Melvyn Bragg and guests discuss the way perceptions of the importance of mathematics have fluctuated in the 20th century, the nature of mathematical ability, and what mathematics can show us about how life began, and how it might continue. Galileo wrote “this grand book the universe… is written in the language of mathematics”. It was said before Galileo and has been said since and in the last decades of the 20th century it is being said again, most emphatically. How important is maths in relation to other sciences at the end of the twentieth century - will it ever be made redundant, or is it increasingly crucial to our understanding of the world and ourselves? What insight can it give us into the origins of life, and the functioning of our brains, and what does it mean to say that maths has become more ‘visual’?With Ian Stewart, Professor of Mathematics and Gresham Professor of Geometry, University of Warwick; Brian Butterworth, Professor of Cognitive Neuroscience, University College, London.

Melvyn Bragg and guests discuss artificial intelligence. Can we create a machine that creates? Some argue so. And is consciousness, as we are, with headaches and tiffs and moods and small pleasures and sore feet - often all at the same time - capable of taking place in a machine? Artificial intelligence machines have been growing much more intelligent since Alan Turing’s pioneering days at Bletchley in World War Two. Its claims are now very grand indeed. It is 31 years since Stanley Kubrick and Arthur C Clarke gave us HAL - the archetypal thinking computer of the film 2001: A Space Odyssey. But are we any nearer to achieving the thinking, feeling computer? Or is it just a dream - and should it remain as one?With Igor Aleksander, Professor, Imperial College London and inventor of Magnus - a neural computer which he says is an artificially conscious machine; John Searle, Professor of Philosophy, University of California and one of only two people in the world to invent an argument, the Chinese Room Argument, which destroys the plausibility of the idea of conscious machines.

Melvyn Bragg examines the future of gene therapy and advances in evolutionary biology. Are we continuing to evolve? If so, what are the signs and if not, why not? And those apes, so very very near us in genetic kinship, why are they so far away in so much else, and will they ever evolve? And is evolution necessarily progression? If so, does our apparent lack of evolution mean lack of progress? Also on the evolutionary front, could electronic devices discover the means of self-replication, and what will that mean for us? The march of the life sciences after the discovery of DNA accelerates by the year but what are the implications?With Professor John Maynard Smith evolutionary biological theorist and Emeritus Professor of Biology at the University of Sussex; Colin Tudge, writer, journalist and Research Fellow at the Centre for Philosophy.

Melvyn Bragg and guests discuss the role of animals in humankind's search for knowledge. Since the Greek physician Galen used pigs for anatomical studies in the 2nd century, animals have been used by scientists to further human knowledge. Yet few, if any subjects in this country, raise such violent feelings and passions as animals and their place in our society. With the growing politicisation of animal rights, it is a subject which is increasing in intensity. Do animals have rights and do our needs permit us to use them still to enhance our own lives in the twentieth century? Is it still necessary to experiment on animals for the good of humankind? Or is that morally unacceptable and barbaric - particularly in the light of new research into animal consciousness?With Colin Blakemore, Professor of Physiology, Oxford University, President of the British Association for the Advancement of Science, Fellow of the Royal Society and targeted in the 1980s by animal welfare activists protesting at his research methods; Dr Lynda Birke, biologist, teacher at Lancaster and Warwick Universities, and previously worked for 7 years in animal behaviour at the Open University.

Melvyn Bragg and guests discuss the history of thought about space, and examines whether cyberspace has introduced a new concept of space in our world or if its roots are in Einsteinian physics. It would have seemed extraordinary to Dante or Newton, from their different perspectives, that at the end of the 20th century there would be learned scholars who would find no place for religion in the great schemes of thought and belief. In the 20th century our notions of physical space have been revolutionised. Einstein said that space was not a separate entity; we’ve probed and explored the outer reaches of our physical space with space flight, powerful telescopes and theoretical physics. But in the last 20 years, with the birth of the Internet, a virtual form of space has been introduced to us - cyberspace - where people can meet and communicate ideas; you sit at home, punch the keys and you can rove all over the world - the keyboard becomes a magic carpet. But does cyberspace introduce a new concept of space in our world? Or does it really have its roots in Einsteinian physics and even in Medieval theologyAccording to the science writer Margaret Wertheim, cyberspace - life on the surfing internet - gives us not only virtual reality, but a soul. Dr John Polkinghorne, the distinguished physicist and ordained priest in the C of E, is not happy with this news, but he does believe that religion is not destroyed by the new technology, and that latest theories in physics reinforce it. With The Reverend Dr John Polkinghorne, Fellow of Queen’s College, Cambridge and Canon Theologian of Liverpool; Margaret Wertheim, science writer and author of The Pearly Gates of Cyberspace: A History of Space from Dante to the Internet.

Melvyn Bragg and guests discuss the history of our ideas about the formation of language. The psychologist George Miller worked out that in English there are potentially a hundred million trillion sentences of twenty words in length - that’s a hundred times the number of seconds since the birth of the universe. “Language”, as Chomsky put it, “makes infinite use of finite media”. “Language”, as Steven Pinker puts it, “comes so naturally to us that it’s easy to forget what a strange and miraculous gift it is”. “All over the world”, he writes, “members of our species spend a good part of their lives fashioning their breath into hisses and hums and squeaks and pops and are listening to others do the same”. Jean Jacques Rousseau once said that we differ from the animal kingdom in two main ways - the use of language and the prohibition of incest. Language and our ability to learn it has been held up traditionally as our species’ most remarkable achievement, marking us apart from the animals. But in the 20th century, our ideas about how language is formed are being radically challenged and altered. With Dr Jonathan Miller, medical doctor, performer, broadcaster, author and film and opera director; Steven Pinker, cognitive scientist, Professor of Psychology and Director of the Centre for Neuroscience, Massachusetts Institute of Technology, California.

Melvyn Bragg and guests discuss the relevance of psychoanalysis at the end of the 20th century. It’s 100 years since Sigmund Freud, the founder of psychoanalysis, a term which he coined, published The Interpretation of Dreams. Sixty years after his death, Freud’s influence and the influence of that book, has been felt in the 20th century in everything from the arts, history and anthropology, to of course psychology and even science. Dreams have inspired political speeches, songs, and seduction, captivating and fascinating mankind since time immemorial. For Sigmund Freud, they were the key to unlocking the working of the unconscious. But at the end of the 20th century, has psychoanalysis become too fractured and too insistent on privileging the past over the present to go forward into the future? Has it failed to develop and adapt to an age increasingly dominated by science? With Dr Juliet Mitchell, psychoanalyst, Fellow of Jesus College, Cambridge, Department of Political and Social Sciences; Adam Phillips, psychoanalyst and author of The Beast in the Nursery.

Melvyn Bragg and guests discuss ageing. In 1900, 1% of the world’s population were over 65. In the 1990s nearly 8% are. By the year 2020, nearly 1/5th of the world’s population will be over 65 - the figure rises to 25% in the UK. We are now living longer than at any time in our history. How much do economic factors, rather than biological factors, determine what ageing really means and our attitude to it? And what are the ethical, economic and biological implications of living longer?Tom Kirkwood, is an expert on the science of ageing and he brings to bear a close study of how the ageing process is being arrested and speculates on the very great age some of us could and will reach. He has said: “Today’s older people are the vanguard of an extraordinary revolution in longevity that is radically changing the structure of society and altering our perceptions of life and death. The price for this success - and make no mistake it is a success - is that we now face the challenge of ageing.”Alan Walker is an expert in the sociology of ageing and he takes in the whole context, especially the economic dimension. With Professor Alan Walker, social gerontologist, advisor to the UN’s programme on Ageing and has chaired the European Commission’s observatory on Ageing and Older People; Professor Tom Kirkwood, Britain’s first professor of Biological Gerontology, University of Manchester and President of the British Society for Research into Ageing.

Melvyn Bragg and guests discuss the implications of the developments in genetic engineering. Out of the city of Cambridge in the mid century came DNA and out of Edinburgh at the end of the century came the cloning of Dolly the sheep. These two facts might well do more to change the world literally, and our view of the world, than anything else that has happened at any time. Genetics have become the conversation of our day and with the Human Genome Project lumbering towards completion, its power grows. But are the consequences likely to be destructive and will what we think of as a human being, a personality, or even a person, change uncomfortably and irredeemably? With Grahame Bulfield, geneticist, honorary professor, Edinburgh University and Director of the Roslin Institute, Edinburgh; Bryan Appleyard, features writer for The Sunday Times and author of Brave New Worlds: Genetics and the Human Experience.

Melvyn Bragg and guests marvel at our brains and discuss how at the end of a century of research we still understand so little about how they work.Developments in the understanding of the brain represent one of the major leaps forward in science in the 20th century, and the research is gathering pace and intensity. It’s a subject which captures the imagination, particularly the search for consciousness whatever that might be, and brings together some of the newest technology and the oldest belief systems. What a piece of work is the brain - a grain-of-sand-sized piece contains one hundred thousand neurons, two-million axons and one billion synapses which all talk to each other. How far we have got with our understanding of the brain and what can it tell us about ourselves and the world we live in?With Professor Susan Greenfield, director of the Royal Institution, Professor of Pharmacology, Oxford University and Professor of Physics at Gresham College; Professor Vilayanur Ramachandran, Professor of Neuroscience and Psychology, Director of the Brain Perception Laboratory, University of California in San Diego and Professor at the Salk Institute.