Early examples of software in mathematical knowledge management
Modern computers are often employed to do the calculations needed for
mathematics, whether numerical or symbolic. There are also roles for software in
mathematical knowledge management (MKM). Much is now possible by developing new
tools to help access our literature. Three simple initial examples of MKM roles
will be considered here. The first is software applied to the Mathematical
Subject Classification (MSC). The MSC is jointly maintained by Mathematical
Reviews and Zentralblatt für Mathematik, now better known for their online
databases MathSciNet and zbMATH. The current MSC2010 has its own web site,
http://msc2010.org, used in preparing the 2010 revision. MSC2010 information is
offered there as a MediaWiki and in SKOS, as well as in RDF/XML, Turtle,
N-Triples, TriX, and JSON. SKOS (Simple Knowledge Organization System) is a W3C
(World Wide Web Consortium) standard. Management of even a largely hierarchical
classification like the MSC (i.e. it's tree-like with some additional
relationships) and its multilingual nature (there are Russian, Chinese and
Italian translations so far) make for software challenges.
Another special aspect of the MSC is the inclusion of some mathematical
formulas. Mathematical expressions are properly encoded in MathML, a second
example. MathML (Mathematics Markup Language) is also a standard from the W3C,
now in its third edition, and to become an ISO standard. MathML is an XML
vocabulary developed to support mathematical publication in the modern
information world. As such MathML specifies a class of labelled rooted planar
trees, but the details are significant. Its purpose is to capture both
presentational aspects and some of the semantics of mathematics, so MathML is in
the tradition of the efforts at pasigraphy reported at the first ICM in 1897,
and also harks back to Leibniz's {\it calculus ratiocinator}.
The third example of software in the service of mathematical knowledge is the
use of programs to analyze the nature of our subject as represented by its
literature. Possibly the oldest consideration of this sort is the Erdos number,
which comes from the co-authorship graph of mathematical papers. Later and more
thorough analyses have been done of other networks representing mathematics'
publications, whether in terms of co-authorship or co-citation, or in relation
to subject areas (using the MSC). Further studies have begun, leading to such
modern topics as persistent homology and A-theory. Machine processing of the
corpus of mathematics as a natural language has also started. Analysis of the
use of formulas depends on a standard notation such as MathML. Finally let it be
pointed out that the MSC and MathML are already extensively used in such places
as Wikipedia, PlanetMath, and the EuDML as well as essentially in the publishing
world, MathSciNet and zbMATH. The easing of access to recorded mathematical
knowledge offered a possible Global DML (or World DML), and even use of MSC and
MathML in swMATH, make clear that mathematical knowledge management, even in its
primitive present form, can aid further development of mathematics. The examples
above are just starting points.