Ontologies and languages for representing mathematical knowledge on the Semantic Web

Mathematics is a ubiquitous foundation of science, technology, and engineering. Specific areas of mathematics, such as numeric and symbolic computation or logics, enjoy considerable software support. Working mathematicians have recently started to adopt Web 2.0 environments, such as blogs and wikis, but these systems lack machine support for knowledge organization and reuse, and they are disconnected from tools such as computer algebra systems or interactive proof assistants. We argue that such scenarios will benefit from Semantic Web technology. Conversely, mathematics is still underrepresented on the Web of [Linked] Data. There are mathematics-related Linked Data, for example statistical government data or scientific publication databases, but their mathematical semantics has not yet been modeled. We argue that the services for the Web of Data will benefit from a deeper representation of mathematical knowledge. Mathematical knowledge comprises structures given in a logical language - formulae, statements (e.g. axioms), and theories -, a mixture of rigorous natural language and symbolic notation in documents, application-specific metadata, and discussions about conceptualizations, formalizations, proofs, and (counter-) examples. Our review of vocabularies for representing these structures covers ontologies for mathematical problems, proofs, interlinked scientific publications, scientific discourse, as well as mathematical metadata vocabularies and domain knowledge from pure and applied mathematics. Many fields of mathematics have not yet been implemented as proper Semantic Web ontologies; however, we show that MathML and OpenMath, the standard XML-based exchange languages for mathematical knowledge, can be fully integrated with RDF representations in order to contribute existing mathematical knowledge to the Web of Data. We conclude with a roadmap for getting the mathematical Web of Data started: what datasets to publish, how to interlink them, and how to take advantage of these new connections.