Conceptual Integration Networks

[Expanded web version, 10 February 2001]

Gilles Fauconnier & Mark Turner

The web page for research on conceptual integration is http://blending.stanford.edu

Published in Cognitive Science, 22(2) 1998, 133-187.

Abstract

Conceptual integration—"blending"—is a general cognitive operation on a par with analogy, recursion, mental modeling, conceptual categorization, and framing. It serves a variety of cognitive purposes. It is dynamic, supple, and active in the moment of thinking. It yields products that frequently become entrenched in conceptual structure and grammar, and it often performs new work on its previously entrenched products as inputs. Blending is easy to detect in spectacular cases but it is for the most part a routine, workaday process that escapes detection except on technical analysis. It is not reserved for special purposes, and is not costly.

In blending, structure from input mental spaces is projected to a separate, "blended" mental space. The projection is selective. Through completion and elaboration, the blend develops structure not provided by the inputs. Inferences, arguments, and ideas developed in the blend can have effect in cognition, leading us to modify the initial inputs and to change our view of the corresponding situations.

Blending operates according to a set of uniform structural and dynamic principles. It additionally observes a set of optimality principles.

Contents

I. Introduction

II. An illustration

III. The network model of conceptual integration

IV. Applications

V. Advanced aspects of the network model

VI. Optimality principles

VII. Additional dimensions of conceptual integration

VIII. Summary and further results

IX. Conclusion

I. Introduction

Much of the excitement about recent work on language, thought, and action stems from the discovery that the same structural cognitive principles are operating in areas that were once viewed as sharply distinct and technically incommensurable. Under the old view, there were word meanings, syntactic structures, sentence meanings (typically truth-conditional), discourse and pragmatic principles, and then, at a higher level, figures of speech like metaphor and metonymy, scripts and scenarios, rhetoric, forms of inductive and deductive reasoning, argumentation, narrative structure, etc. A recurrent finding in recent work has been that key notions, principles, and instruments of analysis cut across all these divisions and in fact operate in non-linguistic situations as well. Here are some of them:

Frames structure our conceptual and social life. As shown in the work of Fillmore, Langacker, Goldberg, and others, they are also, in their most generic, and schematic forms, a basis for grammatical constructions. Words are themselves viewed as constructions, and lexical meaning is an intricate web of connected frames. Furthermore, although cognitive framing is reflected and guided by language, it is not inherently linguistic. People manipulate many more frames than they have words and constructions for.

Analogical mapping, traditionally studied in connection with reasoning, shows up at all levels of grammar and meaning construction, such as the interpretation of counterfactuals and hypotheticals, category formation , and of course metaphor, whether creative or conventional.

Reference points, focus, viewpoints, and dominions are key notions not only at higher levels of narrative structure, but also at the seemingly micro-level of ordinary grammar, as shown convincingly by Langacker 1993, Zribi-Hertz 1989, Van Hoek 1997, Cutrer 1994, among others.

Connected mental spaces account for reference and inference phenomena across wide stretches of discourse, but also for sentence-internal multiple readings and tense/mood distributions. Mappings at all levels operate between such spaces, and like frames they are not specifically linguistic. (Fauconnier 1997, Dinsmore 1991, Cutrer 1994, Fauconnier and Sweetser, 1996).

Connectors and conceptual connections also operate at all levels, linking mental spaces and other domains for coreference, for metonymy (Nunberg 1978), and for analogy and metaphor (Turner 1991, Sweetser 1990).

There are other notions that apply uniformly at seemingly different levels, such as figure/ground organization (Talmy 1978), profiling, or pragmatic scales.Running through this research is the central cognitive scientific idea of projection between structures. Projection connects frames to specific situations, to related frames, and to conventional scenes. Projection connects related linguistic constructions. It connects one viewpoint to another and sets up new viewpoints partly on the basis of old. It connects counterfactual conceptions to non-counterfactual conceptions on which they are based. Projection is the backbone of analogy, categorization, and grammar.

In the present study, we show that projection typically involves conceptual integration. There is extensive previous research on varieties of projection, but not on conceptual integration. Empirical evidence suggests that an adequate characterization of mental projection requires a theory of conceptual integration. We propose the basis for such a theory and argue that conceptual integration—like framing or categorization—is a basic cognitive operation that operates uniformly at different levels of abstraction and under superficially divergent contextual circumstances. It also operates along a number of interacting gradients. Conceptual integration plays a significant role in many areas of cognition. It has uniform, systematic properties of structure and dynamics.

The nature of mapping between domains has enjoyed sustained attention as a central problem of cognitive science, and voluminous literatures have developed in this area, including studies by those who call their subject "analogy" or "similarity" (e. g., Hofstadter 1985, 1995a, Mitchell 1993, French 1995, Keane, Ledgeway, and Duff 1994; Holyoak and Thagard, 1989, 1984; Forbus, Gentner, and Law, 1994; Gentner 1983, 1989; Holland, Holyoak, Nesbett, and Thagard, 1986), studies by those who call their subject "metaphor" (e.g., Lakoff and Johnson 1980; Lakoff and Turner 1989; Sweetser 1990; Turner 1987; Indurkhya 1992; Gibbs 1994) and studies that consider cross-domain mapping in general (e.g., Fauconnier 1997, Ortony 1979a, 1979b, Glucksberg and Keysar 1990, Turner 1991).

Our immediate goal is not to take a stand on issues and problems of cross-space mappings. Those issues are many and the debates over them will continue and will be further enriched, we hope, by taking blending into consideration. What we will be suggesting is that models of cross-space mapping do not by themselves explain the relevant data. These data involve conceptual integration and multiple projections in ways that have typically gone unnoticed. Cross-space mapping is only one aspect of conceptual integration, and the existing body of research on the subject overlooks conceptual integration, which it is our intention to foreground and analyze here. As we move through the data that crucially involves both cross-space mapping and conceptual integration, we will remark that much of it is neither metaphoric nor analogical. [1]

We take it as an established and fundamental finding of cognitive science that structure mapping and metaphorical projection play a central role in the construction of reasoning and meaning. In fact, the data we analyze shows that such projections are even more pervasive than previously envisioned. Given the existence and key role of such mappings, our focus is on the construction of additional spaces with emergent structure, not directly available from the input domains.

We also rely on another fundamental finding of cognitive science, the capacity for mental simulation, as demonstrated in Johnson-Laird (1983), Kahneman (1995), Grush (1995), Schwartz and Black (1996), Barsalou (1996) among others. In our analysis, the simulation capacity assists in the on-line elaboration of blended spaces ("running the blend"). There is the added twist that simulation can operate on mental spaces which need not have potential real world reference.

Our methodology and argumentation take the following form. Since the cognitive process of conceptual integration has been largely overlooked, it is useful to give evidence for its operation in a wide variety of areas. Since conceptual integration has uniform structural and dynamic properties, it is important to reveal this uniformity behind the appearance of observational and functional diversity. We proceed analytically and empirically, by showing that central inferences, emotions, and conceptualizations, not explained in currently available frameworks, are accounted for elegantly by the conceptual integration model. The argumentation often takes the following specific form: a particular process of meaning construction has particular input representations; during the process, inferences, emotions and event-integrations emerge which cannot reside in any of the inputs; they have been constructed dynamically in a new mental space—the blended space—linked to the inputs in systematic ways. For example, "They dug their own financial grave" draws selectively from different and incompatible input frames to construct a blended space that has its own emergent structure and that provides central inferences. In this case, the blended space has become conventional.

The diversity of our data (of which only a small sample appears in the present paper) is necessary to support our claim for generality. (In showing that cell division is a basic process, it is necessary to study it for many kinds of cells. In arguing that natural selection is a general principle, it is necessary to exemplify it for widely different organisms and species.) In arguing that conceptual integration is a basic cognitive operation, we must show that it operates in many different kinds of cases.

Conceptual blending is not a compositional algorithmic process and cannot be modeled as such for even the most rudimentary cases. Blends are not predictable solely from the structure of the inputs. Rather, they are highly motivated by such structure, in harmony with independently available background and contextual structure; they comply with competing optimality constraints discussed in section VI, and with locally relevant functional goals. In this regard, the most suitable analog for conceptual integration is not chemical composition but biological evolution. Like analogy, metaphor, translation, and other high-level processes of meaning construction, integration offers a formidable challenge for explicit computational modeling.

Special cases of conceptual blending have been discussed insightfully by Koestler (1964), Goffman (1974), Talmy (1977), Fong (1988), Moser and Hofstadter (ms.), and Kunda, Miller and Clare (1990). Fauconnier (1990) and Turner (1991) also contain analyses of such phenomena. All these authors, however, take blends to be somewhat exotic, marginal manifestations of meaning. We will show here that the process is in fact central, uniform, and pervasive.

The data and analysis we consider here suggest many psychological and neuropsychological experiments (Coulson 1997), but in the present work our emphasis is on the understanding of ecologically valid data. Research on meaning, we suggest, requires analysis of extensive ranges of data, which must be connected theoretically across fields and disciplines by general cognitive principles.

We start our report with an effective but somewhat idealized example of blending, in order to illustrate the issues and terminology. We then outline the general process of conceptual integration and the systematic dynamic properties of blends. We work through some case-studies in a variety of areas. Section VI presents the competing optimality principles under which conceptual integration operates.

II. An illustration

The riddle of the Buddhist monk

Consider a classic puzzle of inferential problem-solving (Koestler 1964):

A Buddhist monk begins at dawn one day walking up a mountain, reaches the top at sunset, meditates at the top for several days until one dawn when he begins to walk back to the foot of the mountain, which he reaches at sunset. Making no assumptions about his starting or stopping or about his pace during the trips, prove that there is a place on the path which he occupies at the same hour of the day on the two separate journeys.

Our demonstration of the power of blending is likely to be more effective if the reader will pause for a moment and try to solve the problem before reading further. The basic inferential step to showing that there is indeed such a place, occupied at exactly the same time going up and going down, is to imagine the Buddhist monk walking both up and down the path on the same day. Then there must be a place where he meets himself, and that place is clearly the one he would occupy at the same time of day on the two separate journeys.

The riddle is solved, but there is a cognitive puzzle here. The situation that we devised to make the solution transparent is a fantastic one. The monk cannot be making the two journeys simultaneously on the same day, and he cannot "meet himself." And yet this implausibility does not stand in the way of understanding the riddle and its solution. It is clearly disregarded. The situation imagined to solve the riddle is a blend: it combines features of the journey to the summit and of the journey back down, and uses emergent structure in that blend to make the affirmative answer apparent. Here is how this works.

Mental space. In our model, the input structures, generic structures, and blend structures in the network are mental spaces. Mental spaces are small conceptual packets constructed as we think and talk, for purposes of local understanding and action. Mental spaces are very partial assemblies containing elements, and structured by frames and cognitive models. They are interconnected, and can be modified as thought and discourse unfold. Mental spaces can be used generally to model dynamical mappings in thought and language. Fauconnier (1994), Fauconnier (1997), Fauconnier & Sweetser (1996).

In our diagrams, the mental spaces are represented by circles, elements by points (or sometimes icons) in the circles, and connections between elements in different spaces by lines. The frame structure recruited to the mental space is represented either outside in a rectangle or iconically inside the circle.

Input spaces. There are two input spaces. Each is a partial structure corresponding to one of the two journeys.

Figure 1

d1 is the day of the upward journey, and d2 the day of the downward journey. a1 is the monk going up, a2 is the monk going down.

Cross-space mapping of counterpart connections. There is a partial cross-space mapping between the input spaces. The cross-space mapping connects counterparts in the input spaces. It connects the mountain, moving individual, day of travel, and motion in one space to the mountain, moving individual, day, and motion in the other space.

Figure 2

Generic space. There is a generic space, which maps onto each of the inputs. The generic space contains what the inputs have in common: a moving individual and his position, a path linking foot and summit of the mountain, a day of travel. It does not specify the direction of motion or the actual day. (At this point in our exposition, it will not be clear why our model needs a generic space in addition to a cross-space mapping. Later, we will argue that powerful generic spaces can themselves become conventional and serve as resources to be drawn on in attempts to build new cross-space mappings in new integration networks.)

Figure 3

Blend. There is a fourth space, the blend. In the blend, the two counterpart identical mountain slopes are mapped onto a single slope. The two days of travel, d1 and d2, are mapped onto a single day d' and therefore fused. While in the generic space and each of the input spaces there is only one moving individual, in the blend there are two moving individuals. The moving individuals in the blend and their positions have been projected from the inputs in such a way as to preserve time of day and direction of motion, and therefore the two moving individuals cannot be fused. Input 1 represents dynamically the entire upward journey, while Input 2 represents the entire downward journey. The projection into the blend preserves times and positions. The blend at time t of day d' contains a counterpart of a1 at the position occupied by a1 at time t of d1, and a counterpart of a2 at the position occupied by a2 at time t of day d2.

Blended Space

Figure 4

Selective projection. The projection of structure to the blend is selective. For example, the calendrical time of the journey is not projected to the blend.

Emergent structure. The blend contains emergent structure not in the inputs. First, composition of elements from the inputs makes relations available in the blend that did not exist in the separate inputs. In the blend but in neither of the inputs, there are two moving individuals instead of one. They are moving in opposite directions, starting from opposite ends of the path, and their positions can be compared at any time of the trip, since they are traveling on the same day, d'.

Second, completion brings additional structure to the blend. This structure of two people moving on the path can itself be viewed as a salient part of a familiar background frame: two people starting a journey at the same time from opposite ends of a path. By completion, this familiar structure is recruited into the blend. We know, from "common sense," i.e. familiarity with this background frame, that the two people will necessarily meet at some time t' of their journey. We do not have to compute this encounter afresh; it is supplied by completion from a pre-existing familiar frame. There is no encounter in the generic space or either of the inputs, but there is an encounter in the blend, and it supplies the central inference.

Importantly, the blend remains hooked up to the Inputs, so that structural properties of the blend can be mapped back onto the Inputs. In our example, because of the familiarity of the frame obtained by completion, the inference that there is a meeting time t' with a common position p is completely automatic. The mapping back to the input spaces yields:

Figure 5

Since the projection of individuals into the blend preserves positions on the path, we "know" through this mapping that the positions of a1 and a2 are the "same" at time t' on the different days, simply because they are the same, by definition, in the frame of two people meeting, instantiated in the blend by their counterparts a1' and a2'.

It is worth emphasizing that the pragmatic incongruity in the blend of the same person traveling in two opposite directions and meeting himself is disregarded, because the focus of the problem is the meeting point and its counterparts in the Input spaces. Blends are used cognitively in flexible ways. By contrast, in examples we discuss later, similar incongruities in the blend get highlighted and mapped back to the Inputs for inferential and emotional effect. Incongruity makes blends more visible, but blends need not be incongruous—incongruity is not one of their defining characteristics.

Notice also that in this blend, some counterparts have been fused (the days, the path on the different days, and the corresponding times on different days), others have been projected separately (the monk on the way up, the monk on the way down, the directions of motion). Projection from the Inputs is only partial—the specific dates of the journeys are not projected, nor the fact that the monk will stay at the top for a while after his upward journey. But the blend has new "emergent" structure not in the Inputs: two moving individuals whose positions can be compared and may coincide, and the richer frame of two travelers going in opposite directions on the same path and necessarily meeting each other. This emergent structure is crucial to the performance of the reasoning task.

Rather amazingly, the Buddhist monk blend shows up in real life. Hutchins (1995) studies the fascinating mental models set up by Micronesian navigators to sail across the Pacific. In such models, it is the islands that move, and virtual islands may serve as reference points. Hutchins reports a conversation between Micronesian and Western navigators who have trouble understanding each other's conceptualizations. As described in Lewis (1972), the Micronesian navigator Beiong succeeds in understanding a Western diagram of intersecting bearings in the following way:

He eventually succeeded in achieving the mental tour de force of visualizing himself sailing simultaneously from Oroluk to Ponape and from Ponape to Oroluk and picturing the ETAK bearings to Ngatik at the start of both voyages. In this way he managed to comprehend the diagram and confirmed that it showed the island's position correctly. [the etak is the virtual island, and Ngatik is the island to be located.]

Previous insightful work by Kahneman (1995), Schwartz and Black (1996), Barsalou (1996), has emphasized the role of imaginative mental simulation and depiction in making inferences about physical scenarios. In the riddle of the Buddhist Monk, the physical system we are interested in consists of the sequence of the monk's departing, traveling up the hill, reaching the top, waiting, departing, traveling down the hill, and reaching the bottom. Imagining a mental depiction of this scenario does not solve the riddle, but representing it isomorphically as two input spaces to a blend and imagining a mental depiction of that blend does indeed create an event of encounter in the blend which points to a solution, not for the blend, but for the input spaces and therefore identically for the original scenario. Mental simulation, in this case, depends indispensably upon conceptual blending to provide the effective scenario to begin with.

III. The network model of conceptual integration

In this section, we present the central features of our network model, keyed to the illustration we have just given. In section V, we present advanced aspects of the model.

The network model is concerned with on-line, dynamical cognitive work people do to construct meaning for local purposes of thought and action. It focuses specifically on conceptual projection as an instrument of on-line work. Its central process is conceptual blending.

Figure 6

Mental spaces. The circles in Figure 6 represent mental spaces. In the monk example, there are four mental spaces: the two inputs, the generic, and the blend. There are also background frames recruited to build these mental spaces, such as the background frame of two people approaching each other on a path. This is a minimal network. Networks in other cases of conceptual integration may have yet more input spaces and even multiple blended spaces.

Cross-space mapping of counterpart connections. In conceptual integration, there are partial counterpart connections between input spaces. The solid lines in Figure 6 represent counterpart connections. Such counterpart connections are of many kinds: connections between frames and roles in frames; connections of identity or transformation or representation; metaphoric connections, etc. In the monk example, the monks, paths, journeys, days, and so on are counterparts.

Generic space. As conceptual projection unfolds, whatever structure is recognized as belonging to both of the input spaces constitutes a generic space. At any moment in the construction, the generic space maps onto each of the inputs. It defines the current cross-space mapping between them. A given element in the generic space maps onto paired counterparts in the two input spaces.

Blending. In blending, structure from two input mental spaces is projected to a third space, the "blend." In the monk example, the two input spaces have two journeys completely separated in time; the blend has two simultaneous journeys. Generic spaces and blended spaces are related: blends contain generic structure captured in the generic space, but also contain more specific structure, and can contain structure that is impossible for the inputs, such as two monks who are the same monk.

Selective projection. The projection from the inputs to the blend is typically partial. In Figure 6, not all elements from the inputs are projected to the blend.

There are three operations involved in constructing the blend: composition, completion, and elaboration.

Composition. Blending composes elements from the input spaces, providing relations that do not exist in the separate inputs. In the monk riddle, composition yields two travelers making two journeys. Fusion is one kind of composition. Counterparts may be brought into the blend as separate elements or as a fused element. Figure 6 represents one case in which counterparts are fused in the blend and one case in which counterparts are brought into the blend as distinct entities. In the monk example, the two days in the inputs are fused into one day in the blend, but the two monks from the inputs are brought into the blend as distinct entities.

Completion. Blends recruit a great range of background conceptual structure and knowledge without our recognizing it consciously. In this way, composed structure is completed with other structure. The fundamental subtype of recruitment is pattern completion. A minimal composition in the blend can be extensively completed by a larger conventional pattern. In the monk example, the structure achieved through composition is completed by the scenario of two people journeying toward each other on a path, which yields an encounter.

Elaboration. Elaboration develops the blend through imaginative mental simulation according to principles and logic in the blend. Some of these principles will have been brought to the blend by completion. Continued dynamic completion can recruit new principles and logic during elaboration. But new principles and logic may also arise through elaboration itself. We can "run the blend" indefinitely: for example, the monks might meet each other and have a philosophical discussion about the concept of identity. Blended spaces can become extremely elaborated.

Emergent structure. Composition, completion, and elaboration lead to emergent structure in the blend; the blend contains structure that is not copied from the inputs. In Figure 6, the square inside the blend represents emergent structure.

IV. Applications

The Debate with Kant

The monk example presents a salient and intuitively apparent blend, precisely because of its pragmatic anomaly. But our claim is that blends abound in all kinds of cases that go largely unnoticed. Some are created as we talk, others are conventional, and others are even more firmly entrenched in the grammatical structure. We discuss in Fauconnier & Turner 1996, the situation in which a contemporary philosopher says, while leading a seminar,

I claim that reason is a self-developing capacity. Kant disagrees with me on this point. He says it's innate, but I answer that that's begging the question, to which he counters, in Critique of Pure Reason, that only innate ideas have power. But I say to that, what about neuronal group selection? He gives no answer.

In one input mental space, we have the modern philosopher, making claims. In a separate but related input mental space, we have Kant, thinking and writing. In neither input space is there a debate. The blended space has both the modern philosopher (from the first input space) and Kant (from the second input space). In the blend, the additional frame of debate has been recruited, to frame Kant and the modern philosopher as engaged in simultaneous debate, mutually aware, using a single language to treat a recognized topic.

The debate frame comes up easily in the blend, through pattern completion, since so much of its structure is already in place in the composition of the two inputs. Once the blend is established, we can operate cognitively within that space, which allows us to manipulate the various events as an integrated unit. The debate frame brings with it conventional expressions, available for our use. We know the connection of the blend to the input spaces, and the way in which structure or inferences developed in the blend translate back to the input spaces.

A "realist" interpretation of the passage would be quite fantastic. The philosophy professor and Kant would have to be brought together in time, would have to speak the same language, and so on. No one is fooled into thinking that this is the intended interpretation. In fact, using a debate blend of this type is so conventional that it will go unnoticed.

And yet, it has all the criterial properties of blending. There is a Cross-space mapping linking Kant and his writings to the philosophy professor and his lecture. Counterparts include: Kant and the professor, their respective languages, topics, claims, times of activity, goals (e.g. search for truth), modes of expression (writing vs. speaking).

There is Partial projection to the blend: Kant, the professor, some of their ideas, and the search for truth are projected to the blend. Kant's time, language, mode of expression, the fact that he's dead, and the fact that he was never aware of the future existence of our professor are not projected.

There is Emergent Structure through Composition: we have two people talking in the same place at the same time. There is Emergent Structure through Completion: two people talking in the same place at the same time evoke the cultural frame of a conversation, a debate (if they are philosophers), an argument. This frame, the debate frame, structures the blend and is reflected by the syntax and vocabulary of the professor (disagrees, answer, counters, what about, ...).

This example allows us to observes that blends provide Integration of Events: Kant's ideas and the professor's claims are integrated into a unified event, the debate. Looking back now to the monk example, we see that the blend in that case integrated into a single scenario various events of uncertain relation spread out over time. Blends provide a space in which ranges of structure can be manipulated uniformly. The other spaces do not disappear once the blend has been formed. On the contrary, the blend is valuable only because it is connected conceptually to the inputs. The monk blend tells us something about the inputs. The debate with Kant tells us something about the inputs.

Complex numbers

Conceptual projection enables us to extend categories to cover new provisional members. The blended space that develops during such a projection merges the original category with its new extension. When categories are extended permanently, it is the structure of this blend that defines the new category structure, thus carving out a novel conceptual domain. The history of science, and of mathematics and physics in particular, is rich in such conceptual shifts. (See Fauconnier & Turner 1994; Lakoff & Núñez in preparation; Lansing, personal communication.) It is customary to speak of models either replacing or extending previous models, but the pervasiveness and importance of merging may have been underestimated.

Consider as an example the stage of mathematical conceptual development at which complex numbers became endowed with angles (arguments) and magnitudes. Square roots of negative numbers had shown up in formulas of sixteenth-century mathematicians and operations on these numbers had been correctly formulated. But the very mathematicians who formulated such operations, Cardan and especially Bombelli, were also of the opinion that they were "useless," "sophistic," and "impossible" or "imaginary." Such was also the opinion of Descartes a century later. Leibniz said no harm came of using them, and Euler thought them impossible but nevertheless useful. The square roots of negative numbers had the strange property of lending themselves to formal manipulations without fitting into a mathematical conceptual system. A genuine concept of complex number took time to develop, and the development proceeded in several steps along the lines explained above for analogical connections and blending.

The first step exploited the preexisting analogical mapping from numbers to one-dimensional space. Wallis is credited with having observed—in his Algebra (1685)—that if negative numbers could be mapped onto a directed line, complex numbers could be mapped onto points in a two-dimensional plane, and he provided geometrical constructions for the counterparts of the real or complex roots of ax2 + bx + c = 0 (Kline 1980). In effect, Wallis provided a model for the mysterious numbers, thereby showing their consistency, and giving some substance to their formal manipulation. This is of course a standard case of extending analogical connections; geometric space is a source domain partially mapped onto the target domain of numbers. The mapping from a single axis is extended to mapping from the whole plane; some geometric constructions are mapped onto operations on numbers. Notice that neither the original mapping nor its extension requires more than two domains. We do not need a generic space, since there is no assumption in work like Wallis's that numbers and points in a plane share properties at some higher level of abstraction. The necessary structure is already present in the conceptual domain of two-dimensional space because it already contains the notion of distance which is expressed directly by means of numbers. (Of course, this source domain has a conceptual history of its own. We argue elsewhere that in fact it is itself the product of a non-trivial conceptual blend.) Nor does it involve a blend; numbers and points remain totally distinct categories at all levels. Although the mapping proposed by Wallis showed the formal consistency of a system including complex numbers, it did not provide a new extended concept of number. As Morris Kline reports, Wallis's work was ignored: it did not make mathematicians receptive to the use of such numbers. In itself, this is an interesting point. It shows that mapping a coherent space onto a conceptually incoherent space is not enough to give the incoherent space new conceptual structure. It also follows that coherent abstract structure is not enough, even in mathematics, to produce satisfactory conceptual structure: In Wallis's representation, the metric geometry provided abstract schemas for a unified interpretation of real and imaginary numbers, but this was insufficient cognitively for mathematicians to revise their domain of numbers accordingly.

In the analysis developed here, the novel conceptual structure in the mathematical case of numbers is first established within a blended space. In the blend, but not in the original inputs, it is possible for an element to be simultaneously a number and a geometric point, with cartesian coordinates (a,b) and polar coordinates (r,q). In the blend, we find interesting general formal properties of such numbers, such as

(a, b) + (a', b') = (a+a', b+b')

(r, q) x (r', q') = (rr', q + q')

Every number in this extended sense has a real part, an imaginary one, an argument, and a magnitude. By virtue of the link of the blend to the geometric input space, the numbers can be manipulated geometrically; by virtue of the link of the blend to the input space of real numbers, the new numbers in the blend are immediately conceptualized as an extension of the old numbers (which they include by way of the mapping). As in Wallis's scheme, the mapping from points on a line to numbers has been extended to a mapping from points in a plane to numbers. This mapping is partial from one input to the other—only one line of the plane is mapped onto the numbers of the target domain—but it is total from the geometric input to the blend: all the points of the plane have counterpart complex numbers. And this in turn allows the blend to incorporate the full geometric structure of the geometric input space.

Figure 7

Interestingly, when a rich blended space of this sort is built, an abstract generic space will come along with it. Having the three spaces containing respectively points (input 1), numbers (input 2), complex point/numbers (blend) entails a fourth space with abstract elements having the properties "common" to points and numbers. The relevant abstract notions in this case are those of "operations" on pairs of elements. For numbers, the specific operations (in the target domain) are addition and multiplication. For points in the plane, the operations can be viewed as vector transformations—vector addition, and vector composition by adding angles and multiplying magnitudes. In the blended space of complex numbers, vector addition and number addition are the same operation, because they invariably yield the same result; similarly, vector transformation and number multiplication are conceptually one single operation. But such an operation can be instantiated algorithmically in different ways depending on which geometric and algebraic properties of the blend are exploited. [2]

In the generic space, specific geometric or number properties are absent. All that is left is the more abstract notion of two operations on pairs of elements, such that each operation is associative, commutative, and has an identity element; each element has under each operation an inverse element; and one of the two operations is distributive with respect to the other. Something with this structure is called by mathematicians a "commutative ring."

The emergence of the concept of complex numbers with arguments and magnitudes displays all the criterial properties of blending. There is an initial cross-space mapping of numbers to geometric space, a generic space, a projection of both inputs to the blend, with numbers fused with geometric points, emergent structure by completion (arguments and magnitudes), and by elaboration (multiplication and addition reconstrued as operations on vectors).

The blend takes on a realist interpretation within mathematics. It constitutes a new and richer way to understand numbers and space. However, it also retains its connections to the earlier conceptions provided by the Input spaces. Conceptual change of this sort is not just replacement. It is the creation of more elaborate and richly connected networks of spaces.

Under our account, then, the evolution and extension of the concept of number includes a four-space stage at which the concept of complex number is logically and coherently constructed in a blended space, on the basis of a generic space structured as a commutative ring. (That generic space is not consciously conceptualized as an abstract domain when the full-blown concept of complex number gets formed. It becomes a conceptual domain in its own right when mathematicians later study it and name it.) The abstract and mathematical example of complex numbers supports the functioning of many-space conceptual projection, with its blended and generic spaces, [3] and confirms that we are dealing with an aspect of thought that is not purely linguistic or verbal. It highlights the deep difference between naming and conceptualizing; adding expressions like ÷-1 to the domain of numbers, and calling them numbers, is not enough to make them numbers conceptually, even when they fit a consistent model. This is true of category extension in general.

Digging your own grave

Coulson (1997) examines remarkable elaborations of the metaphor "to dig one's own grave." Consider the familiar idiomatic version of the metaphor. "You are digging your own grave" is a conventional expression typically used as a warning or judgment, typically implying that (1) you are doing bad things that will cause you to have a very bad experience, and (2) you are unaware of this causal relation. A conservative parent who keeps his money in his mattress may express disapproval of an adult child's investing in the stock market by saying, "you are digging your own grave."

At first glance, what we have here is a straightforward projection from the concrete domain of graves, corpses, and burial to abstract domains of getting into trouble, unwittingly doing the wrong things, and ultimate failure. Failing is being dead and buried; bad moves that precede and cause failure are like events (grave-digging) that precede burial. It is foolish to facilitate one's own burial or one's own failure. And it is foolish not to be aware of one's own actions, especially when they are actions leading to one's very extinction.

But a closer look reveals extraordinary mismatches between the purported source and target of this metaphor. The causal structure is inverted. Foolish actions cause failure, but grave-digging does not cause death. It is typically someone's dying that prompts others to dig a grave. And if the grave is atypically prepared in advance, to secure a plot, to keep workers busy, or because the person is expected to die, there is still not the slightest causal connection from the digging to the dying. In the exceptional scenario in which a prisoner is threatened into digging his own grave, it is not the digging that causes the death, and the prisoner will be killed anyway if he refuses. The intentional structure does not carry over. Sextons do not dig graves in their sleep, unaware of what they are doing. In contrast, figurative digging of one's own grave is conceived as unintentional misconstrual of action. The frame structure of agents, patients, and sequence of events is not preserved. Our background knowledge is that the "patient" dies, and then the "agent" digs the grave and buries the "patient." But in the metaphor, the actors are fused and the ordering of events is reversed. The "patient" does the digging, and if the grave is deep enough, has no other option than to die and occupy it. Even in the unusual real life case in which one might dig one's own grave in advance, there would be no necessary temporal connection between finishing the digging and perishing. The internal event structure does not match. In the target, it is certainly true that the more trouble you are in, the more you risk failure. Amount of trouble is mapped onto depth of grave. But again, in the source there is no correlation between the depth of a person's grave and their chances of dying.

Now recall the rationale often proposed for metaphor: Readily available background or experiential structure and inferences of the source are recruited to understand the target. By that standard, and in view of the considerable mismatch, digging one's own grave should be a doomed metaphor. In fact, it's a very successful one.

This paradox dissolves when we consider, in addition to the two input spaces, the blended space. In metaphoric cases, such as this one, the two inputs are the "source Input" and the "target Input." The blend in digging one's own grave inherits the concrete structure of graves, digging, and burial, from the source Input. But it inherits causal, intentional, and internal event structure from the target Input. They are not simply juxtaposed. Rather, emergent structure specific to the blend is created. In the blend, all the curious properties noted above actually hold. The existence of a satisfactory grave causes death, and is a necessary precondition for it. It follows straightforwardly that the deeper the grave, the closer it is to completion, and the greater the chance for the grave's intended occupant to die. It follows that in the blend (as opposed to the Input source), digging one's grave is a grave mistake, since it makes dying more probable. In the blend, it becomes possible to be unaware of one's very concrete actions. This is projected from the target Input, where it is indeed fully possible, and frequent, to be unaware of the significance of one's actions. But in the blend, it remains highly foolish to be unaware of such concrete actions; this is projected from the source Input. And it will project back to the target Input to produce suitable inferences (i.e. highlight foolishness and misperception of individual's behavior).

We wish to emphasize that in the construction of the blend, a single shift in causal structure, the existence of a grave causes death, instead of death causes the existence of a grave, is enough to produce emergent structure, specific to the blend: undesirability of digging one's grave, exceptional foolishness in not being aware of it, correlation of depth of grave with probability of death. The causal inversion is guided by the target, but the emergent structure is deducible within the blend from the new causal structure and familiar common-sense background knowledge. This point is essential, because the emergent structure, although "fantastic" from a literal interpretation point of view, is supremely efficient for the purpose of transferring the intended inferences back to the target Input, and thereby making real-world inferences. This emergent structure is not in the Inputs—it is part of the cognitive construction in the blend. But, also, it is not stated explicitly as part of the blend. It just follows, fairly automatically, from the unstated understanding that the causal structure has been projected from the target, not from the source.

The integration of events in the blend is indexed to events in both of the input spaces. We know how to translate structure in the blend back to structure in the inputs. The blend is an integrated platform for organizing and developing those other spaces. Consider a slightly fuller expression, "with each investment you make, you are digging your grave a little deeper." In the target Input, there are no graves, but there are investments; in the source Input, the graves are not financial, but one does dig; in the blend, investments are simultaneously instruments of digging, and what one digs is one's financial grave. A single action is simultaneously investing and digging; a single condition is simultaneously having finished the digging and having lost one's money. Digging your own grave does not kill you, but digging your own financial grave does cause your death/bankruptcy.

Such blends can of course be elaborate, as in Seana Coulson's example from an editorial in the UCSD Guardian:

The U.S. is in a position to exhume itself from the shallow grave that we've dug for ourselves.

In this blend, the digger is identical to the body buried, which can exhume itself. This is impossible for the source Input, but possible for the target Input, where a nation can be in bad conditions but try to get out of them. In the blend, the ease of exhuming is related to the depth of the grave. This logic is available from both source and target Inputs: the shallower the grave, the easier the exhumation; the less bad the conditions, the easier it is to improve them. As in "you are digging your own grave," the actor is responsible but unaware, his actions were unwise, he is culpable for not recognizing that his actions were unwise, and the consequences of those actions are undesirable.

Pattern completion is at work in developing this blend. In recent U.S. history, there have been many disparate events, only some caused by actors, only some caused by American actors, and almost none caused by any single actor. Nonetheless, the blend asks us to integrate those many disparate target events, by blending them with a template, available to the blend from the source Input, of a single integrated action by a single actor, namely, digging as done by a digger. To do so, we must construct in the target a single entity, "the United States," that is causal for those many disparate events, which are in turn causal for current conditions in the United States. In the blend, the United States is a person, whom we want to convince to begin the process of self-exhumation.

Analogical counterfactuals

Consider an analogical counterfactual of the type studied by Fauconnier (1990, in press):

"In France, Watergate would not have harmed Nixon."

Uncontroversially, understanding this counterfactual includes building a generic space that fits both American politics and French politics. It includes a leader who is elected, who is a member of a political party, and who is constrained by laws. This skeletal generic space fits the space of American politics and French politics so well and intricately that it is natural for someone to project a great deal more skeletal information from American politics into the generic space on the assumption that it will of course apply to French politics.

The rhetorical motive for saying, "In France, Watergate wouldn't have done Nixon any harm" is exactly to stop someone from projecting certain kinds of information to the generic space on the assumption that it applies to French politics. The speaker lays down a limit to this projection by constructing a specific, counterfactual, and pragmatically anomalous blend.

Into this blend, the speaker has projected information associated with President Nixon and the Watergate break-in. Nixon and Watergate and so on are brought into the blend with only skeletal properties, such as being a president who breaks laws in order to place members of a political party at a disadvantage. It may be that such information in fact in no way belongs to French politics, that something like Watergate has in fact never happened in French politics. No matter, it can be imported to the blend from the "Nixon in America" input. Additionally, from the "France" input, we can project to the blend French cultural perspectives on such an event.

This counterfactual blended space operates according to its own logic. In this counterfactual blend, an illegal act directed with the knowledge of the elected leader against the opposing political party leader will not cause the public outrage associated with Watergate. For this central inference to take place, we must have both the nature of the event from the "Nixon in America" input and the general cultural attitudes from the "France" input. The blend is not a side-show or curiosity or merely an entertaining excrescence of the projection. It is the engine of the central inferences.

The criterial properties of blending are apparent: cross-space mapping of the Input U.S. and France spaces; generic politics space; selective projection—Nixon and Watergate on the one hand, the frame of French politics on the other; emergent structure:

—composition provides a Watergate-like event in France;

—elaboration includes the explicit predication that the president is not harmed.

Finally, there is projection back to the Inputs: France has features that the U. S. does not have.

Clearly, in the case of such an analogical counterfactual, the construction of meaning cannot be mistaken as an attempt to impose structure from the one input onto the other. In fact, this particular analogical counterfactual is trying to do exactly the opposite. It is trying to make clear in just what areas information projected from one input cannot be imposed on the other. Moreover, its purpose is to illuminate not only the nature of the "France" Input, bu