Human communication displays a striking contrast between the diversity of languages and the universality of the principles underlying their use in conversation. Despite the importance of this interactional base, it is not obvious that it heavily imprints the structure of languages. However, a deep-time perspective suggests that early hominin communication was gestural, in line with all the other Hominidae. This gestural phase of early language development seems to have left its traces in the way in which spatial concepts, implemented in the hippocampus, provide organizing principles at the heart of grammar.