In a programme first broadcast in 2016, Melvyn Bragg and guests discuss Zeno of Elea, a pre-Socratic philosopher from c490-430 BC whose paradoxes were described by Bertrand Russell as "immeasurably subtle and profound." The best known argue against motion, such as that of an arrow in flight which is at a series of different points but moving at none of them, or that of Achilles who, despite being the faster runner, will never catch up with a tortoise with a head start. Aristotle and Aquinas engaged with these, as did Russell, yet it is still debatable whether Zeno's Paradoxes have been resolved.With Marcus du Sautoy Professor of Mathematics and Simonyi Professor for the Public Understanding of Science at the University of OxfordBarbara Sattler Lecturer in Philosophy at the University of St Andrewsand James Warren Reader in Ancient Philosophy at the University of CambridgeProducer: Simon Tillotson

Melvyn Bragg and guests discuss the history of the idea of Sovereignty, the authority of a state to govern itself and the relationship between the sovereign and the people. These ideas of external and internal sovereignty were imagined in various ways in ancient Greece and Rome, and given a name in 16th Century France by the philosopher and jurist Jean Bodin in his Six Books of the Commonwealth, where he said (in an early English translation) 'Maiestie or Soveraigntie is the most high, absolute, and perpetuall power over the citisens and subiects in a Commonweale: which the Latins cal Maiestatem, the Greeks akra exousia, kurion arche, and kurion politeuma; the Italians Segnoria, and the Hebrewes tomech shévet, that is to say, The greatest power to command.' Shakespeare also explored the concept through Richard II and the king's two bodies, Hobbes developed it in the 17th Century, and the idea of popular sovereignty was tested in the Revolutionary era in America and France. With Melissa Lane Class of 1943 Professor of Politics at Princeton UniversityRichard Bourke Professor in the History of Political Thought at Queen Mary University of Londonand Tim Stanton Senior Lecturer in the Department of Politics at the University of YorkProducer: Simon Tillotson.

Melvyn Bragg and guests discuss the Muses and their role in Greek mythology, when they were goddesses of poetry, song, music and dance: what the Greeks called mousike, 'the art of the Muses' from which we derive our word 'music.' While the number of Muses, their origin and their roles varied in different accounts and at different times, they were consistently linked with the nature of artistic inspiration. This raised a question for philosophers then and since: was a creative person an empty vessel into which the Muses poured their gifts, at their will, or could that person do something to make inspiration flow? WithPaul Cartledge Emeritus Professor of Greek Culture and AG Leventis Senior Research Fellow at Clare College, University of CambridgeAngie Hobbs Professor of the Public Understanding of Philosophy, University of SheffieldAndPenelope Murray Founder member and retired Senior Lecturer, Department of Classics, University of WarwickProducer: Simon TillotsonImage: 'Apollo and the Muses (Parnassus)', 1631-1632. Oil on canvas. Nicolas Poussin (1594-1665).

Melvyn Bragg and guests discuss Simone de Beauvoir. "One is not born, but rather becomes, a woman," she wrote in her best known and most influential work, The Second Sex, her exploration of what it means to be a woman in a world defined by men. Published in 1949, it was an immediate success with the thousands of women who bought it. Many male critics felt men came out of it rather badly. Beauvoir was born in 1908 to a high bourgeois family and it was perhaps her good fortune that her father lost his money when she was a girl. With no dowry, she pursued her education in Paris to get work and in a key exam to allow her to teach philosophy, came second only to Jean Paul Sartre. He was retaking. They became lovers and, for the rest of their lives together, intellectual sparring partners. Sartre concentrated on existentialist philosophy; Beauvoir explored that, and existentialist ethics, plus the novel and, increasingly in the decades up to her death in 1986, the situation of women in the world. With Christina Howells Professor of French and Fellow of Wadham College at the University of OxfordMargaret Atack Professor of French at the University of LeedsAnd Ursula Tidd Professor of Modern French Literature and Thought at the University of ManchesterProducer: Simon Tillotson.

A moral theory that emphasises ends over means, Utilitarianism holds that a good act is one that increases pleasure in the world and decreases pain. The tradition flourished in the eighteenth and nineteenth centuries with Jeremy Bentham and John Stuart Mill, and has antecedents in ancient philosophy. According to Bentham, happiness is the means for assessing the utility of an act, declaring "it is the greatest happiness of the greatest number that is the measure of right and wrong." Mill and others went on to refine and challenge Bentham's views and to defend them from critics such as Thomas Carlyle, who termed Utilitarianism a "doctrine worthy only of swine."WithMelissa Lane The Class of 1943 Professor of Politics at Princeton UniversityJanet Radcliffe Richards Professor of Practical Philosophy at the University of OxfordandBrad Hooker A Professor of Philosophy at the University of ReadingProducer: Simon Tillotson.

Melvyn Bragg and his guests discuss the life and work of Al-Ghazali, a major philosopher and theologian of the late 11th century. Born in Persia, he was one of the most prominent intellectuals of his age, working in such centres of learning as Baghdad, Damascus and Jerusalem. He is now seen as a key figure in the development of Islamic thought, not just refining the theology of Islam but also building on the existing philosophical tradition inherited from the ancient Greeks.With:Peter Adamson Professor of Late Ancient and Arabic Philosophy at the LMU in MunichCarole Hillenbrand Professor of Islamic History at Edinburgh and St Andrews UniversitiesRobert Gleave Professor of Arabic Studies at the University of ExeterProducer: Victoria Brignell.

Melvyn Bragg and his guests discuss Adam Smith's celebrated economic treatise The Wealth of Nations. Smith was one of Scotland's greatest thinkers, a moral philosopher and pioneer of economic theory whose 1776 masterpiece has come to define classical economics. Based on his careful consideration of the transformation wrought on the British economy by the Industrial Revolution, and how it contrasted with marketplaces elsewhere in the world, the book outlined a theory of wealth and how it is accumulated that has arguably had more influence on economic theory than any other.With:Richard Whatmore Professor of Modern History and Director of the Institute of Intellectual History at the University of St AndrewsDonald Winch Emeritus Professor of Intellectual History at the University of SussexHelen Paul Lecturer in Economics and Economic History at the University of SouthamptonProducer: Thomas Morris.

Melvyn Bragg and guests discuss phenomenology, a style of philosophy developed by the German thinker Edmund Husserl in the first decades of the 20th century. Husserl's initial insights underwent a radical transformation in the work of his student Martin Heidegger, and played a key role in the development of French philosophy at the hands of writers like Emmanuel Levinas, Jean-Paul Sartre, Simone de Beauvoir and Maurice Merleau-Ponty.Phenomenology has been a remarkably adaptable approach to philosophy. It has given its proponents a platform to expose and critique the basic assumptions of past philosophy, and to talk about everything from the foundations of geometry to the difference between fear and anxiety. It has also been instrumental in getting philosophy out of the seminar room and making it relevant to the lives people actually lead. GUESTSSimon Glendinning, Professor of European Philosophy in the European Institute at the London School of Economics Joanna Hodge, Professor of Philosophy at Manchester Metropolitan University Stephen Mulhall, Professor of Philosophy and Tutor at New College at the University of Oxford Producer: Luke Mulhall.

Melvyn Bragg and his guests discuss the philosophy of truth. Pontius Pilate famously asked: what is truth? In the twentieth century, the nature of truth became a subject of particular interest to philosophers, but they preferred to ask a slightly different question: what does it mean to say of any particular statement that it is true? What is the difference between these two questions, and how useful is the second of them?With:Simon Blackburn Fellow of Trinity College, University of Cambridge, and Professor of Philosophy at the New College of the HumanitiesJennifer Hornsby Professor of Philosophy at Birkbeck, University of LondonCrispin Wright Regius Professor of Logic at the University of Aberdeen, and Professor of Philosophy at New York UniversityProducer: Victoria Brignell and Luke Mulhall.

Melvyn Bragg and guests discuss Zen. It's often thought of as a form of Buddhism that emphasises the practice of meditation over any particular set of beliefs. In fact Zen belongs to a particular intellectual tradition within Buddhism that took root in China in the 6th century AD. It spread to Japan in the early Middle Ages, where Zen practitioners set up religious institutions like temples, monasteries and universities that remain important today.GUESTSTim Barrett, Emeritus Professor in the Department of the Study of Religions at SOAS, University of LondonLucia Dolce, Numata Reader in Japanese Buddhism at SOAS, University of LondonEric Greene, Lecturer in East Asian Religions at the University of BristolProducer: Luke Mulhall.