Wednesday, February 18, 2004

A touch of Kant 

Yesterday, I went to listen to a lecture on Immanuel Kant organised by the university of Twente . To my surprise it was very crowded and I had to sit on the floor. Kant died 200 years ago and there should be some attention to him over the year. from what I understood, his ideas about the knowability of the world are still supprisingly fresh.

One of his prime ideas is that the way we talk about the world is rooted in experience and the rules that govern reality. For example every material entity can only exist at a certain place and a certain time. However our mind can abstract from this in that it can live with the idea of a space and time (or post Einstein space-time) which has an existence independent of the entities that are in it. Once we have made this induction and having figured out the rules of the game, we can start imagining things that could happen, rather than things that did happen. Moreover since other people also share the experience of living in a space at some time we can at some point communicate soundly about such an abstract idealised concept, talking about its properties even if it only could or should have an instantiation, instead of actually existing.

Now if you try to do knowledge representation you are faced with a similar situation. You have to develop concepts to organise the world. The concepts have rules and relationships between them, which allows you to argus about them. You often want the relationships to model realworld relationships between instances of those concepts, but typically you would want to talk and argue about instances relationships that could rather than do exist.
The reason is that existence proofs are difficult (ask any mathematician), infeasible (go query every database in the world), or you are just preparing for the future. Therefore in knowledge representation you end up with having abstract notions, that need not have any instatiations, or whose precise instantiantions you don't care about.

If you read up on RDF and OWL you realise that the RDF and OWL classes are very much of the could exist categories rather than must exist. This is as it should be. Unfortunately the formal underpinning of those languages in the model and syntax is on the level of set theory. Superficially this makes sense. The rdfs:Class Car seems mathematically modeled on the set of all cars. Also the rdfs:Property numberOfWheels seems like a relation between the set of cars and the non negative integers. Well we certainly want to be able to talk about next years cars too, so it is more like all cars that could ever be or all things with an engine and four wheels that you can park in front of your house. Now we get into trouble. The rdfs:Class Car looks more and more like the set of all sets that gives you the Russell Paradox showing that there is no such thing as the set of sets. There seem to be a lot of restrictions on OWL (Description logic version) and RDF to avoid Russels Paradox, but still, there seems to be something conceptually fishy with modelling classes as sets and having everything in a fixed universe. Sets are equal if the instances are equal and this is just not a useful model for the classes that you model. They seem really more like well classes. Dont get me wrong though, a universe a fixed set of all "things" which also contain the subsets of all the sets in the universe is a sound thing, and it is used with virtuosity by no less than Grothendieck for example.

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© Copyright 2004-2006 Rogier Brussee.
These are my personal views and do not necessarily reflect those of my employer.