Semantics as meeting of minds I propose a semantic theory that is not construed as a mapping of language to the world, but a mapping between individual meaning spaces. The meanings of linguistic entities are established via a meeting of minds. The concepts in the minds of communicating individuals are modeled as regions in conceptual spaces. If concepts are convex, it will be possible for interactors to agree on a joint meaning even if they start out from different representational spaces. I argue that the semantic processes involved in pointing are essentially the same as those in spoken communication, and that the development of communication abilities can be seen as a transition from pointing in physical space to pointing in mental spaces. Using combinations of physical (visual) space with mental spaces, my model shows that there is a continuum in development, and that purely verbal communication may arise from a bootstrapping process grounded on gestures.