Gödel and the limits of logic by John Dawson in Plus Magazine
“Melvyn Bragg and his guests discuss the history of logic. Logic, the study of reasoning and argument, first became a serious area of study in the 4th century BC through the work of Aristotle. He created a formal logical system, based on a type of argument called a syllogism, which remained in use for over two thousand years. In the nineteenth century the German philosopher and mathematician Gottlob Frege revolutionised logic, turning it into a discipline much like mathematics and capable of dealing with expressing and analysing nuanced arguments. His discoveries influenced the greatest mathematicians and philosophers of the twentieth century and considerably aided the development of the electronic computer. Today logic is a subtle system with applications in fields as diverse as mathematics, philosophy, linguistics and artificial intelligence.With:A.C. GraylingProfessor of Philosophy at Birkbeck, University of LondonPeter MillicanGilbert Ryle Fellow in Philosophy at Hertford College at the University of OxfordRosanna KeefeSenior Lecturer in Philosophy at the University of Sheffield.Producer: Thomas Morris.”
“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.”
“With a surprising new proof, two young mathematicians have found a bridge across the finite-infinite divide, helping at the same time to map this strange boundary.
The boundary does not pass between some huge finite number and the next, infinitely large one. Rather, it separates two kinds of mathematical statements: “finitistic” ones, which can be proved without invoking the concept of infinity, and “infinitistic” ones, which rest on the assumption — not evident in nature — that infinite objects exist.”