Michael Trott and Eric Weisstein: Position Statement
It is common in the modern technological world for non-mathematicians such as physicists, biologists, engineers, and computer scientists to require and use mathematics during the course of their research and development. Because many excellent textbooks and numerous online resources exist which adequately expound many areas of modern mathematics, it is easier than ever before to acquire a practical working knowledge in these areas. Unfortunately, for portions of mathematics that are neither mainstream nor popular enough to warrant extensive exposition online or in modern and easily accessible books, it is still often necessary to resort to older (or even original) literature. Furthermore, because one of the unique features of mathematics is that its results are timeless, such older results are often still relevant and useful today.
Making use of mathematical results effectively requires three steps:
- (1) finding relevant literature citations;
- (2) obtaining printed or--ever more commonly--electronic copies of the literature; and
- (3) reading, understanding, and applying the actual result.
1) Non-mathematicans wanting to use mathematics typically undertake the step of identifying potentially relevant articles and books using Mathematical Reviews, Zentralblatt, or (increasingly) Google Research. With abstracts and summaries of the cataloged/indexed articles and books available, these tools allow a certain preselection. Of course, at this stage it is not yet known if the selected literature will contain the particular result of interest.
2) After identifying potentially relevant citations, the literature itself must be retrieved. While conventional libraries are the classical means of accessing the literature, more and more scientists now have access to electronic versions of modern works and papers. Unfortunately, obtaining older literature is often still a relatively complicated and time-consuming step, often involving inter-library exchange and the associated long wait times.
3) To understand the results (theorems, algorithms, etc.) contained in an article, a scientist must often effectively read the paper in its entirety from start to finish in order to locate and learn the notations and assumptions. Worse yet, because articles quote earlier articles, one often has to go back to the second step to retrieve these earlier articles in order to fully understand the work under consideration. This last step makes the process recursive, and often scientists do not have the time, patience, or resources to carry it out beyond one or two iterations.
It is obvious that the existence of World Heritage Digital Mathematics Library will represent a giant step forward for step 2. Despite great progress over the last two decades in the digitization and free access of the older mathematical literature, it is often still difficult to retrieve copies of all potentially relevant materials without a subscription service and/or dedicate library staff. Through the WHDML, scientists from all fields and all countries would have access to a vast array of mathematical results.
Steps 1 and 3 above require the existence and functioning of tools for search and computability. Furthermore, these two aspects are not independent since powerful search tools must carry out computations in order to find/select relevant material. It is certainly true that the last 30 years have shown a dramatic progress towards making parts of mathematics computable, with general computing systems (including computer algebra and generalizations such as Mathematica) becoming an integral ingredient of quantitative science. Additionally, automated theorem proving software (such as Theorema) continues to become ever more powerful, though the domains in which such software can operate usefully remain somewhat limited. Despite such progress, existing software is still not powerful enough to allow most of today's mathematical knowledge to be codified and semantically understood. While making all of mathematical knowledge available in a computable form is an ambitious goal, this goal cannot be realized within the near future as constructive implementation of most abstract mathematical knowledge would require substantial investment in human resources and software development (on the order of at least hundreds of person-years).
Fortunately, even without full computability, implementation of a certain level of computational look-up which would allow for a genuine semantic search of mathematical knowledge is feasible in the shorter term. Current collections of mathematical knowledge that allow for a certain amount of mathematical search include the NIST Digital Library of Mathematical Functions (http://dlmf.nist.gov) and the Wolfram Functions Site (http://functions.wolfram.com). Through proper metatags (say in form of partially computable structures) it should be possible to implement a powerful mathematical search in many areas of mathematics other than just the theory of special functions. Such search could encompass not just keywords and key mathematical structures that occur in a theorem/algorithm, but also conditions that make a theorem/algorithm applicable (such as certain convergence conditions or descriptions of geometric regions). Such a search would allow a user to quickly narrow down the information that the scientist is looking for.
The search for concrete quantitative information through an unstructured search is one of the key features of Wolfram|Alpha (http://www.wolframalpha.com). For non-mathematical queries, Wolfram|Alpha supports a flexible free-form natural language interface. Wolfram|Alpha also contains a fair bit of mathematical knowledge that can be queried either natural language, conventional math syntax, or a combination of the two. Simple examples include:
(In fairness, it should be noted that most of these results are based on querying of databases with some instantiation rather that ab initio computational queries of abstract data structures--at least within the current system.)
While such semantic encoding of key conditions and results of theorems still requires a fair bit of human curation and tagging, it can be done in an extensible syntactic manner that produces broader coverage with fewer person-years of labor than could be obtained if attempting to implement full computability. Within a current grant from the Sloan foundations, our group is currently working on a prototype to demonstrate the feasibility of this approach for a small subfield of mathematics of historical and practical interest, namely continued fractions.
