## Sunday, November 10, 2013

### On the ontological nature of incidence structures

Lets consider the following functional dependencies structure #{(#{} #{}) (#{0} #{0}) (#{0} #{1}) (#{1} #{0}) (#{1} #{1}) (#{0 1} #{0 1}) (#{0 1} #{0}) (#{0 1} #{1}) (#{0 1} #{}) (#{0} #{}) (#{1} #{})}. Should this structure be classified as a set, a relation over a set, or a relation over the power set of a set? Depending upon how this structure is classified it could have a size of 2, 4, or 11.

My recently line of thinking is that the underlying set of an incidence structure should be metadata and most ontological questions will be answered by asking about the overlying set itself which means that this set would still qualify as a transitive binary relation regardless of what it is defined over. I am not sure that this is the right approach but this seems to make sense for now.

For non-incidence structures like measure spaces, metric spaces, and rings there is no need to deal with the problem of different underlying sets. With this approach I believe I am on track to classifying most mathematical structures though I haven't exactly worked out exactly how to classify state spaces yet.