Melvyn Bragg and guests Robert Hoyland, Robert Irwin and Hugh Kennedy discuss the life and ideas of the 14th-century Arab philosopher of history Ibn Khaldun.Ibn Khaldun was a North African statesman who retreated into the desert in 1375. He emerged having written one of the most important ever studies of the workings of history.Khaldun was born in Tunis in 1332. He received a supremely good education, but at 16 lost many of his family to the Black Death. His adult life was similarly characterised by sharp turns of fortune. He built a career as a political operator in cities from Fez to Granada. But he often fared badly in court intrigues, was imprisoned and failed to prevent the murder of a fellow statesman. In 1375, he withdrew into the Sahara to work out why the Muslim world had degenerated into division and decline. Four years later, he had completed not only a history of North African politics but also, in the book's long introduction, one of the great studies of history. Drawing on both regional history and personal experience, he set out a bleak analysis of the rise and fall of dynasties. He argued that group solidarity was vital to success in power. Within five generations, though, this always decayed. Tired urban dynasties inevitably became vulnerable to overthrow by rural insurgents.Later in life, Ibn Khaldun worked as a judge in Egypt, and in 1401 he met the terrifying Mongol conqueror Tamburlaine, whose triumphs, Ibn Khaldun felt, bore out his pessimistic theories.Over the last three centuries Ibn Khaldun has been rediscovered as a profoundly prescient political scientist, philosopher of history and forerunner of sociology - one of the great thinkers of the Muslim world.Robert Hoyland is Professor of Islamic History at the University of Oxford; Robert Irwin is Senior Research Associate of the School of Oriental and African Studies at the University of London; Hugh Kennedy is Professor of Arabic in the School of Oriental and African Studies at the University of London.

Melvyn Bragg and guests Raymond Geuss, Esther Leslie and Jonathan Rée discuss the Frankfurt School.This group of influential left-wing German thinkers set out, in the wake of Germany's defeat in the First World War, to investigate why their country had not had a revolution, despite the apparently revolutionary conditions that spread through Germany in the wake of the 1918 Armistice. To find out why the German workers had not flocked to the Red Flag, Theodor Adorno, Max Horkheimer, Walter Benjamin and others came together around an Institute set up at Frankfurt University and began to focus their critical attention not on the economy, but on culture, asking how it affected people's political outlook and activities. But then, with the rise of the Nazis, they found themselves fleeing to 1940s California. There, their disenchantment with American popular culture combined with their experiences of the turmoil of the interwar years to produce their distinctive, pessimistic worldview. With the defeat of Nazism, they returned to Germany to try to make sense of the route their native country had taken into darkness. In the 1960s, the Frankfurt School's argument - that most of culture helps to keep its audience compliant with capitalism - had an explosive impact. Arguably, it remains influential today.Raymond Geuss is a professor in the Faculty of Philosophy at the University of Cambridge; Esther Leslie is Professor in Political Aesthetics at Birkbeck College, University of London; Jonathan Rée is a freelance historian and philosopher, currently Visiting Professor at Roehampton University and at the Royal College of Art.

Melvyn Bragg and guests John Mullan, Karen O'Brien and Barbara Taylor discuss the life and ideas of the pioneering British Enlightenment thinker Mary Wollstonecraft.Mary Wollstonecraft was born in 1759 into a middle-class family whose status steadily sank as her inept, brutal, drunken father frittered away the family fortune. She did what she could to protect her mother from his aggression; meanwhile, her brother was slated to inherit much of the remaining fortune, while she was to receive nothing.From this unpromising but radicalising start, Wollstonecraft's career took a dizzying trajectory through a bleak period as a governess to becoming a writer, launching a polemical broadside against the political star of the day, witnessing the bloodshed of the French Revolution up close, rescuing her lover's stolen ship in Scandanavia, then marrying one of the leading philosophers of the day, William Godwin, and with him having a daughter who - though she never lived to see her grow up - would go on to write Frankenstein.But most importantly, in 1792, she published her great work, A Vindication of the Rights of Woman, which marks her out as one of the great thinkers of the British Enlightenment, with a much stronger, more lasting influence than Godwin. The Vindication was an attempt to apply the Enlightenment logic of rights and reason to the lives of women. Yet it was not a manifesto for the extension of the vote or the reform of divorce law, but a work of political philosophy. And surprisingly, as recent scholarship has highlighted, it was infused with Rational Dissenting Christianity, which Wollstonecraft had absorbed during her time as a struggling teacher and writer in north London.John Mullan is Professor of English at University College, London; Karen O'Brien is Professor of English at the University of Warwick; Barbara Taylor is Professor of Modern History in the School of Humanities and Social Sciences at the University of East London.

Melvyn Bragg and guests Serafina Cuomo, John O'Connor and Ian Stewart discuss the ideas and influence of the Greek mathematician Pythagoras and his followers, the Pythagoreans.The Ancient Greek mathematician Pythagoras is probably best known for the theorem concerning right-angled triangles that bears his name. However, it is not certain that he actually developed this idea; indeed, some scholars have questioned not only his true intellectual achievements, but whether he ever existed. We do know that a group of people who said they were followers of his - the Pythagoreans - emerged around the fifth century BC. Melvyn Bragg and his guests discuss what we do and don't know about this legendary figure and his followers, and explore the ideas associated with them. Some Pythagoreans, such as Philolaus and Archytas, were major mathematical figures in their own right. The central Pythagorean idea was that number had the capacity to explain the truths of the world. This was as much a mystical belief as a mathematical one, encompassing numerological notions about the 'character' of specific numbers. Moreover, the Pythagoreans lived in accordance with a bizarre code which dictated everything from what they could eat to how they should wash. Nonetheless, Pythagorean ideas, centred on their theory of number, have had a profound impact on Western science and philosophy, from Plato through astronomers like Copernicus to the present day.Serafina Cuomo is Reader in Roman History at Birkbeck College, University of London; John O'Connor is Senior Lecturer in Mathematics at the University of Saint Andrews; Ian Stewart is Emeritus Professor of Mathematics at the University of Warwick.

Melvyn Bragg and guests AC Grayling, Beatrice Han-Pile and Christopher Janaway discuss the dark, pessimistic philosophy of Arthur Schopenhauer.As a radical young thinker in Germany in the early 19th century, Schopenhauer railed against the dominant ideas of the day. He dismissed the pre-eminent German philosopher Georg Hegel as a pompous charlatan, and turned instead to the Enlightenment thinking of Immanuel Kant for inspiration. Schopenhauer's central idea was that everything in the world was driven by the Will - broadly, the ceaseless desire to live. But this, he argued, left us swinging pointlessly between suffering and boredom. The only escape from the tyranny of the Will was to be found in art, and particularly in music. Schopenhauer was influenced by Eastern philosophy, and in turn his own work had an impact well beyond the philosophical tradition in the West, helping to shape the work of artists and writers from Richard Wagner to Marcel Proust, and Albert Camus to Sigmund Freud.AC Grayling is Professor of Philosophy at Birkbeck College, University of London; Beatrice Han-Pile is Professor of Philosophy at the University of Essex; Christopher Janaway is Professor of Philosophy at the University of Southampton.

Melvyn Bragg discusses the life, works and enduring influence of the medieval philosopher and theologian St Thomas Aquinas with Martin Palmer, John Haldane and Annabel Brett. St Thomas Aquinas' ideas remain at the heart of the official doctrine of the Roman Catholic Church today and inform philosophical debates on human rights, natural law and what constitutes a 'just war'.Martin Palmer is Director of the International Consultancy on Religion, Education and Culture; John Haldane is Professor of Philosophy at the University of St Andrews; Annabel Brett is Lecturer in History at Gonville and Caius College, Cambridge.

Melvyn Bragg discusses Logical Positivism, the eye-wateringly radical early 20th century philosophical movement. The Logical Positivists argued that much previous philosophy was built on very shaky foundations, and they wanted to go right back to the drawing board. They insisted that philosophy - and science - had to be much more rigorous before it started making grand claims about the world. The movement began with the Vienna Circle, a group of philosophically-trained scientists and scientifically-trained philosophers, who met on Thursdays, in 'Red Vienna', in the years after the First World War. They were trying to remould philosophy in a world turned upside down not just by war, but by major advances in science. Their hero was not Descartes or Hegel but Albert Einstein. The group's new doctrine rejected great swathes of earlier philosophy, from meditations on the existence of God to declarations on the nature of History, as utterly meaningless. When the Nazis took power, they fled to England and America, where their ideas put down new roots, and went on to have a profound impact.Melvyn is joined by Barry Smith, Professor of Philosophy at the University of London; Nancy Cartwright, Professor of Philosophy at the London School of Economics; and Thomas Uebel, Professor of Philosophy at Manchester University.

Patricia Fara, Stephen Pumfrey and Rhodri Lewis join Melvyn Bragg to discuss the Jacobean lawyer, political fixer and alleged founder of modern science Francis Bacon.In the introduction to Thomas Spratt's History of the Royal Society, there is a poem about man called Francis Bacon which declares 'Bacon, like Moses, led us forth at last, The barren wilderness he past, Did on the very border stand Of the blest promis'd land, And from the mountain's top of his exalted wit, Saw it himself, and shew'd us it'.Francis Bacon was a lawyer and political schemer who climbed the greasy pole of Jacobean politics and then fell down it again. But he is most famous for developing an idea of how science should be done - a method that he hoped would slough off the husk of ancient thinking and usher in a new age. It is called Baconian Method and it has influenced and inspired scientists from Bacon's own time to the present day.

Melvyn Bragg and guests discuss The School of Athens – the fresco painted by the Italian Renaissance painter, Raphael, for Pope Julius II’s private library in the Vatican. The fresco depicts some of the most famous philosophers of ancient times, including Aristotle and Plato, engaged in discussion amidst the splendour of a classical Renaissance chamber. It is considered to be one of the greatest images in Western art not only because of Raphael’s skill as a painter, but also his ability to have created an enduring image that continues to inspire philosophical debate today. Raphael captured something essential about the philosophies of these two men, but he also revealed much about his own time. That such a pagan pair could be found beside a Pope in private tells of the complexity of intellectual life at the time when classical learning was reborn in what we now call the Renaissance.With Angie Hobbs, Associate Professor in Philosophy at the University of Warwick; Valery Rees, Renaissance scholar and senior member of the Language Department at the School of Economic Science; Jill Kraye, Professor of the History of Renaissance Philosophy and Librarian at the Warburg Institute at the University of London

Melvyn Bragg and guests discuss the 19th century American writer and philosopher, Henry David Thoreau. Anti-slavery activist and passionate environmentalist, Thoreau was above all a champion of self-reliance and individualism. He was also a champion of the simple life, a lover of nature and an enemy of the modern who lived alone in a log cabin in the woods away from society. In his seminal work, Walden, published in 1854, he wrote: “I went to the woods because I wished to live deliberately, to front only the essential facts of life and not, when I came to die, discover that I had not lived. I wanted to live deep and suck out all the marrow of life.” Thoreau has become emblematic of one version of American values and his work has been an inspiration to politicians and writers alike, from Martin Luther King to Gandhi, Yeats and Tolstoy. Yet in many ways Thoreau remains a mystery, a man of contradictions who advocated self-sufficiency but was happy to let his mother do his washing and cook his meals.With Kathleen Burk, Professor of American History at University College London; Tim Morris, Lecturer in American Literature at the University of Dundee; Stephen Fender, Honorary Professor in English Literature at University College London.