The evolution of language has been linked in the recent research to the evolution of a number of different capacities, from the theory of mind to the type-recursive computation. In this paper, I examine the possibility that language has evolved from the capacity of spatial computation. Similarities, but also certain differences, between the two capacities are outlined and discussed, including the following. From the aspect of neuro-cognitive science, it cannot stay unnoticed that some of the central computations both in the language faculty and in the spatial cognition are located in the same brain area - the hippocampus. On the cognitive side, direct counterparts of the central components of the language faculty can be identified within the domain of spatial cognition. In particular, this is argued for the recursive computation and its categorial base, for the use of two types of information, the descriptive and the geometric, in establishing reference, for the process of update of a mental representation of the relevant context based on the sensory input, and for several other aspects. Since humans and other vertebrates have spatial cognitive capacities of approximately the same nature and complexity, this narrows down the set of possible answers to the question what distinguishes humans and their language faculty from the cognitive capacities present in other species. The hypothesis proposed is that this difference is three-fold, and involves: 1) the domain-general use of the otherwise similar computational capacities as opposed to the use in animals which is bound to the spatial domain, and perhaps one or two others; 2) the serialization of the computations of the descriptive and the geometric means of reference in humans, resulting in a combined aggregate information, as opposed to a strict separation in other animals and 3) the increased use and importance of the update of the relevant mental representation of the context by a group of humans rather than just an individual.