In this paper, we offer a Piagetian perspective on the construction of the logico-mathematical schemas which embody our knowledge of logic and mathematics. Logico-mathematical entities are tied to the subject's activities, yet are so constructed by reflective abstraction that they result from sensorimotor experience only via the construction of intermediate schemas of increasing abstraction. The ‘axiom set’ does not exhaust the cognitive structure (schema network) which the mathematician thus acquires. We thus view ‘truth’ not as something to be defined within the closed ‘world’ of a formal system but rather in terms of the schema network within which the formal system is embedded. We differ from Piaget in that we see mathematical knowledge as based on social processes of mutual verification which provide an external drive to any ‘necessary dynamic’ of reflective abstraction within the individual. From this perspective, we argue that axiom schemas tied to a preferred interpretation may provide a necessary intermediate stage of reflective abstraction en route to acquisition of the ability to use formal systems in abstracto.