Large portions of the book involve detailed discussions of mathematics, physics, and philosophy. Most of these discussions use fictional Arbran terminology, but treat ideas from actual science and philosophy. Stephenson acknowledges the work of author Julian Barbour as the source for much of this material.[2]
A major theme of the novel is the many-worlds interpretation of quantum mechanics based on a directed acyclic graph, which accounts for the various "worldtracks" and "narratives" explored by Fraa Orolo and manipulated by Fraa Jad.[2] Another major theme is the recurring philosophical debate between characters espousing mathematical Platonic realism (called "Halikaarnians" in the novel and associated with Incanters) and characters espousing nominalism (called "Procians" in the novel and who are the Rhetors).
Stephenson cites the work of Roger Penrose as a major influence on the novel. Specific ideas from Penrose's work include: the idea that the human mind operates in certain fundamental ways as a quantum computer, espoused in Penrose's The Emperor's New Mind; Platonic realism as a philosophical basis for works of fiction, as in stories from Penrose's The Road to Reality; and the theory of aperiodic tilings, which appear in the Teglon puzzle in the novel.[2] Stephenson also cites as an influence the works of Kurt Gödel and Edmund Husserl, both of whom the character Durand mentions by name in the novel.[2]
Much of the Geometers' technology seen in the novel reflects existing scientific concepts. The alien ship moves by means of nuclear pulse propulsion.
As an appendix to the novel, Stephenson includes three "Calca", discussions among the avout of purely philosophical or mathematical content. The first is a discussion of a cake-cutting procedure corresponding to the geometric problem of "doubling the square" presented in Plato's Meno. The second presents configuration spaces (called "Hemn spaces" in the novel) as a way of representing three-dimensional motion. The third discusses a "complex" Platonic realism, in which several realms of Platonic ideal forms (called the "Hylaean Theoric Worlds" in the novel) exist independently of the physical world (called the "Arbran Causal Domain" in the novel). The mathematical structure of a directed acyclic graph is used to describe the way in which the various realms can influence one other, and even the physical world can function as part of the realm of ideal forms for some worlds "downstream" in the graph.
