CONCEPTUAL AND REFERENTIAL MEANING OF MATHEMATICS

As has been elsewhere analysed (Chassapis, 1997), each mathematical concept acquires its meaning by a particular mathematical theory in which it is embedded. This is its conceptual meaning, assigned by the propositions of a particular mathematical theory. For example, the concept of addition for natural numbers is

defined recursively by the Peano axioms (i) a + 0 = a, and (ii) a + Sb = S(a + b), where Sa is the successor of a. Accordingly in set theory addition is defined by the cardinality of the disjoint union and in any other kind of mathematical structure is defined in terms of its propositions. Mathematical concepts, however, are used to describe or are “applied” to the real world on the basis of a mapping between them and real world situations. Every such mapping is indispensably mediated by a class of non-mathematical concepts that circumscribes the real world situations to which the mathematical concept is applied as well as by a set of pertinent linguistic or more generally symbolic expressions that signify these concepts. These non-mathematical concepts and their symbolic expressions assign another non-mathematical meaning and simultaneously specify a reference of the mathematical concept to a particular description or application. This meaning is the contextual or the referential meaning of a mathematical concept. Number addition, for example, may be used to describe or may be applied to a class of real world situations circumscribed by concepts of change, combine or compare. Particular instances of these concepts (e.g., the growth of a quantity as an instance of change, the union of two quantities as an instance of combination or the difference of two quantities as an instance of comparison) together with their signifying linguistic expressions attribute to the mathematical concept of addition various referential meanings, according to the case, which are not in their every aspect identical.

The referential meaning of a mathematical concept beyond its practical sources is related to the values associated by the people employing it in their everyday activities and/or by various communities using mathematics applied to their practices. This second aspect is socially and historically determined, since one mathematical concept can be valued in one context and de-valued in another, while its value in the same context can change over time due to social changes. Adopting the view that real world situations acquire their meanings by the implicated human activities which are always meaningful since intentional, it may be claimed that any real world situation and its representations bear meanings that are never value-free nor ideologically neutral. One step further, it may be claimed that the selected real world situations and consequently the associated references of the mathematical concepts to specific aspects of the real world used as examples, applications, questions or problems-to-be-solved in the teaching of school mathematics are never value-free and bear - in any feasible case - a more or less definite, even if not clear, ideological orientation. They thus assign to the mathematical concepts analogous, ideologically oriented, referential meanings. The ideological orientations of the referential meanings assigned to number addition, for instance, when applied to, and interpreted as describing, a growth process of profit in a situation of commercial dealings or a growth process of nuclear waste in a situation of environmental pollution are not identical. The two situations highlight different aspects of human activity, and implicitly emphasize different attitudes and patterns of thinking towards human activities, support different life values and ultimately transmit different social ideologies.

From this point of view, school mathematics, just as many other school subjects, may not be considered as an ideologically and hence socially neutral subject of knowledge, derived from a similarly neutral scientific mathematical activity. It has to be conceived as a school subject composed of selected mathematical topics bearing ideologically oriented referential meanings assigned to mathematical concepts and tools by their selected mappings in selected applications to selected real world situations.