I start with a reassessment of selected linguistic traditions. Since they were developed to meet needs other than those I want them to serve, a careful reexamination is not out of place.

Empirical characteristics of two work languages

The first issue I shall take up is simply whether they are able to handle the language variety in which most computer systems must function, namely the languages of people at work, and in order to do so we need to know what a “work language” is.

The car repair shop

The typical Danish car repair shop is a small scale enterprise, consisting of the owner and a couple of skilled workers, possibly an apprentice and a bookkeeper, as well.

Most of the work can be performed by one worker alone, so cooperation is seldom a physical necessity. Planning and execution of the work procedure is integrated. The mechanic performs both tasks. Personal relationships between employer and employee survive, such as the master-journeyman relationship. Normally, a small stock of tools is used: hammers, screwdrivers, nippers, etc.

In the early eighties, I investigated communication in a small repair shop, and the following pages are based on this project. The research was done in collaboration with one of the mechanics employed in the garage. A taperecorder placed on a bench recorded the conversation that took place during work. (Everybody in the garage knew about the recording.) The mechanic also produced written descriptions of whole working days.

The analysis of the data took 1 1/2 years. The mechanic was employed as an assistant teacher in an advanced course at the university.

In Part I of this book I presented a general framework for understanding language and computers, and in Part II showed how to apply this framework to analysis and design of computer systems. What remains to be done now is to broaden the scope of our analysis, and show how the same framework can be used for describing and designing computer systems in a practical context of work. In order to do that I need a concrete example, and I shall use data from the 1986 project at the Postal Giro Office in Stockholm, “Professional language in change”, mentioned in Section 1.1.1.

Before I start on this last stage of the journey, I think it would not be out of place to make a short summary of what has been accomplished so far.

Although the theoretical framework of the book is not natural science but a structuralist tradition of semiotics and linguistics, the project of the book is a part of an ongoing paradigm change in computer science itself: from seeing the computer system as a self-sufficient mathematical object, the focus is gradually being shifted to the relations between system and work context. An important practical motivation for this shift is simply that it hopefully will enable us to construct computer systems that meet the needs of the users in a better way. However, the shift is also theoretically motivated by the structuralist framework, since structuralism is characterized by focusing on relations, not on objects, as the real existing phenomena.

What is computer semiotics?

Semiotics is the science of signs and their life in society. A sign is anything that stands for something else for someone.

Semiotics treats all kinds of signs: verbal language, pictures, literature, motion pictures, theater, body language. Computer semiotics is a semiotic discipline that studies the nature and use of computer-based signs.

The motivation for the discipline comes from the nature of computer systems: although in many respects computer systems can be conceived as tools in analogy with typewriters, pencils, brushes and filing cabinets, they differ from these tools by not primarily existing or being used as physical objects, but as signs. The pencil of the drawing program is no real pencil that can be used to chew on, it is merely stands for a pencil, represented by a collection of pixels on the screen. Computer systems resemble other media by primarily acting as carriers of meaning.

Even if computer systems basically are symbolic tools, this symbolic nature has only attracted serious attention in recent years.

The reason for this is threefold: from being a tool only for specialists, computers have now been integrated into many occupations, and therefore the demands for interpretability have risen. The mode of operation and the meaning of the data must be easy to grasp for the secretary or the manager whose time should not be spent deciphering cryptic codes but writing agendas or making decisions. A good interface has become a necessary prerequisite for a good system.

We have now looked at work language, the language variety that must be central to our concerns, and made a preliminary list of the descriptive shortcomings of classical structuralism with respect to this language type.

In the next, constructive sections I will describe those parts of the tradition I wish to keep, and propose changes in other parts that make the theory better suited for the purposes at hand. Although the next few sections are concerned with language proper, I shall append short indications of how the concepts described can be used to understand computer systems, so that readers primarily interested in computers may judge if reading on is still worth the effort.

After presenting the key concepts (Section I.2.1), I show in Section I.2.2 how they can be applied to analysis of work processes and organizations, in particular the car repair and the Postal Giro examples from I.1.

Section I.2.3 extends the framework further to deal with computer systems. Computer systems are interpreted as media for human communication, and as a consequence of this, the theoretical focus is shifted from the system itself to the relations between system and user (Section I.2.4).

Basic concepts for describing symbolic acts

One of the basic points of structuralism is that units of language are defined relationally.

Computer-based signs

I start my investigation of computer-based signs from the viewpoint of the expression plane, partly because the computer's means of expression are different from other media, partly because the history of linguistics tells us that expression analysis is easier than content analysis. What are the characteristic features of computer-based signs, and in which respects do they differ from other kinds of signs we know?

Handling, transient, and permanent features

The prototypical computer-based sign is composed of three classes of features:

• A handling feature of a computer-based sign is produced by the user and includes key-press, mouse and joystick movements that cause electrical signals to be sent to the processor. Handling features articulate user actions.

• A permanent feature of a computer-based sign is generated by the computer. It is a property of the sign that remains constant throughout the lifetime of a sign token, serving to identify the sign by contrasting it to other signs. Permanent features articulate system states into parts.

• A transient feature of a computer-based sign is also generated by the computer, but unlike permanent features, it changes as the sign token is used. It does not contrast primarily to other signs, but only internally in the same sign, symbolizing the different states in which the sign referent can be. Transient features articulate system transformations.

As our first example, let us look at a version of the game Breakout. The system displays a paddle, a ball and a brick wall. The ball bounces back and forth, and the player must hit it with the paddle.

The first part of the book ended with a sketch of the computer system as a medium, and the purpose of this second part is to turn this sketch into a more detailed portrait. What are important properties of computer systems from this perspective, and how should systems be structured?

It should come as no surprise that the interface, defined in Section 1.2.3.3 as a collection of computer-based signs, and its functions relative to the linguistic environment and working context reign supreme in this approach. The basic function of a system is to generate processes in which an interface can be expressed, just as the only reason for having the costumes and wings in a theater is the experience they may contribute to giving the audience. Although this view on design is relatively new in computer science, it exists under the name of user centered design.

This view makes it difficult to maintain the tradition of separating functionality from the interface. Traditionally, functionality is what can be done with the system, while the interface is the manner in which it is done. Thus, two word processors have different functionality if one allows the user to make an automatic table of contents while the other does not. They have the same interface if operations are done in the same way, for example if text is selected by dragging the mouse over the desired piece of text, whereas their interface is different if one uses the mouse while the other requires the user to type line and character numbers in to select text.

In this chapter and in Chapter 5 we shall present many mathematical applications of computer graphics. In order to draw the line somewhere (pardon the pun) we shall restrict ourselves to the mathematics associated with the plane and in particular with curves in the plane. There is a whole other realm, just as fascinating, connected with objects, such as curves, polyhedra and surfaces, in three-dimensional space. Nevertheless what the computer actually draws is, in these cases too, a curve or system of curves in the plane – that of the screen. The extra complications come from taking a three-dimensional object and associating with it a curve or system of curves – for example its outline when seen from a distance, or a sequence of such outlines or a sequence of plane sections of the object. We touch on this in a discussion of the swallowtail surface in Section 4.14.

The computer can (with our help) draw curves and collections of curves which are just too complicated to attempt by hand. For some purposes a rough sketch of a curve does very well – you will probably have drawn many such sketches by hand, and we are certain that the art of curve-sketching by hand is still an art well worth acquiring. However, for some other purposes, such as the illustration or discovery of facts connected with the differential geometry of curves and families of curves (not to mention surfaces), accurate drawings are essential.

This book is intended for anyone who has some mathematical knowledge and a little experience with programming a microcomputer in BASIC (or any other language). The book shows how simple programs can be used to do significant mathematics.

To spell out our mathematical prerequisites in more detail, some of the chapters assume no more mathematical knowledge than whole numbers, but for the most part we assume some calculus, and the rudiments of algebra (polynomials, equations) and trigonometry (sines, cosines and tangents). Thus, British readers with A or A/S level in mathematics and American readers with a Freshman calculus course behind them will, we hope, have little difficulty in following most of the mathematics here. We have, naturally, included some material for the more mathematically sophisticated reader: those sections requiring closer inspection are, appropriately, printed in smaller type. (We hope that readers who do not immediately recognise the small type material will be intrigued rather than frightened. Surely one of the charms of mathematics is the glimpses it affords of mysterious and fascinating territory which is for the moment just out of reach.)

As for programming, the knowledge we assume is very small, and most programs are given full listings in the text. It seems to us that a very effective way to learn programming is to use it to solve interesting mathematical problems. We have regarded the mathematics as the pre-eminent interest, and have not tried too hard to make the sample programs beautiful or elegant, or even particularly efficient.

Sir Isaac Newton is certainly one of the greatest scientists to have ever lived. He is generally reckoned to have been one of the three most outstanding mathematicians of all time, along with Archimedes and Gauss, and his discoveries in physics are unrivalled in their width and influence. What was Newton's secret? How did he achieve as much as he did? Obviously there is no simple answer, but Newton had one secret, which he guarded jealously, and which he believed to be vital. It was ‘Data aequatione quotainque fluentes quantitoe involuente fluxions invenire et vice versa’ or in English ‘solve differential equations’.

Nowadays this ‘secret’ is entirely unremarkable; we are all aware that many processes and phenomena in the world are governed by differential equations. The very fact that Newton's secret is now common knowledge clearly indicates its worth and power. Of course his secret was rather hard won; he did have to invent differential equations before pronouncing his dictum concerning solving them!

In this chapter we shall see what the microcomputer can do for those intending to follow Newton's advice. Our eventual viewpoint will be considerably more modern than Newton's. It turns out that in certain circumstances solving differential equations is not as useful as watching them.

Differential equations and tangent segments

Much of science is devoted to the problems of predicting the future Differential behaviour of some physical system or other. Often the underlying equations and physical law will describe the rate at which the system evolves; what tangent segments we require is a description of how it evolves.

One of the most important and useful ways in which mathematics can help us to solve problems is by the solution of equations. ‘Let x be the length of the piece of string; then x satisfies the equation x2 – 2x - 3 = 0 and solving the equation gives x = 3.’ We are sure that you have solved many problems using equations; unfortunately all but the simplest equations cannot be solved exactly.

There are two reasons for this. In the first place even for a quadratic equation, unless the solutions are rational numbers (as in the above example), there is a square root such as √2 to be evaluated, and this cannot be done exactly. The decimal expansion does not terminate or recur, so we must be satisfied either with the formal ‘√2’ or with an approximation to so-many decimal places.

The second reason is more profound. Exact formulae analogous to the famous quadratic formula do exist for equations of degrees 3 and 4 – of course these formulae involve cube roots and so on, so are open to the same difficulty as we noted above for quadratics. On the other hand no algebraic formula exists at all for equations of degree 5 or more! In a precise sense, the equation x5 - 6x + 3 = 0 cannot be solved algebraically at all. This is a difficult statement and has an even more difficult proof, in which computers won't help in the least.