Well, intersection is just a disambiguation page, but the categories for intersection (set theory) are good starting points for queries. Maybe I want to look at all concepts that are in set theory and calculus. Or maybe I want to see all mathematical concepts except for those in set theory and then sort them by the date of the first publication that described them. I'd argue that these are potentially common scenarios that we want to make easier for everyone.
I agree that there is no single perfect/master/universal ontology. Sure, our minds are all rooted around our perceptions. If I say "dad" it conjures different images in each of our heads. But if I say "Tom Cruise" our mental images are much more similar. So there are large portions of our internal ontologies and mental representations that we share, which are generally what we put in an encyclopedia for our culture. Cultural differences are certainly fascinating and often follow linguistic barriers. The Pirahã people don't have numbers, just one, two, and many, and their language can be whistled. In standard psychology, a typical hurdle for self-awareness in children and animals is the ability to find a spot painted on one's head by using a mirror. To try and imagine an extreme, if I was the first bear to use Wikipedia I might want to make things that can and cannot be eaten as the fundamental categories. Who knows? You can certainly view the current category system as a graph with only one type of edge (a is a member of b, or equivalently, b contains a). Having loops just means, for example, that the graph can't be a tree, which isn't inherently bad. It just means that you have to alter the definition of some concepts, like root, to fit a broader variety of possible structures.
> From: paul@ontology2.com
> To: wikidata-l@lists.wikimedia.org
> Date: Tue, 7 May 2013 13:50:10 +0000
> Subject: Re: [Wikidata-l] Question about wikipedia categories.
>
> Statistical methods can deal with black swans, but you've got to get
> away from normal distributions and also model the risk that your model is
> wrong.
>
> Since training sets come from the same place sausage comes from,
> training sets in machine learning rarely teach the algorithm the correct
> prior distribution of the class. Punch a new prior into the system and it
> will perform much better.
>
> Some kinds of sampling biases can be somewhat overcome. Involvement of
> multiple people smoothes out individual bias. (Kurzweil's project of
> stealing a human soul with a neural network is already being scoops by
> projects that are stealing statistical models of many souls.)
>
> Language zone Wikipedias are obviously biased towards the viewpoint of
> people in that language zone. Mostly that's a good thing, because a
> Chinese knowledge base that reflected an Anglophone bias would seem
> unnatural to Chinese speakers.
>
> And that's the point. Useful systems don't "eliminate bias" but they
> are given the bias that they need in order to do their job.
>
> I agree categories are most useful when they are the categories you
> need. The toolbox above can help you estimate these with precision so high
> that it's difficult to measure.
>
> Arnold S isn't the best case for categories because humans,
> bodybuilders, places, chemicals and such are well ontologized. Look at
> the collection that comes up for the word "Intersection",
>
> http://en.wikipedia.org/wiki/Intersection
>
> Most of these are connected to the larger mass through just a few
> categories that would be hard to express as restriction types. Wikipedia
> is reasonable to require concepts to have a category because really, if you
> want to assert something exists and can't find some category that this thing
> is a member of, I wouldn't be so sure that this thing exists.
>
> I'm not sure if there is anything I can't do with the current situation,
> but bear in mind that I'm going to look at DBpedia, Wikidata and Freebase
> facts too and be willing to do data cleaning processing and hand cleaning of
> results that I cannot accept. It's a tricky and somewhat expensive process
> (though it's cheaper than conventional ontology construction), so cleaner
> data makes this process cheaper and quicker and available to more end users
> personalized to their own needs to define the categories they need.
>
>
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