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.

Melvyn Bragg and guests discuss the music of the spheres, the elegant and poetic idea that the revolution of the planets generates a celestial harmony of profound and transcendent beauty. In Shakespeare’s The Merchant of Venice the young Lorenzo woos his sweetheart with talk of the stars: “There’s not the smallest orb which thou behold’stBut in his motion like an angel sings,Still quiring to the young-eyed cherubins;Such harmony is in immortal souls;But whilst this muddy vesture of decayDoth grossly close it in, we cannot hear it.”The idea of music of the spheres ran through late antiquity and the medieval period into the Renaissance and its echoes could be heard in astrology and astronomy, in theology, and, of course, in music itself. Influenced by Pythagoras and Plato, it was discussed by Cicero, Boethius, Marcello Ficino and Johannes Kepler It affords us a glimpse into minds for which the universe was full of meaning, of strange correspondences and grand harmonies.With Peter Forshaw, Postdoctoral Fellow at Birkbeck, University of London; Jim Bennett, Director of the Museum of the History of Science at the University of Oxford and Angela Voss, Director of the Cultural Study of Cosmology and Divination at the University of Kent, Canterbury.

Melvyn Bragg and guests delve into the dark world of genetics under Joseph Stalin in discussing the career of Trofim Lysenko. In 1928, as America lurched towards the Wall Street Crash, Joseph Stalin revealed his master plan - nature was to be conquered by science, Russia to be made brutally, glitteringly modern and the world transformed by communist endeavour.Into the heart of this vision stepped Trofim Lysenko, a self-taught geneticist who promised to turn Russian wasteland into a grain-laden Garden of Eden. Today, Lysenko is a byword for fraud but in Stalin’s Russia his outlandish ideas about genetic inheritance and evolution became law. They reveal a world of science distorted by ideology, where ideas were literally a matter of life and death. To disagree with Lysenko risked the gulag and yet he destroyed Soviet Agriculture and damaged, perhaps irreparably, the Soviet Union’s capacity to fight and win the Cold War. With Robert Service, Professor of Russian History at the University of Oxford; Steve Jones, Professor of Genetics at University College London; Catherine Merridale, Professor of Contemporary History at Queen Mary, University of London.

Melvyn Bragg and guests discuss the strange mathematics of probability where heads or tails is a simple question with a far from simple answer. Gambling may be as old as the hills but probability as a mathematical discipline is a relative youngster. Probability is the field of maths relating to random events and, although commonplace now, the idea that you can pluck a piece of maths from the tumbling of dice, the shuffling of cards or the odds in the local lottery is a relatively recent and powerful one. It may start with the toss of a coin but probability reaches into every area of the modern world, from the analysis of society to the decay of an atom. With Marcus du Sautoy, Professor of Mathematics at the University of Oxford; Colva Roney-Dougal, Lecturer in Pure Mathematics at the University of St Andrews; Ian Stewart, Professor of Mathematics at the University of Warwick

Melvyn Bragg and guests discuss the history of ideas about the human brain. Since time immemorial people have puzzled over the brain and its functions. In the 5th century BC the Greek physician Hippocrates confidently asserted:“Men ought to know that from the brain and from the brain only arise our pleasures, joys, laughter and jests, as well as our sorrows, pains, grieves and tears.” This might suggest that people have never doubted the importance of the brain, but for Aristotle the heart was the ruler of the body and the seat of the soul. Only in the 17th century, with new scientific advances, did the true importance of the brain begin to be appreciated. In 1669 the Danish anatomist, Nicolaus Steno, still lamented that, “the brain, the masterpiece of creation, is almost unknown to us.”How far have our perceptions of how the brain works and what it symbolises changed over the centuries? And, in amongst the matter or our little grey cells, are we still searching for our souls? With Vivian Nutton, Professor of the History of Medicine at University College London; Jonathan Sawday, Professor of English Studies at the University of Strathclyde; Marina Wallace, Professor at the University of the Arts, London, Central St Martin’s College of Art and Design

Melvyn Bragg and guests discuss Newton’s Laws of Motion. In 1687 Isaac Newton attempted to explain the movements of everything in the universe, from a pea rolling on a plate to the position of the planets. It was a brilliant, vaultingly ambitious and fiendishly complex task; it took him three sentences. These are the three laws of motion with which Newton founded the discipline of classical mechanics and conjoined a series of concepts - inertia, acceleration, force, momentum and mass - by which we still describe the movement of things today. Newton’s laws have been refined over the years – most famously by Einstein - but they were still good enough, 282 years after they were published, to put Neil Armstrong on the Moon. With Simon Schaffer, Professor in History and Philosophy of Science at the University of Cambridge and Fellow of Darwin College; Raymond Flood, University Lecturer in Computing Studies and Mathematics and Senior Tutor at Kellogg College, University of Oxford; Rob Iliffe, Professor of Intellectual History and History of Science at the University of Sussex.

Melvyn Bragg and guests discuss the 19th century mathematician Ada Lovelace. Deep in the heart of the Pentagon is a network of computers. They control the US military, the most powerful army on the planet, but they are controlled by a programming language called Ada. It’s named after Ada Lovelace, the allegedly hard drinking 19th century mathematician and daughter of Lord Byron. In her work with Charles Babbage on a steam driven calculating machine called the Difference Engine, Ada understood, perhaps before anyone else, what a computer might truly be. As such the Difference Engine is the spiritual ancestor of the modern computer.

Ada Lovelace has been called many things - the first computer programmer and a prophet of the computer age – but most poetically perhaps by Babbage himself as an ‘enchantress of numbers’.

With Patricia Fara, Senior Tutor at Clare College, Cambridge; Doron Swade, Visiting Professor in the History of Computing at Portsmouth University; John Fuegi, Visiting Professor in Biography at Kingston University.

Melvyn Bragg and guests will be leaving the studio, the planet and indeed, the universe to take a tour of the Multiverse. If you look up the word ‘universe’ in the Oxford English Dictionary you will find the following definition: “The whole of created or existing things regarded collectively; all things (including the earth, the heavens, and all the phenomena of space) considered as constituting a systematic whole.” That sounds fairly comprehensive as a description of everything, but for an increasing number of physicists and cosmologists the universe is not enough. They talk of a multiverse – literally many universes – to explain aspects of their theory, the character of the universe and the riddle of our existence within it. Indeed, compared to the scope and complexity of the multiverse, the whole of our known reality may be as a speck of sand upon a beach.The idea of a multiverse is still controversial, some argue that it isn’t even science, because it is based on an idea that we may never be able to prove or even see. But what might a multiverse be like, why are physicists and cosmologists increasingly interested in it and is it really scientific to discuss the existence of universes we may never know anything With Martin Rees, President of the Royal Society and Professor of Cosmology and Astrophysics at the University of Cambridge; Fay Dowker, Reader in Theoretical Physics at Imperial College; Bernard Carr, Professor of Mathematics and Astronomy at Queen Mary, University of London

Melvyn Bragg and guests discuss how the science of plate tectonics revolutionised our understanding of the planet on which we live. America is getting further away from Europe. This is not a political statement but a geological fact. Just as the Pacific is getting smaller, the Red Sea bigger, the Himalayas are still going up and one day the Horn of Africa will be a large island. This is the theory of plate tectonics, a revolutionary idea in 20th century geology that claimed the continents of Earth were dancing to the music of deep time. A dance of incredible slowness, yet powerful enough to throw up the mountains and pour away the oceans.Plate tectonics, the idea that the earth’s surface moved on a carpet of molten magma, constituted a genuine scientific revolution in geology. It explained why mountains appeared and why earth quakes occurred; it explained the curious distribution of fossils across the globe and finally solved the age old conundrum of why continents such as Africa and South America appeared to fit together like a giant jigsaw puzzle. Plate tectonics has made geologists, and many more besides, profoundly re-think what the Earth was, how it worked and how it related to all the things in it. With Richard Corfield, Visiting Senior Lecturer in Earth Sciences at the Open University; Joe Cann, Senior Fellow in the School of Earth and Environment at the University of Leeds; Lynne Frostick, Director of the Hull Environment Research Institute and Professor of Physical Geography at the University of Hull

Melvyn Bragg and guests talk about blood, black bile, yellow bile and phlegm. These are the four humours, a theory of disease and health that is among the most influential ideas aver conceived. According to an 11th century Arabic book called the Almanac of Health, an old man went to the doctor complaining of a frigid complexion and stiffness in winter. The doctor, after examining his condition, prescribed a rooster. Being a hot and dry bird it was the perfect tonic for a cold and rheumatic old man. This is medicine by the four humours. The idea that the body is a concoction of these four essential juices is one of the oldest on record. From the Ancient Greeks to the 19th century it explained disease, psychology, habit and personality. When we describe people as being choleric, sanguine or melancholic we are still using the language of the humours today. It also explains why, in the long and convoluted history of medical practice, pigeon livers were an aphrodisiac, blood letting was a form of heroism (and best done in spring) and why you really could be frightened to death. The theory was dismantled from the 17th century but in its belief that the mind and body are intimately connected and that health requires equilibrium the humours retain an influence to this dayWith David Wootton, Anniversary Professor of History at the University of York; Vivian Nutton, Professor of the History of Medicine at University College London; Noga Arikha, Visiting Fellow at the Institut Jean-Nicod in Paris