n. 1. a systematic statement of the principles underlying some fragment of mathematics. It is often not clear whether what is described as a theory is an informal, intuitive description of a subject or its subsequent mathematization. Often, the formalization is merely partially undertaken, with no more than a tacit presumption of the possibility of its completion. Sometimes, however, the prior conception of what should constitute such a formal theory proves unrealizable, or the process of mathematical modelling itself shows the pre-theoretical notions to be untenable; see, for example, naive set theory. 2. rigorously, a formal language, together with its axioms (see axiom) and rules of inference. Such a system generates a set of truths, the theorems, but it cannot itself refer to the truth of its own sentences.