Josephson was born in Cardiff, Wales, to Jewish parents, Mimi (née Weisbard, 1911–1998) and Abraham Josephson. He attended Cardiff High School, where he credits some of the school masters for having helped him, particularly the physics master, Emrys Jones, who introduced him to theoretical physics. In 1957, he went up to Cambridge, where he initially read mathematics at Trinity College, Cambridge. After completing Maths Part II in two years, and finding it somewhat sterile, he decided to switch to physics.