In this book “The Computer and the Brain” von Neumann observed that the brain works efficiently using low arithmetical precision and very few logical steps. From this he made the remarkable conjecture that the brain’s internal logical system was non-standard. In this paper we prove von Neumann’s conjecture by considering a topological surface spin network brain model. Two theoretical arguments are presented. Both of which lead to testable predictions that suggest that the internal logical system of the brain is not standard as its logic does not obey the classical logical law of excluded middle of classical logic, which states that a logical proposition must be either true or false. We also show how the brain might function as a quantum computer when retrieving memories. This result explains how the brain can function efficiently with low arithmetical precision and few logical steps as a quantum computer requires very few steps to solve a problem. Observational support for the arguments and the reason for the existence of a quantum memory structure are discussed. The basic properties of the model used are explained. Thus we prove that the surface chosen for the model exactly captures the topological connectivity of the brain, sketch how that the model can produce all observed brain excitations using its global features and show that it has a natural way to store the information carried by signals as memory in a quantum spin structure.