Compared to half duplex communications, full duplex communications can significantly improve link capacity. However, in a large scale wireless network such as a wireless mesh network, the capacity gain from full duplex communications has not been fully investigated. To this end, a metric of network capacity called transmission capacity is studied in this paper for a full duplex wireless network. It captures the maximum transmission throughput in a unit area, subject to a certain outage probability. The key challenge of deriving transmission capacity is to characterize the aggregate interference of the typical link in a full duplex wireless network, which is completely different from that in a half duplex wireless network. In this paper, stochastic geometry is employed to model the network topology as a Thomas cluster point process and then the aggregate interference is characterized as a shot-noise process. Based on these models, the transmission capacity is derived. Analytical results show that under the same network density the distribution of aggregate interference in a full duplex wireless network is more dispersed than that in a half duplex wireless network. Comparisons of transmission capacity between a full duplex network and a half duplex network reveal that the capacity gain from full duplex communications is limited due to severe aggregate interference. This result implies that self-interference cancellation alone cannot ensure scalable full duplex wireless networking.