The STEM sciences – Science, Technology, Engineering and Mathematics – are moving to bigger and bigger
developments, ultimately facing problems such as the one-brain-barrier [
In the following I will use mathematics as a working example. Of the STEM sciences it is one with the most well-understood structure. I expect that the observations and results will generalize to all STEM sciences.
We observe the following aspects of mathematical practices: (i) taking
existing mathematical knowledge and using it to make inferences ; (ii) guid- Tabulation
ing a reader by writing narrating texts about a specific topic; (iii)
generating a large set of information – such as return values of a particularly Ontology
interesting function – and tabulating it inside some form of database; and Narration Inference
(iv) making computations. We see these four as forming central aspects
of mathematics. [
To support each of these practices, different kinds of computer-provided mathematical services are required. Services commonly work on objects ranging from low-level mathematical objects, such as numbers, via complex objects, such as specific mathematical groups, to objects on a higher meta-level, such as mathematical computations. Any particular service might treat specific objects as blackboxes – knowing that it is an object but not knowing any of its’ internal structure – or as a white box – being fully aware of the semantics of the objects and being able to infer all its’ properties. We call services treating objects as the former shallow mathematical services and those as the latter deep mathematical services.
Deep services also have to take context into account. For example, in certain cases an integer might only be an integer, but in other cases the same integer might represent the value of a specific property of an object. This leads to the orthogonal distinction between local services – those taking only the current document into account – and global services – those taking an entire ontology of background knowledge or corpus of documents into account.
Consider on one hand a service that takes as input a simple mathematical term within a document, and then navigates to the document defining this term. This service is global (because it takes the corpus as a whole into account) and shallow (because it does not rely on the internal structure of the object). Consider on the other hand a service that converts all physical quantity expressions (e.g. 10 khm ) from the imperial system to the metric system. This service is local (because it only works on a particular expression) and deep (because for each quantity expression it takes into account its’ internal structure consisting of value and units).
Services that are shallow and local are easy to implement – requiring only application of standard software engineering practices to quickly be usable on a researchers machine – whereas deep and global services require more infrastrucure. To be able to support these more complex services, a highly semantic, highly structured ontology is needed to bind the different mathematical aspects together.
The classification of aspects of ‘doing mathematics’ and – based on that – the services sketched above give Copyright © by the paper’s authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). a plausible, albeit theoretical, model for designing a system providing computer support for mathematics. To validate this model, I want to build a concrete scalable implementation. This leads to the following research questions: RQ1 Can an information system provide infrastructure to jointly support all four tetrapodal aspects of mathematics? RQ2 Is it possible to support community-based workflows for authoring, maintaining, curating, and visualization of tetrapodal mathematics?
A positive answer to question RQ1 – backed by an implementation – will demonstrate the feasibility of our model. The aim of Question RQ2 is to guide any implementation towards practical applicability. In particular, it is insufficient to build a system that is merely applicable to a specific subset of documents; instead we want to aim to make it broadly accessible to the community of mathematics as a whole. We expect these particular workflows to cause a need in an implementation to support both deep and global services.
Unlike the above, the answer to Question RQ3 is not yet clear. At first glance, one would expect any such ontology to be fully formal, defining every part of mathematics in great detail. However, it might be possible to re-write mathematical services to not need a complete formalization of every object in question. As an alternative a flexi-formalized ontology – defining only core concepts and providing human-readable descriptions for the rest – might be sufficient1.
Another possible requirement on the ontology might be to ensure that it is not system specific. In particular,
it is possible for it to remain findable and accessible by other researchers, to further system interoperability?
These desired properties are also known as FAIR [
I want to base my system on MathHub, a portal for active mathematical documents, an archive for flexiformal
mathematics, and a frontend to mathematical databases initially conceived by Michael Kohlhase, Florian Rabe
and others in [
MathHub was initially implemented by Mihnea Iancu and others during the scope of his PhD [
I want to base my work to answer the research questions on MathHub by developing the system further. On a technical level, I want to overcome the two limitations above, and then focus on the following workflow-oriented work packages that concretly instantiate the research questions: WP1 Which tetrapodal Active Document Services can be developed and efficiently integrated into MathHub that make the system more user-friendly? WP2 Which processes are efficient at keeping MathHub Content up-to-date and simultaneously scale to large amounts of data?
Drupal.org – Community plumbing. url: http://drupal.org (visited on 02/14/2015).
1There are already some ideas how such an ontology might look like in detail. We refer the interested reader to [