Multilingual and Multicultural Aspects

Since different types of context may influence the choice of a mathematical notation and language, state-of-the-art technologies need to be employed to derive localized presentations of the mathematical fragments in digital documents from language-independent semantic representations of the content of its subject material. Among the many factors that influence the way mathematics is written, the following are always taken into account:

  • semantics of the mathematical operation e.g., the kind of multiplication (the notation for cross-product differs from that of scalar product of vectors) level of mathematical sophistication of the target audience e.g., using × notation for simple multiplication in elementary school, but "invisible" multiplication in secondary and higher education or expressing formulae in natural language instead of using a compact symbolic presentation
  • typographic conventions arising from the specific area of study or from the geographic location e.g. using (a,b) or ]a,b[ for denoting the open interval between two points a and b, or using i or j for the imaginary unit depending whether it occurs in complex analysis or in electrical engineering
  • individual stylistic choices (e.g. the use of a mirrored capital E, , vs. a big Or notation, for existential quantification, with corresponding changes in layout positions for the respective parts of the quantified expression)
  • cultural or linguistic distinctions (e.g. tan vs. tg in different parts of the world, and the different notations for the greatest common divisor in different languages gcd in English, ggT in German, mcd in Spanish, MCD in Italian, and so on in different languages, all abbreviating the respective languages' translation).
  • choice between formal or informal rendering (e.g. “f where x is an element of S” vs. “ ?x:S.f”)
  • choice between different rendering modes (e.g. visual vs. aural)

Content markup, in particular mathematical information in MathML and OpenMath provides semantically rich, high-quality representations of knowledge at a level of abstraction that is suitable for electronic communication and processing. Typically using XML, this representation of mathematical content is designed to be independent of the cultural, regional and notational influence which is so important when presenting mathematics in a written or spoken form. The XML representation is intended for automated processing by software tools but at the same time it maintains a certain degree of human readability. Both MathML and OpenMath emphasize the distinction between content markup and presentation, the former providing a set of primitives for typesetting mathematical notation in the markup language called MathML Presentation. It is during "rendering" that, for instance, the MathML content representation of the concept of greatest common divisor is localized and presented in English as gcd(12,27), in German as ggT(12,27), and in Spanish as mcd(12,27) . Since the content representation can be unambiguously evaluated by computational software whereas the presentation can be inferred depending on the client location, this XML verbose format is the preferred format for storing digital material containing mathematics. Thus, the mathematical content markup provides, on the one hand, a standard format that is rich enough to allow the invocation of any computational software for performing mathematical manipulations, and, on the other hand, a language-independent format that can be fed to natural language generation software and to stylesheets for producing the mathematical vernacular and notations used in the final presentation.