Basics of the ontology of mathematics

Our mathematical ontology will start with some structures that are distributive lattice ordered like sets, multisets, and magnitude numbers. We can group the notions of sets and multisets together as unordered collections. Sets can be considered special classes of multisets, or at least unordered collections, whose elements are all distinct. An ontology of multisets can be constructed which includes sets as a member as well as other classes like equal multisets, whose members are all the same, singular multisets who have only one element, unique multisets, order two multisets, powerful multisets, regular multisets, and so on. Then classes of sets are special elements of this ontology.

Likewise, an ontology of magnitude numbers could be constructed. The most obvious thing to include within this ontology is the rational numbers. Then different computable subclasses of the rational numbers can be included hierarchically, and there is a whole theory of classes of magnitude numbers based upon total order theory which I can talk about more later. Scattered sets of rational numbers include integers, natural numbers, negative integers, unit fractions, half integers, and so on. Dense sets of rational numbers include intervals like the set of rational numbers from zero to one.

Nested types of this structures can be constructed to get sets of sets, sets of multisets, multisets of sets, and multisets of multisets as well as multisets of numbers and sets of numbers. Additive partitions for example can be classified as multisets of positive integers. Then additive divisions can be classified as the intersection of additive partitions and equal multisets. One thing that is noticeably missing from this ontology is the notion of an ordered pair or a list. Based upon our multiset theoretic foundation, lists will be classified as structured multisets.



Anything that is a structure other then an unordered collection like a set or a multiset is a structural multiset of some kind. It follows that relations are sets of structural multisets and particularly sets of ordered multisets. Given a relation we can also produce some multiset of multisets from the relation by getting the multiset of underlying multisets of the structures in the relation. This is only a set of multisets in the special case in which the relation is fully antisymmetric. This is a greater degree of distinctiveness then simply being a relation rather then a multirelation.

The classification of different types of lists and sequences will proceed from the fact that they are ordered multisets. Each different type of multiset will produce a different type of ordered multiset corresponding to it. An ordered additive paritition for example is often called a composition. An ordered family is a distinct list of sets and an ordered multifamily is any list of sets. Different classes of ordered multisets emerge from different types of relations, but of course, there are also special types of sequences based upon their own definition. Monotone sequences for example are a special type of sequence whose classification is not based upon the underlying multiset. This produces an ontology of sets, multisets, lists, and magnitude numbers.

Eventually relations and set systems can be used to construct the different types of structured sets like rings, fields, topologies, metrics, measures, graphs, hypegraphs, categories, pointed sets, partially ordered sets, and so on. Elements can include any of the elements of these different abstract structures, like complex numbers as elements of a particular field. This would lead to a much more comprehensive ontology of mathematics. Notice that here I focused mainly on numbers, sets, multisets, and lists and nested collections of them. That is because these are the basic structures needed to create symbolic expressions. The only thing missing is text related structures like characters, strings, and symbols and then we can describe all the data structures used in symbolic expressions. This is necessary because ultimately a significant portion of the structures encountered like polynomials will be defined symbolically rather then literally.