1.1 Language and mind

We do not study language, or a language, only for the sake of knowing more about language or languages, but because language, any language, tells us something about some of the workings of the human mind, a view that is now common among cognitive linguists. This is not quite as straightforward as it might seem and needs a little clarification. The term ‘mind’ is deliberately chosen: studying language does not tell us about the brain, at least only in an extremely indirect manner. Investigating language tells us something about the sorts of things the human mind does or can do. I certainly do not mean that mind is independent of brain, but investigating language does not tell us directly about neurons and neuronal networks. And by investigating language, I do not mean investigating languages or investigating examples of the use of language in attested contexts.

Language is a tool for stimulating silent conceptualisation, and I do not mean only conscious conceptualisation. What sort of conceptualisations is language capable of stimulating? If we look into this question we are going to find, if we are lucky, only a small part of the conceptual activity that takes place in our minds. The term ‘conceptualisation’ is being used here in preference to ‘meaning’, since this term is often connected to the view that language elements (lexical items, constructions) have meaning in some independent language system in the mind. However, along with other cognitive linguists, I am adopting the view that there is no independent linguistic–semantic inventory, but rather that there is conceptualisation that can happen independently of language, but which language accesses. I also take the view that the particular structure of a language – its lexis and grammar – does not privilege or make more accessible or delimit some particular kinds of conceptualisation. That is, I think there is no clear evidence for the strong Whorf hypothesis – that, for example, a classifier language will make speakers more sensitive to, or restrict speakers to, certain kinds of conceptualisation. This is not to say that some empirical work does not give convincing evidence of certain limited effects (Lucy Reference Lucy1992, Levinson Reference Levinson1996, Reference Levinson2003), but these effects appear not to be extensive or not sufficient to make us think that linguistic meaning is something separate from a universal conceptualising ability.¹

This view is of course different from the cognitive view inspired by Chomsky and his followers, according to which language is autonomous, modular, interfaced with a semantic component. I am not, however, throwing out the idea that the language system may be modular in some sense – but it is not sealed off or encapsulated (in the sense of Fodor Reference Fodor1983): rather, it is linked inextricably to non-linguistic cognitive abilities.

It is also different from denotational semantics, which is why I use the term ‘semantics’ throughout this book sparingly, usually to refer to the natural concepts associated with lexical items, that is, their ‘semantic frame’ (Fillmore Reference Fillmore1982b). Since the 1980s cognitive scientists and linguists have realised that the meanings of words and constructions in languages are in some way built up on the basis of our embodied experience of physical space (Lakoff and Johnson Reference Lakoff and Johnson1980, Reference Lakoff and Johnson1999, Johnson Reference Johnson1987, Pinker Reference Pinker1997). Pinker sums up some of the conceptual elements that are important for the geometrical approach:

‘Space’ and ‘force’ are crucial to the approach developed in the present book, because they can be, and commonly are, formalised in geometric terms, as vectors. The nature of the linkage between spatial cognition and the structures of language is similar to metaphorical linkage: spatial, motor and visual systems have some sort of counterparts in the linguistic–communicative system. For example, humans can attend to detail or take in a gross gestalt, and linguistic constructions and lexical items also permit this kind of alternating focus. Depth perception enables us to judge one object as more distant from the self than another object opposed to another: similarly, linguistic structures can place information in the foreground or background that is, topicalise some information relative to other information (cf. Talmy Reference Talmy2000 [1983], Langacker Reference Langacker and Davis1995, Croft and Cruse Reference Croft and Cruse2004). Spatial expressions, and the spatial perceptual–cognitive systems, provide a source domain for linguistic expressions of time, in many, perhaps all, languages (Lakoff and Johnson Reference Lakoff and Johnson1980, Reference Lakoff and Johnson1999, Haspelmath Reference Haspelmath1997, Boroditsky Reference Boroditsky2000, Evans Reference Evans2004, Evans and Chilton Reference Evans and Chilton2010).

Such views as these are not uncommon amongst linguists. However, the apparatus that I am proposing is an unusual one in linguistics, though it is hinted at in many places, as will become clear by way of the references that I shall give as we move along. Moreover, the formal apparatus itself, as will be seen, is one that springs from natural foundations – namely, the fact (I am assuming it is a fact) that spatial location and orientation are experiences that organisms must cognise for their survival. We shall see, using the apparatus that I will outline, that these concepts of physical space can be used for other kinds of fundamental concept that languages use as their bedrock and about which they enable humans to communicate among one another. This idea is not new in cognitive linguistics (for example, Lakoff and Johnson Reference Lakoff and Johnson1980, Reference Lakoff and Johnson1999, Langacker Reference Langacker1987, Reference Langacker1991, Talmy Reference Talmy2000; cf. also ‘localists’, for example, Anderson Reference Anderson1971, Lyons Reference Lyons1977) or even outside it (cf. Jackendoff, for example, Reference Jackendoff1976, Reference Jackendoff2002a, Frawley Reference Frawley1992). What I'm suggesting is that there is a derived conceptual foundation that uses spatial cognition of various kinds, and that language in turn uses it as the most fundamental requirement for communication. Actually, this means that language is not simply a ‘window’ on the mind, but that investigating language and conceptualisation together, using the apparatus I describe below, will tell us something about both. It is perhaps not surprising that we need to view language through different kinds of apparatus and that indeed language may, after all, have components and modules that that require different instruments of investigation.

1.2 Formalisation

Linguistics is bedevilled by a proliferation of formalised notations and diagrams, the most systematic of which are bound up with propositional and predicate calculus and other elements of mathematical logic and reasoning. Others are ad hoc, intuitive and iconic. A few have origins in Euclidean geometry.³

1.3 Using geometry

Whereas Croft, and to a lesser extent Langacker, borrow elements of geometry to model meanings of linguistic expressions, the present book starts at a more fundamental level. To begin with, I attempt to apply geometry to the sorts of meaning that one would expect geometry to be well equipped to describe – spatial meanings. Then, building on the insight of many linguists that spatial meanings are somehow crucial for many kinds of more abstract meaning, I look into the potential of geometry for describing apparently non-spatial meanings, which may nonetheless be derived from, or linked to, spatial conceptualisation. This in turn means taking seriously certain elements of Euclidean (and Cartesian) geometry and developing out of them a descriptive model of language-based conceptual space. The elementary geometry of coordinate systems, vectors and transformations then becomes a heuristic for exploring linguistic–conceptual space as well as a descriptive model-building project.

The theoretical ideas that I develop in the present book draw on insights and methods found in the broad approaches outlined in the preceding sections, as well as many other meaning-based approaches to language. What I am putting forward is a research framework that investigates the applicability of coordinate vector geometry for the description and perhaps explanation of certain kinds of linguistic meaning. I am not suggesting that all linguistic meaning can be described or explained in this way, only that the range of lexical–constructional meanings that can be described in this way is wider than linguists may have thought.

There is nothing new about the use of coordinate geometry to account for spatial meanings, for example in prepositions and deictic expressions. Bühler, in writing of the ‘deictic field’, clearly has Cartesian coordinate geometry in mind. His description of what he calls ‘the here-now-I system of subjective orientation’ begins starkly:

Bühler's idea that deixis is one fundamental aspect of human language and has something to do with self, space and time leads naturally to geometric modelling. It is in some ways also the starting point for the Deictic Space Theory that I shall outline in this book. However, the space in question will be three-dimensional and the three axes will be defined conceptually in ways that are distinct from Bühler's ideas.⁵ Bühler also spoke of ‘displacement’⁶ of the deictic centre away from the here and now of utterance (Bühler Reference Bühler and Goodwin1990 [1934]: 136–57). As will be seen in later chapters such deictic displacements can be modelled geometrically as transformations of coordinate systems. In more recent linguistic research, spatial expressions that relate objects to reference points other than self, particularly in spatial prepositions, have also been treated in explicitly geometric terms.

Highly abstract geometry and topology have been used to model language phenomena by Thom (Reference Thom, Haeflinger and Narasimham1970) and Petitot (e.g. Reference Petitot, Port and van Gelder1995). Gärdenfors (Reference Gärdenfors2000) argues for the geometrical representation of conceptualisation. Gallistel (Reference Gallistel1990) argues that low-dimensioned geometries are fundamental in the vertebrate nervous system. Of particular relevance is the empirical work of Paillard and Jeannerod amongst other neuroscientists, which shows that visuo-spatial processing and actions such as reaching and grasping depend on the brain's ability to work with egocentric coordinates systems whose origin is located at different parts of the body (shoulder, hand, retina, for example) and in different sensory modalities (Paillard Reference Paillard and Paillard1991, Jeannerod Reference Jeannerod1997).⁷ This work builds on other important empirical work (Ungerleider and Mishkin Reference Ungerleider, Mishkin, Ingle and Mansfield1982, Goodale and Milner Reference Goodale and Milner1992, Milner and Goodale Reference Milner and Goodale1995) that indicates two pathways in visual processing, one relating to the identification of objects and the other locating objects in body-centred frames of reference. Such finding are highly suggestive both for investigating the linguistic phenomenon of deixis and for using a geometrical approach to linguistic description.

While the work mentioned so far focuses on egocentric coordinate systems, experimental findings of O'Keefe and Nadel (Reference O’Keefe and Nadel1978) point to the brain's ability to represent allocentric and absolute maps of an organism's spatial environment, as noted by Levinson (Reference Levinson2003: 9–10). Levinson argues strongly for the need to recognise non-egocentric systems in the spatial semantics of certain languages. However, there need be no conflict between the two research emphases. It would appear likely that both egocentric (deictic) coordinate systems and non-egocentric systems are instantiated in neural structures and mental representations, including linguistic ones.

I shall not attempt to explore this body of scientific work further, nor attempt to apply any but the simplest geometrical concepts. Sufficient has, I hope, been said to justify the scientific appropriateness of a geometric approach in cognitive linguistics.

1.4 Space, situation and deixis

In the study of spatial semantics the most precise use of geometry, in particular of coordinate systems, has been Levinson (Reference Levinson2003). As will be clear in Chapter 2, geometrical descriptions of linguistic expressions referring to physical space provide a foundation for the present book, though the aim is to move into an abstract conceptual geometry that can handle linguistic expressions and constructions that are not spatial in any obvious sense.⁸

This book is not concerned with spatial expressions as such – and it is important to emphasise this at the outset. But there is now a rich body of research into the precise nature of spatial meaning in language and the precise nature of the relationship between language and spatial cognition (see Evans and Chilton Reference Evans and Chilton2010 for a sample of recent theoretical and experimental work). In this work, while geometric aspects (or geometric descriptions) of spatial cognition are well established, geometry is not the whole story. In the case of spatial prepositions, which have the most obvious links to spatial cognition, it is now widely recognised that for spatial prepositions a coordinate geometry alone is not adequate, though it is needed and is in some sense fundamental. Many linguists and psycholinguists have shown the crucial role played also by the shape, presence or absence of surface contact and the function of reference objects. What is loosely called ‘topology’ is also relevant (e.g. for prepositions like in which involve a concept of bounded space), although topology is indeed a kind of geometry (Herskovits Reference Herskovits1986, Vandeloise Reference Vandeloise1991, Carlson-Radvansky et al. Reference Carlson-Radvansky, Covey and Lattanzi1999, Coventry et al. Reference Coventry, Prat-Sala and Richards2001, Tyler and Evans Reference Tyler and Evans2003). It is also the case that when prepositions (e.g. in front of) locate objects relative to a reference point, it is not actually a point that is at issue, but a spatial region around the reference landmark. Such conceptualised regions are affected by the function and shape of the landmark itself (see for example Carlson Reference Carlson2010).

But coordinate systems (or reference frames) remain the foundation of the cognitive system underlying spatial orientation. The crucial point is that a rather simple geometrical formalism is an economical way, and indeed a natural way, of describing fundamental spatial meanings. On a more abstract level, what coordinate geometry enables us to do is to analyse ‘point of view’ or ‘perspective’, both in a physical–spatial and in various abstract sense. As is now recognised (see for example Tomasello Reference Tomasello1999), one of the reasons why language is so complex and apparently redundant if we look at it from a truth-conditional position may well be that language ‘is designed’ to permit viewpoint alternation. Euclidean geometry is a well-understood way to analyse space the way humans experience it. This book is proposing that it is also a natural way to describe non-spatial linguistic meanings that are nonetheless in some way rooted in human spatial experience.

Once we introduce the geometry of coordinate systems it is a natural step to introduce vectors – mathematical objects conventionally visualised as arrows that have (i) direction and (ii) magnitude. In a coordinate system the position of a point can be given by the length and direction of a vector from the origin to the point. This approach can be used for explicating the denotation of certain prepositions, including in front of, by specifying a vector space in which all vectors have the same origin in some coordinate system (O'Keefe Reference O’Keefe1996, Reference O’Keefe2003, O'Keefe and Burgess Reference O’Keefe and Burgess1996, Zwarts Reference Zwarts1997). This space will be the ‘search domain’ within which an object can be said to be, for example, ‘in front of John’.

Cognitive linguists often assert that linguistic meaning is situated and/or embodied (e.g. Lakoff and Johnson Reference Lakoff and Johnson1999, Croft and Cruse Reference Croft and Cruse2004, and, from a philosopher's perspective, Johnson Reference Johnson1987). To be situated means, for human language users, that our use of language will always assume and/or refer to the place and time of speaking, and will take a perspective on the surrounding physical environment. The most striking way in which situatedness is built into the ‘design’ of language is the phenomenon of deixis. The etymological connection with the human gesture of pointing is revealing in the cognitive perspective, suggesting as it does the embodied nature of linguistic expression. Pointing itself is a way of establishing joint attention (as Tomasello emphasises: Reference Tomasello1999, Reference Tomasello2008) and can only work because the direction of the extended arm has an origin in the body of the pointer, who in turn has a spatial location relative to the interlocutor. In linguistic expression, the origin is the speaker, and a pointing gesture may or may not be present. Crucially, deixis is not solely a spatial matter. The definition offered by Lyons gives the essence of the concept but mentions explicitly only the further aspect of time:

Spatial and temporal deixis, as reflected in words like here, there, now and then and demonstratives such as English this and that, may indeed be the most frequent and salient deictic expressions in languages. Spatial deixis, however, may be more fundamental, because it is capable of metaphorical projection into abstract domains. Linguists have also added distinct, but related, deictic categories. Person deixis, implicit in Lyons's definition, is expressed by pronouns, particularly the first and second pronouns, whose referential meaning switches according to the speaker. Social ‘distance’ and status have also been held to be forms of deixis.⁹

But there is another dimension of deixis that may be considered fundamental. In the prototypical situation of utterance the persons, objects, events, processes and activities being talked about are not only positioned relative to the speaker, by the speaker. Situated speakers also take an epistemic stance. They assess the likelihood of the existence of persons and objects, the likelihood of the occurrence of events, the plausibility of persons and objects having certain properties, having certain thoughts and intentions, and so forth. And speakers do this by reference to what they take to be real, i.e. known to them individually. Pragmatically no utterance is modally innocent,¹⁰ whether modal expressions are present or not in the utterance. An utterance without modal morphemes is taken to be an assertion of a known (to the speaker) reality. Expressions of degrees of certainty and possibility (expressions such as might, may, must, etc.) are thus relative to a fixed reality point in the speaker's mind and can be regarded as deictic. A deictic definition of modality is outlined in Frawley (Reference Frawley1992). The epistemic dimension is the third fundamental dimension in the three-dimensional space I shall develop in Chapter 2.

These three dimensions are always ‘anchored’ at the speaking self or in the mind of the speaking self. It is only when we use language in context – that is, in discourse – that we produce deixis. For this reason, I have sometimes called the theory outlined in this book Discourse Space Theory (DST), but because deixis is the overarching concept of the theory it makes better sense now to call it Deictic Space Theory. Throughout the book I shall simply refer to it as DST.

The three-dimensional space of DST is an abstract space. However, the theory is grounded in the embodied cognition of physical space, Euclidean three-dimensional space as it is, or may be, represented by the human nervous system (cf. O'Keefe and Nadel Reference O’Keefe and Nadel1978, Gallistel Reference Gallistel1990, Paillard Reference Paillard and Paillard1991, O'Keefe Reference O’Keefe1996, Reference O’Keefe2003, O'Keefe and Burgess Reference O’Keefe and Burgess1996, Jeannerod Reference Jeannerod1997, Gärdenfors Reference Gärdenfors2000). This approach has two important implications. The first is that in order to model and explore the way language is grounded in the experiential situatedness of human individuals, we need a notation with which to work. The second implication is that any such notation will need to be well motivated, by which is meant here that it will need to be more or less directly linked with the way we model the behaviour of human cognition and perception in general.

The main aim of this book is to propose that rather simple vector geometry in coordinate systems allows us to model some fundamental properties of human language – or, to be more precise, the kinds of conceptualisations that human languages prompt their speakers and hearers to use in communication. The aim is not to account for all properties of human language, but to explore those aspects that the geometrical approach illuminates, and these properties we will expect to be those that are adapted to the situated use of language.

One way of seeking to justify the proposal is to investigate the extent of its efficiency for descriptively capturing the linguistic data – the usual procedure for any proposed linguistic theory (or any other theory, for that matter). However, it has long been argued that this is not enough, and in cognitive linguistics, we should want our descriptive apparatus to go some way towards interfacing with the descriptive approaches of neighbouring sciences, such as cognitive science and neuroscience.

This does not mean that cognitive linguists need to be specialists in two disciplines, but it does mean that we should try to propose descriptive means that can be assessed by the appropriate specialists. One way to do this is to use the common language of mathematics, in the present instance the mathematics of space, namely, geometry. The chapters that follow try to demonstrate that there are advantages in thinking in terms of coordinate systems, one of the key advantages being that this enables us see the individual cogniser and speaker as situated at the intersection of different dimensions. What the DST model itself consists of is an abstract three-dimensional space in which we use geometric (also known as Euclidean) vectors¹¹ to represent (i) spatial locations (position vectors), (ii) movements (translations) and (iii) forces. The use of vectors in linguistics is also not entirely new. It is position vectors that are used by some authors to model (at least partially) the spatial regions picked out by spatial prepositions (Zwarts Reference Zwarts1997, Reference Zwarts2003, Reference Zwarts2005, Zwarts and Winter Reference Zwarts and Winter2000). O'Keefe, a cognitive neuroscientist, uses vectors for describing spatial prepositions, which he links with the neural mechanisms of location and navigation (O'Keefe and Nadel Reference O’Keefe and Nadel1978, O'Keefe Reference O’Keefe1996, Reference O’Keefe2003, O'Keefe and Burgess Reference O’Keefe and Burgess1996; also Burgess Reference Burgess2002, Burgess et al. Reference Burgess, Jeffery and O’Keefe2003). Zwarts (Reference Zwarts2010) uses force vectors to model certain types of spatial preposition that involve more than relative position and Wolff uses force vectors to model causative verbs, making more mathematically explicit Talmy's (Reference Talmy2000 [1988]) iconic drawings illustrating ‘force dynamics’ (Wolff and Zettergren Reference Wolff, Zettergren, Gray and Schunn2002, Wolff Reference Wolff2007).

It is important to make clear from the outset that I am talking here not only of lexical expressions that conventionally encode meanings having to do literally with physical positions, movements and forces. The meanings of the words and constructions that we shall investigate are related metaphorically to physical spatial meanings and many are usually thought of as ‘abstract’, if not as meaningless or purely functional. In fact, the three-dimensional conceptual space in which we shall operate is itself derived metaphorically from humanly cognised physical space. The idea that abstract meanings are metaphorically related to basic physical spatial ones is scarcely a new idea in cognitive linguistics, but the point here will be to use elementary geometry to describe the spatial and trans-spatial meanings at issue. This exploration will bring together and offer a unified account of, an unexpectedly diverse range of lexical and grammatical phenomena.

A property of coordinate systems, which we shall make use of, is that they can be transformed: that is, the origin can be shifted relative to the base set of coordinates. There may be limitations on what we can do in this way. In general, however, adopting the type of geometric formalism just outlined does manage to integrate the notions of situatedness, embodiment and speaker–hearer orientation, by taking, as its starting point, the speaker in a physical orientation to physical space as perceived and conceived by human beings.

As already stated, the theory is built on the idea that not only any and all utterances and their associated conceptualisation are relativised to a minimal communication pair (speaker and hearer) but also that they integrate three conceptual dimensions. These dimensions are scalar: attentional distancing from S, temporal distancing from S and epistemic-modal distancing from S. Here and throughout the book S stands for speaker, self and the experiencing subject. The first dimension, referential distancing, is based on the perceptual and cognitive fact that in binocular vision, we see objects in perspective, as closer or more distant, foregrounded or backgrounded. Moreover, we can apply attentional cognition to these objects. Similarly, in conceptualising entities we are talking about, whether they are physical and visually perceptible or not, we can conceptualise such entities as conceptually closer and/or of greater attentional interest. The second dimension is temporal distancing from S. I will not attempt to resume the entire history of thinking about time, in philosophy, theology, physics and linguistics. More will be said in Chapter 2, but to anticipate now, the temporal dimension is framed in DST as relative distance from the now of S. There is an important area of discussion that will be touched on in Chapter 2 concerning the relationship between linguistically mediated conceptualisations of time, non-linguistic conceptualisation of time and ‘real’ time, the time of the space-time continuum studied by physicists (see for example the important work of Jaszczolt Reference Jaszczolt2009). Suffice it here to note that DST claims that all utterances are not only events that occur in real time, but also always represent states, processes and events that are time-anchored relative to the time of utterance by S. The third dimension is epistemic distancing relative to S. The claim here is that epistemic cognition is as fundamental as the other two dimensions and is also necessarily involved in the conceptualisations associated with any and all utterances that represent states, processes and events. The epistemically modal dimension is modelled in DST in a particular manner that will be explained in more detail Chapter 2. Essentially, the idea is that what is closer to the cognising subject is more evident, more certain, while what is more distant is less certain, even counterfactual.¹² The claim is, again, that in modal cognition is always present in the conceptualisations associated with any and all utterances that represent happenings and states of affairs. Indeed, the three dimensions are integrated in such conceptualisations.