In mathematics, clear, precise language using only previously defined concepts is essential for communicating results whose meaning is unambiguous. Scientific thought cannot do without such rigour. But the choice of words is not neutral when mathematical concepts are introduced and defined. Some mathematicians exploit the ambiguity of the terms they choose, even constructing a whole syntactic system that extends everyday language in troubling ways. This can affect how well the general public understands the results, subtly introducing a level of meta-comprehension, or even under-comprehension. Yet, as Maurice Allais suggests, "any author who uses mathematics should always make a point of stating in ordinary language the meaning of the assumptions made and the results obtained. The more abstract the theory, the more pressing this obligation becomes."
Jargon and everyday language --------------------------
Vocabulary moves in both directions. Scientific jargon makes its way into everyday language. The mathematical concepts most readily understood by non-specialists are those emphasised in secondary education. Every term in Euclidean geometry has become part of ordinary language, including the wholly abstract notions of point, line, plane, parallelism and perpendicularity. The terms equation, polynomial, sine, cosine and tangent are also used, though less often (the latter can give rise to misleading uses of the word "slope": in ordinary language, a 100% slope represents an angle of 90°, but in trigonometry it represents an angle of 45°). It is therefore impossible to partition all the elements of mathematical language into two disjoint subsets, one containing words and expressions from everyday language and the other technical terms. These two subsets depend on the context and the speakers' sociocultural background, and beautifully illustrate the notion of "fuzzy" sets—a term whose everyday and mathematical meanings plainly do not coincide.
It is common for an everyday term to be imported into mathematical language to denote a new and generally abstract concept. This can happen when adjectives become nouns. Thus the various sets of numbers have become sets of "naturals," "integers," "rationals," "reals" or "complexes," with the noun "numbers" omitted. In group theory, two distinguished elements are "the identity" and "the absorbing element." In analysis, derivative functions and antiderivatives become simply "derivatives" and "antiderivatives."
Nouns, too, can shift in meaning. This is true of terms such as "group" and "field," which in mathematical language denote structures equipped with operations (groups) and, in the eyes of novices, generally consisting of numbers in the ordinary sense (fields), and possessing certain properties. Of course, "set" is itself being used here in its mathematical sense, which differs from its everyday meaning. In this context, some statements are clear and admit only a technical reading.