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 chats with Martin Palmer, a director focused on religion and culture, John Haldane, a moral philosophy expert, and Annabel Brett, a political thought academic. They delve into St. Thomas Aquinas's revolutionary impact on medieval philosophy by merging Aristotelian and Christian thought. The discussion also explores his five proofs for God’s existence, his views on divine governance, and how his philosophy influences modern ideas of human rights and social authority, revealing his lasting legacy in church doctrine.

Explore the radical ideals of Logical Positivism, a movement questioning the shaky foundations of traditional philosophy. From the Vienna Circle's influence post-WWI to rejecting metaphysical notions for empirical knowledge, discover the impact on reshaping modern philosophy. Delve into the legacy of logical empiricism, ethical language analysis, and the enduring relevance of scientific rigor in shaping contemporary thought.

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.

Melvyn Bragg and guests discuss the consolation of Philosophy. In the 6th century AD, a successful and intelligent Roman politician called Boethius found himself unjustly accused of treason. Trapped in his prison cell, awaiting a brutal execution, he found solace in philosophical ideas - about the true nature of reality, about injustice and evil and the meaning of living a moral life. His thoughts did not save him from death, but his ideas lived on because he wrote them into a book. He called it The Consolation of Philosophy. The Consolation of Philosophy was read widely and a sense of consolation is woven into many philosophical ideas, but what for Boethius were the consolations of philosophy, what are they more generally and should philosophy lead us to consolation or lead us from it?With AC Grayling, Professor of Philosophy at Birkbeck College, University of London; Melissa Lane, Senior University Lecturer in History at the University of Cambridge and Roger Scruton, Research Professor at the Institute for the Psychological Sciences.

Melvyn Bragg and guests discuss one of the most important works of political philosophy ever written - Aristotle’s ‘Politics’. Looking out across the city states of 4th century Greece Aristotle asked what made a society good and developed a language of ‘oligarchies’, ‘democracies’ and ‘monarchies’ that we still use today. Having witnessed his home town of Stagira destroyed by Philip of Macedon, Aristotle tried to establish a way of preserving a good society in dangerous times. How should it be governed and who should be allowed to live in it? Thomas Hobbes, Jean-Jacques Rousseau, Thomas Aquinas and Niccolo Machiavelli, to name but a few, have all asked the same questions and come up with wildly differing answers.Aristotle’s conclusions range across the role of wealth and the law, across men, women and slaves, education and leisure. They are far reaching, influential and, at times, deeply unpalatable. But they are also answers to questions that have not and will not go away. With Angie Hobbs, Associate Professor of Philosophy at the University of Warwick; Paul Cartledge, AG Leventis Professor of Greek Culture at the University of Cambridge and Annabel Brett, Senior Lecturer in History at the University of Cambridge.

Melvyn Bragg and guests discuss an iconic piece of 20th century maths - Gödel’s Incompleteness Theorems. In 1900, in Paris, the International Congress of Mathematicians gathered in a mood of hope and fear. The edifice of maths was grand and ornate but its foundations, called axioms, had been shaken. They were deemed to be inconsistent and possibly paradoxical. At the conference, a young man called David Hilbert set out a plan to rebuild the foundations of maths – to make them consistent, all encompassing and without any hint of a paradox. Hilbert was one of the greatest mathematicians that ever lived, but his plan failed spectacularly because of Kurt Gödel. Gödel proved that there were some problems in maths that were impossible to solve, that the bright clear plain of mathematics was in fact a labyrinth filled with potential paradox. In doing so Gödel changed the way we understand what mathematics is and the implications of his work in physics and philosophy take us to the very edge of what we can know.With Marcus du Sautoy, Professor of Mathematics at Wadham College, University of Oxford; John Barrow, Professor of Mathematical Sciences at the University of Cambridge and Gresham Professor of Geometry and Philip Welch, Professor of Mathematical Logic at the University of Bristol.