A Repository of Mathematical Examples and Counterexamples

Abstract

The World Wide Web contains a huge amount of mathematical resources. Many of these are documents that reproduce electronically books, articles, lecture notes, etc. By using the Web one can access such documents more quickly than the corresponding paper versions. The Web contains also non classical mathematical resources, such as applets, on-line calculators etc. that do not have paper analogue and which allow new effective ``mathematical experiences''.

Our project explores the possibilities of using the web for selecting and glueing both classical and non classical mathematical minimal documents ("atoms") in dinamically generated complex documents. This is technically possible by

i) archiving atomic documents in XML format, validated against appropriate DTD's which specify, among other things, their glueing interface.
ii) selecting arrays of XML documents, resulting from certain queries, parsing them, and piping SAX events. iii) applying XSL to transform SAX events to PDF.

Following theses ideas, we have already realized a dynamic repository of exercises and multiple choice quizzes, expecially designed for Teaching purposes, as described in [Enrico Rogora and Paolo Roselli Archiving, delivering and reusing Electronic Teaching Resources] and [Enrico Rogora and Andrea Sterbini Multiple Choice Quiz in a web based teaching environment] and [http://didnet.mat.uniroma1.it/] . In this paper we discuss the design principles for building a repository of mathematical examples and counterexamples with all services for producing, storing, delivering, updating and combining these "mathematical atoms". This project addresses different issues than the previous one; the diffusion and use of mathematical knowledge among a scientific community. In particular, it aims at providing a broad set of examples and counterexamples to be screened when searching for hints to solve, through mathematics, problems related to biology, chemistry, computer science, etc. It will also be possible to access a broad set of typical errors and misunderstanding, carefully discussed, in order to prevent from misusing standard mathematical tools.

A central issue of this project is the design of detailed indexes, organized by topics, examples, and application fields (biology, chemistry, computer science, etc.). An important feature of these indexes is the fact that thay can be updated by users themselves, which are alloweded to point out new relations among `atoms', and to enrich the repository with their observations.

A project like this can be effective only if the whole scientific community is allowded to contribute, (by adding new examples, appending remarks and pointing at new relations). Contributions can only be given through the web, giving rise to a new kind of scientific activity, totally different from the production of books and articles.