Language that Adds Up

By Wray Herbert

Computer software currently in development will transform the way mathematics is presented and performed on the World Wide Web.

Illustration by Lung-I LoWilliam Coleman has a vision of how college chemistry will be taught in the future. In classrooms today, he explains, the fundamentals of introductory chemistry, inorganic chemistry, and physical chemistry—the courses he teaches to his Wellesley College students—are stored and distributed in textbooks, just like the seminal texts of 19th-century Russian literature, existential philosophy or American urban history. But unlike literary and historical texts, which are archived for analysis, he says, much of chemistry is not static.

By its very nature chemistry, and indeed any discipline that uses mathematics as its primary language, is meaningful only when it is manipulated. It requires motion. Coleman would not do away with chemistry textbooks; many facts about the behavior of molecules and electrons and muons can and must be explained in words. But he anticipates a day when the equations and illustrations of chemical behavior that make up much of chemistry and chemistry education are removed from textbooks and stored on the World Wide Web. That way, he hopes, students will come to see the chemical behavior of the world as a dynamic phenomenon. They will in an abstract sense be able to interact with the chemical world, manipulating variables such as temperature and the number of molecules, and see the chemical world in action.

That day is not today. As dynamic and interactive as the World Wide Web may seem when compared with print media (especially to those raised exclusively on print), it is really not dynamic at all; indeed, it is essentially a world of text, just displayed on a screen without the familiar conventions of paper and pages. When images and equations do appear on the World Wide Web, they are essentially artwork that has been technically “embedded” in the text. There is no possibility for manipulation or interaction. This is actually one of the great ironies of the Internet, experts note.

This now ubiquitous electronic medium routinely used by school kids today to research term papers, check cinema listings, and stay in touch with friends was originally created by scientists for scientists. Its purpose was to share scientific ideas more quickly and efficiently than through journals and meetings. Yet the pioneers of the Internet and later the designers of the World Wide Web never gave this powerful medium the capability to communicate in the lingua franca of the scientific world, mathematics. Even today, when scientists do communicate with one another over the Internet by e-mail, for example, they're communicating in words; equations may appear in specialized notation, but basically they are rendered in the symbols of written language.

All this may be changing, now or in the near future. Mathsoft Engineering & Education, Inc., a software and education company based in Cambridge, Massachusetts, has been developing a product called MathML (for “mathematics mark-up language”). Currently, the content of the World Wide Web uses a standard called HTML (for “hypertext markup language”), the invisible, technical specifications that allow web pages to appear as they do: It lays down the technical rules underlying such familiar conventions as capitalization and paragraphing. MathML would do the same thing for mathematics online. It would spell out in technical language such familiar arithmetical notations as:

+ - / =


as well as the concepts and notations underlying trigonometry, calculus, numerical analysis, and other branches of advanced mathematics.

MathML is fully supported by Mathsoft's software for applied mathematics, called Mathcad, and is the only standard so far to be approved by the World Wide Web Consortium (W3C), the group made up of representatives from industry and academia that is working to get agreement on a math standard for the Web. Mathcad and MathML—or some similar application—will one day transform the way mathematics is presented and performed online.

But it takes time for the market to sort itself out, especially the market for highly specialized math capabilities. For example MathML could go a long way toward making Professor Coleman's vision of future chemistry instruction a reality. But working chemists themselves will likely require more specialized applications that build on the basic power of MathML.

Indeed, Coleman has been conducting an online survey of working chemists to determine the kinds of specialized math they require for their work, and he's finding that chemists have very specialized mathematical needs. The same is true of genomics researchers, economists, and astrophysicists. Ultimately a complete and successful Web-based mathematics capability will have to address these specialized interests.

But for now, the first goal is to get the mathematics community, broadly defined, to agree on a standard for Web-based mathematics applications. MathML has a leg up on becoming the universal standard because of a partnership that Mathsoft has recently formed with IBM. Under the terms of the deal, Mathsoft becomes a “preferred partner” of the computer powerhouse, working to integrate Mathcad and MathML with IBM's technical browser, called Techexplorer Hypermedia Browser. Mathsoft will be licensing one million copies of the browser as part of the partnership. The integration of Mathcad and MathML with IBM's browser will not only improve the interaction between students and professors in traditional educational settings, it will also enhance the burgeoning area of distance learning, or “e-learning.” Says Mathsoft's CEO Chris Randles: “MathML not only makes possible new authoring tools for creating such content conveniently, but it also provides the means to deliver a dynamic, interactive math Web experience to virtually all 250 million Web users from within their browsers.”


BUILDING THE RAMP


Bringing mathematical capability to the World Wide Web actually involves three distinct but complementary projects, according to Angel Diaz, a computer scientist at IBM and co-chair of the mathematics working group at the World Wide Web Consortium. The first task is to figure out (and standardize) how mathematics actually looks online. Although the problem of “presentation” may seem fairly straightforward, it's actually quite complicated to insert a complex equation for instance,

into a document designed for words, sentences, and paragraphs. Indeed, books have been written on the subject of mathematics typography. Risers and descenders in math are different from risers and descenders in regular text, so special programming is needed so equations don't look strange on the screen and page.

The main competition for MathML in terms of presentation is a product called TEX (pronounced “tech”— a mathematics typesetting language that has been used for 20 years as the standard for presenting math in print. As well known as TEX is in the world of math, however, it is not widely used beyond that niche and will offer little competition for web applications.

And if the problem of presentation is complicated, the second stage in creating a useable mathematics mark-up language is even more complex, according to Diaz and other experts. This is what mathematics software designers call the problem of meaning, or semantics. “You cannot tell from the looks of an equation what it means,” explains Diaz. He offers the example of

The same meaning could be (and often is) expressed as,

so a sophisticated mathematics software program needs to recognize not only standardized mathematics notation but also its inherent meaning, which is not always an exact match. It's a semiotics problem, and as the pioneering semiotician Charles Sanders Peirce said in the mid-19th century, mathematical notation and mathematical sense are not identical; close, but not identical. The challenge for MathML is to account for that semantic misalignment in its computer language.
Finally, MathML or any competing computing language must include what Diaz calls “the cool stuff,”or what is more technically known as “math-specific interactivity.” This is the “enlivening language” that will allow users to not just read an equation but to work it, to take one denominator and replace it with another and observe the results. Such interactivity would allow specialized math and science software to integrate more smoothly on the Web. For example, Barry MacKichan, CEO of MacKichan software in Washington state, plans in the future to use MathML as a tool to integrate his software products with existing computer language systems. Or in the case of Professor Coleman's chemistry classes, to have his students manipulate the conditions of an experiment to see how the changes affect the outcome. It is for the world of mathematics and science the equivalent of a highly sophisticated spreadsheet, which can make computations based on ever-changing variables.

Up until now, what scientists have had to do when they want to communicate with one another is to write the mathematics algorithms, equations, and so forth in one of many software languages and hope that the recipients have the same software. But of course they might not, as any HTML user knows who has received an “attachment” that their computer was unable to decipher and display. Ultimately, explains Diaz, MathML aims to be the mathematics player in the burgeoning world of so-called “Web services.” When Web services are fully functional, every florist will display its inventory and costs on the World Wide Web in an identical language, so consumers can compare and contrast. Similarly, everyone posting content that involves mathematics will be using the same language.

To accomplish these ambitious goals, the underlying instructions for MathML are what computer scientists describe as “highly verbose.” That is to say, the language is intentionally over expressed, which makes it easier for the computer to use.

In the end, though, that is really the intent of MathML, explains Terry Rochford, vice-president of product development at Mathsoft. Just as nine out of ten licensed automobile drivers do not need to know how a transmission works in order to get from place to place, for most MathML users the technicalities will always be invisible. In fact, the typical user of MathML, if and when it penetrates the market, will not be a computer scientist or even a chemistry student at Wellesley, but the average person making a credit card transaction. As Chris Randels says, “Mathematics permeates everyday life.”

Wray Herbert is a freelance writer living in suburban Washington, D.C.