Mathematicians’ culture.Mathematicians form a community with a mathematical
culture, with sets of concepts and prior knowledge, methods, problems, criteria of
truth and validity, methodology and rules, and values, which are shared to a varying
degree. A number of authors have explored the culture and values of mathematics,
including Bishop (1988), Davis and Hersh (1980) and Wilder (1974, 1981). Here we
will undertake a more limited inquiry, restricted to the different realms of discourse
and knowledge of mathematicians, and their associated values. The analysis given here
is three-fold, proposing that mathematicians operate with knowledge on the three
levels of the syntax, semantics and pragmatics of mathematics. This is based on the
classificatory system of Charles Morris (1945) who distinguishes these three levels in
language use. In his sytem the syntax, semantics and pragmatics of a language refer to
the formal rule system (grammar and proof), the system of meanings and interpretations,
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and the nexus of human rules, purposes and decisions concerning language use,
respectively. In constructing this system, Morris added to the formal logical levels of
syntax and semantics a further level of pragmatics, inspired by pragmatism.