5 ms·

OntoMathPro Ontology – A hub for math knowledge

59 points by nzhiltsov 12y ago

Ih8registering2 12y ago
Stuff like this reminds me of Doug Lenat's AM (automatic mathemetician) from back in the day, written in everyone's favorite, Lisp.
- nzhiltsov 12y ago
  To be precise, AM is related to so-called 'computer mathematics'. In this project, we do not aim at modelling mathematics for theorem provers or similar things. Our applications (e.g. keyword-based formula search, education, information extraction) are introduced in the papers.
murbard2 12y ago
I'm really glad this is happening. I looked for such a thing a few months ago and couldn't find it. This morning, the authors mentioned it on my math.stackexchange question. 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.
- nzhiltsov 12y ago
  Hi, nice to here. I'm a co-author. Let me clarify some of your points. 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.links.dbpedia.nt 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 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 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.
  - murbard2 12y ago
    3) My bad, I don't know why I read it with an S... strange. 6) No, OWL is very powerful, my point is about the choice of vocabulary... For instance, a markov chain is not an "element of probability theory" in the same way as it is a "probabilistic model" or a "stochastic process".
    - nzhiltsov 12y ago
      Well, your suggestion? Could you pick up the right parent element in the ontology?
      - vdimarco 12y ago
        As I understand it, this is a good example highlighting one major challenge for Linked Data: There can be many ways to describe a concept.
        nzhiltsov 12y ago
        I would say this is by no means a flaw of Linked Data (the Semantic Web approach). There is an essential duality in many domains, including mathematics. For example, we can adhere different approaches to describe the math theory as a whole from the following points of view: 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:)
hackuser 12y ago
Applying semantic web technologies to mathematics sounds interesting, but this project otherwise is mysterious to me. It would be much appreciated if someone would describe what this is, what purposes it is designed to serve, what its scope of knowledge is and for what audience (e.g., all mathematics? for graduate-level users?). Is it ready to use or an experiment, and what tools do I need to utilize it? (Some background on the web of data would be interesting too.) Perhaps I'm overlooking the obvious or I'm just uninformed in this particular area.
- nzhiltsov 12y ago
  All right, I think we shall answer all these fair questions on the website. For motivation, please feel free to peruse our research papers that are publicly available from the website. As we note, [1-2] addresses to the Semantic Web community, [3] explains in terms that are likely close to mathematicians.
- nzhiltsov 12y ago
  Concerning scope, we focus on professional-level mathematics (Pro suffix emphasizes that). It is ready to use, and we describe a formula search mashup (https://github.com/CLLKazan/MathSearch https://github.com/CLLKazan/MathSearch) built atop of the ontology. On GitHub, you may find references to a published RDF dataset, which was extracted automatically from a test collection of our university's math journal. 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.
- nzhiltsov 12y ago
  For now, we measure coverage on our test collection of scholarly papers from our university's journal, and it is sufficient only for it. However, there are concepts from the following fields of mathematics: geometry (pretty advanced, believe me), analysis, number theory, set theory, algebra, logics, discrete mathematics, differential equations, numerical analysis, theory of computation, probability theory and statistics.
- 12y ago
amelius 12y ago
This is really cool, but I was hoping that they were going to formalize mathematics. For example, like this ([1]) team is doing. Mathematics should primarily read like code, not (only) like prose. [1] http://www.cs.ru.nl/~herman/FormMath.html http://www.cs.ru.nl/~herman/FormMath.html
- nzhiltsov 12y ago
  Yep, there is a lot of cool stuff on formalizing maths rigorously, called informally 'computer mathematics'. We give a brief overview of this work in our papers. Moreover, for the interested reader, I would suggest paying close attention to univalent foundations of mathematics (http://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations.html 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.

59 points by nzhiltsov 12y ago

Stuff like this reminds me of Doug Lenat's AM (automatic mathemetician) from back in the day, written in everyone's favorite, Lisp.

nzhiltsov 12y ago
To be precise, AM is related to so-called 'computer mathematics'. In this project, we do not aim at modelling mathematics for theorem provers or similar things. Our applications (e.g. keyword-based formula search, education, information extraction) are introduced in the papers.

murbard2 12y ago

I'm really glad this is happening. I looked for such a thing a few months ago and couldn't find it. This morning, the authors mentioned it on my math.stackexchange question. 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.

nzhiltsov 12y ago
Hi, nice to here. I'm a co-author. Let me clarify some of your points. 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.links.dbpedia.nt 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 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 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.
- murbard2 12y ago
  3) My bad, I don't know why I read it with an S... strange. 6) No, OWL is very powerful, my point is about the choice of vocabulary... For instance, a markov chain is not an "element of probability theory" in the same way as it is a "probabilistic model" or a "stochastic process".
  - nzhiltsov 12y ago
    Well, your suggestion? Could you pick up the right parent element in the ontology?
    - vdimarco 12y ago
      As I understand it, this is a good example highlighting one major challenge for Linked Data: There can be many ways to describe a concept.
      - nzhiltsov 12y ago
        I would say this is by no means a flaw of Linked Data (the Semantic Web approach). There is an essential duality in many domains, including mathematics. For example, we can adhere different approaches to describe the math theory as a whole from the following points of view: 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:)

hackuser 12y ago

Applying semantic web technologies to mathematics sounds interesting, but this project otherwise is mysterious to me. It would be much appreciated if someone would describe what this is, what purposes it is designed to serve, what its scope of knowledge is and for what audience (e.g., all mathematics? for graduate-level users?). Is it ready to use or an experiment, and what tools do I need to utilize it? (Some background on the web of data would be interesting too.) Perhaps I'm overlooking the obvious or I'm just uninformed in this particular area.

nzhiltsov 12y ago
All right, I think we shall answer all these fair questions on the website. For motivation, please feel free to peruse our research papers that are publicly available from the website. As we note, [1-2] addresses to the Semantic Web community, [3] explains in terms that are likely close to mathematicians.
nzhiltsov 12y ago
Concerning scope, we focus on professional-level mathematics (Pro suffix emphasizes that). It is ready to use, and we describe a formula search mashup (https://github.com/CLLKazan/MathSearch https://github.com/CLLKazan/MathSearch) built atop of the ontology. On GitHub, you may find references to a published RDF dataset, which was extracted automatically from a test collection of our university's math journal. 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.
nzhiltsov 12y ago
For now, we measure coverage on our test collection of scholarly papers from our university's journal, and it is sufficient only for it. However, there are concepts from the following fields of mathematics: geometry (pretty advanced, believe me), analysis, number theory, set theory, algebra, logics, discrete mathematics, differential equations, numerical analysis, theory of computation, probability theory and statistics.
12y ago

amelius 12y ago

This is really cool, but I was hoping that they were going to formalize mathematics. For example, like this ([1]) team is doing. Mathematics should primarily read like code, not (only) like prose. [1] http://www.cs.ru.nl/~herman/FormMath.html http://www.cs.ru.nl/~herman/FormMath.html

nzhiltsov 12y ago
Yep, there is a lot of cool stuff on formalizing maths rigorously, called informally 'computer mathematics'. We give a brief overview of this work in our papers. Moreover, for the interested reader, I would suggest paying close attention to univalent foundations of mathematics (http://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations.html 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.