The Future of Mathematical Text: A Proposal for a New Internet Hypertext for Mathematics
The Internet has transformed the practice of mathematical writing, and mathematical texts of all kinds are moving online. But the fundamental change to come in mathematical publication is not just moving print forms to electronic documents, but recreating mathematics in a new architecture: a hypertext that reflects the deep unity and universality of mathematics, that can grow and diversify as mathematics changes. It is argued that hypertext is a natural representation of mathematical thought, with its deep interconnection of ideas, the need for constant revision, and the multiplicity of viewpoints. The design of a hypertext must take into account how mathematics is structured and how it is understood: its internal consistency, the need for preparation and review, and the importance of strong tool support for reading and writing text. A high-level design is proposed, combining structured and network hypertext, with a simple link and editorial structure, and design issues concerning language representation, medium- and high-level structure, editorial policy, administration, and technology are examined. A hypertext of sufficient quality and usability will powerfully influence how mathematics is taught, communicated and used, in the classroom, in the workplace and in research. This change in how the Internet is used is not primarily technical: it is the extension of current technology towards a new goal. Gaps and weaknesses of the design are discussed, as well as possible solutions, and a plan to implement the hypertext on the Web is developed.
The Internet has become the standard means of exchange of mathematical writing, and over the past two decades we have seen electronic journals, preprint servers, journal databases and archives, newsgroups, mailing lists, self-publication, conference proceedings, online encyclopedias, and so on, all appearing on the Web. The lively profusion that is the Web has changed mathematics as well as everything else.
The Web has a growing role in the literature of mathematics, as publications move online. A central resource of the research mathematician, the Mathematical Reviews, is now primarily an Internet operation [MSN]. An initiative of the National Science Foundation, the National Science Digital Library [NSDL] has sought to improve the educational resources for mathematics on the Web. Electronic journals are becoming commonplace, and research articles from print journals are routinely distributed online. Libraries are using digital archives of mathematics journals like JSTOR [JST] in place of the print versions. A new project, of great importance to mathematics, is the systematic digitization of all mathematical journals (Ewing 2002).
Looking at mathematics as something that is learned, used, explored, and savored, the Web today is a most uneven source of information, where writings of distinction and wit are next to trash. But even if we could have an unending choice of the best mathematical papers, books, and databases, we would fall short of the full potential of the Web.
For mathematics has a deep interconnectedness, an inherent organization, in detail and as a whole. Mathematical work cannot be understood in isolation. Without this context, mathematics is a library in another language, extensive and impenetrable. Even for experts, learning new mathematics requires preparation and review.
Learning the context of mathematical ideas, and placing new ideas within a larger framework, is part of the mathematics education and the working activity of mathematicians. Each mathematician builds an internal network of mathematical ideas and extends them in new directions. The great potential of mathematics online is to make this synthesis of ideas explicit, to present mathematical work within its full context. The great potential is to find not just a particular text, but a way to understand the text, a guide to prerequisite work, to examples and applications, to a network of ideas that the reader can explore following individual needs and interests. The natural form of mathematical text is as a hypertext that reflects the internal unity and increasing growth and specialization of mathematics.
It is proposed to build a large-scale hypertext of mathematics, an interconnected network of Web pages but with a consistent structure and coherence, an evolving representation of the entire edifice of mathematics. The hypertext links together the actual concepts and structures of mathematics, theorems and proofs, examples and commentary, conjectures and programs. This paper defines a basic hypertext structure, prior to an implementation, and discusses some design issues, goals and possibilities.
In some sense, the whole of mathematics is a dynamic text. Mathematics is in a constant state of flux. New results extend or challenge the existing framework; older proofs are simplified, generalized, refined. New theories emerge; old theories are reorganized from new perspectives. The dynamic content and organization of mathematics fit naturally in a hypertext framework. It can make explicit the connections between new work and old, between one theory and another.
Mathematics, in the words of Halmos (1983), is "an amazingly unified intellectual structure". Most mathematical research fits into some larger program. This organization is an essential part of our understanding and assimilation of mathematics. It gives the coherence to mathematical work from different individuals, different research groups, in different journals and languages. It has a continuous meaning over time. Theories may lay dormant for 20 years or 50 years and then revive. A hypertext can represent the organic unity of mathematical thought better than any single publication. It can describe the content-based organization of mathematical work, at a level of detail that we have not seen before.
At the same time, mathematics has had an astonishing growth, and has become increasingly diverse and specialized. The context of new work, the learning needed to follow the reasoning and place it with related work, has become more extensive and more narrowly focused. Mathematical writing is always starting over and reviewing, in textbooks, survey articles, seminars, and lecture notes. Older work is fundamental in every area of mathematics, but the context of older work also becomes more remote and difficult, as notation, structure, and priorities change over time. A hypertext provides a means to bring the scattered pieces of mathematics into connected units. The mathematics are related, not by linking one paper to another, but by connecting the mathematical ideas: theorems, proofs, conjectures, examples, applications, and commentary.
Finally, the hypertext can have a size and richness greater than even the largest publications, and it can have global accessibility through the Internet.
What do we want the hypertext to look like?
A good place to work
The intended user is experienced in mathematics and looking to learn new material or to review partly forgotten mathematics. This user must have confidence that the exposition is clear and efficient, that the links are coherent and understandable, that all major topics are covered, that the proofs are well-written and correct. On any subject covered, the text must be usable as the primary learning source for that subject. The standard set by the best mathematics books is almost frighteningly high, and cannot be met universally in so large a system. But the quality must be high enough overall to win the continued interest and attention needed to sustain the hypertext.
Mathematics in context
Mathematical texts are placed within larger structures and organizations. For most texts, the user can find one or more paths that will lead to the text, containing background needed to understand the text, commentaries and summaries, alternative proofs and examples.
Ease of text entry and editing
We seek the simplest possible means to enter text, formulas, diagrams, and sketched pictures. It is more important that entry is easy and natural, than that the rendered output is of high quality. The text represents a blackboard, not a finished paper. The ease of writing into the system was a feature of the early KMS hypertext system (Akscyn et al. 1988) that we want to adopt here. The user should have a variety of methods to input and edit text, and can select an interface close to what he/she is already familiar with.
Collaborative construction and use
The text should have a number of readers, contributors, and editors. It is intended as a public place, built, used, and owned by mathematicians. It should manage easily communications, updates and corrections, joint writing, classroom uses, personal notes and linkages. The growth and extent of the hypertext will depend on its support by mathematicians and on its usefulness to the mathematical community.
Breadth of material
The text should eventually incorporate at least a detailed introduction to every branch and sub-branch of mathematics. This is a very large piece of work, but I believe that this broad coverage will be essential for the usefulness and sustainability of this text. The growth of the text, in depth and complexity, must be balanced by the increasing range of subjects covered.
The project of building this hypertext will take many years, and its hardware and software environment will evolve considerably over its lifetime. The design of the hypertext must have a semantic simplicity and durability, to minimize any changes needed to maintain the system in a new environment.
A hypertext structure can potentially be a great benefit to reading, writing, and managing mathematical text. Hypertext was first proposed as a superior representation of a domain of knowledge (Nelson 1974). What does mathematics as a domain of knowledge impose on the design of a hypertext?
Mathematics is full of linear and hierarchical structure. Groups, topologies, and graphs are extensively classified, and it is rare that you can reorder the steps of a proof. Mathematical results often fall in a line, proving stronger theorems with increasing technique and complexity of proof. Many papers in mathematical journals are built upon earlier papers and prove something new, a leaf node on the current tree of related results. A hypertext must represent this linear and hierarchical structure as well as more associative forms of linking.
Size of text pages
Mathematical text is often dense and tightly knit, and should not be cut into little pieces. The units of text should be longer and more internally cohesive than in most hypertext designs. It must be easy to jump back and forth within a long piece of text, to help understand how the material fits together.
Extra help is needed to orient and navigate a hypertext, particularly a mathematics text. Reading mathematics well is a slow and strenuous activity. The link structure must be semantically very simple, used in moderation, and as flexible as possible; a minimal distraction from the work at hand.
One of the themes of the hypertext literature is that the reader is a co-author of the text (Bieber 2000). This point of view is particularly apt for mathematics. Most of us learn mathematics by rewriting it. We learn the proofs, to understand, to solve a problem, to explore, to apply to a new situation, to try out a new viewpoint, to connect with what we already know. At every level, this rewriting is an essential part of learning mathematics.
So the tools to write mathematical text, to create pages in the hypertext, are of particular importance for the use of the hypertext, for a true reading of the text. These tools to edit text and formulas, to draw sketches and diagrams, will have independent uses, for writing papers, personal notes, class preparation. They can in turn form contributions to the hypertext.
A wide variety of mathematical texts and systems are on the Web today, too many to summarize here. Some serve a teaching purpose; others a research program; others are expressions of individual interest. To select two fine examples: The MacTutor History of Mathematics [MAC] has selections on mathematical history and biography, and Lane [LAN] writes about statistics.
Two systems on the Web today have considerable mathematical content and are examples of how a large-scale mathematical text may be built: The Online Encyclopedia of Integer Sequences [OEIS], developed by Sloane, and Weisstein's World of Mathematics [WEIS].
Sloane (1973) published a collection of integer sequences, of initial terms of sequences defined by different rules. The database includes references to the literature, rules of construction, and links to related sequences. It has grown into a database of over 80000 sequences, with a support group and a sophisticated search program for matching a given sequence to a sequence in the database (Sloane 2003).
The World of Mathematics began around 1990 as a personal collection of pieces of mathematics: definitions, theorems, formulas. With support from Wolfram Research, Inc., it has grown to over 10,000 items and has a collection of well-designed animations.
Although different in purpose and scope, it is worthwhile to look at some common characteristics:
simple, consistent structure
a broad mathematical range
long development time
detailed references to the mathematics literature
a focused gathering of material in its initial stages, mostly by a single individual
more contributions after the main structures were built
In addition, in both systems there is a fine sense of openness and fun. The casual browser is welcome, and puzzles and curiosities have a place. But they do not restrict their scope to suit a particular level of education. Nothing is omitted because it is too "hard".
These remarkable systems, which need a more thorough account than I can give here, are models and inspiration for the mathematical hypertext.
The hypertext proposed here is defined on the Web as a series of interconnected Web pages. It has two parts: the center text, and the open text. Anyone may contribute a page in the open text. The design outlined here is for the center text, which has a qualification procedure for changes and additions.
The organization is a blend of structured hypertext (book texts) and network hypertext (core text), with static links. The core text pages are guides and commentary. They are compactly written and densely linked, an internal monologue of mathematics. They are primary means of moving from topic to topic. The book texts have a roughly hierarchical structure, the main means of developing proofs and theories.
At the highest level, the text is structured around threads: collections of pages, book texts, and core texts that have a common viewpoint and presentation. A navigational map of the current state of the text is provided to guide the user of the text. More expansive pages, called room texts, will be present throughout the hypertext as entry points within the text.
The design relies on editorial distinctions more than formal structure to communicate the overall shape of the hypertext. Books and core text are distinguished by their function within the system, and authors are free to structure them as they like. This flexibility allows some leeway as the hypertext is built and allows formal structure to be added later as needed.
The design takes a conservative approach towards visual displays and computations, insisting that all technologies are securely embedded in the text. In fact, very little is asked of the varieties of hypermedia, except to display the natural structure of mathematics.
To discuss the question of language, of how the text is represented internally in the system, it will be useful to distinguish the different components of a mathematical language system:
The base language of characters, e.g. Unicode
Extensions of the base language with mathematical symbols
Further extensions to create rendered constructs, such as formulas, matrices, commuting diagrams, and other displays
The input/edit language
The internal representation of text
The display rendering system
Extensions for semantic information: for example, in a product a*b, is this a group operation, and if so, is it commutative or not?
We look to design the system from first principles, to set out what is needed first before trying to fit the system into one technology or another. It is clear, though, that we will use or borrow from TEX, LaTeX, and MathML.
The system should be able from the start to represent multiple languages. The use of Unicode is an obvious choice for this generality. The system should distinguish, as TEX does, between language text and mathematical text. For example, the mathematical forms for Greek letters are more pronounced than for Greek letters used in Greek writing. The language of forms should be structured like LaTeX.
The system must provide methods to create new symbols, to use symbols in nonstandard contexts, and to create new syntactic forms. Mathematicians cannot give up the right to invent, recycle, and abuse notation.
A major decision is what semantic information should be represented in the text. We propose that the internal language represent only the syntax of mathematical expressions, not the semantics. In the language of MathML, only the presentation mathematical markups are recorded, not the content markups [SAN]. Mathematical expressions have few basic forms, but potentially a vast number of interpretations. For a system devoted to mathematical writing and proofs, where easy entry and editing is essential, adding semantic information is unrealistic, and inflexible. We seek instead a straightforward syntactic representation, with no extra structure.
For example, the expression a+b should be represented internally as something very close to a+b. The actual meaning depends on the context: addition of integers, or real numbers, or elements in an arbitrary abelian group, or complex-valued functions, or operators, or subspaces of a vector space, or random variables, or ordinals, order types, or cardinals, or one of several other interpretations that the reader may think of on his/her own. Not all of these interpretations have a computational meaning, or can interface usefully with other mathematical systems.
The representation of mathematical text in a Web page should be in MathML. However, MathML and even LaTeX are difficult to edit, and we look for a very simple language of forms (such as division, matrices, and so on) that can be mechanically translated into MathML.
The editor should work on a representation of the text that looks as it is rendered, and should have methods to open or close a form or symbol. When closed, the form is displayed as it is rendered; when open, the form is an input expression that creates the form, with the operands in rendered form.
So the editor must handle text partly in input expressions, and partly in rendered displays. It must be able to translate automatically from the internal representation language to the input/entry language. It should have specialized interfaces for more complex constructs, such as commuting diagrams or matrix displays, that can be opened and closed as other forms.
In this design, the input/edit language and the internal representation language are distinct. For example, a Chinese ideogram will be represented internally by its Unicode number, but it could be entered, as many Chinese text systems do, by typing some multi-letter code, by menu selection, or both. Further, there is no reason to prefer one system of input/editing to another. By using editors other that the one just described, the user may input the integral of sin(x) by typing a TEX-like expression, or using drag-and-drop icons, or reading some external document, or even, one day, by handwriting. There is, however, one standard internal representation. Users may edit the text using their own preferred input/entry system, without knowing how the text was entered in the first place.
Since the internal representation is no longer directly tied to an editor, the representation language should be as simple and transparent as possible, easy to read and to program. It must retain its meaning as the editor and other portions of the system evolve.
The rendering of expressions will be done by the browser reading a MathML file. Within the editor, the rendering will follow the conventions of TEX, with some simplifications. There should be facilities for printing selected parts of the text, and producing a readable LaTeX file.
The most important graphics tool in the hypertext is an editable line drawing. Following the example of a blackboard, the drawing records lines made by sweeps of the mouse. The lines can be edited one by one, or rearranged in a sequence. The lines may be free-form, or constrained in direction, or constrained to end at a particular point. In addition to lines, arbitrary symbols or text may be placed in the drawing. A drawing may be opened, edited, and closed like a mathematical form.
The drawing and editing must be clear and easy to use. Ideally, a rough sketch should take no more work with the tool than by hand drawing.
In addition, the system should have capabilities for various types of plots, block diagrams, images, and so on. These aspects of the system are familiar in other tools. These are not the focus of the system, but necessary capabilities for completeness. Really sophisticated displays belong outside the system.
The main unit of the hypertext is the page, which contains mathematical text, drawings and static links to other pages - a Web page with additional structure.
In the hypertext literature, a number of different link types and linking strategies have been discussed. Should links be embedded in the text or implicit, built-in or generated, single or bidirectional or multiple-endpoint, elaborately typed or simply typed or no types at all? (Weinreich et al. 2001)
Our approach here is to concentrate on the semantic foundation of the text, the presentation of mathematics ideas and proofs, and on the reader who intends to work through the mathematics. We should start with a link structure that is very simple, visually unobtrusive and easy to understand, both for authors and readers. We will rely on the text, and placement of the link within the text, to determine the meaning of the link. We will rely on editorial structures - such as books, core text and threads - to organize the text, rather than on individual links.
The advantage of this approach is its flexiblity and simplicity. It is easy to learn and use links, easy to standardize in multi-authored text, and easy to plan and administer texts. It gives the author more control in how to shape a text. If it does not realize the full power of hypertext, it can serve as a base for alternative linking strategies.
For navigating the text, we propose two link types: an inner link, and an outer link. The inner link goes to a text segment that returns to the source of the link; the outer link goes to a new page. In addition the system will support ordinary URLs and invisible continuation links that connect separate files in a single page.
Weinreich et al. (2001) calledthe visualization of link markers "an underestimated problem" and compared alternative link visualizations, such as underlining, overlays and special symbols. We propose that the inner and outer link each have a special symbol, such as a solid arrow, that is about the same size as a capital letter. This will fit in well with the visual appearance of the text, and will appear the same on printed and online versions of the text.
At the start of any page referenced by an outer link, there is a special symbol followed by a title line. This symbol can open and close in the editor like a mathematics display form, and contains administration information and any data needed to read the page, such as special fonts.
There is currently no enforced structure in the hypertext beyond what is described above. But within the text there are systems of pages with their own style conventions that guide the user through the text:
A text is a collection of pages, linked by inner and continuation links, and is the main organizational unit of the hypertext.
The core text consists of pages of dense, informal discussion, with many links. It is a kind of interior monologue, with a tight linkage of ideas, modeling the way we actually think about mathematics.
For mathematical theory that we understand well, the pieces of the mathematics - definitions, examples, theorems, proofs - seem to be represented in our minds by condensed linkages of ideas, jumps from step to step, that have to be expanded when we write out a proof, or prepare a lecture, or explain to someone else.
Of course, there is no reason to assume that different people "represent" mathematics internally the same way, and every indication that a personal meditation on any mathematical topic will vary with individual taste, training and experience. The core text should be a mathematical "representation" for an ideal person. It is more extensive than our personal viewpoints, since it can incorporate vast amounts of mathematics, and it is also less condensed than our personal viewpoints, because it must be organized in a way that other people can understand. The core text is a working synthesis of the ideas at hand. To get a balanced, comprehensive text, we will need guidance from experienced individuals within each field.
The core text will be hard to follow, but will get easier as the mathematics becomes understood. In fact, the ability to read the core text with understanding is a measure of how well the subject has been learned.
Each core text page is about a particular limited subject, such as a theorem or a method. Core pages can launch a book (see below), or conclude a book, or provide a dense commentary about some theorem in a book. Core pages are intended to be everywhere within the system, splitting off as the subject changes, a running commentary and guide, a means of moving to another topic and searching within a topic. They are the binding that holds the disparate pieces of the hypertext together as a whole.
Here is an outline of a core text page on the Weierstrass Approximation Theorem, which states that any continuous real-valued function on a closed interval can be uniformly approximated by polynomial functions:
Statement of the theorem, with short standard proof
Bernstein's proof, links to Bernstein polynomials
Examples of why the hypotheses are needed, e.g. f(x) = 1/x on (0,1].
Examples of polynomial convergence that are easy to prove, e.g. for real analytic functions
Several examples of typical application
Topological viewpoint, other dense sets in C[a,b]
Topological generalization, link to Stone-Weierstrass Theorem
Bishop's Theorem and related generalizations
Muntz Theorem and generalization
Local approximations, links to page on Taylor's Theorem
Relation to Fourier series, Fejer's Theorem, Chebyshev polynomials
Failure for complex functions, links to another page on convergence of analytic functions and the Weierstrass double series theorem
Proof for quaternionic functions
Best polynomial approximations, links to approximation theory
Rate of approximation for Lipschitz functions and smooth functions
Polynomial approximations in L1
Best approximations in L2, link to Legendre polynomials
Polynomial approximations with additional constraints:
point constraints, derivative constraints
links to Lagrange/Newton interpolation, Hermite interpolation
Other classes of functions besides polynomials, such as exponential polynomials
Multiplicative version for functions whose range is the unit interval
Generalizations to n dimensions, on several topics
Books are long texts modeled after typical mathematics books. A well-written book is the easiest way to learn mathematics. A book text should have the organization, uniform notation and systematic exposition of a book.
There are some differences between ordinary math books and book texts within the hypertext. Most mathematics books start with a chapter on preliminaries, introducing notation and background material. In the hypertext, this comes from the context in which the book is placed. Instead, a book text might start with a page of links to related topics, to help the reader who came here looking for something else. In the hypertext, a book text will need a clear goal or set of goals, a definite sense of where the book ends and how far you have to go to get there. Many math books will cover the basics of a topic, and then branch off into applications, refinements, and related topics. In the hypertext, a book text is more tree-like than linear, linking to other books and to core texts.
A room is a text with an open, discursive style, a relief from the density and difficulty of the core texts. It is modeled after a room in a museum: pictures, exhibits, models, recreations, histories, pages for random browsing and serendipity. Rooms are the most public face of the hypertext, the starting points for exploration within.
Within a particular topic, the material is organised - book texts, core texts, miscellaneous pages and structures - into several threads, larger units of text with a similar perspective. Most new pages would be integrated into one or another of these threads. The threads may be interlinked at many points, but each could be read independently. This idea of an organizing path through text has appeared in different forms in the hypertext literature, beginning with Bush's (1945) trails.
Here is an example of how the thread structure might be used in Galois theory. If we were to prove and interconnect all the important theorems of Galois theory, for example, we would have an incomprehensible mass. Without architecture, the theory could never be learned. Instead, we present Galois theory in several threads:
A simple direct exposition of the main theorems, using the minimum prerequisites.
The approach of a modern algebraist: foundations in category theory, Galois cohomology, Krull topology, and so on. This text would connect with abstract algebra that extends beyond Galois theory. It could branch in several directions: extensions to rings and algebras, applications to algebraic-differential equations, determination of the Galois groups among the rationals, and so on.
A historical thread, presenting the work of Ruffini, Abel, Lagrange, Galois, Dedekind.
At the highest level, we can organize the text by topic, using the American Mathematical Society's Mathematical Subject Classification [AMS], or a library catalog classification. A hypertext allows for a dynamic high-level organization, to add topics and meld topics together. But it is difficult to see how to make these capabilities useful. A new categorization will be artificial without considerable reorganization of the material underneath.
The hypertext should not attempt a comprehensive architecture. The task seems too difficult for mathematics today. An alternative is to advance the text by smaller topics: by classes of theorems, new problems or new structures, which is closely aligned with how we create and observe changes in mathematics. These topics are the subject of core text pages and are interlinked in the text. From this collection of topics we can dynamically build a map of topics - called the terrain - that is the working guide to the system. The form of this map is a kind of imaginary landscape, marking the internal structures: book texts and core texts, rooms and threads, that can help the reader navigate the text. It can also guide the planned expansion of the text itself.
The hypertext exists on the Internet, and is divided between the center and open text. The open text may be contributed by anyone, much as a Web page can be created by anyone. All that is required is a format check and some kind of notification procedure for updates. It is up to its owner to revise, update links, determine access rights (it may be wholly private), claim copyright, and so on. The center text will have a qualification process for new text and for revisions on older text.
The core text is intended to be the primary means of navigation through the system. A search for a solution to some mathematical problem will often be qualified and complex, and will require a learning process by the user to understand what is relevant and whether a particular theory applies. By reading through a network of core pages, of discussion that leads and branches from topic to subtopic, the reader gains some of the background needed to search the book texts, and the outside literature, more capably.
The core text is important in this architecture as a means of synthesizing related mathematical ideas. However, as a search mechanism it has limited usefulness. It is labor intensive, it must grow with the text, and it cannot be expected to answer all questions. Also, it cannot be relied on to search outside the center text.
The system will need ways of searching outside the given organization of text, to visually represent the results of the search, and to let the user select and redirect the search interactively. These daunting tasks will not be attempted in the early development of the text. Here we describe a few partial steps toward a solution.
The simple global search for words and patterns has proven to be useful in all kinds of document systems, and should be available here. It will be useful to extend the patterns to mathematic formulas, for example, all integrals of a certain type. However, the connections in mathematical text are more idea-driven than word-driven. Words mutate in meaning as a subject evolves, and the terminology is notoriously unhelpful. Some information retrieval techniques, such as keyword matching, may be tried here. Mathematical constructs have a fairly precise meaning, and the center text has a controlled submission procedure.
There should be tool support to display and navigate the terrain map, showing your place among interlocking threads, a sort of global positioning system for the text. The terrain map will extend with each addition, but the large-scale outlines will be slow to change.
Finally, mathematics is not facts or data, and the main problem is not access. A page of mathematics can be inaccessible, even when it is right in front of you. A system where you push a button and get the right page won't help you if you cannot understand the page. The goal of the hypertext is not just access, but comprehension.
The style of presentation should be what we use for our best students: keep a brisk pace, use modern standard notation, get to the important ideas quickly, tour the literature, train the intuition.
The visual consistency of the text, the similarity from page to page, is as important in a hypertext as in an ordinary book. In navigating a hypertext, as in flipping through the pages of a book, a consistent presentation helps to orient the reader.
Thüring et al. (1995) analysed the comprehension of hypertext documents, leading to eight design principles to improve local and global coherence. Among these principles are the preservation of context with links, and the organization of text into larger units. For this hypertext, it will be editorial policy, the shape of the written text, that must carry the burden of making the text coherent.
Except in core text, there should be sparing use of links on most pages. Attempting to link every definition, theorem, etc., quickly leads to an unreadable text. Some pages will be long, and naturally linear pieces of text should be presented as such.
In general the value of a link depends on how much context is inherited (Thüring et al. 1995). An inner link especially must retain as much context as possible. The text linked by an inner link should be of moderate size, with a consistent policy on what and when to link.
In changing context with an outer link, we will often want to "soften" the outer link by placing inner-linked text as a preview of the outer linked text. For example, in a proof that applies the Hahn-Banach Theorem, an outer link to a core text page on the Hahn-Banach theorem may be too abrupt, an overload of information that the reader doesn't need. It is better to have an inner link to text that explains what the Hahn-Banach theorem says and how it is used in the particular context of the proof, limited to what the reader needs at that moment. From this inner-linked text there can be an outer link to the Hahn-Banach Theorem. This policy preserves local coherence and gives the reader a choice.
In a linked hypertext, one can have an arbitrary number of continuations and asides. Besides what follows in the text, there is what is linked. What material should be linked together?
Three criteria are proposed: Link what is important, what is accessible, what is beautiful.
The main theorems and conjectures should always be close at hand when beginning a subject. For example, an extremely important application of the Central Limit Theorem is in statistical sampling and hypothesis testing. Text explaining these applications should be easy to find within the text on the Central Limit Theorem.
Proofs and applications that are easy to understand, or that avoid high-powered tools, should be available close to the introduction of a subject. To continue the example: the Central Limit Theorem is usually proved by Fourier analysis, but the earliest versions of the theorem, proved by DeMoivre and Laplace, require only algebra. A core page on the Central Limit Theorem should explain this connection and link to the DeMoivre-Laplace Theorem.
The most interesting and beautiful parts of the subject should always be close at hand. The Central Limit Theorem is part of a fascinating array of mathematics: Weiner processes and Brownian motion, the Erdos-Kac theorem, and many other topics.
Theories may be related at different levels and depth. To solve a polynomial equation, we get different theories if we want a numerical solution, or a solution in an expression with radicals, or a solution in integers. These theories are not closely related, in our current understanding, On the other hand, mathematics abounds in deep relationships between seemingly unrelated theories. Both sorts of linking should be provided in the hypertext. The deep links illuminate a subject and help us understand it; the shallow links - change a question slightly and get a different theory - guide the user in searching for a specific topic.
Texts are separated in to several distinct threads, each of which pursues a topic at a particular level and with a particular internal organization.
Another reason to separate topics is if their study requires a distinct preparation and a distinct point of view, even if they are roughly about the same thing. A course on computational group theory will require a different set of tools than the usual courses of finite group theory. They are connected, and should be connected in the hypertext. But the context and aims are different, and the user should be able to read about one and not the other. Another example: work in nonstandard analysis is sufficiently different from usual analysis that the mathematics is not usefully interconnected in detail. We want a multiplicity of viewpoints without cluttering the clear presentation of basic mathematical ideas. It is important that each piece of mathematics is placed in its most useful context.
Topics may be separated because of a different direction and emphasis. Mathematical recreations have connections to all sorts of mathematics, but they have a different criterion of beauty and interest. Expositions of geometry, as it developed in Greece, China and Japan, should be included in the text but should not crowd out a modern presentation of geometry.
Another point of separation is in the language. The center text is intended to be in English, but there is every reason to develop hypertexts in other languages. Portions of the center text can be dual-language, to help individuals read mathematics in different languages.
With some trepidation, I list some models for the mathematical writing of the hypertext, all books of extremely high accomplishment. Readers will doubtless know other examples that could be included here:
Hua (1982) Introduction to Number Theory
The broad coverage of important results, and the multi-threaded approach, repeating topics at increasing levels of depth and complexity, are goals for the hypertext.
Mathematical Society of Japan (1980) Encyclopedic Dictionary of Mathematics
A comprehensive introduction to modern mathematics.
Rudin (1974) Real and Complex Analysis
Widely used for its clarity and efficiency.
Feller (1968) An Introduction to Probability Theory and Its Applications
A vast amount of material presented in an interesting and comprehensible fashion.
Zwillinger (1989) Handbook of Differential Equations
A broad concise summary of a vast subject, with a consistent point of view toward applications. This book could be a model of a thread within a larger text on differential equations.
A hypertext structure offers a more fluid organization than a mathematics book, but not an easier task: the demands for producing a well-thought-out architecture, an interesting exposition, and painstaking accuracy are just as great.
The subjects of the hypertext should include at least a basic text to each area of mathematics, and should extend where needed to mathematical models and theories in the sciences. Besides mathematics texts, it can have scientific biography and history, catalogs of examples and counter-examples, fact sheets, tables and bibliographies.
The range of subject matter, then, is extremely broad; I believe a narrower text is harder to sustain. A broad text will not only gain more attention, but is closer to a long-term solution. As an analogy, consider the development of Unicode, an encoding for most of the world's language scripts. At its beginning, it seemed an impossibly large task. Yet one reason for its steady progress is that we know a smaller solution won't work in the long term. I believe that, like certain induction proofs, the more ambitious program is easier to make succeed.
Clearly, a project this large is an enormous task, and must be a long-term, collaborative effort.
In terms of depth, the level of difficulty should be somewhere between the calculus sequence in undergraduate mathematics, and research mathematics. Expositions of basic calculus, linear algebra, and ordinary differential equations are common, and there is no need to add to this work here. College textbooks are aimed at beginning students, and often show considerable effort and skill in explaining concepts and in developing technique. But from a more advanced viewpoint, the presentation can be bland, repetitive, over-explained, and badly architected, a poor model for this hypertext. Yet within these subjects there is a wealth of interesting work in mathematical journals, old and new, that is not incorporated in the textbooks. This is a good place to start the hypertext.
The exclusion of research-level work is not a strict rule, but simply a recognition that the hypertext can never keep up with the enormous literature.
Finally, the hypertext should largely avoid philosophy and controversy. The text is only possible because of the universality of mathematics and the collegiality of mathematicians. Calls for change, priority disputes, professional debates, should be met with a polite silence, and the battles for the hearts and minds of mathematicians should be fought elsewhere.
The computer system supporting the hypertext can do a lot more than turn pages and display formulas. It can calculate, plot, search databases, draw models and animation, compile user-defined programs, draw phase planes, knot diagrams, and scatter plots. Let us call the collection of these functions the technology of the system. What do we want the technology to do?
In a mathematical hypertext, the first principle is restraint. For every advantage we can get from a programmed display, there are drawbacks, sometimes serious ones, to take into account.
The first problem is with cognitive overload. Reading mathematics, which is hard work under the best circumstances, can become unbearable under the weight of menus, pop-up windows and help manuals.
The distraction could also affect the development of the text. The construction of software tools could become the main focus of the text, instead of advancing the mathematical writing.
Systems for symbolic mathematics, visualization of data, statistical analysis, etc., are essential tools in many areas of mathematics. But with mathematical software, like any software, the most rapidly developing tools are also the most perishable. The effort to keep up with the field, and the administrative problems of maintaining these tools, are further distractions from the purposes of the text.
In spite of all these problems, we will absolutely need to build technology within the text. The approach suggested for the hypertext should include:
programmed interfaces to external systems
internal tools for simple, local, one-time applications.
The hypertext should have interface tools to link to all kinds of external systems: MAPLE and MATHEMATICA, group theory and number theory tools, numerical software, automatic theorem provers, and so on. The interface may define MathML files, for example, that are understood from both systems.
The programmed interfaces should not require the user to add information into the mathematical text. The interface itself should provide any context information needed, perhaps with user input within the interface.
The system must be able to export text and graphics in various formats, including PostScript, PDF, TEX and LaTeX, and to import text in TEX, LaTeX or MathML. Eventually, the system should support the document structuring in LaTeX and the customizable rendering of TEX.
The development of internal tools should follow a bottom-up approach, tied very closely to the textual explanation. These tools are small "toy" programs, panels with a couple of buttons that the user can figure out immediately. After the display or animation is shown, or rerun a few times, then the point is demonstrated: we are through with the tool, and move on.
For example, a text on functional iteration to find a solution to an equation may be demonstrated with an animated plot that runs a few canned functions from a user-selected starting point. A display may take a few examples of matrices of real numbers and show the steps of putting them into row echelon form. Another display may generate a random permutation and show its cycle structure. A class of real functions may be introduced with a set of plots that the user can manipulate.
We may compare the technology in this approach to the building of bridges on an interstate highway. The large bridges are central, crowded, and expensive. But most bridges on an interstate are small, unobtrusive, and so easy to cross that the driver barely notices them. They are simply part of the road.
The user comprehension of all these one-time tools should be enhanced by common interface conventions. All tools should employ a common stable programming method that is easy to understand and modify. The programming itself should extend the pedagogic purpose of the hypertext. The tools should make no heavy demands on the computational environment.
The requirement to fit new work into a common structure will present new challenges to editorial staff. Here we look at some of the issues and possibilities.
The hypertext exists on the Internet, and is divided between the center text and the open text. The center text, which is described in this paper, will have a qualification process for new text and for revisions on older text.
The hypertext should move smoothly and transparently between the center text, the open text and a personal text. The user should be able to set up personal links from the public text to a private text.
We can distinguish three different tasks in assembling the hypertext, which may or may not be done by different people.
This editor decides the outline of threads, book texts, core texts, and miscellaneous pages for a particular subject area.
As for any science journal, the referee judges whether the work is qualified for publication, finds errors and improvements, and assists in revising work to meet a professional standard. The size of contributions may vary considerably, from a simple correction to an entire book.
The contributed work must be merged into the hypertext, revised to fit in the existing structure, and linked into multiple contexts.
The idea of a structural editor already occurs in practice. The lectures of Hilbert or of Artin have been written up, then reworked and extended by many mathematicians. The original outline may not change significantly for decades.
What differs in the function of the referee is that the refereed text is not the final version: there is no final version. The text can be modified as it is placed into the hypertext, and is subject to continual revision. Also, the sense that the text has an author is changed, because the author's version is not the version in the hypertext.
The text must be scrupulous in crediting mathematical results. Also, some way must be found to recognize the work of individual contributors, without responsibility for later changes that they may not approve.
The text will be copyrighted by its support organization, to protect against misuse, but is open to any academic or scholarly purpose. The intent is a public text, for public use. The project will need to find long-term institutional support, perhaps a consortium of mathematical societies and universities.
The hypertext can provide a detailed context for reference to published work. It can make individual work more visible and approachable, by providing the specific background material needed to understand this work.
The hypertext can also provide an ideal setting for older papers, for collected works. Annotations within the hypertext can connect the work to current results and perspectives.
The vast extent of the mathematical literature in fact poses problems little discussed among the professional concerns: mathematics today has an inadequate grasp of its past, of its abundance. In response to an overwhelming literature, mathematicians specialize more narrowly, follow current research trends, create self-contained packages to prove a few theorems in depth. Much significant work, by almost any definition of significance, has a smaller and smaller base of people who know the work, and in each generation, some of it is lost.
Perhaps this is inevitable: that in mathematics, as in music or poetry, more deserves to survive than is actually remembered.
The hypertext as envisioned here could extend the individual's reach in mathematics and the useful lifetime of work. It can contain the mathematical context sufficiently to introduce any complex mathematical ideas, whether active or not, integrated with the rest of mathematics in its current form.
The surest way to find the design flaws of a system is to start building and using that system. But we can anticipate some problems almost immediately, and can try to mitigate them from the start.
The terminal is not a comfortable workplace for reading and writing mathematics. Most of us would not want to read a novel online, much less work through a long mathematical proof.
Part of the answer lies in the improvement in human/computer interfaces. It will surely be possible one day to have screens for reading and writing, as light, portable, and flexible as a sheet of paper, capable of recording and recognizing your handwriting and your speech.
Today, the strain in reading computer screens is not the only source of this problem. In a study comparing reading online and paper documents (O'Hara and Sellen 1997), the critical issues are not the readability of the screen, but the advantages of paper text over online text for reading: for annotations, quick navigation, and spatial layout.
The problem of reading online underscores the need for the simplest interfaces in reading and writing text. It also underscores the essential role of the writing tools for mathematical text in developing familiarity with the hypertext and building a readership. The user who puts together a personal hypertext - for lecture notes and class assignments, scratch notes that get pulled together into a single text, copying interesting bits from the hypertext or from the Web, drafts of papers and personal notes - this user will become comfortable reading mathematics from the hypertext.
The feeling of a hypertext is that of infinite expanse, of text beyond what any one of us can know. The editorial structures like book texts are ways to restore a sense of boundedness to the text, to keep things to a human size. But the design calls for a fairly dense text, which can cause fatigue and a sense of sameness.
There is a large literature on the visualization of hypertext and the use of physical metaphors to help orient and guide the reader. Kim and Hirtle (1995) use metaphors such as differentiated regions, guided tours, landmark nodes, and travel histories to reduce the user's disorientation and increase the sense of place within the text. Dieberger and Frank (1998) use the metaphor of a city to organize a large information space.
The design proposes a structure of information, but we will also need a visual geography to communicate the structure to the reader. For example, the reader should know without changing the display if he/she is reading an inner-linked page, and how far to the end of the book. The visual design of the text must have more variety and more structure than proposed here. The text will need places to slow down, to stop, to wonder, to play, to escape, and to relax before proceeding.
Inevitably, the hypertext will reflect the interests of the major contributors, leading to a spotty, incomplete text. A careful plan of expansion is needed to balance the text as it develops.
On any mathematical subject, there are portions that are frequently presented in classrooms, lectures and textbooks, because the material is easy to understand and important. Also, on any subject, there are portions that are at least as important or useful, but are harder to present and so are not found in the textbooks. For example, any book covering derivatives of a real variable will have L'Hôpital's rule, but most will not cover Baire's theorem on the continuous points of a derivative, because Baire category is a little more subtle. This tendency for textbooks to cover easily presented material gives newcomers an unbalanced view of a subject and its literature.
An analogous problem has faced historians like Arnold Toynbee who attempt to write a world history. Due to the sheer size of the historical literature, one is forced to rely on secondary sources, and the gaps and distortions in that literature may be magnified in the finished product.
In building the hypertext, the project must present mathematics from a variety of sources, and must seek advice from reseachers in each topic on the essential subjects to cover.
This is a real and essential difficulty with hypertext: a certain amount of repetition is inevitable.
In a study of a hypertext in journalism (Huesca et al. 1999), readers were not kind in their assessment of hypertext reporting as prose: "fractured", "choppy", "jumbly", "butchered", "disconnected", "cut and pasted", etc.
A thorough editorial practice will be needed to reinforce the continuity in writing. One method, mentioned above, is to interpolate an inner-linked text before providing an outer link. Another policy, assuming repetition is ineveitable, is to at least make it easier to recognize, by using standard notation, names, and idioms for mathematical concepts.
Inevitably, a text this large will not be updated as frequently or as carefully as it needs to be. Even work that is frozen in time, like classic papers, will need to have links and annotations revised periodically.
Fortunately for this project, a well-written mathematical text will retain its value and relevance. The best books of a decade ago are essential resources, even in rapidly developing fields. On the other hand, in some subjects like numerical analysis, a text will be little used if it is out of date.
In planning the expansion of the text, we may need to compromise on the breadth and depth of subject matter and to avoid writing on certain topics until there is a sufficient support group to develop and maintain the text.
The range of application for this hypertext is as broad and ubiquitous as mathematics itself. Let us allow a certain license and pretend the future will go the author's way. What changes will be brought about with the mathematical hypertext?
Within ten years time, the text will be a well-established self-sustaining operation, integrating mathematical work from all sources. It will serve as the primary mathematical reference on the Internet.
The first use of the text will be in education, in mathematics and science. The text is an instruction in mathematics, not just for the standard courses but everywhere. In university teaching, sections of the text may be used as a foundation for new studies, outside the mold of the standard textbooks. It will be especially valuable for advanced work, for courses preparing for research, and as a research resource. It will become an essential part of online courses in math and science, and a growing resource for mathematics in developing countries. In turn, the work of teachers and students will feed back into the text, in filling gaps, working out problems, correcting errors, annotating and rewriting the proofs, and developing multiple-language texts.
Outside the university, the text will see increasing use in the technical professions. In engineering, for example, specialized training and retraining has become a fact of life, perhaps a permanent feature of the modern economy. The hypertext will be a standard resource for retraining, inside and outside the workplace. The text will become a resource for technical documentation of all kinds. In science experiments, in statistical analyses of medical protocols, in the design of telecommunications networks, in the whole array of applications requiring mathematics, the hypertext can be a central resource for mathematics. The benefits it can provide are the benefits of a hypertext: explanation and commentary, links to related work, to background material, to worked examples, and to the literature outside the text.
Within ten years the text will not be finished but its basic outline and shape will be evident. We will begin to see increased growth in the open text, the collection of independent texts that use and link up with the hypertext. The text will be a force in standardizing the tools and linking in mathematical writing. Individual writings, papers, lecture notes, personal files, seminar presentations will use a common framework. This framework will support the publication and dissemination of mathematical work, formal and informal, into independent hypertexts. In time, the open text could become more abundant and more important than the center text, which would serve an administrative and support role for the open text.
Within the research community, the text will serve as a resource of increasing depth and sophistication. It will be used in scholarly publications, as a standard reference for mathematical work. Research papers will begin to be published as hypertext, linked within the text as well as in hypertext journals. In the recent phenomena of large-scale proofs, of computationally intensive mathematics, a hypertext will become the standard way to write, manage and communicate the material. Research mathematicians will use it as a means of searching for work at a very detailed level, of finding analogies between one problem and another. Finally, a great deal of the researcher's time is spent in reading and learning mathematics, sometimes outside the individual specialty. The hypertext cannot and should not replace the research library, but it can be a guide and service within the library, of growing power and importance.
The support work needed for this hypertext will gain in acceptance and professional recognition, a necessary part of maintaining the infrastructure of mathematics to keep the enterprise going. Writing and editing contributions will become recognized professional activities. The development of semantic maps of each subject, of explicit classification of concepts and theory, will be seen as essential work, a basis for organizing the current literature, the common view of how things fit together.
Perhaps the most important benefit of the hypertext is as a stimulant to mathematical research and scholarship. The very act of building the text will bring together a host of unsolved problems, of obscurities to clear up, of the undocumented "folklore" known to experts but not to the rest of us, of independently discovered techniques that can be unified and strengthened, of the uncharted boundaries of a theorem or proof.
The development of this hypertext makes modest demands on technology. An important foundational step is the emergence of MathML as a standard in mathematical communication over the Internet. Tools to support interactive mathematical writing, text editing for mathematical formulas, and sketches have been slow to arrive and are still incomplete. The improvement in editing mathematical text and sketches, the integration of TEX and MathML, etc., are messy but highly solvable problems, addressed in a number of current proposals and products [DES, GAR, GOO, GUR, SAN].
The technology needed to start building the text exists today, and as the project is so broad in scope, one can start almost anywhere. The following outline suggests how the project can unfold over time.
Develop a prototype text
Writing a portion of the text, of reasonable depth and completeness, is a necessary first step in proving the design and the viability of the project as a whole.
The author has started a text of Web pages, using public domain tools: the presentation portion of MathML and the ITEX authoring tools developed by Gartside [GAR]. The text is on linear algebra, a subject with connections to many parts of mathematics, to numerical work, and to a wide range of applications. This seems a good proving ground for the design of the hypertext.
Develop a set of metrics on the quality of the design
A test of the usability, reader comprehension and convenience of the design is needed at an early stage, to make adjustments before the text gets too large.
Mathematical review of the text
Each section will need periodic review by mathematical specialists for accuracy, clarity and balance of coverage.
Build connections with online mathematical libraries and with emerging Web standards
A number of initiatives are building digital libraries in mathematics and science generally. The project must work to align with these efforts, and look to combine with existing projects.
Publicize the hypertext and look for contributors
The role of contributors will change as the project grows. Initially we seek a small, dedicated group of contributors to build a working system, to alter the design when it runs into problems, and to build the tool support needed to write and manage the text. We will look for a broader set of contributors when the hypertext is a functioning, well-established system, and when there is stronger tool support for writing, managing and collaborating on text.
Develop semantic maps for various subject areas
For example, a survey of books and journals in general topology could produce a connected placement of definitions, theorems, conjectures, examples and counter-examples. This would form the basis of the threads, books, and core text in general topology.
Develop plans for growth and acquisition of material
Each subject covered by the text will need a plan of growth, similar to developing a collection in a library. The success of the project will depend on detailed and realistic planning. Part of the plan will be an in-depth semantic map of a subject area, lists of papers, books, and other resources for material, specifications of the threads through the material, outlines of the book and room texts, a list of target results that the text must cover, a validation of the plan by mathematicians in the field, an editorial policy, the recruiting of contributors, a timeline, and (at least initially) a sharp limit on depth.
Look for sponsorship and develop a business model for sustainability
The appropriate means of sustaining the text depends on the reception it receives in the mathematical and educational community, in quality and service to mathematics.
The proposal sketched here is a design for a mathematical hypertext, a set of requirements for the tools to support the text, and a look at the journey needed to create that text. It stands between the details of a software and editorial system and the long-term possibilities that a growing hypertext can offer mathematics. For the text itself can never be completed; it can only expand, until it stops from exhaustion or is replaced by a better system.
This kind of hypertext has tremendous potential as a teaching tool, repository, and communicator of mathematics. It is a representation of the whole of mathematics as we actually know it, a city of ideas and personalities, and it will surely lead to new mathematics and to new ways of looking at mathematical work.
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Bush, V. (1945) "As we may think". The Atlantic Monthly, 176, 1, July, 101-108 http://www.theatlantic.com/unbound/flashbks/computer/bushf.htm
Ewing, J. (2002) "Twenty centuries of mathematics: digitizing and disseminating the past mathematical literature". Notices of the American Mathematical Society, 49, 771-777 http://www.ams.org/notices/200207/fea-ewing.pdf
Huesca, R., Dervin, B., et al. (1999) "Inverted pyramids versus hypertexts: a qualitative study of readers' responses to completing narrative forms". Proceedings of the 82nd annual convention of the Association of Education in Journalism and Mass Communication, New Orleans, LA, August http://www.trinity.edu/rhuesca/hyperaej.html
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Weinreich, H., Obendorf, H. and Lamersdorf, W. (2001) "The look of the link: concepts for the user interface of extended hyperlinks". Proceedings of the 12th ACM Conference on Hypertext and Hypermedia (Hypertext '01), Aarhus, Denmark
Robert Mayans got his BA in mathematics at Yale University and MA and PhD in mathematics at the University of Michigan. He is now an assistant professor of mathematics at Fairleigh Dickinson University in Madison, New Jersey. He worked for 16 years at Bell Laboratories as a developer and system engineer, in circuit pack CAD systems and wired equipment design. He invites mathematicians of all persuasions to join in building this hypertext.