There has been considerable optimism recently, among philosophers and neuroscientists, concerning the prospect for major discoveries about the neurophysiological basis of mind. The support for this optimism has been extremely abstract and general. I wish to present some grounds for pessimism. That type of self-understanding may encounter limits which have not been generally foreseen: the personal, mentalist idea of human beings may resist the sort of coordination with an understanding of humans as physical systems, that would be necessary to yield anything describable as an understanding of the physical basis of mind. I shall not consider what alternatives will be open to us if we should encounter such limits. I shall try to present grounds for believing that the limits may exist – grounds derived from extensive data now available about the interaction between the two halves of the cerebral cortex, and about what happens when they are disconnected. The feature of the mentalist conception of persons which may be recalcitrant to integration with these data is not a trivial or peripheral one, that might easily be abandoned. It is the idea of a single person, a single subject of experience and action, that is in difficulties. The difficulties may be surmountable in ways I have not foreseen. On the other hand, this may be only the first of many dead ends that will emerge as we seek a physiological understanding of the mind.