Our proposed research addresses both of the Big Questions in the program call: (1) What are the limits of mathematics in advancing human knowledge? (2) What have the difficulties of AI taught us about the nature of mind and intelligence? One major recent scientific advance has been Embodiment ? the realization that scientific understanding of mind entails modeling of the human brain and how it evolved to control a physical body in a social community. This is the big answer to question (2) and has been understood informally for decades. But there has been a conflict between traditional Math and AI models of mind and the manifest embodiment of intelligence. The ICSI/Berkeley NTL group has developed methods that treat the embodied mind with rigor, using formalisms that are both tractable and realizable as connectionist and thus as neural systems. This entails new formalizations of events, asynchrony, and cooperation and the central idea of simulation semantics, which links to the neural level. Our answer to question (1) is that appropriate formalization is possible and is necessary for an adequate science of embodied mind and intelligence. The two supplements describe the background and our continuing work, with references.To date, NTL has only been applied to individual minds and we propose to extend it to communities. Our formalization of language communities (LCs) is based on the social construction of knowledge and addresses the deep issue of public meaning in an embodied theory of semantics. The key insight divides language understanding into an analysis phase, common to an LC, and an intrapersonal simulation phase, with constructions as the interface. The proposed research focuses on integrating the existing formalizations of analysis(ECG) and simulation(CPRM) and extending these to more complex issues like metaphorical language. This should result in an open integrated system for further research and insights into the nature of public and private language.