OntoMathPro Ontology – A hub for math knowledge
ontomathpro.org
ontomathpro.org
For me, one of the motivation it to be able to draw dependency graphs to understand each concept clearly.
However, I have a feeling that some of this was extracted automatically from Wikipedia and needs curation. For instance "Markovian chain" has, as a superclass, "Elements of probability theory". Hum no. "Markovian chain" is NOT a kind of "Elements of probability theory". You could argue it is an element of probability theory (which is a rather vague term), but then the plural would have to be dropped. I also haven't seen any example of multiple inheritance, which would be absolutely critical to describe most interesting mathematical objects.
In general, since this is using a description language, the semantic of inheritance seems like a weak choice to describe the often more subtle relationships that exist between mathematical objects.
1) Indeed, we consider visualization of graph dependencies in OntoMathPro as an important application for learning. Given sufficient coverage of relationships between concepts, it can provide a helpful context for any non-trivial term.
2) No, the ontology was constructed collaboratively and manually from scratch, and Wikipedia was just one of the used resources. BTW, overlapping between the math part of Wikipedia (or DBpedia as we think in terms of Linked Data) and OntoMathPro is saved in the mapping file (https://github.com/CLLKazan/OntoMathPro/blob/master/external...), which was extracted automatically afterwards.
3) Concerning "ElementS of Probability Theory", could you please provide class URI you are talking about? Because I can see only this relevant one: E2406 http://ontomathpro.org/ontology/E2406___599545262.html which has the proper name (without 's').
4) We do allow and have multiple inheritance. Please see E1892 Differential Equation, which is a sub-class of both E1891 Equation and E2688 Element of Differential Equations. I believe there are more subtle examples in the ontology (can't remember exactly for now).
5) About ontology engineering principles, if you are interested in, please peruse our research papers (especially, [2]), in which we elaborate our modeling principles.
6) I can't agree about 'weakness' of the chosen language. OWL 2 is quite expressive to provide non-trivial logical rules and properties. For examples, some of them are already in place: P5 'see also' property is transitive and symmetric. Surely, we can't describe the precise semantics of mathematics (we would have to have a more expressive language than mathematics itself according to Popper's methodology). But we don't need it to build fascinating applications atop of the existing ontology, as our work hopefully shows.
BTW, I'd suggest using our mailing list further to keep these valuable discussions in the proper place: https://groups.google.com/forum/#!forum/ontomathpro
Next, you can submit pull requests in GitHub with suggesting improvements, then we can discuss them there.
No offense to HN, it's just to make our life easier.
1) Classical (N. Burbaki's approach): Kantor's set theory and logic 2) Constructive: where we are standing on constructive (intuitionistic) logic 3) Univalent foundations of mathematics (a novel approach).
Even if we stick to the 1st approach only (as we did for the ontology), there are also many dualities (alternative definitions), if we apply, say, terminology from geometry or, alternatively, from set theory while describing the same math objects.
Anyway, I think the methodology, we are working on during this project, should clarify many such hidden aspects. And we expect that it will be valuable for the modern math theory itself. So, let's collaborate:)
Mathematics should primarily read like code, not (only) like prose.
Moreover, for the interested reader, I would suggest paying close attention to univalent foundations of mathematics (http://www.math.ias.edu/~vladimir/Site3/Univalent_Foundation...) introduced by Vladimir Voevodsky. It disrupts classical derivation of math results rooted in Kantor's set theory and logics, and provides a theoretical framework that is much more convenient for computerizing.
Unlike them, we follow the different, less theoretical and more pragmatic, approach.
Perhaps I'm overlooking the obvious or I'm just uninformed in this particular area.
However, I should warn that the ontology is a mature, but ongoing work (and it is meant to be, frankly speaking, since we follow the crowdsourcing methodology), and the quality of labels (especially, English) or coverage of fields of mathematics should be constantly improved.
Indeed, we suffer from the absence of a web UI that visualizes the ontology neatly. But this is due to the lack of such tools for any large RDF graphs.