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Conceptual spaces
Peter Gärdenfors
Cognitive Science, Lund University
Peter.Gardenfors@lucs.lu.se
Abstract: The dominating models of information processes have been based on symbolic
representations of information and knowledge. During the last decades, a variety of non-
symbolic models have been proposed as superior. The prime examples of models within the
non-symbolic approach are neural networks. However, to a large extent they lack a higher-
level theory of representation.
In this paper, conceptual spaces are suggested as an appropriate framework for non-symbolic
models. Conceptual spaces consist of a number of ’quality dimensions’ that often are derived
from perceptual mechanisms. It will be outlined how conceptual spaces can represent various
kind of information and how they can be used to describe concept learning. The connections
to prototype theory will also be presented
1. THE PROBLEM OF MODELING REPRESENTATIONS
Cognitive science has two overarching goals. One is explanatory: By studying the
cognitive activities of humans and other animals, one formulates theories of different
aspects of cognition. The theories are tested by experiments or by computer
simulations. The other goal is constructive: By building artifacts like chess-playing
programs, robots, animats, etc, one attempts to construct systems that can
accomplish various cognitive tasks. For both kinds of goals, a key problem is how
the representations used by the cognitive system are to be modeled in an appropriate
way.
Within cognitive science, there are currently two dominating approaches to the
problem of modeling representations. The symbolic approach starts from the
assumption that cognitive systems should be modelled by Turing machines. On this
view, cognition is seen as essentially involving symbol manipulation. The second
approach is associationism, where associations between different kinds of information
elements carry the main burden of representation. Connectionism is a special case of
associationism, which models associations by artificial neuron networks. Both the
symbolic and the associationistic approaches have their advantages and
disadvantages. They are often presented as competing paradigms, but since they
attack cognitive problems on different levels, I shall argue later that they should
rather be seen as complementary methodologies.
However, there are aspects of cognitive phenomena for which neither symbolic
representation nor connectionism seem to offer appropriate modelling tools. In this
article, I will advocate a third form of representing information that is based on
using geometrical structures rather than symbols or connections between neurons.
Using these structures similarity relations can be modeled in a natural way. The
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notion of similarity is crucial for the understanding of many cognitive phenomena. I
shall call my way of representing information the conceptual form since I believe that
the essential aspects of concept formation are best described using this kind of
representations.
Again, conceptual representations should not be seen as competing with symbolic or
associationist (connectivist) representations. Rather, the three kinds can be seen as
three levels of representations of cognition with different scales of resolution.
I shall outline a theory of conceptual spaces as a particular framework for
representing information on the conceptual level. A conceptual space is built up
from geometrical representations based on a number of quality dimensions. The
emphasis of the theory will be on the constructive side of cognitive science.
However, I believe that it also can explain several aspects of what is known about
representations in various biological systems.
2. QUALITY DIMENSIONS
One notion that is severely downplayed in symbolic representations is that of
similarity. I submit that judgments of similarity are central for a large number of
cognitive processes. Judgments of similarity reveal the dimensions of our
perceptions and their structures. For many kinds of dimensions it will be possible to
talk about distances. The general assumption is that the smaller the distance is
between the representations of two objects, the more similar they are. In this way,
the similarity of two objects can be defined via the distance between their
representing points in the space. Thus conceptual spaces provide us with a natural
way of representing similarities. In general, the epistemological role of the
conceptual spaces is to serve as a tool in sorting out various relations between
perceptions.
As introductory examples of quality dimensions one can mention temperature,
weight, brightness, pitch and the three ordinary spatial dimensions height, width
and depth. I have chosen these examples because they are closely connected to what
is produced by our sensory receptors (Schiffman 1982). The spatial dimensions
height, width and depth as well as brightness are perceived by the visual sensory
system, pitch by the auditory system, temperature by thermal sensors and weight,
finally, by the kinesthetic sensors. However, there are also quality dimensions that
are of an abstract non-sensory character.
The primary function of the quality dimensions is to represent various “qualities” of
objects.1 They correspond to the different ways stimuli are judged to be similar or
different. In most cases, judgments of similarity and difference generate an ordering
relation of stimuli (Clark 1993, p. 114). For example, one can judge tones by their
pitch that will generate and ordering of the perceptions. The dimensions form the
“framework” used to assign properties to objects and to specify relations between
them. The coordinates of a point within a conceptual space represent particular
instances of each dimension, for example a particular temperature, a particular
weight, etc.
1In traditional philosophy, following Locke, a distinction between “primary” and “secondary”
qualities is often made. This distinction corresponds roughly to the distinction between “scientific”
and “phenomenal” dimensions to be presented in the following section.
3
The quality dimensions are taken to be independent of symbolic representations in
the sense that we and other animals can represent the qualities of objects, for
example when planning an action, without presuming an internal language or
another symbolic system in which these qualities are expressed. In other words, the
dimensions are the building blocks of representations on the conceptual level.
When the explanatory aim of cognitive science is in focus, the quality dimensions
should be seen as theoretical entities used as a modeling factor in describing
cognitive activities of organisms. When constructing artificial systems, the
dimensions function as the framework for the representations used by the systems.
The notion of a dimension should be understood literally. It is assumed that each of
the quality dimensions is endowed with certain geometrical structures (in some cases
they are topological or orderings). As a first example to illustrate such a structure, the
dimension of “weight” which is one-dimensional with a zero point, and thus
isomorphic to the half-line of non-negative numbers. A basic constraint on this
dimension that is commonly made in science is that there are no negative weights.2
0
Figure 1. The weight dimension
In previous writings on conceptual spaces, I have used the example of the perceptual
color space to illustrate a more structured set of quality dimensions (Gärdenfors
1990, 1991, 2000). However, we can also find related spatial structures for other
sensory qualities. For example, consider the quality dimension of pitch, which is
basically a continuous one-dimensional structure going from low tones to high. This
representation is directly connected to the neurophysiology of pitch perception.
Apart from the basic frequency dimension of tones, it is possible to identify some
further structure in the mental representation of tones. Natural tones are not simple
sinusoidal tones of only one frequency, but constituted of a number of higher
harmonics. The timbre of a tone, which is a phenomenal dimension, is determined
by the relative strength of the higher harmonics of the fundamental frequency of
the tone. An interesting perceptual phenomenon is “the case of the missing
fundamental.” If the fundamental frequency is removed by artificial methods from a
complex physical tone, the phenomenal pitch of the tone is still perceived as that
corresponding to the removed fundamental.3 Apparently, the fundamental
frequency is not indispensable for pitch perception, but the perceived pitch is
determined by a combination of the lower harmonics.
Thus, the harmonics of a tone are essential for how it is perceived. This entails that
tones that share a number of harmonics will be perceived to be similar. The tone
that shares the most harmonics with a given tone is its octave, the second most
similar is the fifth, the third most similar is the fourth and so on. This additional
“geometric” structure on the pitch dimension, which can be derived from the wave
structure of tones, provides the foundational explanation for the perception of
musical intervals.4
2However, it is interesting to note (cf. Kuhn 1970) that during a period of phlogiston chemistry, the
scientists were considering negative weights in order to evade some of the anomalies for the theory.
3See e .g. Gabrielsson (1981), pp. 20-21.
4For some further discussion of the structure of musical space see Gärdenfors (1988), Sections 7-9.
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For another example of sensory space representations let me only mention that the
human perception of taste appears to be generated from four distinct types of
receptors: salt, sour, sweet, and bitter. Thus the quality space representing tastes
could be described as a 4-dimensional space. One such model was put forward by
Henning (1961), who suggested that phenomenal gustatory space could be
described as a tetrahedron (see figure 2). Actually, Henning speculated that any taste
could be described as a mixture of only three primaries. This means that any taste
can be represented as a point on one of the planes of the tetrahedron, so that no taste
is mapped onto the interior.
Saline
Sour
Bitter
Sweet
Figure 2. Henning’s taste tetrahedron
However, there are other models that propose more than four fundamental tastes.5
The best model of the phenomenal gustatory space remains to be established. This
will involve sophisticated psychophysical measurement techniques. Suffice it to say
that the gustatory space quite clearly has some non-trivial geometrical structure. For
instance, we can meaningfully claim that the taste of a walnut is closer to the taste of
a hazelnut than to the taste of popcorn in the same way as we can say that the color
orange is closer to yellow than to blue.
It should be noted that some quality “dimensions” have only a discrete structure,
that is, they merely divide objects into disjoint classes. Two examples are
classifications of biological species and kinship relations in a human society. One
example of a phylogenetic tree of the kind found in biology is shown in figure 3.
Here the nodes represent different species in the evolution of, for example, a family
of organisms, where nodes higher up in the tree represent evolutionarily older
(extinct) species.
time
(evolution)
Figure 3. Phylogenetic tree
5See Schiffman (1982), chapter 9 for an exposition of some such theories.
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The distance between two nodes can be measured by the length of the path that
connects them. This means that even for discrete dimensions one can distinguish a
rudimentary geometrical structure. For example, in the phylogenetic classification of
animals that mirrors evolutionary branchings, it is meaningful to say that rats and
whales are more closely related than whales and fish.
3. PHENOMENAL AND SCIENTIFIC INTERPRETATIONS OF DIMENSIONS.
In order to separate different uses of quality dimensions it is important to introduce
a distinction between a phenomenal (or psychological) and a scientific interpretation.
The phenomenal interpretation concerns the cognitive structures (perceptions,
memories, etc) of humans or other organisms. The scientific interpretation, on the
other hand, treats dimensions as a part of a scientific theory.
The distinction is relevant in relation to the two goals of cognitive science presented
above. When the dimensions are seen as cognitive entities, that is, when the goal is
to explain natural cognitive processes, their geometrical structure should not be
determined by scientific theories that attempt at giving a “realistic” description of
the world, but by psychophysical measurements that determine the structure of how
our perceptions are represented. Furthermore, when it comes to providing a
semantics for a natural language, it is the phenomenal interpretations of the quality
dimensions that are in focus.
On the other hand, when we are constructing an artificial system, the function of
sensors, effectors and various control devices are in general described in terms of
scientifically modeled dimensions. For example, the input variables of a robot may
be a small number of physically measured magnitudes, like brightness, delay of a
radar echo, or the pressure from a mechanical grip. With the aid of the programmed
goals of the robot, these variables can then be transformed into a number of
physical output magnitudes as, for example, the voltages of the motors controlling
the left and the right wheels.
To give an example of the distinction, consider color. The distinction introduced here
is supported by Gallistel (1990, p. 518-519) who writes:
The facts about color vision suggest how deeply the nervous system may be committed to
representing stimuli as points in descriptive spaces of modest dimensionality. It does this even
for spectral compositions, which does not lend itself to such a representation. The resulting
lack of correspondence between the psychological representation of spectral composition and
spectral composition itself is a source of confusion and misunderstanding in scientific
discussions of color. Scientists persist in refering to the physical characteristics of the stimulus
and to the tuning characteristics of the transducers (the cones) as if psychological color terms
like red, green, and blue had some straightforward translation into physical reality, when in
fact they do not.
Gallistel’s warning against confusion and misunderstanding of the two types of
representation should be taken seriously. It is very easy to confound what science
says about the characteristics of reality and our perception of it. In this article, it is
the phenomenal representation that will be in focus.
A conceptual space can now be defined as a collection of one or more quality
dimensions. However, the dimensions of a conceptual space should not be seen as
totally independent entities, but they are correlated in various ways since the
properties of the objects modeled in the space co-vary. For example, the ripeness
and the color dimensions co-vary in the space of fruits.
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4. ON THE ORIGINS OF QUALITY DIMENSIONS
In the previous sections, I have given some examples of quality dimensions from
different kinds of domains. There seems to be different types of dimensions so a
warranted question is: Where do the dimensions come from? I do not believe there
is a unique answer to this question. In this section, I will try to trace the origins of the
different kinds of quality dimensions.
Firstly, some of the quality dimensions seem to be innate or developed very early in
life. They are to some extent hardwired in our nervous system, as for example the
sensory dimensions presented above. This probably also applies to our
representations of ordinary space. Since domains of this kind are obviously
extremely important for basic activities like finding food, avoiding danger, and
getting around in the environment there is evolutionary justification for the
innateness assumption. Humans and other animals who did not have a sufficiently
adequate representation of the spatial structure of the external world were
disadvantaged by natural selection.
The brain of humans and animals contains topographic areas mapping different
kinds of sense modalities onto spatial areas. The structuring principles of these
mappings are basically innate, even if the finetuning is established during the
development of the human or animal. The same principles seem to govern most of
the animal kingdom. Gallistel (1990, p. 105) argues:
[…] the intuitive belief that the cognitive maps of “lower” animals are weaker than our own is
not wellfounded. They may be impoverished relative to our own (have less on them) but they
are not weaker in their formal characteristics. There is experimental evidence that even insect
maps are metric maps.
Quine notes that something like innate quality dimensions is needed to make
learning possible:
Without some such prior spacing of qualities, we could never acquire a habit; all stimuli
would be equally alike and equally different. These spacings of qualities, on the part of men
and other animals, can be explored and mapped in the laboratory by experiments in
conditioning and extinction. Needed as they are for all learning, these distinctive spacings
cannot themselves all be learned; some must be innate. (Quine 1969:123)
However, once the process has started, new dimensions can be added by the
learning process.6 One kind of examples comes from studies of children’s cognitive
development. Two-year-olds can represent whole objects, but they cannot reason
about the dimensions of the object.
Learning new concepts is, consequently, often connected with expanding one's
conceptual space with new quality dimensions. For example, consider the
(phenomenal) dimension of volume. The experiments concerning “conservation”
performed by Piaget and his followers indicate that small children have no separate
mental dimension of volume; they confuse the volume of a liquid with the height of
the liquid in its container. It is only at about an age of five years that they learn to
represent the two dimensions separately. Similarly, three- and four-year-olds
6It must be noted that it is impossible to draw a sharp distinction between innate and learned
quality dimensions, since many sensory dimensions are structurally prepared in the neural tissue at
birth, but require exposure to sensory experiences to lay out the exact geometrical structure of the
mapping.
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confuse high with tall, big with bright, etc (Carey 1978). Smith (1989, p. 146-47) argues
that
working out a system of perceptual dimension, a system of kinds of similarities, may be one of
the major intellectual achievements of early childhood. […] The basic developmental notion is
one of differentiation, from global syncretic classes of perceptual resemblance and magnitude
to dimensionally specific kinds of sameness and magnitude.
Still other dimensions may be culturally dependent.7 Take “time,” for example: In
some cultures time is conceived to be circular – the world keeps returning to the
same point in time and the same events occur over and over again; and in other
cultures it is hardly meaningful at all to speak of time as a dimension. A sophisticated
time dimension, with the full metric structure, is needed for advanced forms of
planning and coordination with other individuals, but is not necessary for the most
basic activities of an organism. As a matter of fact, the standard Western conception
of time is a comparatively recent phenomenon (see Toulmin and Goodfield (1965)).
The examples given here indicate that many of the quality dimensions of human
conceptual spaces are not directly generated from sensory inputs. This is even
clearer when we use concepts based on the functions of artifacts or the social roles of
people in a society. Even if we do not know much about the geometrical structures
of these dimensions, it is quite obvious that there is some non-trivial such structure.
This has been argued by Marr and Vaina (1982) and Vaina (1983), who give an
analysis of functional representation where functions of an object are determined in
terms of the actions it allows.
Culture, in the form of interaction between people, may in itself generate constraints
on conceptual spaces. For example, Freyd (1983) puts forward the intriguing
proposal that conceptual spaces may evolve as a representational form in a
community just because people have to share knowledge:
There have been a number of different approaches towards analyzing the structures in
semantic domains, but what these approaches have in common is the goal of discovering
constraints on knowledge representation. I argue that the structures the different semantic
analyses uncover may stem from shareability constraints on knowledge representation.
[…]
So, if a set of terms can be shown to behave as if they are represented in a three-dimensional
space, one inference that is often made is that there is both some psychological reality to the
spatial reality (or some formally equivalent formulation) and some innate necessity to it. But it
might be that the structural properties of the knowledge domain came about because such
structural properties provide for the most efficient sharing of concepts. That is, we cannot be
sure that the regularities tell us anything about how the brain can represent things, or even
“prefer”to, if it didn't have to share concepts with other brains” (1983, pp. 193-194)
Here Freyd hints at an economic explanation of why we have conceptual spaces: they
facilitate the sharing of knowledge.
Finally, some quality dimensions are introduced by science. Witness, for example,
Newton's distinction between weight and mass, which is of crucial importance for the
development of his celestial mechanics, but which hardly has any correspondence in
human perception. To the extent we have mental representations of the masses of
objects in distinction to their weights, these are not given by the senses but have to
7I don’t claim that my typology of the origins of quality dimensions is exclusive, since, in a sense, all
culturally dependent dimensions are also learned.
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be learned by adopting the conceptual space of Newtonian mechanics in our
representations.
The most drastic changes in science occur when the underlying conceptual space is
changed. I believe that most of the “paradigm shifts” discussed by Kuhn (1970) can
be understood as shifts of conceptual spaces. I do not see any principal difference
between this kind of change and the change involved in the development of a child's
conceptual space. Introducing the distinction between “height” and “volume” is the
same kind of phenomenon as when Newton introduced the distinction between
“weight” and “mass.” That distinction is nowadays ubiquitous in physics, even
though there is only scant sensory support for it.
The conceptual space of Newtonian particle mechanics is, of course, based on
scientific (theoretical) quality dimensions and not on phenomenal (psychological)
dimensions. The quality dimensions of this theory are ordinary space (3-D
Euclidean), time (isomorphic to the real numbers), mass (isomorphic to the non-
negative real numbers), and force (3-D Euclidean space). Once a particle has been
assigned a value for these eight dimensions, it is fully described as far as Newtonian
mechanics is concerned. In this theory, an object is thus represented as a point in an
8-dimensional space.
5. CONCEPT FORMATION DESCRIBED WITH THE AID OF CONCEPTUAL
SPACES
In more abstract terms, a conceptual space S consists of a class D1, ... Dn of quality
dimensions. A point in S is represented by a vector v = <d1,...,dn> with one index for
each dimension. Each of the dimensions is endowed with a certain topological or
metrical structure. The purpose of this section is to show how conceptual spaces can
be used to model concepts.
A first rough idea is to describe a concept as a region of a conceptual space S, where
'region' should be understood as a spatial notion determined by the topology and
metric of S. For example, the point in the time dimension representing 'now' divides
this dimension, and thus the space of vectors, into two regions corresponding to the
concepts 'past' and 'future'. But the proposal suffers from a lack of precision as
regards the notion of a 'region'. A more precise and powerful idea is the following
criterion where the topological characteristics of the quality dimensions are utilized
to introduce a spatial structure on concepts:
Criterion P: A natural concept is a convex region of a conceptual space.
A convex region is characterized by the criterion that for very pair of points v1 and
v2 in the region all points in between v1 and v2 are also in the region. The
motivation for the criterion is that if some objects which are located at v1 and v2 in
relation to some quality dimension (or several dimensions) both are examples of the
concept C, then any object that is located between v1 and v2 on the quality
dimension(s) will also be an example of C. I shall argue later that this criterion is
psychologically realistic. Criterion P presumes that the notion of betweenness is
meaningful for the relevant quality dimensions. This is, however, a rather weak
assumption which demands very little of the underlying topological structure.
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Most concepts expressed by simple words in natural languages are natural concepts
in the sense specified here. For instance, I conjecture that all color terms in natural
languages express natural concepts with respect to the psychological representation
of the three color dimensions. In other words, the conjecture predicts that if some
object o1 is described by the color term C in a given language and another object o2
is also said to have color C, then any object o3 with a color that lies between the
color of o1 and that of o2 will also be described by the color term C. It is well-known
that different languages carve up the color circle in different ways, but all carvings
seems to be done in terms of convex sets. Strong support for this conjecture can be
found in Berlin and Kay (1969), although they do not treat color terms in general but
concentrate on basic color terms. On the other hand, the reference of an artificial
color term like 'grue' (Goodman 1955) will not be a convex region in the ordinary
conceptual space and thus it is not a natural concept according to Criterion P.8
Another illustration of how the convexity of regions determines concepts and
categorizations is the phonetic identification of vowels in various languages.
According to phonetic theory, what determines a vowel are the relations between
the basic frequency of the sound and its formants (higher frequencies that are
present at the same time). In general, the first two formants F1 and F2 are sufficient
to identify a vowel. This means that the coordinates of two-dimensional space
spanned by F1 and F2 (in relation to a fixed basic pitch F0) can be used as a fairly
accurate description of a vowel. Fairbanks and Grubb (1961) investigated how
people produce and recognize vowels in 'General American' speech. Figure 4
summarizes some of their findings.
Figure 4. Frequency areas of different vowels in the two-dimensional space generated by the first two
formants. (From Fairbanks and Grubb 1961)
8For an extended analysis of this example, see Gärdenfors (1989).
10
The scale of the abscissa and ordinate are the logarithm of the frequencies of F1 and
F2 (the basic frequency of the vowels was 130 cps). As can be seen from the
diagram, the preferred, identified and selfapproved examples of different vowels
form convex subregions of the space determined by F1 and F2 with the given
scales.9 As in the case of color terms, different languages carve up the phonetic space
in different ways (the number of vowels identified in different languages varies
considerably), but I conjecture again that each vowel in a language will correspond
to a convex region of the formant space.
An important thing to note in this example is that identifying F1 and F2 as the
relevant dimensions for vowel formation is a phonetic discovery. We had the
concepts of vowels already before this discovery, but the spatial analysis makes it
possible for us to understand several features of the classifications of vowels in
different languages.
Criterion P provides an account of concepts that is independent of both possible
worlds and individuals and it satisfies Stalnaker's desideratum that a concept "...
must be not just a rule for grouping individuals, but a feature of individuals in virtue
of which they may be grouped" (Stalnaker 1981, p. 347). However it should be
emphasized that I only view the criterion as a necessary but perhaps not sufficient
condition on a natural concept. The criterion delimits the class of concepts that are
useful for cognitive purposes, but it may not be sufficiently restrictive.
6. RELATIONS TO PROTOTYPE THEORY
Describing concepts as convex regions of conceptual spaces fits very well with the so
called prototype theory of categorization developed by Rosch and her collaborators
(Rosch 1975, 1978, Mervis and Rosch 1981, Lakoff 1987). The main idea of prototype
theory is that within a category of objects, like those instantiating a concept, certain
members are judged to be more representative of the category than others. For
example robins are judged to be more representative of the category 'bird' than are
ravens, penguins and emus; and desk chairs are more typical instances of the
category 'chair' than rocking chairs, deck-chairs, and beanbag chairs. The most
representative members of a category are called prototypical members. It is well-
known that some concepts, like 'red' and 'bald' have no sharp boundaries and for
these it is perhaps not surprising that one finds prototypical effects. However, these
effects have been found for most cocnepts including those with comparatively clear
boundaries like 'bird' and 'chair'.
In traditional philosophical analyses of concepts, based on truth-functions or
possible worlds it is very difficult to explain such prototype effects (see Gärdenfors
1991). Either an object is a member of the class assigned to a concept (relative to a
given possible world) or it is not and all members of the class have equal status as
category members. Rosch's research has been aimed at showing asymmetries
among category members and asymmetric structures within categories. Since the
9A selfapproved vowel is one that was produced by the speaker and later approved of as an example
of the intended kind. An identified sample of a vowel is one that was correctly identified by 75% of
the observers. The preferred samples of a vowel are those which are "the most representative
samples from among the most readily identified samples" (Fairbanks and Grubb 1961, p. 210)
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traditional definition of a concept neither predicts nor explains such asymmetries,
something else must be going on.
In contrast, if concepts are described as convex regions of a conceptual space,
prototype effects are indeed to be expected. In a convex region one can describe
positions as being more or less central. For example, if color concepts are identified
with convex subsets of the color space, the central points of these regions would be
the most prototypical examples of the color. In a series of experiments, Rosch has
been able to demonstrate the psychological reality of such 'focal' colors. For another
illustration we can return to the categorization of vowels presented in the previous
section. Here the structure of the subjects' different kinds of responses show clear
prototype effects.
For more complex categories like 'bird' it is perhaps more difficult to describe the
underlying conceptual space. However, if something like Marr and Nishihara's
(1978) analysis of shapes is adopted, we can begin to see how such a space would
appear.10 Their scheme for describing biological forms uses hierarchies of cylinder-
like modeling primitives. Each cylinder is described by two coordinates (length and
width). Cylinders are combined by determining the angle between the dominating
cylinder and the added one (two polar coordinates) and the position of the added
cylinder in relation to the dominating one (two coordinates). The details of the
representation are not important in the present context, but it is worth noting that
on each level of the hierarchy an object is described by a comparatively small
number of coordinates based on lengths and angles. Thus the object can be
identified as a hierarchically structured vector in a (higher order) conceptual space.
Figure 5 provides an illustration of the hierarchical structure of their representations.
10This analysis is expanded in Marr (1982), Ch. 5. A related model, together with some psychological
grounding, is presented by Biederman (1987).
12
Figure 5. Representing shapes by cylinders. (From Marr and Nishihara 1978)
It should be noted that even if even if different members of a category are judged to
be more or less prototypical, it does not follow that some of the existing objects
must represent 'the prototype'. If a concept is viewed as a convex region of a
conceptual space this is easily explained, since the central member of the region (if
unique) is a possible individual in the sense discussed above (if all its dimensions are
specified) but need not be among the existing members of the category. Such a
prototype point in the region need not be completely described as an individual, but
is normally represented as a partial vector, where only the values of the dimensions
that are relevant to the concept have been determined. For example, the general
shape of the prototypical bird would be included in the vector, but its color or age
would presumably not.
It is possible to argue in the converse direction too and show that if prototype
theory is adopted, then the representation of concepts as convex regions is to be
expected. Assume that some quality dimensions of a conceptual space are given, for
example the dimensions of color space, and that we want to partition it into a
number of categories, for example color categories. If we start from a set of
prototypes p1, ..., pn of the categories, for example the focal colors, then these
should be the central points in the categories they represent. One way of using this
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information is to assume that for every point p in the space one can measure the
distance from p to each of the pi's. If we now stipulate that p belongs to the same
category as the closest prototype pi, it can be shown that this rule will generate a
partitioning of the space that consists of convex areas (convexity is here defined in
terms of an assumed distance measure). This is the so-called Voronoi tessellation, a
two-dimensional example of which is illustrated in figure 6.
p5
p1
p2
p4
p3
p6
Figure 6. Voronoi tessellation of the plane into convex sets.
Thus, assuming that a metric is defined on the subspace that is subject to
categorization, a set of prototypes will by this method generate a unique
partitioning of the subspace into convex regions. Hence there is an intimate link
between prototype theory and the analysis of this article where concepts are
described as convex regions in a conceptual space.
As a concrete instance of this technique, Petitot (1989) applies Voronoi categorization
to explain some aspects related to the categorical perception of phonemes. In
particular, he analyses the relations between the so-called stop consonants /b/, /d/,
/g/, /p/, /t/, /k/. The relations between these consonants are expressed with the
aid of two dimensions: one is the voiced-unvoiced dimension, the other is labial-
dental-velar dimensions which relates to the place of articulation of the consonant.
Both these dimension can be treated as continuous. Figure 7 shows how he
represents the boundaries between the six consonants.
Figure 7. A Voronoi model of the boundaries of stop consonants. (From Petitot (1989), p. 69).
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As an example of the information contained in this model, he points out (Petitot
1989, p. 68):
The geometry of the system of boundaries can provide precious information about the
hierarchical relations that stop consonants maintain with each other. The fact that in the model
of Massaro and Oden, the domains of /p/ and /d/ are adjacent, whereas those of /b/ and /t/
are separated, indicates that the contrast between /b/ and /t/ is much greater than that
between /p/ and /d/.
7. CONCLUSION
The main purpose of this article has been to present the core of the theory of
conceptual spaces. In this connection an important question is: what kind of theory is
the theory of conceptual spaces? Is it an empirical, normative, computational,
psychological, neuroscientific, or linguistic theory?
As was stated in section 1, cognitive science has two predominant goals: to explain
cognitive phenomena and to construct artificial systems that can solve various
cognitive tasks. The theory of conceptual spaces is presented as a framework for
representing information. It should be seen as a theory that complements the symbolic
and the connectionist models and forms a bridge between these forms of
representation.
The primary aim is to use the theory of conceptual spaces in constructive tasks. In
previous work, I have shown how it can be used in computational models of concept
formation (Gärdenfors 1992) and induction (Gärdenfors 1990, 1993) and that it is useful
for representing the meanings of different kinds of linguistic expressions in a
computational approach to semantics.
The borderline between constructive and explanatory uses of conceptual spaces is
not sharp. When, for example, constructing the representational world of a robot, it
is often worthwhile to take lessons from how biology has solved the problems in
the brains of humans and other animals. Conversely, the construction of an artificial
system that can successfully solve a particular cognitive problem may provide clues
to how an empirical investigation of biological systems should proceed.
Consequently, there is a spiraling interaction between constructive and explanatory
uses of conceptual spaces.
This article has been asking questions about the geometry of thought. With the aid
of the notion of conceptual spaces I have provided an analysis of concepts. A key
notion is that of a natural concept that is defined in terms of well-behaved regions of
conceptual spaces – a definition that crucially involves the geometrical structure of
the various domains.
I my opinion, a conceptual level of representation should play a central role within
the cognitive sciences. After having been dominant for many years, the symbolic
approach was challenged by connectionism (which is nowadays broadened to a
wider study of dynamical systems). However, for many purposes, the symbolic
level of representation is too coarse, and the connectionist too fine-grained. In
relation to the two goals of cognitive science, I submit that the conceptual level will
add significantly to our explanatory capacities when it comes to understanding
cognitive processes, in particular those connected with concept formation and
language understanding.
15
Where do we go from here? The main factor preventing a rapid development of
different applications of conceptual spaces is the lack of knowledge about the
relevant quality dimensions. It is almost only for perceptual dimensions that
psychophysical research has succeeded in identifying the underlying geometrical
and topological structures (and, in rare cases, the psychological metric). For example,
we only have a very sketchy understanding of how we perceive and conceptualize
things according to their shapes.
When the structure of the dimensions of a particular domain is discovered, this often
leads to fruitful research. For example, the development of the vowel space that was
presented in section 5 led to a wealth of new results in phonetics and a deeper
understanding of the speech process.
Thus, those who want to contribute to the research program should start hunting
for the hidden conceptual spaces. Even if results may not be easily forthcoming, they
are sure to have repercussions in other areas of cognitive science as well.
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