Mathematical knowledge is largely communicated by way of a specialized set of documents (e.g. e-mails, letters, pre-prints, journal articles, and textbooks). These employ special notational conventions and visual representations to convey the mathematical knowledge reliably and efficiently.
When marking up mathematical knowledge, one always has the choice whether to mark up the structure of the document itself, or the structure of the mathematical knowledge that is conveyed in the document. Even though in most documents, the document structure is designed to help convey the structure of the knowledge, the two structures need not be the same. To frame the discussion we will distinguish two aspects of mathematical documents. In the
we organize the mathematical knowledge by its function, and do not care about a way to present it to human recipients. In the
we are interested in the structure of the argument that is used to convey the mathematical knowledge to a human user.