The Method of Abstraction - CiteSeerX

Luciano Floridi and Jeff W. Sanders

The Method of Abstraction

Abstract: A method is proposed for phenomenological or conceptual analysis, based on making explicit the level of abstraction of discourse. The result formalises an approach that is traditional in science, but has hitherto found little application in philosophy. The constituents of the method are ‘observables’ collected together and moderated by predicates restraining their ‘behaviour’. The resulting collection of sets of observables is called a ‘gradient of abstractions’ and it formalises the minimum consistency conditions that the chosen abstractions must satisfy. Two useful kinds of gradient of abstraction – disjoint and nested – are identified. The case is made that in any discrete (as distinct from analogue) domain of discourse a complex phenomenon may be explicated in terms of simple approximations organised together in a gradient of abstractions. Thus the method replaces, for discrete disciplines, the differential and integral calculus, which form the basis for understanding the complex analogue phenomena of science and engineering. The method is demonstrated on the Turing test, the concept of agenthood, the definition of emergence and the notion of artificial life, and its place in philosophy is considered. It is hoped that our treatment will promote the use of the method in certain areas of the humanities and especially in philosophy.

Keywords: conceptual scheme, level of abstraction, observables, philosophy of information, Turing test, typed variables

1 Introduction If we are asking about wine, and looking for the kind of knowledge which is superior to common knowledge, it will hardly be enough for you to say ‘wine is a liquid thing, which is compressed from grapes, white or red, sweet, intoxicating’ and so on. You will have to attempt to investigate and somehow

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Luciano Floridi and J.W. Sanders explain its internal substance, showing how it can be seen to be manufactured from spirits, tartar, the distillate, and other ingredients mixed together in such and such quantities and proportions. (Gassendi’s Fifth Set of Objections to Descartes’ Meditations)

Can a complex system always be approximated more accurately at finer and finer levels of abstraction, or are there systems which can simply not be studied in that way? Perhaps the mind, or society, to name two typical examples, are not susceptible to such techniques. That is the defining issue of the recent area of plectics [18] and this paper makes no attempt to resolve it. Rather, we clarify that method of approximation at finer levels of abstraction (the method of abstraction), we provide a rigorous formulation of it and we argue for its indispensability in the study of discrete systems. In studying the characteristics of artificial agents (e.g. [15]) in cyberspace one is presented with a choice: either to acknowledge that such characteristics, including even the properties defining agenthood, are not well defined, or that they depend on the level of abstraction at which they are studied. Although conservativeness would counsel the former, that (negative) approach would be at the expense of a rich theory with potentially interesting applications in the foundations of Information Ethics. So it seems worth exploring the consequences of predicating a theory (of agents, to name but one example) on the level of abstraction. In summary, whether or not a property holds (in particular the definition of agenthood) depends on the level of abstraction of discourse. Sustained study has shown that view not to suffer the flaws of relativism or perspectivism. The purpose of this paper is to present it from scratch, making the claim that, particularly for the study of discrete systems (like cyberspace), the method of abstraction is as important as the method of differential and integral calculus has been for the study of analogue systems. The paper is methodological and provides applications in various, disparate, areas both conceptual and phenomenological. In the former case it facilitates and clarifies study; in the latter it may also aid synthesis. For in the task of top-down construction, an entity is constructed in several passes, starting from an initial approximation which is subsequently iteratively refined until an acceptable result is pro-


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duced. That process may be viewed as a succession of exact constructions at progressively less abstract levels of observation. It is in that sense that the method of abstraction underpins top-down construction. For example an article may be written ‘top-down’. The author may determine in advance to start simply with the title. On the second pass the section headings will be decided, presumably reflecting the organisation of material and order of exposition. On the next pass each section is broken into subsections; after that a note is made of the content of each paragraph; and finally sentences are produced. Of course some feedback occurs between passes, but the benefits of that approach are the overall consistency and modularity into components which can be developed independently of each other after a certain stage in the development. Some (hyper) text systems nowadays support that approach to composition. Each pass corresponds to a complete article, at a certain level of abstraction. At the first level of abstraction only the title is visible; and at that level the result of the first pass is indistinguishable from a finished product, since in both cases only the title is observable. At the second level of abstraction the title and the section headings are visible; and so on. There is no difference, in this discrete domain, between the writer’s construction of an approximation to the final article at a certain pass and an exact or completed article at that level of abstraction. That explains the importance of the method of abstraction for discrete systems. The succession of levels of abstraction we call a ‘gradient of abstractions’. That example illustrates the important case we call a ‘nested’ (or linear) gradient of abstractions. To ensure the well-definedness of the levels and gradient of abstraction in any application, and to avoid any hint of relativism, some degree of precision is required. Such precision is always possible in discrete systems (where, by definition, only finitely many alternatives are possible for the outcome of each observable), but may not be always possible in analogue systems. That is the reason for the unresolved nature of the fundamental problem of plectics in the case of analogue or hybrid systems, and the reason for our much more positive stance on applicability of the method of abstraction to discrete systems.

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Luciano Floridi and J.W. Sanders

If the method of abstraction does indeed have wide application, indications of its use – inchoate at the very least – should be evident in various areas. Part of the reason for the extent of examples in this paper is to follow up that point. Not surprisingly it is common to argue that complex systems are best studied at different levels of analysis (see, for example, [32, 23, 31, 40, 44, 11, 2, 37, 4, 18, 25, 47, 12]). One major success of the general method, from which we have derived inspiration, has been the area of Formal Methods in Computer Science [20, 22]. There, indisputably complex information systems are constructed to meet precise and demanding specifications; and once constructed, existing systems are understood and modified in the on-going task of (software) maintenance. The levels of abstraction are made precise, and at each level a mathematical specification is formulated of the system under construction or study. Techniques are developed and used to ensure that the product at one level of abstraction meets its more abstract specification, and that those techniques are incremental: the validity of the final implementation follows by combining the validity of each incremental synthesis. We have already mentioned that the fundamental definition of the method of abstraction is that of a ‘gradient of abstractions’. Embodied in it are the conditions required for the incremental synthesis, analysis or observation of successively finer levels of abstraction. The definition is founded on the concepts of ‘typed variable’ (a variable constrained to be of a certain type), ‘observable’ (an interpreted typed variable), ‘level of abstraction’ (a collection of observables), ‘behaviour’ (constrained observable values) and ‘moderated level of abstraction’ (level of abstraction predicated on behaviour). Those definitions are provided in Section 2, where they are accompanied by a variety of simple examples designed to familiarise the reader with the methodology. In Section 3 several substantial examples are analysed, to indicate how the method of abstraction may be applied in different areas. We consider the definition of agenthood, the Turing test, the definition of emergence, the definition of artificial life, and we discuss the consistency of the method of abstraction with the notion of quantum observation from Physics, and with the concept of decidable process from Computability.


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We are aware that we are proposing a new tool, while philosophers typically attempt to eliminate tools. Quine eliminates the analyticsynthetic distinction; Davidson eliminates conceptual schemes. In Section 4 we make the case for the method of abstraction in the face of concepts like levels of organisation, levels of explanation, conceptual schemes and pluralism, and we confront the problems of relativism and realism. Although some degree of rigour underlies the definitions on which the method of abstraction is founded, only the rudiments of mathematical notation are presupposed – the main concepts are introduced here without assuming any previous knowledge.

2 Definitions and preliminary examples In this section we define the simple concepts fundamental to the paper: typed variable, observable, level of abstraction (LoA), behaviour and gradient of abstraction (GoA). Some elementary examples are introduced in order to illustrate the content and variety of the definitions. More elaborate and probing examples are considered in section 3, once the new vocabulary has been established. 2.1 Typed variable A variable is a symbol that acts as a place-holder for an unknown or changeable referent. A ‘typed variable’ is to be understood as a variable qualified to hold only a declared kind of data. Definition. A typed variable is a uniquely-named conceptual entity (the variable) and a set, called its type, consisting of all the values that the entity may take. When notation is required, we shall write x: X to mean that x is a variable of type X. Two typed variables are regarded as equal if and only if their variables have the same name and their types are equal as sets. A variable that is not capable of being assigned well-defined values is said to constitute an illtyped variable (see example 2 in section 2.3).

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In positing a typed variable an important decision is being made about how its component variable is to be conceived. We shall be in a position to appreciate that fact after the next definition. 2.2 Observable The notion of an ‘observable’ is common in science, occurring whenever a (theoretical) model is constructed. The manner in which the features of the model correspond to the situation being modelled is usually left implicit in the process of modelling. In our approach it is important to make that correspondence explicit. We settle on the phenomenological word ‘system’ to stand for the object of study. This may indeed be what would normally be described as a system in science or engineering, but it may also be a domain of discourse, of analysis or of conceptual speculation. Definition. By an observable we mean an interpreted typed variable, that is, a typed variable together with a statement of what feature of the system under consideration it represents. Two observables are regarded as equal if and only if their typed variables are equal, they model the same feature and, in that context, one takes a given value if and only if the other does.

Being an abstraction, an observable is not necessarily meant to result from quantitative measurement or even perception. Although the ‘feature of the system under consideration’ might be empirical and physically perceivable, it might alternatively be an artefact of a conceptual model, constructed entirely for the purpose of analysis. We are not here concerned with inferential problems associated with the simultaneous observation of several system parameters or of the interaction between the observer and the observed, a point to which we return in section 3.5. (Our approach, in summary, is to model such situations using observables of the kind defined here but constrained appropriately.) An observable, being a typed variable, has specifically determined possible values. In particular: Definition. An observable is called discrete if and only if its type is finite: it has only finitely many possible values; otherwise it is called analogue.


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In this paper we are interested in observables as a means of describing behaviour at a quantified (though seldom numerical) level of abstraction; in general several observables will be employed. But before defining the relevant terms, let us consider some simple examples of the two concepts introduced so far. 2.3 Examples 1. Suppose we wish to study some physical human attributes. To do so we, in Oxford, introduce a variable, h, whose type consists of rational numbers. The typed variable h becomes an (analogue) observable once we decide that the variable h represents the height of a person, using the Imperial system (feet and parts thereof). To explain the definition of equality of observables, suppose that our colleague, in Rome, is also interested in observing human physical attributes and defines the same typed variable but decrees that it represents height in metres and parts thereof. Our typed variables are the same, but as observables they differ: for a given person the two variables take different representing values. This example shows the importance of making precise the interpretation by which a typed variable becomes an observable. 2. Consider next an example of an ill-typed variable. Were we to be interested in the rôles played by people in some community, we could not introduce an observable standing for those beauticians who depilate just those people who don’t depilate themselves (modernising a standard example [43]). For such a variable would not be well typed, by the standard and straightforward case analysis. Similarly, each of the standard antinomies [24] reflects an ill-typed variable. Of course the modeller is at liberty to choose whatever type befits the application and if that involves a potential antinomy then the appropriate type might turn out to be a non-well-founded set [3]. But in this paper we shall operate entirely in standard naive set theory.

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3. Suppose we were interested in wine. Observables relating to tasting wine include the attributes that commonly appear on ‘tasting sheets’: ‘nose’ (representing bouquet), ‘legs’ or ‘tears’ (viscosity), ‘robe’ (peripheral colour), ‘colour’, ‘clarity’, ‘sweetness’, ‘acidity’, ‘fruit’, ‘tannicity’, ‘length’ and so on, each with a determined type (in spite of a recent trend towards numeric values, these have not been standardised and so we leave to the reader the pleasant task of contemplating appropriate types; for a secondary source of inspiration we refer to tasting-related entries in [41]). If two wine tasters choose different types for, say, ‘colour’ (as is usually the case) then the observables are different, despite the fact that their variables have the same name and represent the same feature in reality. Indeed, as they have different types they are not even equal as typed variables. Information about how wine quality is perceived to vary with time – how the wine ‘ages’ [42] – is important for the running of a cellar. An appropriate observable is the typed variable a which is a function associating to each year y:Years a perceived quality a(y):Quality, where the types Years and Quality may be assumed to have been previously defined. Thus a is a function from Years to Quality, written a : Time → Quality . This example demonstrates that types are in general constructed from more basic types, and that observables may correspond to operations, taking input and yielding output. Indeed an observable may be of arbitrarily complex type. 4. The definition of an observable reflects a particular view or attitude towards the entity being studied. Most commonly it corresponds to a simplification, in which case nondeterminism, not exhibited by the entity itself, may arise. The method is successful when the entity can be understood by combining the simplifications. Let us consider an example. In observing a game of chess we would expect to record the moves of the game (done by recording the history of the game: move by move the state of each piece on the board is recorded – in English algebraic notation – by rank and file, as are recorded the piece being moved and the consequences of the move). Other observables


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might include the time taken per move; the body language of the players; and so on. Let us consider, for the sake of argument, a grossly simplified observation. Suppose we were able to view the chessboard by looking just along files (the columns stretching from player to player); we are unable see the ranks or the individual squares. Files cannot sensibly be attributed a colour black or white, but each may be observed to be occupied by a set of pieces (namely those that appear along that file), identified in the usual way (king, queen and so on down to pawn). With this view a move may be observed by the effect it has on the file of the piece being moved. For example a knight moves one or two files either left or right from its starting file. A bishop is indistinguishable from a rook which moves along a rank; and a rook which moves along a file appears to remain stationary. Whether or not a move results in a piece being captured, appears to be nondeterministic. Whilst the ‘underlying’ game is virtually impossible to reconstruct, each state of the game and each move (i.e. operation on the state of the game) can be ‘tracked’ with this dimensionally impoverished family of observables. If a second view is then taken corresponding instead to rank, and the two views combined, then the original game of chess can be recovered (since each state is determined by its rank and file projections, and similarly for each move). The two disjoint observations together reveal the underlying game. 5. The degree to which a type is appropriate depends on its context and use. For example, to describe the state of a traffic light in Rome we might decide to consider an observable colour of type {red, amber, green} that corresponds to the colour indicated by the light. That option abstracts the length of time for which the particular colour has been displayed, the brightness of the light, the height of the traffic light, and so on. That is why the choice of type corresponds to a decision about how the phenomenon is to be regarded. To specify such a traffic light for the purpose of construction, a more appropriate type would comprise a numerical measure of wavelength. In Oxford, the type of colour would be a little more complex, since – in addition to red, amber and green – red and am-

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ber are displayed simultaneously for part of the light’s cycle. So, an appropriate type would be {red, amber, green, red-amber}. 6. The design of a database is a special case of the definition of a collection of observables. In a database an observable is called a key, its correspondence with reality is left implicit (though it is usually reflected in the name) and its type is inferred from either the values it takes or from its declaration in the database programming language. The type ‘finite string of characters’ is frequently used, often being the most appropriate concrete method of description. Examples include names, addresses and such like. So too with an observable in general: it is sometimes preferable not to provide in advance all the possible outcomes (i.e. a type) but simply to define its type to consist of all finite sequences of characters, with each value equal to a character string. That definition reflects a decision that, although the observable is well-typed, its actual type is not of primary concern. 7. The method proposed in this paper will seem familiar not only to scientists. With the passing of the classical view that a literary text has the absolute view determined by its author, deconstruction promotes the revealing of other, sometimes implicit, values embodied in a text. As Fish ([13], page 327) writes: Interpretation is not the art of construing but the art of constructing. Interpreters do not decode poems; they make them.

In that art the interpreter’s view is determined by a collection of observables, documented in order to make explicit a view taken. To consider a popular example [49], when interpreting the line A rose is a rose is a rose

we might choose observables that relate to the semantics of the text, or that reflect syntax only, the rhythmic nature of the words. We might further choose observables that permit the reader to make a comparison with the rhythms of a Flamenco dancer, and so forth.


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Thus a reading of Shakespeare might be based on observables that relate to literary style; or to history; or economics; or social (in particular gender) issues; or theatrical concerns; and so on. A collection of observables provides a particularly clear way of acknowledging and documenting the different views and how they interact. Part of the early appeal of deconstruction [45, 46] was that, by revealing values implicit in the text, interesting behaviour appeared as emergent (in a sense to be made explicit later): it required the discovery of new observables to make it explicable. It is hardly surprising that the use of observables has powerful application in textual analysis, in view of its close relationship with the science of anthropology: Doing ethnography is like trying to read (in the sense of ‘construct a reading of’) a manuscript […] [17]

2.4 Level of Abstraction, LoA Any collection of typed variables can, in principle, be combined into a single ‘vector’ observable whose type is the Cartesian product of the types of the constituent variables. In the wine example, the type Quality might be chosen to consist of the Cartesian product of the types Nose, Robe, Colour, Acidity, Fruit and Length. The result would be a single, more complex, observable. In practice, however, such vectorisation is unwieldy since to express a constraint on just some of the observables would require projection notation to single out those observables from the vector. Instead we base our approach on a collection of observables. Since the very choice of a collection of observables characterises an approach to some phenomenon, it warrants a name. Definition. A level of abstraction, LoA, is a finite but non-empty set of observables. No order is assigned to the observables, which are expected to be the building blocks in a theory characterised by their very definition. A LoA is called discrete [respectively analogue] if and only if all its observables are discrete [respectively analogue]; otherwise it is called hybrid.

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In considering wine, different LoAs are appropriate for different purposes. To evaluate a wine the ‘tasting LoA’, consisting of observables like those mentioned in the previous section, would be relevant. For the purpose of ordering wine, a ‘purchasing LoA’ would be appropriate, containing observables like ‘maker’, ‘region’, ‘vintage’, ‘supplier’, ‘quantity’, ‘price’, and so on; but here the ‘tasting LoA’ would be irrelevant. For the purpose of storing and serving wine, the ‘cellaring LoA’ would be relevant, containing observables for ‘maker’, ‘type of wine’, ‘drinking window’, ‘serving temperature’, ‘decanting time’, ‘alcohol level’, ‘food matchings’, ‘quantity remaining in the cellar’, and so on. The traditional sciences tend to be dominated by analogue LoAs, the humanities and information science by discrete LoAs and mathematics by hybrid LoAs. We are about to see why the resulting theories are fundamentally different. 2.5 Behaviour The definition of observables is only the first step in studying a system at a given level of abstraction. The second step consists of deciding what relationships hold between the observables, a task we support in this section by introducing the concept of system ‘behaviour’. We shall see that it is the fundamentally different ways of describing behaviour in analogue and discrete systems which accounts for the differences in the resulting theories. Not all values exhibited by combinations of observables in a LoA may be realised by the system being modelled. For example if the four traffic lights at an intersection are modelled by four observables, each representing the colour of a light, the lights can not in fact all be green together. In other words, the combination in which each observable is green cannot be realised in the system being modelled, although the types chosen allow it. Similarly the choice of types corresponding to a rank-and-file description of a game of chess allow the white queen and the white king to be placed on any square, whereas in fact they cannot occupy the same square.


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Some technique is therefore required to describe those combinations of observable values that are allowed. The most general method is simply to describe the allowed combinations of values. Such a description is determined by a predicate, whose allowed combinations of values we call the ‘system behaviours’. Definition. A behaviour of a system at a given LoA is defined to consist of a predicate whose free variables are observables at that LoA. The substitutions of values for observables that make the predicate true are called the system behaviours. A moderated LoA is defined to consist of a LoA together with a behaviour at that LoA.

To pursue the previous examples, in reality human height does not take arbitrary rational values: it is positive and bounded above by (say) nine feet. The variable h representing height is therefore constrained to reflect reality by defining its behaviour to consist of the predicate 0