**Abstract. **This paper forms the substance of a course of lectures given at the International Summer School in Computer Programming at Copenhagen in August, 1967. The lectures were originally given from notes and the paper was written after the course was finished. In spite of this, and only partly because of the shortage of time, the paper still retains many of the shortcomings of a lecture course. The chief of these are an uncertainty of aim—it is never quite clear what sort of audience there will be for such lectures—and an associated switching from formal to informal modes of presentation which may well be less acceptable in print than it is natural in the lecture room. For these (and other) faults, I apologies to the reader. There are numerous references throughout the course to CPL [1–3]. This is a programming language which has been under development since 1962 at Cambridge and London and Oxford. It has served as a vehicle for research into both programming languages and the design of compilers. Partial implementations exist at Cambridge and London. The language is still evolving so that there is no definitive manual available yet. We hope to reach another resting point in its evolution quite soon and to produce a compiler and reference manuals for this version. The compiler will probably be written in such a way that it is relatively easy to transfer it to another machine, and in the first instance we hope to establish it on three or four machines more or less at the same time. The lack of a precise formulation for CPL should not cause much difficulty in this course, as we are primarily concerned with the ideas and concepts involved rather than with their precise representation in a programming language.

**Preliminaries:**

*1.1. Introduction*

Any discussion on the foundations of computing runs into severe problems right at the start. The difficulty is that although we all use words such as ‘name’, ‘value’, ‘program’, ‘expression’ or ‘command’ which we think we understand, it often turns out on closer investigation that in point of fact we all mean different things by these words, so that communication is at best precarious. These misunderstandings arise in at least two ways. The first is straightforwardly incorrect or muddled thinking. An investigation of the meanings of these basic terms is undoubtedly an exercise in mathematical logic and neither to the taste nor within the field of competence of many people who work on programming languages. As a result the practice and development of programming languages has outrun our ability to fit them into a secure mathematical framework so that they have to be described in ad hoc ways. Because these start from various points they often use conflicting and sometimes also inconsistent interpretations of the same basic terms.

A second and more subtle reason for misunderstandings is the existence of profound differences in philosophical outlook between mathematicians. This is not the place to discuss this issue at length, nor am I the right person to do it. I have found, however, that these differences affect both the motivation and the methodology of any investigation like this to such an extent as to make it virtually incomprehensible without some preliminary warning. In the rest of the section, therefore, I shall try to outline my position and describe the way in which I think the mathematical problems of programming languages should be tackled. Readers who are not interested can safely skip to Section 2.

*1.2. Philosophical considerations*

The important philosophical difference is between those mathematicians who will not allow the existence of an object until they have a construction rule for it, and those who admit the existence of a wider range of objects including some for which there are no construction rules. (The precise definition of these terms is of no importance here as the difference is really one of psychological approach and survives any minor tinkering.) This may not seem to be a very large difference, but it does lead to a completely different outlook and approach to the methods of attacking the problems of programming languages.

The advantages of rigor lie, not surprisingly, almost wholly with those who require construction rules. Owing to the care they take not to introduce undefined terms, the better examples of the work of this school are models of exact mathematical reasoning. Unfortunately, but also not surprisingly, their emphasis on construction rules leads them to an intense concern for the way in which things are written—i.e., for their representation, generally as strings of symbols on paper—and this in turn seems to lead to a preoccupation with the problems of syntax. By now the connection with programming languages as we know them has become tenuous, and it generally becomes more so as they get deeper into syntactical questions. Faced with the situation as it exists today, where there is a generally known method of describing a certain class of grammars (known as BNF or context-free), the first instinct of these mathematicians seems to be to investigate the limits of BNF—what can you express in BNF even at the cost of very cumbersome and artificial constructions? This may be a question of some mathematical interest (whatever that means), but it has very little relevance to programming languages where it is more important to discover better methods of describing the syntax than BNF (which is already both inconvenient and inadequate for ALGOL) than it is to examine the possible limits of what we already know to be an unsatisfactory technique. This is probably an unfair criticism, for, as will become clear later, I am not only temperamentally a Platonist and prone to talking about abstracts if I think they throw light on a discussion, but I also regard syntactical problems as essentially irrelevant to programming languages at their present stage of development. In a rough and ready sort of way it seems to me fair to think of the semantics as being what we want to say and the syntax as how we have to say it. In these terms the urgent task in programming languages is to explore the field of semantic possibilities. When we have discovered the main outlines and the principal peaks we can set about devising a suitably neat and satisfactory notation for them, and this is the moment for syntactic questions.

But first we must try to get a better understanding of the processes of computing and their description in programming languages. In computing we have what I believe to be a new field of mathematics which is at least as important as that opened up by the discovery (or should it be invention?) of calculus. We are still intellectually at the stage that calculus was at when it was called the ‘Method of Fluxions’ and everyone was arguing about how big a differential was. We need to develop our insight into computing processes and to recognise and isolate the central concepts—things analogous to the concepts of continuity and convergence in analysis. To do this we must become familiar with them and give them names even before we are really satisfied that we have described them precisely. If we attempt to formalise our ideas before we have really sorted out the important concepts the result, though possibly rigorous, is of very little value—indeed it may well do more harm than good by making it harder to discover the really important concepts. Our motto should be ‘No axiomatisation without insight’.

However, it is equally important to avoid the opposite of perpetual vagueness. My own view is that the best way to do this in a rapidly developing field such as computing, is to be extremely careful in our choice of terms for new concepts. If we use words such as ‘name’, address’, ‘value’ or ‘set’ which already have meanings with complicated associations and overtones either in ordinary usage or in mathematics, we run into the danger that these associations or overtones may influence us unconsciously to misuse our new terms—either in context or meaning. For this reason I think we should try to give a new concept a neutral name at any rate to start with. The number of new concepts required may ultimately be quite large, but most of these will be constructs which can be defined with considerable precision in terms of a much smaller number of more basic ones. This intermediate form of definition should always be made as precise as possible although the rigorous description of the basic concepts in terms of more elementary ideas may not yet be available. Who when defining the eigenvalues of a matrix is concerned with tracing the definition back to Piano’s axioms?

Not very much of this will show up in the rest of this course. The reason for this is partly that it is easier, with the aid of hindsight, to preach than to practice what you preach. In part, however, the reason is that my aim is not to give an historical account of how we reached

the present position but to try to convey what the position is. For this reason I have often Preferred a somewhat informal approach even when mere formality would in fact have been easy.