FP6

Overview of Mathematical Knowledge Management Network

Mathematics is one of the oldest areas of knowledge, and Euclid's Elements is one of the oldest encapsulations of mathematical knowledge. Nevertheless, the issue of the management of this ever-growing corpus of knowledge has attracted little attention, even though several groups of people have become convinced of its necessity. The primary objective of this network is to bring together the communities who could benefit from improved management of mathematical knowledge. These communities include: producers of new mathematics whether in academia or industry; teachers of and trainers in mathematics (again in academic or commercial settings); users of mathematics working in other disciplines such as economics, science and technology, or social sciences. This cross-disciplinary input will be used to articulate current and future needs in mathematical knowledge management, and thereby to articulating suitable projects for Framework 6.

Hence the objectives of this Network are:

to analyse the various socio-economic needs of the stakeholders in Mathematical Knowledge Management;
to analyse the scope to which migration tools could let the community move from the current paper-oriented and presentation-oriented view, e.g. LaTeX, to a semantics-oriented view of the universe of mathematical knowledge;
to analyse the current state of formal mathematical tools, their rôle in mathematical knowledge management, and the extent to which they could inter-operate in a mathematical knowledge management framework;
to analyse the requirements from users for a search engine that could consider mathematics, rather than words about mathematics, e.g. ``Bessel's equation'', and also distinguish content from presentation, so that for example the integral of sin(x) over the range [0,1] matches the integral of sin(t) over the range [0,1];
to analyse the various notations present, both nationality-specific and subject-specific, in mathematical presentation, and to relate these back to a common semantic base;

all with a view to having concrete research and development challenges and project(s) to propose under the auspices of the EU's Framework 6 plan.

One could ask ``why does mathematical knowledge need to be managed''? There are a variety of answers to this question.

Mathematical Knowledge is important to industry and to society. The invention of public-key cryptography showed that no part of mathematics was inapplicable. At first these applications used Number Theory, Gauß's ``Queen of Mathematics'', but now hyper-elliptic curves are proposed as the basis of crypto-systems, and companies have been established to exploit elliptic curves, lattices and doubtless other parts of ``pure'' mathematics.
Mathematical Knowledge is far too vast to be understood by one person. Odlyzko [1] estimated that the total amount of mathematics that has been published doubles every ten-fifteen years.
Mathematicians and users of mathematics are changing the way they deal with knowledge. As the distinguished experimental mathematician Peter Borwein said recently [2] ``I stopped reading mathematics journals some years ago. ... MathSciNet replaced browsing''. However, MathSciNet does not cover many uses of mathematics and mathematical notation, nor does it support searching by mathematical content. Equally, many people's first action on looking for knowledge is to try a search engine.
The consequences of not managing knowledge can be expensive. The Ariane-5 launch disaster, was due to a faulty piece of code which coerced a floating-point variable into a short integer. This coercion was valid on Ariane-4, but not with the new parameters of Ariane-5. The formula showing that the coercion was valid for Ariane-4 (and which, given the Ariane-5 parameters, would have given ``false'') was present in the comments of the code. So the mathematical knowledge was there. Unfortunately it was not explicit, and it was not managed.
Mathematical Knowledge needs to be managed for the whole range of e-Education initiatives in mathematical education.
The changes in Mathematical Knowledge need to be managed. Papers, and even respected reference works such as Abramowitz & Stegun [3] contain errors, and so errata are published. It is currently very hard in the paper-oriented world for the reader to discover the existence of these errata.

One could also ask ``why does the issue of Mathematical Knowledge Management need to be researched -- is it not just a subset of Knowledge Management in general''?

Mathematical Knowledge is difficult to manage: even the simple case of a database of formulae is a non-trivial task. 33 years ago, a survey [4] showed that error rates in major published integral tables ranged up to 26%.
Mathematical Knowledge is inherently different from its printed representation. Very different printed representations can mean the same thing and, conversely, the same printed representation can mean different things in different contexts. Nevertheless, and unlike many other areas of knowledge, there are fundamental semantics which can be applied, and, at a deep rather than a presentational level, there is formalisable knowledge which can be managed.
Mathematical Knowledge comes in many varieties. Some formulae are statements of theorems in a totally formal system with a machine-verified proof, some are, or more likely occur in, the statements of theorems in an informal system with an informal proof, others occur in conjectures, or as hypotheses, or with other rôles. Even when one is dealing with the first category, which might be thought to be very accessible to computerised knowledge management, one has to ask of the proof ``in which logic?''.
There are also advantages to treating Mathematical Knowledge Management as a special case. Mathematics is a field that is well structured and has a good classification scheme [5]. Some members of the consortium also have an interest in the representation of legal knowledge, and believe that Mathematical Knowledge Management might have something to contribute to this area in particular, as well as the more general spin-offs from this research.

Before one can talk about ``Knowledge Management'', one has to talk about ``Knowledge Representation''. Fortunately, in this respect Mathematical Knowledge Management is not starting from zero. OpenMath is a standard for representing mathematical formulae, allowing them to be exchanged between computer programs, stored in databases, or published on the worldwide web. While the original designers were mainly developers of computer algebra systems, it is now attracting interest from other areas of scientific computation and from many publishers of electronic documents with a significant mathematical content. There is significant representation from the OpenMath community in this consortium.

Unfortunately, not all items of mathematical knowledge are simple formulae: a typical theorem of calculus might be ``If (a_n) and (b_n) are two convergent sequences, then (a_n + b_n) is also a convergent sequence.'' This can only be converted into a single formula by unpacking all the definitions, which would lead to massive growth in the formulae, as well as unreadability, as one tackled more complicated enunciations. The OMDoc mechanism is an attempt to extend OpenMath to cover these sorts of concepts and contexts. It is clear that more work will need to be done on Knowledge Representation.

Ever since the pioneering AUTOMATH [6] project, there have been significant projects aimed at automating or semi-automating mathematical knowledge and proof. Some of this has led to theorem-provers, such as COQ, Isabelle, THEOREMA, VSE and Omega, others to mathematical representations such as MIZAR, also represented in the consortium.

References

Odlyzko,A.M., Tragic loss or good riddance? The impending demise of traditional scholarly journals. Intern. J. Human-Computer Studies 42 (1995) pp. 71-122. A condensed version appeared in Notices Amer. Math. Soc. 42 (1995) pp. 49-53, reprinted in Deutsche Math. Ver. Mitteilungen, no. 1, 1995, pp. 19-24. Reprinted in Electronic Publishing Scholarly Publishing: The Electronic Frontier (ed. R.P. Peek & G.B. Newby), M.I.T. Press, 1995, pp. 91-101.
Times Higher Education Supplement 1514 (23 November 2001).
Abramowitz,M. & Stegun,I., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. US Government Printing Office, 1964. 10th Printing December 1972.
Klerer, M. & Grossman,F., Error Rates in Tables of Indefinite Integrals, Industrial Math. 18 (1968) pp. 31-62.
Mathematics Subject Classification: a joint effort between Mathematical Reviews and Zentralblatt für Mathematik.
de Bruijn,N.G., The mathematical language AUTOMATH, its usage and some of its extensions. In: Proc. Symp. Automated Deduction, Springer Lecture Notes in Mathematics 125, Springer-Verlag, 1968.

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