As it is well known, Linked Opend Data (LOD) and computational ontologies have great success in the fields of Life Sciences (Biology, Medicine, etc). See e.g. the big LS-cluster at .
However, I wonder why mathematics are – in comparison – covered only sparsely by ontologies or LOD.
Indicators (to the best of my current knowledge):
- https://lod-cloud.net/datasets?search=mathematics retrieves only one result. This links to http://msc2010.org/mscwork/ which seems outdated and contains several broken (404) links.
- Since http://ksl-web.stanford.edu/knowledge-sharing/papers/engmath.html (Gruber and Olsen) there seems to be no attempt for ontological modelling of mathematics as a whole (or at least a significant portion of it).
- https://www.ebi.ac.uk/ols/ontologies gives only one hit for "math" (in browser search)
- there is no "math*" tag on https://lov.linkeddata.es/dataset/lov/vocabs?&tag_limit=0 (but there is "biology" or "geography" or "geometry")
Probably there is some (machine-processable) formalization of mathematical knowledge but it seems almost disconnected from the "semantic web" and LOD-bubble.
Questions:
Very interesting question.
It is the community of experts in a field of knowledge that makes an effort to use semantic web technologies to improve the interpretation of knowledge in their field, particularly in web pages or quite simply to create a common vocabulary ith broad consensus to avoid any ambiguity in the communication, these datasets are interpretable by the machines and can be enriched for the inference of new facts.
I think that the question should be asked differently, why mathematicians do not use semantic web technologies in their work or their projects? An expert in this field could enlighten us.
Alternatively, maybe see it as a niche to tap into.
Perhaps a cultural issue for the math discipline as it happens in others.
We have developed recently a foundational ontology (ThingFO) in the framework of a 4-layered ontological architecture (here is the article for both Conference Paper Analyzing the Usefulness of ThingFO as a Foundational Ontolo...
), which can be useful for any science. From ThingFO, we have developed core, top-domain and low-domain ontologies.I guess the point is not just the semantic web and the semantic processing, but first the big issue of the knowledge representation of the scientific world, which can be useful for mathematics, informatics, medicine, etc., etc.
About 10 years ago a colleague of mine did some research in the direction of integrating mathematics and the semantic web in the course of his PhD. You may have a look at his thesis:
https://opus.jacobs-university.de/frontdoor/index/index/docId/434
Or a more dedicated paper regarding your question:
https://www.academia.edu/2811153/Ontologies_and_languages_for_representing_mathematical_knowledge_on_the_Semantic_Web
My stance on the question is that description logics (as logical foundation of formal ontologies) is relatively poor in expressiveness as compared to what can be formalized in mathematics with more powerful logics. You may have a look at research on formal methods in this regard - for instance here: http://hets.eu/
Yet, this might not be what you are after, since it requires more knowledge in formal methods than an average consumer of linked data may have. May be the question is: What would be your intention with mathematics as linked data?
Thank you all for your views and contributions so far.
Luis Olsina Are you aware of any ontology which uses ThingFO and contains moderately abstract mathematical concepts like polynomial or complex number or anything similar? Also what about the other foundational ontologies? I only looked into BFO up to now but it seems not to be targeted towards such abstract subjects.
Immanuel Normann Thanks for the hint to the works of your colleague and also to hets.eu. I agree that the formal methods (and tools) used there are much more niche knowledge than OWL, SPARQL etc.
In terms of your question regarding my "intention with mathematics as linked data": On application could be to quickly and unambiguously refer to mathematical concepts in research papers. E.g the notion of a "covector field" (also called "1-form" or "differential form of degree 1") is sometimes used in control theory, but not so widespread that familiarity with it could be presupposed among all readers. For the sake of self-containedness its definition is often repeated. This could be prevented if there where an catalogue of mathematical concepts to which one could refer. There, a covector field would be (formally) defined using other (more elementary) concepts. Furthermore it would be possible to automatically identify methods or other concepts which built upon this one. In short: Accessible semantic formalization of mathematics could IMHO help to reduce ambiguity and comprehension hurdles, especially in application fields such as engineering.
Carsten Knoll What you are looking for was 10-15 years ago to some extend subject of research in the field of so called "Mathematical Knowledge Management" (with its own conferences). In fact I considered as part of my doctorial research to make mathematical knowledge retrievable by formalizing it in description logics. My approach at that time would be to map concepts from large mathematical knowledge bases like Mizar to concepts and relations expressed in OWL.
If you take "semantic formalization of mathematics" serious you should be at least aware of math knowledge libraries like Mizar (https://en.wikipedia.org/wiki/Mizar_system) and the vision of the QED manifesto (http://mizar.org/trybulec65/8.pdf). This may at least provide you some insights to the efforts in formal mathematics.
Perhaps, though, your notion of "semantic formalization of mathematics" is less strict and you rather intend to organize namings in thesaurus where e.g. "covector field", "1-form" and "differential form of degree 1" are labels for the same concept. In this case you rather may analyse math dictionaries e.g. Bronstein or the like to extract the key concepts and their most common labels.
Dear Carsten Knoll, I will answer your questions/concerns this week end.
Cheers, Luis
Dear Carsten Knoll. I developed ThingFO two y. ago, so I'm not aware so far that other researchers used it for abstract mathematical concepts like polynomial or complex number as you are looking for. My interest in ontologies started in early 2000. At that moment, we developed a process ontology, and an ontology for metrics and indicators. In the last years we have harmonized semantically those ontologies with ThingFO in the context of the four-layered ontological architecture. Obviously that for indirect metrics and elementary and derived indicators, we use mathematical constructs (formulas, aggregation models), but that is all. These ontologies are specified at the low-domain level.
For MetricsLDO Preprint MetricsLDO v2.0's Terms, Properties, Relationships and Axiom...
For IndicatorsLDO Preprint IndicatorsLDO v2.0's Terms, Properties, Relationships and Ax...
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