Motif is a recurrent word, image, idea or sound which explains the theme/s in a novel.
In 'Sophie's World,' the theme is about the place of philosophy in our lives. The novel seems to suggest that philosophy is a tool which helps us to make sense of our existence, the reason for our existence and the meaning of life itself. We later learn that Sophie's World is the creation of Albert Knag who designed the world as a present for his daughter, Hilde.
The motifs (the recurring ideas and tools of the theme surrounding philosophy) are philosopher/instructor (Alberto Knox), student (Sophie) and curious reader (Hilde). The motifs suggest that we have to be all three in order to succeed in defining the philosophy of life (theme). As students,we need to learn from teachers; we should also read widely in order to attain the the knowledge we need in order to better understand our existence as human beings.
In Sophie's World, Alberto teaches Sophie about Western thought through the philosophers Socrates, Plato and Aristotle, and he also teaches her about Church history, about Democritus and the theory of atoms. He introduces her to the philosophies of Kant, Hume, Descartes, Kierkegaard (father of existentialis), Marx, Freud and Darwin.
Hilde's father, Albert Knag, is the creator of the world that Sophie and Alberto Knox inhabits. They want to break out of what they consider are oppressive restraints in the world that is Knag's creation. Rather than try to understand the paradigm of Knag's world, they choose to rebel, questioning everything that is the construct of this world. In that end, the novel is an existentialist cry for independent freedom.