By
Wray Herbert
Computer software
currently in development will transform the way mathematics is presented
and performed on the World Wide Web.
William
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 19thcentury
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 email, 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 markup 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 Webbased 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 Webbased 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 “elearning.”
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 cochair 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 markup 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 mid19th
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 “mathspecific 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 everchanging 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 socalled “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,
vicepresident 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.
