Encoding Mathematical Language

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Publishing a mathematical paper in a reputable journal is one way of making mathematical knowledge accessible. People whose libraries contain that particular journal (one among tens of thousands of journals) can read that paper, interpret the results and notation in terms of the problem under consideration, and implement any algorithms described. This of course leaves the work (and danger) of correctly inferring the intention behind the content to the reader, often an onerous, error-prone and inefficient task.

Surely it is better if the best algorithms and most useful mathematical knowledge are already implemented in a standard, widely available program primed for immediate use. At best, the mathematician user could safely use the program without knowing all the technical refinements that make the underlying computation possible---though in our view a basic knowledge on the part of the user is indispensable, providing confidence that the algorithms are not being totally umis-used. At worst, it provides a common basis or language for communication and exchange, one which has been carefully crafted for precision, accuracy and consistency, much like the traditional logical formalism behind conventional mathematics. This appears to be what is happening today. A large portion of mathematical knowledge is being encoded into computer algebra systems and into numerical packages for scientific computation. How many mathematicians would be able to write a program which could evaluate the Jacobian elliptic functions efficiently and accurately over the complex plane? But once it's done, then everyone can use it (assuming the right computer with the right software with the requisite license).

It is this encoding of mathematical knowledge that we find the most significant aspect of modern technology. The next most significant is the exchange or communication of that encoded knowledge, for example the act of asking on the Internet for help on a problem---by keyword search or simply a query in the right newsgroup---is also extremely important. There is a tendency to drown in student-type questions (it's considered unethical to ask for outside help on assignments, for example, but it has been known to happen on some newsgroups) but nonetheless an important amount of research communication happens on the Internet.

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omp@cecm.sfu.ca