1. Introduction

• 1.1 Mathematics and its Notation
• 1.2 Origins and Goals
• 1.2.1 The History of MathML
• 1.2.2 The Need for MathML
• 1.2.3 The Uses of MathML
• 1.2.4 Goals of MathML
• 1.3 The Role of MathML on the Web
• 1.3.1 Layered Design of Mathematical Web Services
• 1.3.2 Relation to Other Web Technology
• 1.4 Encoding Notation and Content with MathML

1.1 Mathematics and its Notation

Mathematical notations are constantly evolving as people continue to discover innovative ways of approaching and expressing ideas. Even the commonplace notations of arithmetic have gone through an amazing variety of styles, including many defunct ones advocated by leading mathematical figures of their day [Cajori 1928/1929]. Modern mathematical notation is the product of centuries of refinement, and the notational conventions for high-quality typesetting are quite complicated. For example, at the simplest level, variables, or letters which stand for numbers, are usually typeset today in a special italic font subtly distinct from the text italic, and the spacing around the symbols +, -, x and / is slightly different from that of text, to reflect that by convention multiplication is a higher precedence operation than addition. Slightly more sophisticated is the now common convention of keeping the baselines of superscripts and subscripts aligned in formulas although the base letters may have different heights and depths; in addition, parentheses around subformulas are usually made to grow with the sizes of the expressions they enclose, or to make clearer the grouping within a mathematical expression. Many subfields of mathematics have their own refined notational devices too.

Although notational conventions in mathematics, and printed text in general, can be complicated, they guide the eye and make printed expressions much easier to read and understand. Though we usually take them for granted, we rely on hundreds of conventions such as paragraphs, capital letters, font families and cases, and even the device of decimal-like numbering of sections such as we are using in this document (an invention due to G. Peano, who is probably better known for his axioms for the natural numbers). It is easy to forget how important these aids to comprehension are until one is obliged to read a poorly typeset document. This is apparent in many mathematical documents on the Web today, where there are difficulties in properly displaying even the most basic notations; we must substitute an "x" for a times symbol, and use a slash for the division sign.

However, there is more to putting math on the Web than merely finding ways of displaying traditional mathematical notation in a Web browser. The Web represents a fundamental change in the underlying metaphor for knowledge storage, a change in which interconnectivity plays a central role. It is becoming increasingly important to find ways of communicating mathematics which facilitate automatic processing, searching and indexing, and reuse in other mathematical applications and contexts. With this advance in communication technology, there is once again an opportunity to expand our ability to represent, encode, and ultimately to communicate our mathematical insights and understanding with each other. We believe that MathML is an important step in developing Mathematics on the Web.

1.2 Origins and Goals

1.2.1 The History of MathML

Since its inception, the Web has demonstrated itself to be a very effective method of making information available to widely separated groups of individuals. However, even though the World Wide Web was initially conceived and implemented by scientists for scientists, the capability to include mathematical expressions in HTML is very limited. At present, most mathematics on the Web consists of text with GIF images of scientific notation, which are difficult to read and author.

The World Wide Web Consortium (W3C) has long recognized that lack of support for scientific communication is a serious problem, and Dave Raggett, the author of the HTML 3.0 working draft, made a proposal for HTML Math in 1994. Following  a panel discussion on math at the WWW IV Conference in Darmstadt in April 1995, a group was formed to discuss the problem further. In the intervening two years, this group has grown, and been formally reconstituted as the W3C HTML-Math working group.

The MathML proposal reflects the interests and expertise of a very diverse group. Many contributions to the development of MathML deserve special mention, some of which we touch on here. One such contribution concerns the question of accessibility, especially for the visually handicapped. T. V. Raman is particularly notable in this regard. Neil Soiffer and Bruce Smith from Wolfram Research shared their extensive experience with the problems of representing mathematics in connection with the design of Mathematica 3.0. MathML has benefited from the participation of a number of working group members involved in other math encoding efforts in the SGML community, including Stephen Buswell from Stilo, Stéphane Dalmas from INRIA, Stan Devitt from Waterloo Maple, Angel Diaz, Robert Sutor, and Stephen Watt from IBM. In particular, MathML has been influenced by the OpenMath project, the work of the ISO 12083 working group, and Stilo Technologies' work on a 'semantic' math DTD fragment. Finally, the American Mathematical Society has played a key role in the development of MathML. Among other things, it has provided two working group chairs: Ron Whitney led the group from May 1996 to March 1997, and Patrick Ion, who has co-chaired the group with Robert Miner from The Geometry Center, from March 1997 to the present.

1.2.2 The Need for MathML

The most obvious problems with HTML for mathematical communication are of two types:

Display Problems. Consider the equation . This equation is sized to match the surrounding line in 14pt type on the system where it was authored. Of course, on other systems, or for other font sizes, the equation is too small or too large. A second point to observe is that the equation image was generated against a white background. Thus, if a reader or browser resets the page background to the gray default, the anti-aliasing is wrong. Next, consider the equation . This equation has a descender which places the baseline for the equation at a point about a third of the way from the bottom of the image. One can pad the image like this: , so that the centerline of the image and the baseline of the equation coincide, but this causes problems with the inter-line spacing, which also makes the equation difficult to read. Moreover, center alignment of images is handled in slightly different ways by different browsers, making it impossible to guarantee proper alignment for different clients.

Image-based equations are generally harder to see, read and comprehend than the surrounding text in the browser window. Moreover, these problems become worse when the document is printed. The resolution of the equations will be around 70 dots per inch, while the surrounding text will typically be 300 or more dots per inch. The disparity in quality is judged to be unacceptable by most people.

Encoding Problems. Consider trying to search this page for part of an equation, for example, the "=10" from the first equation above. In a similar vein, consider trying to cut and paste an equation into another application. Using image based methods, neither of these common needs can be adequately addressed. Although the use of ALT text in the document source can help, it is clear that highly interactive Web documents must provide a more sophisticated interface between browsers and mathematical notation. Another problem with encoding mathematics as images is that it requires more bandwidth. By using markup-based encoding, more of the rendering process is moved to the client machine. Markup describing an equation is typically much smaller than an image of the equation.

Some of the display problems associated with including math notation as images could be solved by improving browser image handling. However, even if image handling in browsers were improved, the encoding problems would still remain. In planning for the future, it is clear that making the information contained in mathematical expressions conveniently accessible to other applications will be increasingly important.

1.2.3 The Uses of MathML

The education community is a large and important group that must be able to put scientific curriculum materials on the Web. At the same time, educators often have limited resources of time and equipment, and are severely hampered by the difficulty of authoring technical Web documents. Teachers, for example, need to be able to post notes and exams quickly and easily.

Electronic textbooks are another way of using the Web which will potentially be very important in education. Management consultant Peter Drucker has recently been prophesying the end of big-campus residential higher education and its distribution over the Web [Drucker 1997]. The form of an electronic text will need to be active, allowing links to other scientific software and graphics.

In the research community here are more and more large, online knowledge bases as typified by highly successful preprint servers, like that at Los Alamos started by Paul Ginsparg. This is especially true in some areas of physics and mathematics where academic journal prices have been increasing at an unsustainable rate. In mathematics there are large collections at Duke, MSRI and SISSA, and on the AMS e-MATH server. In addition, databases of information on mathematical research, such as Mathematical Reviews and Zentralblatt für Mathematik, offer millions of records containing math on the Web. In addition, any design for math on the Web must facilitate the maintenance and operation of large document collections, where automatic searching and indexing are important. Because of the large collection of legacy data, especially TeX documents, the ability to convert between existing formats and new formats is also very important to the research community.

Corporate and academic scientists and engineers also use technical documents in their work to collaborate, to record results of experiments and computer simulations, and to verify calculations. For such uses, math on the Web must provide a standard way of sharing information that can be easily read and generated by authors and by software.

Another design requirement is the ability to render mathematical material in other media such as speech or braille, which is extremely important for the visually impaired.

Commercial publishers are also involved with math on the Web at all levels from electronic versions of print books to interactive textbooks to academic journals. Publishers require a method of putting math on the Web that is capable of high-quality output, robust enough for large-scale commercial use, and preferably compatible with their current, usually SGML-based, production systems.

1.2.4 Design Goals of MathML

In order to meet the diverse needs of the scientific community, the HTML-Math Working Group intends to develop an open specification for a mathematical markup language, MathML, to be used with HTML, that:

• Is suitable for teaching, and scientific communication.
• Is easy to learn and to edit by hand for basic math notation, such as arithmetic, polynomials and rational functions, trigonometric expressions, univariate calculus, sequences and series, and simple matrices.
• Is well suited to template and other math editing techniques.
• Insofar as possible, allows conversion to and from other math formats, both presentational and semantic, such as TeX and computer algebra systems. Output formats may include graphical displays, speech synthesizers, computer algebra systems input, other math layout languages such as TeX, plain text displays (e.g. VT100 emulators), and print media, including braille. It is recognized that conversion to and from other notational systems or media may lose information in the process.
• Allows the passing of information intended for specific renderers.
• Supports efficient browsing for lengthy expressions.
• Provides for extensibility, for example through contexts, macros, new rendering schemas or new symbols. Some extensions may necessitate the use of new renderers.

• MathML equations in HTML pages should render properly in popular Web browsers, in accordance with reader and author viewing preferences, and at the highest quality possible given the capabilities of the platform.
• HTML documents containing MathML equations should print properly and at high-quality printer resolutions.
• MathML equations in Web pages should be able to react to mouse gestures, and coordinate communication with other applications through the browser.
• Equation editors and converters should be developed to facilitate the creation of Web pages containing MathML equations.

1.3 The Role of MathML on the Web

1.3.1 Layered Design of Mathematical Web Services

At the same time, it is important for many groups, such as students, to have simple ways to include math in Web pages by hand. Similarly, other groups, such as the TeX community, would be best served by a system which allowed the direct entry of markup languages like TeX directly in Web pages. In order to resolve the contradictory goals of providing more specialized kinds of input and output for specific user communities, while still providing a system of sufficient generality and power, the idea of a layered design architecture naturally emerges.

MathML is designed to be a general and powerful underlying communication layer which is machine-friendly. It is designed to encode complex notational and semantic structure in an explicit, regular, and easy to process way. Sitting on top of the MathML communication layer will be input syntax layers that are designed to be simple to learn, and easy to edit by hand. Many different input syntax layers designed for different user communities can potentially all piggy-back on top of the MathML layer. Equation editors and translators will be used to convert input syntaxes into MathML. Alternatively, renderers may convert input syntaxes directly included in Web pages into MathML on the fly.

One consequence of a layered design architecture is that the core language of MathML is not intended to be particularly well-suited to hand entry. Instead, MathML is designed to facilitate the development of software and input syntaxes that are carefully tailored to the needs of specific user communities, while providing a low-level, standardized format for communication over the Web.

In some ways, MathML is analogous to other low-level, communication formats such as TeX's DVI format, or Adobe's PostScript. You can create a PostScript file in a variety of ways, depending on your needs; experts write and modify them by hand, authors create them with word processors, graphic artists with paint programs, and so on. Once you have a PostScript file, however, you can share it with a very large audience, since devices which render PostScript, such as printers and screen previewers, are widely available.

Similarly, the HTML-Math working group envisions typical users creating MathML documents by using equation editors, converters or other scientific software, or by hand in some cases, according to their needs. A student might prefer to use a menu-driven equation editor that can write out MathML to an HTML file. A researcher might use a computer algebra package that automatically encodes the mathematical content of an expression, so that it can be cut from a Web page and evaluated by a colleague. A journal publisher might typically use a program that converts TeX markup to MathML. Others may prefer to include other math markup languages directly in an HTML page which is translated on the fly into MathML by a specific embedded renderer in a Web browser. Regardless of the method used to create a MathML web page, once it exists, all the advantages of a powerful and general communication layer become available. MathML-compliant renderers can be developed for a variety of purposes including speech, print, embedded web software, and computer algebra. One may expect that eventually MathML can be integrated into other arenas where mathematical formulas occur, such as spreadsheets, statistical packages and engineering tools.

The HTML-Math working group is moving aggressively to ensure that both MathML software and high-level input syntax layers will soon be available. The Working Group plans to produce a proposal for input syntax and macro capability by May, 1998. One proposed short form input syntax has already been developed by Wolfram Research. In addition, two renderers, WebEQ and IBM techexplorer, have announced plans to implement MathML, and both will accept an input syntax based on TeX. In addition, a number of software vendors and other organizations have expressed interest in developing MathML-compliant software, including the American Mathematical Association, IBM, members of the OpenMath consortium, Geometry Technologies, Stilo Technologies, Waterloo Maple, and Wolfram Research.

1.3.2 Relation to Other Web Technology

However, in order to effectively stimulate software development, it is important that MathML interact well with existing software. In particular, MathML has been designed with three kinds of interaction in mind: with existing mathematical markup languages, with HTML extension mechanisms, and with Web browser extension mechanisms.

Existing Mathematical Markup Languages

Extensive work on encoding mathematics has also been done in the SGML community, and SGML-based encoding schemes are widely used by commercial publishers. ISO 12083 is an important layout-based markup language which primarily describes the visual presentation of mathematical notation. Because ISO12083 and its derivatives share many presentational aspects with TeX, and because SGML enforces structure and regularity more than TeX, much of the work in ensuring MathML is compatible with TeX also applies well to ISO12083.

MathML also pays particular attention to compatibility with other mathematical software, and in particular, computer algebra systems. Many of the presentation elements of MathML are derived in part from the mechanism of typesetting boxes. The MathML content elements are heavily indebted to the OpenMath project and the Semantic Maths DTD. The OpenMath project has close ties to both the SGML and computer algebra communities, and has laid a foundation for an SGML-based means of communication between mathematical software packages, among other things.

HTML Extension Mechanisms

One of the goals of XML is to be suitable for use on the Web, and in the context of this discussion it can be viewed as a general mechanism for extending HTML. As its name implies, extensibility is a key feature of XML; authors are free to declare and use new tags and attributes. At the same time, the XML syntax carefully enforces document structure to facilitate automatic processing and maintenance of large document collections. In addition to its advantages, XML has garnered support from major browser vendors as well. Consequently, both on theoretical and pragmatic grounds, it makes a great deal of sense to specify MathML as an XML application, and we have done so.

Browser Extension Mechanisms

A general model for rendering and processing XML extensions to HTML is still being developed by the W3C XML working group. However, broad features of the model are already fairly clear. Style sheets provide the mechanism for specifying the processing model, and embedded objects provide a way of doing the processing. Cascading Style Sheets (CSS) and DSSSL are the main style specification mechanism under consideration, and some combination of these methods will probably be used to bind rendering instructions to XML extensions of HTML.

At present, however, the rendering and style parameters that are recognized by major browsers are geared toward primarily text-based content. Thus, for content such as MathML (or many other kinds of complex structured data) it is necessary to extend native browser capabilities by providing embedded elements to do the rendering. As one popular slogan puts it, XML gives Java something to do. Ultimately, some sort of style sheet mechanism will instruct a browser to use a particular embedded renderer to process MathML and coordinate the resulting output with the surrounding Web page. In order to achieve this kind of full nteraction, however, it will be necessary to define a document object model rich enough to facilitate complicated interactions between browsers and embedded elements. For this reason, the HTML-Math working group is coordinating its efforts closely with the Document Object Model working group.

While work on XML, style sheets, embedded objects, and the document object model is still ongoing, the intent of these efforts is to provide an infrastructure capable of supporting sophisticated markup and rendering applications such as MathML. Moreover, while much remains to be done, enough of this infrastructure is already available to provide a workable, short term solution for the needs of MathML.

1.4 Encoding Notation and Content Structure

The relationship between a mathematical notation and a mathematical idea is subtle and deep. On a formal level, the results of mathematical logic raise profound and unsettling questions about the correspondence between symbolic logic systems and the phenomena they model. At a more intuitive level, anyone who uses mathematical notation knows the difference that a good choice of notation can make; the symbolic structure of the notation suggests the logical structure. For example, the Leibniz notation for derivatives "suggests" the chain rule of calculus through the symbolic cancellation of fractions:

Mathematicians and teachers understand this very well; part of their expertise lies in choosing notation that emphasizes key aspects of a problem while hiding or diminishing extraneous aspects. It is commonplace in math and science to write one thing when technically something else is meant, because long experience shows this actually communicates the idea better at some higher level.

At the same time, mathematical notation is capable of prodigious rigor. Used carefully, mathematical notation is virtually free of ambiguity. Even when mathematical notation is "abused" in the way described in the preceding paragraph, a completely precise description of the underlying idea still usually exists. Of course in practice, the more abstract the subject matter, the more difficult and tedious it becomes to give a full description of the concepts under discussion; typically the context is understood between the author and the audience, and notation is used almost as shorthand.

In many other settings, though, the full, precise meaning of mathematical expressions is apparent to both the author and the reader. Moreover, there is great utility in encoding that precise meaning explicitly in the markup language so that it is available for use by other renderers and processors, from computer algebra systems to voice renderers or even 3D graphics packages.

Given the complex relationship between mathematical notation and ideas, between authors and readers, and the multiplicity of scenarios in which they interact, the question remains, "What should the content of a mathematical markup language for the Web be?" The answer which MathML gives is this:

MathML is a markup language for describing the notational structure and mathematical content of mathematical expressions.

In terms of their ability to describe high quality screen and print rendering, the MathML presentation tags are on a par with TeX. More importantly, because the tags describe notational structure, not visual layout per se, the presentation expression structure is as compatible as possible with the natural underlying mathematical structure.

Consider the notation . Using MathML presentation tags, this might be marked up as:

<MSUP>
  <MROW>
    <MF>(</MF>
      <MROW> 
        <MI>x</MI> 
        <MO>+</MO>
        <MN>2</MN> 
      </MROW> 
    <MF>)</MF>
  </MROW> 
  <MN>2</MN>
</MSUP>

Although a superscript can denote function composition, a derivative, or even a cohomological index, a human reader easily understands from the context that the superscript in the preceding example usually indicates a power. However, making this information explicit facilitates speech rendering and other automatic processing. Ideally, it should be easy to specify simple mathematical operations completely enough to aid speech rendering, etc., and in MathML it is. MathML provides around 50 content tags in addition to the presentation tags. Using these tags, the preceding example can be encoded as

<EXPR> 
  <EXPR>
    <MI>x</MI>
    <PLUS/>
    <MN>2</MN>
  </EXPR> 
  <POWER/> 
  <MN>2</MN>
</EXPR>

<EXPR>
  <EXPR> 
    <MF>(</MF>
    <MI>x</MI>
    <PLUS/> 
    <MN>2</MN> 
    <MF>)</MF> 
  </EXPR>
  <POWER/> 
  <MN>2</MN>
</EXPR>

In cases where the semantic meaning of an expression cannot be unambiguously described with MathML tags, there is a way of binding arbitrary semantic interpretation data and presentation structure together. The author is free to provide semantic data in any form, for example as an OpenMath expression, or a computer algebra system expression. This makes the information available for renderers and processors that know how to take advantage of it, while providing a notation for screen and print renderings.