxford; Robin Wilson, Professor of Pure Mathematics at the Open University and Gresham Professor of Geometry; Jackie Stedall, Junior Research Fellow in the History of Mathematics at Queen's College, Oxford.

Melvyn Bragg and guests discuss artificial intelligence. Can machines think? It was a question posed by the mathematician and Bletchley Park code breaker Alan Turing and it is a question still being asked today. What is the difference between men and machines and what does it mean to be human? And if we can answer that question, is it possible to build a computer that can imitate the human mind? There are those who have always had robust answers to the questions that those who seek to create artificial intelligence have posed. In 1949 the eminent neurosurgeon, Professor Geoffrey Jefferson argued that the mechanical mind could never rival a human intelligence because it could never be conscious of what it did: "Not until a machine can write a sonnet or compose a concerto because of thoughts and emotions felt", he declared "and not by the chance fall of symbols, could we agree that machine equals brain - that is, not only write it but know that it had written it." Yet the quest rolled on for machines that were bigger and better at processing symbols and calculating infinite permutations. Who were the early pioneers of artificial intelligence and what drove them to imitate the operations of the human mind? Is intelligence the defining characteristic of humanity? And how has the quest for artificial intelligence been driven by warfare and conflict in the twentieth century? With Jon Agar, Lecturer in the History and Philosophy of Science, University of Cambridge; Alison Adam, Professor of Information Systems, Salford University; Igor Aleksander, Professor of Neural Systems Engineering at Imperial College, University of London.

Melvyn Bragg and guests discuss the search for the Graviton particle. Albert Einstein said "I know why there are so many people who love chopping wood. In this activity one immediately sees the results". Einstein spent the last thirty years of his life trying to find a theory that would unify electromagnetism with gravity, but success eluded him. The search is still on for a unifying theory of gravitational force and hopes are pinned on the location of the graviton - a hypothetical elementary particle that transmits the force of gravity. But the graviton is proving hard to find. Indeed, the Large Hadron Collider at CERN still won't allow us to detect gravitons per se, but might be able to prove their existence in other ways. The idea of the graviton particle first emerged in the middle of the 20th century, when the notion that particles as mediators of force was taken seriously. Physicists believed that it could be applicable to gravity and by the late 20th century the hunt was truly on for the ultimate theory, a theory of quantum gravity. So why is the search for the graviton the major goal of theoretical physics? How will the measurement of gravitation waves help prove its existence? And how might the graviton unite the seemingly incompatible theories of general relativity and quantum mechanics? With Roger Cashmore, Former Research Director at CERN and Principal of Brasenose College, Oxford; Jim Al-Khalili, Professor of Physics at the University of Surrey; Sheila Rowan, Reader in Physics in the Department of Physics and Astronomy at the University of Glasgow.

Melvyn Bragg and guests discuss the unique properties of asteroids. They used to be regarded as the 'vermin of the solar system', irritating rubble that got in the way of astronomers trying to study more interesting phenomena. It was difficult or even impossible for an observer of asteroids to book time using the world's best telescopes, because they were regarded as unspectacular objects that could tell us little about the origins of the universe. However, that has all changed. It is now thought that asteroids are the unused building blocks of planets, 'pristine material' that has remained chemically unchanged since the creation of the solar system; a snapshot of matter at the beginning of time. At the moment the Japanese probe Hayabusa is 180 million miles away, pinned to the back of the asteroid Itokawa, attempting to gain our first samples of the chemical composition of an asteroid. Why did asteroids fail to form planets? How do they differ from their celestial cousins, the comets? And are either of them likely to create another impact on planet Earth? With Monica Grady, Professor of Planetary and Space Sciences, Open University; Carolin Crawford, Royal Society Research Fellow, University of Cambridge; John Zarnecki, Professor of Space Science, Open University.

Melvyn Bragg and guests discuss the rise of the mammals. The Cenozoic Era of Earth's history began 65 million years ago and runs to this day. It began with the extraordinary 'KT event', a supposed asteroid impact that destroyed the dinosaurs, and incorporates the break up of Pangaea, the enormous landmass that eventually formed the continents we know today. It is known as the 'Age of the Mammals', and it is the period in which warm-blooded, lactating, often furry animals diversified rapidly and spread across the globe on land and in the sea. According to evolutionary theory, what conditions created the opportunity for mammals to thrive? What environmental factors lead to the characteristics they share - and the features they don't? And how did they become the most intelligent class of animals on the planet? With Richard Corfield, Senior Lecturer in Earth Sciences at the Open University; Steve Jones, Professor of Genetics at University College London; Jane Francis, Professor of Palaeoclimatology at the University of Leeds.

Melvyn Bragg and guests discuss the history of magnetism. Pliny the Elder, in his Historia Naturalis, tells a story of a legendary Greek shepherd called Magnes who, while guiding his flock on Mount Ida, suddenly found it hard to move his feet. The nails of his sandals held fast to the rock beneath them, and the iron tip of his crook was strangely attracted to the boulders all around. Magnes had stumbled across the lodestone, or 'Magnetite', and discovered the phenomenon of magnetism. Plato was baffled by this strange force, as were Aristotle and Galen, and despite being used in navigation, supposedly suspended over the body of Mohammed and deployed in the pursuit of medical cures. St Thomas Aquinas thought magnets had souls. it was not until the late 16th century that any serious scientific attempt was made to explain the mystifying powers of the magnet. Descartes developed a particle theory of magnetism but the great Isaac Newton fought shy of the problem of what caused magnets to attract and repelWho pioneered the study of magnetism? What theories did they construct from its curious abilities and how was the power of the magnet brought out of the realm of magic and into the service of science? With Stephen Pumfrey, Senior Lecturer in the History of Science at the University of Lancaster; John Heilbron, Emeritus Professor of History at the University of California, Berkeley; Lisa Jardine, Professor of Renaissance Studies at Queen Mary, University of London.

Melvyn Bragg and guests discuss the KT Boundary. Across the entire planet, where it hasn't been eroded or destroyed in land movements, there is a thin grey line. In Italy it is 1 cm thick, in America it stretches to three centimetres, but it is all the same thin grey line laid into the rock some 65 million years ago and it bears witness to a cataclysmic event experienced only once in Earth's history. It is called the KT Boundary and geologists believe it is the clue to the death of the dinosaurs and the ultimate reason why mammals and humans inherited the Earth.But exactly what did happen 65 million years ago? How was this extraordinary line created across the Earth and does it really hold the key to the death of the dinosaurs?With Simon Kelley, Head of Department in the Department of Earth Sciences, Open University, Jane Francis, Professor of Palaeoclimatology, University of Leeds; Mike Benton, Professor of Vertebrate Palaeontology in the Department of Earth Sciences, University of Bristol.