The proponents of artificial intelligence want to prove that it is only a matter of time before computers will be doing everything that a human mind can do. They take it for granted that pleasure and pain, the appreciation of beauty and humour, consciousness and free will are capacities that a computer will display once the appropriate programs of algorithms have been developed. Some disagree, because although electronic computers can calculate very rapidly, that does not make them understand what they are doing any more than, for example, an abacus does. The author puts forward his view that there is some facet of human thinking that can never be emulated by a machine. He shows the physical and mathematical ideas that are the background to his argument - from Turing machines, algorithms and the Chinese room, via quantum mechanics, cosmology and relativity to the structure of the brain, inspiration and consciousness itself.