The methods of science have proven to be extraordinarily successful in describing the physical behaviour of the world. This success depends, to a large extent, upon the remarkably close accord between mathematics and the laws of physics, at least when the appropriate mathematical theories have been ascertained. This mysterious relationship is related to a second one: how it is that our minds are able to perceive this world of "ideal" mathematical concepts.

If it is purely physical action in the brain that is responsible for human mentality, then it follows that something of a non-computable character resides in Nature\'s laws. This brings us to a third mystery, namely how can it be that a physical object such as a living human brain can actually evoke conscious mentality. Any genuine progress towards understanding this mystery, will, in my opinion, require addressing the two other mysteries also. I shall argue that the natural place to expect the required non-computability is in a yet undiscovered theory bridging the small-scale physics of quantum theory to the large-scale physics of general relativity.

Roger Penrose is the Rouse Ball Professor of Mathematics Emeritus at Oxford University and Gresham Professor of Geometry at Gresham College, London. He has a part-time appointment as Francis and Helen Pentz Distinguished Professor of Physics and Mathematics at Penn State.

A Fellow of the Royal Society of London since 1972, he was elected a Foreign Associate of the National Academy of Sciences in 1998. He has received a number of prizes and awards including the 1988 Wolf Prize(which he shared with Stephen Hawking for their understanding of the universe), the Dannie Heinemann Prize, the Royal Society Royal Medal, the Dirac Medal, and the Albert Einstein prize. His 1989 book The Emperor\'s New Mind became a best-seller and won the 1990 Science Book Prize. In 1994 he was knighted for services to science.