Markup for Mathematical Knowledge
Mathematicians make use of various kinds of documents (e.g. e-mails, letters, pre-prints, journal articles, and textbooks) for communicating mathematical knowledge. Such documents employ specialized notational conventions and visual representations to convey the mathematical knowledge reliably and efficiently. The respective representations are supported by pertinent markup systems like TEX/LATEX.
Even though mathematical documents can vary greatly in their level of presentation, formality and rigor, there is a level of deep semantic structure that is common to all forms of mathematics and that must be represented to capture the essence of the knowledge. As John R. Pierce has written in his book on communication theory [Pie80], mathematics and its notations should not be viewed as one and the same thing. Mathematical ideas exist independently of the notations that represent them. However, the relation between meaning and notation is subtle, and part of the power of mathematics to describe and analyze derives from its ability to represent and manipulate ideas in symbolic form. The challenge in putting mathematics on the World Wide Web is to capture both notation and content (that is, meaning) in such a way that documents can utilize the highly-evolved notational forms of written and printed mathematics, and the potential for interconnectivity in electronic media.
In this chapter, we present the state of the art for representing mathematical documents on the web and analyze what is missing to mark up mathematical knowledge. We posit that there are three levels of information in mathematical knowledge: formulae, mathematical statements, and the large-scale theory structure (constructing the context of mathematical knowledge). The first two are immediately visible in marked up mathematics, e.g. textbooks, the third is largely left to an implicit meta-level of mathematical communication, or the organization of mathematical libraries. We will discuss these three levels in the next sections.
KeywordsMathematical Knowledge Mathematical Object Mathematical Formula Mathematical Document Semantic Element
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