In software engineering the unified modeling language ("UML") is a well established technique for providing overview of complex systems and an efficient means of communicating about them. There are about ten diagrams for different views on the system. These diagrams have tremendously improved the ability to construct large systems by large teams, as one can look at the system at different levels and as one can profit from visualization.
Furthermore the UML models contains all the system's information needed for implementation.
I'm wondering whether a similar method could be useful for doing (and commuicating about) mathematical structures and theories.
Often so many definitions are built one upon the other and so many properties are introduced that it seems to be difficult to have an overview of the "architecture" of a theory.
For example one could have one sort of diagram showing how the mathematical structures are build form each other (e.g. a field built by two groups – with the respective axioms "inherited" – and further "compatibility conditions" between them). In priciple you would be able to track back all structures to the "mother structures" algebraic structure, order structure, topolological structure.) Interestingly the object oriented paradigm used by UML, i.e. encapsulating attributes and methods into classes is somehow similar to the categorical approach in mathematics (encapsulating objects and morphisms).
Another sort of diagram could represent a (part of a) theory by an annotated graph, the nodes/edges of which are the definitions, theorems and proofs and by navigating you see exactly which property/definition is used at which point.
I apologize for the "fuzziness" of the question but I feel a discussion about a sort of visual notation / graphical representation in mathematics could be of interest (perhaps the categorical viewpoint with its unifying force and its diagramms is already what can be achieved, but I think there could be other, complementary ways).
Does anybody know of attempts in this direction? Would you consider such a graphical representation of mathematical structures (in addition to the standard, more linear way of representing things) helpful for communication in research and/or in education?
Best Answer
One could consider UML as a kind of "front end" for category theory. For example, a basic database schema is just a category (and a more interesting database schema is a sketch).
So if UML is just a visual representation of category theory, then your question can be easily answered in the affirmative. Category theory is excellent at modeling all sorts of different math, and visually representing these models is surely useful.