A formal ontology and knowledge base dedicated to mathematical concepts should be constructed. It is immediately evident that the foundation of this ontology should be set theory, but even more generally it should be based upon distributive lattice theory which is the theory of lattices whose elements are most like sets. All the foundational concepts of ontology like classes and relations are essentially set theoretical so ontology and set theory belong together. A major component of this ontology should the ontology of set systems.
An important part of mathematical ontology historically has been the construction of all the concepts of mathematics using set theory. This lead to things like defining ordered pairs as kuratowski pair set systems and defining numbers as Von Neumann ordinals. All of these different classes of sets themselves can be organized hierarchically into an ontology, but so far as I can tell no one else has done that. The best example of a project of this sort is graphclasses but it has a limited scope and isn't based upon set theory or multiset theory.
Mathematics requires a set theoretic ontology of categories of structures and their relations. Well the starting foundation will be sets, multisets, and related structures this ontology can be extended to include all kinds of structural sets and multisets as well as symbolic expressions like polynomials. In doing so, all mathematical constructs can be categorized into an ontology. I will describe how this can be done.
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