Computational fields have been proposed as an effective abstraction to fill the gap between the macro-level of distributed systems (specifying a system’s collective behaviour) and the micro-level (individual devices’ actions of computation and interaction to implement that collective specification), thereby providing a basis to better facilitate the engineering of collective APIs and complex systems at higher levels of abstraction. This approach is particularly suited to complex large-scale distributed systems, like the Internet-of-Things and Cyber-Physical Systems, where new mechanisms are needed to address composability and reusability of collective adaptive behaviour. This work introduces a full formal foundation for field computations, in terms of a core calculus equipped with typing, denotational, and operational semantics. Critically, we apply techniques for formal programming languages to collective adaptive systems: we provide formal establishment of a link between the micro- and macro-levels of collective adaptive systems, via a result of computational adequacy and abstraction for the (aggregate) denotational semantics with respect to the (per-device) operational semantics.