42. What's the Type of an Ontological Mismatch?

Lately I've been thinking about ontological mismatches a lot - even more than I usually might. This is because it's a key aspect of the grant I'm on, but also because "different ways of cutting up and understanding the world lead unavoidably to different ways of taking actions in the world" keeps coming up as a principle both in my side explorations and my personal life. So the topic weighs heavily on my mind.

If we want to understand something so strange, so precise, so important, and so abstract, one way of beginning the search might be to get a better handle on its type. In other words: let's say we have some algorithm we could run or some machine we could employ that would tell us how different some pair of ontologies is. What would information would it need in order to do its job, and what would the output look like?

Tackling the easier one first, let's assume that the output is a (nonnegative) real number: we then have the form so far of O_1 × O_2 → ℝ, where O_1 and O_2 are provisional types corresponding to whatever we're feeding in. For a next step, it might help us to recognize that the two things we're feeding into this algorithm have to, themselves, share a type - that of an ontology, so maybe more like O × O → ℝ - that is, the type signature of a metric, more or less.

But where do ontologies live? What are they made of, and how do they taste? For an initial guess, we might ask ourselves what purpose an ontology serves. An ontology has to encompass some way of splitting up some ambient space of things that might exist as populated by things that do exist, and giving some additional information about the splitting. Two possibilities present themselves: ontologies as labelled partitions of an ambient thingspace T, and ontologies as labelled topologies on T. The former has the property of guaranteeing that everything in the thingspace falls into precisely one ontological category, and the latter has the property of permitting much richer and more general structures of classification, as well as clearly belonging to a known mathematical object - namely 2^T; any or all or none of that might be desirable. And of course we might have somewhat more exotic possibilities, like a self-similar nesting of partitions all the way down, or enforcing some niceness guarantees on the topology, like compactness or Hausdorffness. In either case, the labels on the substructure - the partitions, or the open sets - are of vital importance; even if we have two ontologies with identical structure over the same thingspace, and we assume that whatever's using the ontologies can agree on what labels exist and what they mean, they can still clash on what labels they give to corresponding clusters in the thingspace.

One question that might immediately arise is: "should the function taking two ontologies to their difference be symmetric, or no?" While "yes" seems like the obvious right answer here, it makes more sense to tread carefully, given the wide applicability of the KL-divergence - an asymmetric measure on pairs of probability distributions over spaces. I thus have no strong answer to this question yet, but suspect that given that one ontology can be strictly finer or coarser than another, it might pay for the function to be an asymmetric one. This whole question is part of the larger question of whether some distinct pair of ontologies ought to be accorded distance 0, making it a pseudometric; similarly, the question of whether the measurement ought to be real-valued at all, and not something more complicated like a list of subsets by measure.

For a more pressing question, what are we to do with an unwitnessed ontology mismatch? By this I mean to spotlight the fact that it's not just the ambient thingspace T that we care about, but rather something like ΔT - that is, probability distributions over T - as given by the actual occurrence of possible things from thingspace in the world that whatever's using each of the two ontologies live in. If two ontologies clash in a thingspace and there's no actual thing around to inhabit the disagreement, does the clash make a sound? This may seem like a moot question, but consider that if we widen the thingspace from T to some T', two ontologies with different organizing principles that looked identical on T might actually have wildly different ways of dividing up T'\T; further, this move from T to T' is equivalent to having been in T' the whole time but having T'\T inhabited only after some special moment. This implies that whatever measure of ontological mismatch we settle on should be sensitive to what objects even exist! 

So maybe the type isn't O × O → ℝ but rather O × O × ΔT → ℝ, or equivalently that it's O × O → ℝ but controlled or parametrized or measure-weighted by some element of ΔT. Of course, these are the same up to currying and decurrying. But either way, we might justly be struck by what a type analysis has given us - that it seems most natural for an ontological mismatch to be judged at least partially on whether that mismatch is witnessed, and thus on what possible things actually exist.