In Shadows of the Mind: A Search for the Missing Science of Consciousness (SM) (Penrose 1994), Roger Penrose continues to develop some central themes of his earlier book, The Emperor's New Mind (ENM) (Penrose 1989), but he also strikes off in some entirely new directions. Penrose argues in ENM that human cognition cannot in principle be fully understood in terms of computation. The argument, following an earlier argument of J. R. Lucas (Lucas 1961), was based on Gödel's incompleteness theorem. Penrose returns to this material in Part I of SM and develops it at great length in response to the many criticisms of the argument in ENM. In Part II of SM, Penrose opens a new line of inquiry which is motivated by the Conclusion of Part I. Since he thinks consciousness is a function of the brain but cannot be fully understood in terms of computation, he asks how the brain can perform the needed noncomputational actions. He is not willing to forgo a scientific account of consciousness, and to develop such an account we must look to neuroscience. But how might noncomputational actions arise within scientifically comprehensible physical laws? What physical principles might the brain use? Penrose is driven to the conclusion that these principles must be subtle and largely unknown. He suggests in Part II that neuron signals in the brain may behave as classically determinate events, but synaptic connections between neurons are controlled at a deeper level where there is physical activity at the quantum-classical borderline.