What cognitive mechanisms support the emergence of linguistic conventions from repeated interaction? We present results from a large-scale, multi-player replication of the classic tangrams task, focusing on three foundational properties of conventions: arbitrariness, stability, and reduction of utterance length over time. These results motivate a theory of convention-formation where agents, though initially uncertain about word meanings in context, assume others are using language with such knowledge. Thus, agents may learn about meanings by reasoning about a knowledgeable, informative partner; if all agents engage in such a process, they successfully coordinate their beliefs, giving rise to a conventional communication system. We formalize this theory in a computational model of language understanding as social inference and demonstrate that it produces all three properties in a simplified domain.