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 the importance of the oath in ancient Greece and Rome, The importance of oaths in the Classical world cannot be overstated. Kings, citizens, soldiers, litigants all swore oaths, inviting divine retribution if they proved false to their word. Oaths cemented peace treaties, they obliged the Athenian citizenry to protect their democracy, they guaranteed the loyalty of the Roman army to its Emperor and they underpinned the legal systems of Athens and Rome. And in Homer's epic poem, The Iliad, it is a broken oath to settle the dispute between Menelaus and Paris that leads the Greeks to storm Troy in pursuit of Helen. But how did the Classical world come to understand the oath? Why did oaths come to occupy such a central place in the political, social and legal life of the Athenian State? And what role did oath-making play in the expanding Roman Empire? With Alan Sommerstein , Professor of Greek at the University of Nottingham;Paul Cartledge , Professor of Greek History at the University of Cambridge; Mary Beard , Professor in Classics at the University of Cambridge

Melvyn Bragg and guests discuss the ‘Oresteia’, the seminal trilogy of tragedies by Aeschylus. The composer Richard Wagner recalled the visceral sensations of reading Aeschylus' great trilogy for the first time. "I could see the Oresteia with my mind's eye ... Nothing could equal the sublime emotion with which the Agamemnon inspired me; and to the last word of the Eumenides, I remained in an atmosphere so far removed from the present day that I have never since been really able to reconcile myself with modern literature." Aeschylus' audience were all familiar with the tale of one man's return home from the Trojan War. Homer's Odyssey recounted Odysseus' perilous journey home, the forceful ejection of the suitors from his household and his reunion with wife Penelope and son Telemachus. Aeschylus had a very different tale of homecoming to tell in his Oresteia. Agamemnon arrives home from Troy to a murderous welcome from a vengeful wife and a cycle of atrocities unfolds in his household. The Oresteia has inspired some of the greatest artists and thinkers of the modern world. From Richard Wagner and Friedrich Nietzsche to T.S. Eliot and Simone de Beauvoir – the ‘Oresteia’ has fired the modern imagination.Why did Aeschylus make the family the subject of his bloody revenge tragedy? How did his trilogy make a contribution to the development of Athenian legal institutions? And why has the Oresteia had such a powerful hold over the modern imagination? With Edith Hall, Professor of Greek Cultural History at Durham University; Simon Goldhill, Professor of Greek at the University of Cambridge; Tom Healy, Professor of Renaissance Studies at Birkbeck College, University of London.

Melvyn Bragg and guests discuss ideas of heaven and the afterlife. The great medieval theologian Thomas Aquinas wrote 'that in the end language can only be related to what is experienced here, and given that the hereafter is not here, we can only infer'. Aquinas encapsulated a great human conundrum that has preoccupied writers and thinkers since ancient times: what might heaven be like. And although human language is constrained by experience, this has not stopped an outpouring of artistic, theological and literary representations of heaven. In the early Middle Ages men ascended up a ladder to heaven. In his Divine Comedy, Dante divided heaven into ten layers encompassing the planets and the stars. And the 17th century writer John Bunyan saw the journey of the soul to heaven as a spiritual struggle in his autobiography, The Pilgrim's Progress. But what exactly is heaven and where is it? How does the Protestant conception of the afterlife differ from the Catholic conception? How does one achieve salvation and what do the saved do when they get there? And, if heaven is so interesting, why has western culture been so spellbound by hell? With Valery Rees, Renaissance scholar and senior member of the Language Department at the School of Economic Science; Martin Palmer, Theologian and Director of the International Consultancy on Religion, Education and Culture; John Carey, Emeritus Professor of English Literature at Oxford University.

Melvyn Bragg and guests discuss the Peterloo Massacre in 1819, a defining moment of its age. In 1819 Percy Bysshe Shelley wrote: 'I met Murder on the way He had a mask like Castlereagh Very smooth he looked, yet grim; Seven blood-hounds followed him: All were fat; and well they might Be in admirable plight, For one by one, and two by two, He tossed them human hearts to chew Which from his wide cloak he drew.' As Foreign Secretary, Castlereagh had successfully co-ordinated European opposition to Napoleon, but at home he had repressed the Reform movement, and popular opinion held him responsible for the Peterloo Massacre of peaceful demonstrators in 1819. Shelley's epic poem, The Mask of Anarchy, reflected the widespread public outrage and condemnation of the government's role in the massacre. Why did a peaceful and orderly meeting of men, women and children in St Peter's Field, Manchester turn into a blood bath? How were the stirrings of radicalism in the wake of the Napoleonic Wars dealt with by the British establishment? And what role did the Peterloo Massacre play in bringing about the Great Reform Act of 1832? With Jeremy Black, Professor of History at the University of Exeter; Sarah Richardson, Senior Lecturer in History at the University of Warwick; Clive Emsley, Professor of History at the Open University.

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 great 17th century political philosopher Thomas Hobbes who argued: "During the time men live without a common power to keep them all in awe, they are in that condition which is called war; and such a war as is of every man against every man". For Hobbes, the difference between order and disorder was stark. In the state of nature, ungoverned man lived life in "continual fear, and danger of violent death". The only way out of this "solitary, poor, nasty, brutish, and short" existence, he said, was to relinquish all your freedom and submit yourself to one all powerful absolute sovereign. Hobbes' proposal, contained in his controversial and now classic text, Leviathan, was written just as England was readjusting to life after the Civil War and the rule of Oliver Cromwell. In fact, in his long life Hobbes’ allegiance switched from Charles I to Cromwell and back to Charles II. But how did the son of a poor clergyman end up as the most radical thinker of his day? Why did so many of Hobbes' ideas run counter to the prevailing fondness for constitutionalism with a limited monarchy? And why is he regarded by so many political philosophers as an important theorist when so few find his ideas convincing? With Quentin Skinner, Regius Professor of History at the University of Cambridge; David Wootton, Professor of History at the University of York; Annabel Brett, Senior Lecturer in Political Thought and Intellectual History at Cambridge University.

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 American philosophy of pragmatism. A pragmatist "turns away from abstraction and insufficiency, from verbal solutions, from bad apriori reasons, from fixed principles, closed systems, and pretended absolutes and origins. He turns towards concreteness and adequacy, towards facts, towards action and towards power". A quote from William James' 1907 treatise Pragmatism: A New Name for Some Old Ways of Thinking. William James, along with John Dewey and Charles Sanders Peirce, was the founder of an American philosophical movement which flowered during the last thirty years of the nineteenth century and the first twenty years of the 20th century. It purported that knowledge is only meaningful when coupled with action. Nothing is true or false - it either works or it doesn't. It was a philosophy which was deeply embedded in the reality of life, concerned firstly with the individual's direct experience of the world he inhabited. In essence, practical application was all. But how did Pragmatism harness the huge scientific leap forward that had come with Charles Darwin's ideas on evolution? And how did this dynamic new philosophy challenge the doubts expressed by the Sceptics about the nature and extent of knowledge? Did Pragmatism influence the economic and political ascendancy of America in the early 20th century? And did it also pave the way for the contemporary preoccupation with post-modernism? With A C Grayling, Professor of Applied Philosophy at Birkbeck College, University of London and a Fellow of St Anne's College, Oxford; Julian Baggini, editor of The Philosophers' Magazine; Miranda Fricker, Lecturer in Philosophy at Birkbeck College, University of London.

Melvyn Bragg and guests discuss the religious orders of the Dominicans and the Franciscans, known as the Blackfriars and Greyfriars. "Just as it is better to light up others than to shine alone, it is better to share the fruits of one's contemplation with others than to contemplate in solitude". Thus St Thomas Aquinas described his vocation, not only as a teacher, but also as a Dominican friar and philosopher at the University of Paris. In the 13th century, the religious orders of the Dominicans and the Franciscans were a great force for change in Catholic Europe. They thrived in the emerging towns and cities of the High Middle Ages, leading crusades and changing the way the Church dealt with heretics. They were the evangelists who transformed the Church's preaching of the Christian message to the people. On top of all this, these two orders were also responsible for reconciling Classical and Christian philosophy; their studies of Aristotle paved the way for the Renaissance. They also managed to change the curriculum at the universities of Paris and Oxford. But the Blackfriars and the Greyfriars did not come from the great monasteries of the time; they started out as itinerant preachers surviving upon the charity of the faithful. So how did these two orders come to dominate the spiritual and academic life of the 13th century, and how did they manage to accumulate such huge wealth while professing allegiance to lives of poverty? With Henrietta Leyser, medieval historian and Fellow of St Peter's College, Oxford; Alexander Murray, medieval historian and Emeritus Fellow of University College, Oxford; Anthony Kenny, philosopher and former Master of Balliol College, Oxford.