ABSTRACT
There are basically three different senses of knowing mathematics. The first involves semantics—the internal structure of the objects of mathematical study: embodiments, systems, and families of systems. Second is concerned with syntax—and involves rules which operate within formal systems or directly on them. Third deals with axiomatic theories. This chapter focuses on the various kinds of relationships which exist between syntax and semantics. Special attention is given to the kinds of inference rules involved and to the processes by which rules are combined. Comments are made about the rules involved in conjecture-making, constructing counterexamples, and in model theory and metamathematics. Mathematical knowledge is sufficiently complex that any hope of devising a completely systematic and mechanical way of devising an adequate account from a given mathematical description is undoubtedly unrealistic. The chapter introduces higher order rules which map classes of rules into classes of rules where the individual pairs of rules operate in different systems.
