**Topological
Foundations of Cognitive Science**

**Barry Smith**

This is a revised version of the introductory essay in C. Eschenbach, C. Habel and B. Smith (eds.),

Topological Foundations of Cognitive Science,Hamburg: Graduiertenkolleg Kognitionswissenschaft, 1994, the text of a talk delivered at the First International Summer Institute in Cognitive Science in Buffalo in July 1994.

I shall begin by introducing the concepts at the heart of topology in an informal
and intuitive fashion. Two well-known alternatives present themselves to this
end. These prove to be equivalent from the mathematical point of view, but they
point to distinct sorts of extensions and applications from the perspective of
cognitive science.

**1. The Concept of Transformation**

A first introduction to the basic concepts of topology
takes as its starting point the notion of *transformation. *We note,
familiarly, that we can transform a spatial body such as a sheet of rubber in
various ways which do not involve cutting or tearing.
We can invert it, stretch or compress it, move it, bend it, twist it, or
otherwise knead it out of shape. Certain properties of the body will in general
be invariant under such transformations - which is to say under transformations which are neutral as to shape, size, motion
and orientation. The transformations in question can be defined also as being those which do not affect the possibility of our connecting
two points on the surface or in the interior of the body by means of a
continuous line. Let us provisionally use the term 'topological spatial
properties' to refer to those spatial properties of bodies which are invariant
under such transformations (broadly: transformations which do not affect the *integrity*
of the body - or other sort of spatial structure - with which we begin).
Topological spatial properties will then in general fail to be invariant under
more radical transformations, not only those which involve cutting or tearing,
but also those which involve the gluing together of parts, or the drilling of
holes through a body, or the decomposition of a body into separate constituent
parts.

The property of being a (single, connected) body is a topological spatial
property, as also are certain properties relating to the possession of holes
(more specifically: properties relating to the possession of tunnels and
internal cavities). The property of being a *collection* of bodies and
that of being an *undetached part* of a body, too, are topological spatial
properties. It is a topological spatial property of a pack of playing cards
that it consists of this or that number of *separate* cards, and it is a
topological spatial property of my arm that it is *connected*
to my body.

This concept of topological property can of course be
generalized beyond the spatial case. The class of phenomena structured
by topological spatial properties is indeed wider than the class of phenomena
to which, for example, Euclidean geometry, with its determinate Euclidean
metric, can be applied. Thus
topological spatial properties are possessed also by mental images of spatially
extended bodies. Topological properties are discernible also in the temporal
realm: they are those properties of temporal structures which are invariant
under transformations of (for example) stretching (slowing down, speeding up)
and temporal translocation. Intervals of time, melodies, simple and complex
events, actions and processes can be seen to possess
topological properties in this temporal sense. The motion of a bouncing ball can be said to be topologically isomorphic to another,
slower or faster, motion of, for example, a trout in a lake or a child on a
pogo-stick.

**2. The Concept of Boundary**

A second introduction to the basic concepts of
topology takes as its starting point the intuitive notion of *boundary*.
We begin, once again, with spatial examples. Imagine a solid and homogeneous
metal sphere. We can distinguish, with some intellectual
effort, two parts of the sphere which do not overlap (they have no parts in common):
on the one hand is its *boundary, *its *exterior surface*; on the
other hand is its *interior*, the *difference* between the sphere and
this exterior surface (that which would result if, *per impossibile,
*the latter could be subtracted from the former). Similarly
in the temporal realm we can imagine an interval as being composed of its
initial and its final points together with the *interior *which results
when these points are removed from the interval as a whole.

Define, now, the *complement* *x*¢* *of an entity* x *as that entity
which results when we imagine *x* as having been deleted from the universe
as a whole. The boundary of an entity *x *is from the point of view of
classical mathematical topology also the boundary of the complement of *x.*
We can imagine, however, a variant topology which would
recognize *asymmetric *boundaries, such as we find, for example, in the
figure-ground structure as this is manifested in visual perception. As
Rubin (1921) first pointed out, the boundary of a figure is experienced as a
part of the figure, and not simultaneously as boundary of the ground, which is
experienced as running on behind the figure. Something similar applies also in
the temporal sphere: the beginning and ending of a race, for example, are not
in the same sense boundaries of any complement-entities (of all time prior to
the race, and of all time subsequent to the race) as they are boundaries of the
race itself.

The ideas of boundary and complement can be extended
in a natural way into the conceptual realm. Imagine the instances of a concept
arranged in a quasi-spatial way, as happens for example in familiar accounts of
colour- or tone-space. Suppose that
each concept is associated with some extended region in which its instances are
contained, and suppose further that this is done in such a fashion that the *prototypes*,
the most typical instances, are located in the centre
of the relevant region and the less typical instances located at distances from
this centre in proportion to their degree of
non-typicality. Boundary or fringe cases can now be defined as those cases which are so untypical that even the slightest further
deviation from the norm would imply that they are no longer instances of the
given concept at all. The notion of *similarity* can be
understood in this light as a topological notion.^{(1)}
In the realm of colours, for example, *a *is similar to *b* might be taken to mean that
the colours of *a *and *b* lie so close
together in colour-space that they cannot be
discriminated with the naked eye. A similarity relation is in general symmetric
and reflexive, but it falls short of transitivity, and is thus not an
equivalence relation. This means that it partitions the space of instances not
into tidily disjoint and exhaustive equivalence classes, but rather into
overlapping *circles of similars*. This falling
short of the discreteness and exhaustiveness of partitions of the type which are generated by equivalence relations is
characteristic of topological structures. In some cases
there is a continuous transition from one concept to its neighbouring
concepts in concept-space, as for example in the transition from *red* to *yellow
*which results from a continuous variation of hue. In other cases circles of similars are
separated by gaps. This is so in regard to the
transition from, say, *dog *to *cat* or from *cyclo-octatetrene**
*to *cyclobutadiene**.*

**3. The Concept of Closure**

The two approaches briefly sketched above may be unified into a single system
by means of the notion of *closure*, which we can think of as an operation
of such a sort that, when applied to an entity *x* it results in a whole
which comprehends both *x* and its boundaries. We employ as basis of our
definition of closure the notions of mereology.^{(2)}
Some of the reasons why we shun the set-theoretical instruments employed in
standard presentations of the foundations of topology will be set out below.

We shall employ '*x *£* y*'
to signify '*x *is a proper or improper part of *y*', and '*x *è *y*' to signify the mereological sum of two objects of *x *and y. The
range of our variables is *regions* of one or other type, including
boundaries and punctiform regions. An operation of *closure*
(c) is defined in such a way as to satisfy the
following axioms:

(AC1) *x* £ c(*x*)
(expansiveness)

(each object is a part of its closure)

(AC2) c(c(*x*)) £ c(*x*) (idempotence)

(the closure of the closure adds nothing to the
closure of an object)

(AC3) c(*x* è *y*) = c(*x*)
è c(*y*) (additivity)

(the closure of the sum of two objects is equal to
the sum of their closures)

These axioms were first set out in informal terms by the Hungarian
topologist Friedrich Riesz in 1906, and independently
by the Pole Kazimierz Kuratowski in 1922. The axioms
define a well-known kind of structure, that of a *closure algebra*, which
is the algebraic equivalent of the simplest kind of topological space. Kuratowski's list includes in addition the following axiom,
where '0' would designate a null element (the empty set, in a set-theoretical
formulation):

(AC0) c(0) = 0 (zero)

Here, however, we are interested in a mereological formulation of topology, and because there is no mereological analogue of the empty set this further axiom has no meaning.

Various modifications and weakenings of these
axioms are possible which preserve the possibility of defining analogues of the
standard topological notions of boundary, interior, etc.^{(3)} Thus we may drop the
axiom of additivity, which, as Hammer puts it, might most properly 'be called
the *sterility* axiom. ... it requires that two
sets cannot produce anything (a limit point) by union that one of them alone
cannot produce.'^{(4)}
If we apply our notion of closure to the conceptual sphere, then the closure of
an object can be defined as the smallest circle of similars including this object. That additivity then fails is seen by considering the case where *x *is an
instance of red and *y* an instance of green. The closure of *x *is
the circle of similars including all instances of
red; the closure of *y *is the circle of similars
including all instances of green. The sum of these closures is then strictly
smaller than the closure of *x *è *y*,
which is the circle of similars including all
instances of colour in general.

**4. The Concept of Connectedness**

Let '*x**' *' stand for the mereological complement of *x* and 'Ç' for mereological
intersection. On the basis of the notion of closure we
can now define the standard topological notion of (symmetrical) *boundary*,
b(*x*), as follows:

(DB) b(*x*) := c(*x*) Ç c(*x*¢)
(boundary)

Note that it is a trivial consequence of the definition of boundary here supplied that the boundary of an entity is in every case also the boundary of the complement of that entity.

It is indeed possible to define in standard topological terms an
asymmetrical notion of 'border', as the intersection of an object with the
closure of its complement:

(DB*) b*(*x*) = *x* Ç c(*x*¢) (border)

Moreover, where Kuratowski's axioms were formulated in terms of the single topological primitive of closure, Zarycki showed (1927) that a set of axioms equivalent to those of Kuratowski can be formulated also in terms of the single primitive notion of border, and the same applies, too, in regard to the notions of interior and boundary.

The notion of interior is defined as follows. We first of all introduce '*x - y*' to signify the result
of subtracting from *x *those parts of *x *which overlap with *y*.
We then set:

(DI) i(*x*) := *x*
- b(*x*) (interior)

We may define a *closed object* as an object which
*is identical with its closure. *An *open object, *similarly, is an object which is identical with its interior. The complement
of a closed object is thus open, that of an open
object closed. Some objects will be partly open and partly closed. (Consider
for example the semi-open interval (0,1], which
consists of all real numbers *x *which are greater than 0 and less than or
equal to 1.) These notions can be used to relate the two approaches to topology
distinguished above: topological *transformations* are those transformations which map open objects onto open objects.

A closed object is, intuitively, an independent constituent - it is an object which exists on its own, without the need for any
other object which would serve as its host. But a
closed object need not be *connected* in the sense that we can proceed
from any one point in the object to any other and remain within the confines of
the object itself. The notion of *connectedness*, too, is a topological
notion, which we can define as follows:

(DCn) Cn(*x*) := "*yz*(*x*
= *y* è *z* ® $*w*(*w*
£ (c(*y*) Ç c(*z*)))) (connectedness)

(a connected object is such that all ways of
splitting the object into two parts yield parts whose closures overlap)

The following yields an alternative concept of connectedness
which is useful for certain purposes:

(DCn*) Cn*(*x*) := "*yz*(*x*
= *y* è z ® ($*w*(*w* £ *x* Ù
w £ c(*y*)) ú $*w*(*w*
£ c(*x*) Ù *w* £ *y*)))

(connectedness*)

(a connected object is such that, given any way of
splitting the object into two parts *x *and *y, *either *x *overlaps
with the closure of *y *or *y* overlaps with the closure of *x*)

Neither of these notions is quite satisfactory however. Thus
examination reveals that a whole made up of two adjacent spheres which are
momentarily in contact with each other will satisfy either condition of
connectedness as thus defined. For certain purposes,
therefore, it is useful to operate in terms of a notion of *strong
connectedness* which rules out cases such as this. This latter notion may be defined as follows:

(DSCn) Scn(*x*) := Cn*(i(*x*))

(an object is strongly connected if its interior is
connected*)

**5. Mereotopology vs. Set Theory**

The rationale for insisting on a mereological rather
than a set-theoretic foundation for the axioms and definitions of topology for
our present purposes can be stated as follows. Imagine
that we are seeking a theory of the boundary-continuum structure as this makes
itself manifest in the realm of everyday human experience. The standard
set-theoretic account of the continuum, initiated by Cantor and Dedekind and
contained in all standard textbooks of the theory of sets, will be inadequate
to this purpose for at least the following reasons:

1. The application of set theory to a subject-matter
presupposes the isolation of some basic level of *Urelemente*
in such a way as to make possible a simulation of all structures appearing on
higher levels by means of sets of successively higher types. If, however, as
holds in the case of investigations of the ontology of the experienced world,
we are dealing with mesoscopic entities and with their mesoscopic constituents
(the latter the products of more or less arbitrary real or imagined divisions
along a variety of distinct axes), then there are no *Urelemente*
to serve as our starting-point. ^{(5)}
This idea is, incidentally, at the heart also of Gestalt-theoretical criticisms
of psychological atomism, which in many respects parallel criticisms of
set-theory-induced atomism of the sort presented here.

2. The experienced continuum is in ever cases a concrete, changing phenomenon, a phenomenon existing in time, a whole which can gain and lose parts. Sets, in contrast, are abstract entities, entities defined entirely via the specification of their members.

3. In the absence of points or elements, the experienced continuum does not sustain the sorts of cardinal number constructions imposed by the Dedekindian approach. The experienced continuum is not isomorphic to any real-number structure; indeed standard mathematical oppositions, such as that between a dense and a continuous series, here find no application.

4. Even if points or elements were capable of being isolated in the
experienced continuum, the set-theoretical construction would still be
predicated on the highly questionable thesis that out of unextended
building blocks an extended whole can somehow be constructed.^{(6)}
The experienced continuum, in contrast, is organized not in such a way that it
would be built up out of particles or atoms, but rather in such a way that the
wholes, including the medium of space, come before the parts which these wholes
might contain and which might be distinguished on various levels within them.

Of course, set theory is a mathematical theory of tremendous power, and none
of the above precludes the possibility of reconstructing topological theories
adequate for cognitive-science purposes also on a set-theoretic basis. Standard
representation theorems indeed imply that for any precisely formulated
topological theory formulated in non-set-theoretic terms we can find an
isomorphic set-theoretic counterpart. Even so, however, the reservations stated
above imply that the resultant set-theoretic framework could yield at best a *model*
of the experienced continuum and similar structures, not a theory of these
structures themselves (for the latter are after all *not sets, *in light
of the categorial distinction mentioned under 2.
above).

Our suggestion, then, is that mereotopology will
yield more interesting research hypotheses, and in a more direct and
straightforward fashion, than would be the case should we be constrained to
work with set-theoretic instruments.

**6. Foundations of Cognitive Science**

On the one hand, then, there is topology as a branch of mathematics. Topology
in this mathematical sense has been used by cognitive
scientists in work on the mathematical properties of connectionist networks and
elsewhere. On the other hand there is mereotopology,
a treatment of concepts such as 'region', 'connectedness', 'boundary',
'surface', 'point', 'neighbourhood', 'nearness', and
so on, that is inspired by standard mathematical treatments but which involves
departures from standard mathematical topology in ways designed to meet the
requirements of the subject-matters encountered within specific domains of
cognitive science. Mereotopology as here conceived is
not, however, a matter of loose analogies; rather it is a matter of deviations
from standard topology which can be rigorously defined.

**7. Husserl's Mereotopology**

The idea of using topology as a foundation for cognitive science is not without
precursors. It is above all in the tradition established by Brentano, a
tradition which extends through Carl Stumpf to the Berlin school of Gestalt psychology, that the
most important early contributions are to be found. Two such contributions will be dealt with here, those of Edmund Husserl and Kurt
Lewin. Husserl's *Logical Investigations* (1900/01) contain a formal
theory of part, whole and dependence that is used by
Husserl to provide a framework for the analysis of mind and language of just
the sort that is presupposed in the idea of a topological foundation for
cognitive science.^{(7)}
The title of the third of Husserl's Logical Investigations is "On the
Theory of Wholes and Parts" and it divides into two chapters: "The
Difference between Independent and Dependent Objects" and "Thoughts Towards a Theory of the Pure Forms of Wholes and
Parts". Unlike more familiar theories of wholes and parts, such as those prounded by Lesniewski, and
before him by Bolzano, Husserl's theory does not concern itself merely with
what we might think of as the vertical relations between parts and the wholes which comprehend them on successive levels as we move
upwards towards ever larger wholes. Rather, Husserl's theory is concerned also
with the horizontal relations between the different parts within
a single whole, relations which serve to give unity or integrity to the wholes in question. To put the matter simply: some parts of
a whole exist merely side by side, they can be
destroyed or removed from the whole without detriment to the residue. A whole
all of whose parts manifest exclusively such side-by-sideness
relations with each other is called a heap or aggregate or, more technically, a
purely summative whole (what we referred to above as an *Und-Verbindung*). In many wholes,
however, and one might say in *all* wholes manifesting any kind of unity,
certain parts stand to each other in relations of what Husserl called *necessary
dependence* (which is sometimes, but not always, necessary *interdependence*).
Such parts, for example the individual instances of hue, saturation and
brightness involved in a given instance of colour,
cannot, as a matter of necessity, exist, except in association with their
complementary parts in a whole of the given type. There is a huge variety of
such lateral dependence relations giving rise to correspondingly huge variety
of different types of whole which more standard approaches of 'extensional
mereology'^{(8)}
are unable to distinguish.

The connection between part and whole on the one hand and dependence on the
other may be seen in the fact that every whole can be
regarded as being dependent on its own constituent parts. This thesis may
amount to no more than the trivial claim that every object is such that it
cannot exist unless all the objects which are, at
different times, its parts, also exist at those times. Or it may consist in a
non-trivial thesis to the effect that certain special sorts of objects are such
as to contain special 'integral parts' which must exist at all the times when
these objects exist: their loss (for example the loss of brain or heart in a
mammal) is sufficient to bring about the destruction of the whole. Or, finally, it may be transformed into the metaphysical
thesis of mereological essentialism, i.e. into the
assertion that *every* spatio-temporal object is
dependent in the non-trivial sense upon *all* its parts, so that the ship
ceases to exist (becomes another thing) with the removal of the first splinter
of wood. It is one not inconsiderable advantage of Husserl's theory that it
allows a precise formulation of these and a range of related theses within a
single framework, a framework, furthermore, that is rooted on ideas concerning
part, whole and dependence which are consistent with
our common intuitions. Both Stanislaw Lesniewski, the
founder of mereology, and the linguist Roman Jakobson
applied Husserl's ideas on parts, wholes and categories from the *Logical
Investigations *in different branches of linguistics, in the early
development of categorial grammar and of phonology,
respectively.^{(9)} Thus Jakobson's account of distinctive features is as he himself
admits an application of Husserl's idea of dependent moments from the third
Investigation.

The topological background of Husserl's work makes itself felt already in
his theory of dependence.^{(10)}
It comes to the fore above all however in his treatment of the notion of *phenomenal
fusion*:^{(11)}
the relation which holds between two adjacent parts of an extended totality
when there is no qualitative discontinuity between the two. Adjacent squares on
a chess-board array are not fused together in this
sense; but if we imagine a band of colour that is
subject to a gradual transition from red through orange to yellow, then each
region of this band is fused with its immediately adjacent regions. After
distinguishing dependent and independent contents (for example, in the visual
field, between a colour- or brightness-content on the
one hand, and a content corresponding to the image of a moving projectile, on
the other), Husserl goes on to note that there is in the field of intuitive
data an additional distinction,

between

intuitivelyseparated contents, contentsset in relieffromorseparated off fromadjoining contents, on the one hand, and contents which arefusedwith adjoining contents, or whichflow overinto them without separation, on the other (Investigation III, §8, 449).

He
points out that independent contents

which are what they are no matter what goes on in their neighbourhood, need not have this quite different independence of separateness. The parts of an intuitive surface of a uniform or continuously shaded white are independent, but not separated.

(Loc. cit.)

Such
content Husserl calls 'fused'; they form an 'undifferentiated whole' in the
sense that the moments of the one pass 'continuously' ['*stetig*']
into corresponding moments of the other. (§9, 450)

That Husserl was at least implicitly aware of the topological
aspect of his ideas, even if not under this name, is unsurprising given that he
was a student of the mathematician Weierstrass in
Berlin, and that it was Cantor, Husserl's friend and colleague in Halle during
the period when the *Logical Investigations* were being written, who first
defined the fundamental topological notions of open, closed, dense, perfect
set, boundary of a set, accumulation point, and so on. Husserl
consciously employed Cantor's topological ideas, not least in his writings on
the general theory of (extensive and intensive) magnitudes
which make up one preliminary stage on the road to the third
Investigation.^{(12)}

More generally, it is worth pointing out that the early
development of topology on the part of Cantor and others was part of a wider
project on the part of both mathematicians and philosophers in the nineteenth
century to produce a *general theory of space* - to find ways of
constructing fruitful generalizations of such notions as extension, dimension,
separation, neighbourhood, distance, proximity,
continuity, and boundary. Husserl participated in this project with Stumpf and other students of Brentano such as Meinong.^{(13)} Significantly, the 1906
paper on "The Origins of the Concept of Space", in which Riesz first formulated the closure axioms at the heart of
topology is in fact a contribution to formal phenomenology, a study of the
structures of *spatial presentations*, in which the attempt is made to
specify the additional topological properties which must be possessed by a
mathematical continuum if it is adequately to characterize the continuity and
order properties of our experience of space.

**8. Lewin's Topological Psychology**

Of all the precursors of contemporary applications of topology in cognitive
science, the most notorious is the work on "topological and vector
psychology" of the German Gestalt psychologist Kurt Lewin. It will suffice
for our present purposes merely to illustrate some of the ways in which Lewin
uses topological notions in his *Principles of Topological Psychology *of
1936.

Lewin begins with the opposition *thing* (intuitively: a closed
connected unity) and *region* (intuitively: a space within which things
are free to move). As Lewin points out, what is a thing from one psychological
perspective may be a region from another: 'A hut in the mountain has the
character of a thing as long as one is trying to reach it from a distance. As
soon as one goes in, it serves as a region in which one can move about.' (1936,
p. 116) He then defines the notion of a *boundary zone* *z *between
two disconnected but proximate regions *m *and *n, *as the region,
foreign to *m *and *n, *which has to be crossed
in passing from one to the other. The whole *m + n* + *z *is then *connected* in the topological sense. (1936, p.
121)

The concept of a *barrier* he defines as a boundary zone
which offers resistance to passage of things between one region and
another. Such resistance may be asymmetric; thus it
may be greater in one dimension than in the opposite direction. Barriers effect
the degree of communication between one region and another, or in other words
the *degree of influence* of the *state* of one region on that of
another region. Hence the notion of degree of
influence, too, need not be symmetric: the fact that *a *is in a certain
degree of communication with *b *does not imply that *b *is in
equally close communication with *a.*

Two regions *a *and *b* are said to be
parts of a *dynamically connected region* if a change of state of *a *results
in a change of state of *b*. The notion of *dynamic connectedness*,
too, is by what was said earlier a matter of degree. In fact we can distinguish
a hierarchy of degrees of interlinkage between regions, and here Lewin echoes
discussions in the Gestalt-theoretical literature of the notions of 'strong'
and 'weak' Gestalten (1936, pp. 173f.). A strong Gestalt may
be defined as a complex with a high degree of dynamic connectedness
between its parts. Examples are: an organism, an
electromagnetic field. A weak Gestalt, for example a chess-club or a crowd of
onlookers, has a lesser but still non-zero dynamic connectedness between its
parts, while a purely summative whole (an '*Und-Verbindung*'
in Gestaltist terminology) is such that its separate
units manifest a zero degree of dynamic connectedness. Interestingly, in light
of our discussions of the two alternative motivations underlying topological
theory above, the notions central to Gestalt theory can be defined not merely on the basis of the notion of dynamic connectedness, but
also in terms of structure-preserving transformations.^{(14)}

We have deliberately introduced the basic concepts of Lewin's topological
psychology in a rather general way, abstaining from any specific applications
to psychological matters. The notions are, as Lewin himself sometimes
recognizes, *formal* in the sense that they can be applied
indiscriminately to a wide range of different sorts of material regions. The
general tendency of Lewin's writings, however, is to switch too unthinkingly to
psychological applications, whereby his use of the concepts in question -
concepts of 'barrier', 'path', '(psychological) locomotion', 'dynamic
interdependence (as the main determinant of the topology of the person)',
'tension', 'resistance', 'inhibition', etc. - seems often to remain on the
level of metaphor. Some cognitive scientists may be content never to pass
beyond this level in their investigations. Lewin's critics, however, rightly
drew attention to a certain crucial shortfall in his use of mathematical
notions in his writings. As was correctly pointed out, Lewin rarely makes the mathematical
theory of notions such as connectedness, boundary, separateness, and so on, do
any substantial work within the framework of his investigations. This criticism
was put forward in an influential article by London (1944), an article which did much to thwart the further development of
topological psychology (or of a topologically founded cognitive science) as
Lewin had conceived it. Yet London's critique is in some respects exaggerated,
as is shown by the fact that certain aspects of Lewin's generalizations of
standard topology have since shown themselves to be highly fruitful. These
generalizations include:

1. the recognition that it is possible to construct topology on a
non-atomistic, mereological basis which works in
terms of *wholes (regions) *as well as parts - where London, a defender of
the analytic method at the heart of physical science, holds that for scientific
purposes 'all experience must be dealt with in bits' (p. 279);

2. the systematic employment of the notion of asymmetric boundary, a notion which turns out to be crucial in many cognitive spheres;

3. the employment of topological ideas and methods also in relation to finite domains of objects. London argues (pp. 288f.) that topology makes sense only in infinite domains; as Latecki (1992) and others have shown, however, it is possible to construct in rigorous fashion finitistic systems in which analogues of topological notions can be defined.

More recent developments have demonstrated that it
is possible to go beyond the merely metaphorical employment of topological
concepts in cognitive science and to exploit the formal-ontological properties
of these concepts for theoretical purposes in a genuinely fruitful way.
Topology can serve as a theoretical basis for a unification of diverse types of
psychological facts.^{(15)}
Thus many of Lewin's ideas recall principles of 'force dynamics' worked out in
greater sophistication in the linguistic sphere by Talmy
(1988), and Talmy, along with Petitot
and others, has demonstrated the importance of topology for the understanding
of a variety of different sorts of linguistic structuring.^{(16)}
As Talmy notes, the conceptual structuring effected
by language is illustrated most easily in the case of
prepositions. A preposition such as 'in' is magnitude neutral (in a thimble, in
a volcano), shape neutral (in a well, in a trench), closure-neutral (in a bowl,
in a ball); it is not however discontinuity neutral (in a bell-jar,
in a bird cage). Work on verb-aspect and the mass-count distinction, too, has
profited from a topological orientation.^{(17)}

Topological structures play a central role also in studies
of naive physics, not least in virtue of the fact that even well-attested
departures from true physics on the part of common sense leave the topology and
vectorial orientation of the underlying physical
phenomena invariant:^{(18)}
our common sense would thus seem to have a veridical grasp of the topology and
broad general orientation of physical phenomena even where it illegitimately
modifies the relevant shape and metric properties.

In talking somewhat grandly of 'topological foundations for cognitive science', now, we are contending that the topological approach yields not simply a collection of insights and methods in selected fields, but a unifying framework for a range of different types of research across the breadth of cognitive science and a common language for the formulation of hypotheses drawn from a variety of seemingly disparate fields. Initial evidence for the correctness of this view is provided not just by the scope of the inquiries referred to above, but also by the degree to which in different ways they overlap amongst themselves and support each other mutually.

One rationale behind the idea that the inventory of
topological concepts can yield a unifying framework for cognitive science turns
on the fact that, as has often been pointed out (see e.g. Gibson 1986),
boundaries are centres of salience not only in the
spatial but also in the temporal world (the beginnings and endings of events,
the boundaries of qualitative changes for example in the unfolding of speech
events: cf. Petitot 1989). Moreover,
topological properties are more widely applicable than are those properties
(for example of a geometrical sort) with which metric notions are associated.
Metric features have certainly proved highly useful for the purposes of natural
science. Given the pervasiveness of qualitative elements in
every cognitive dimension, however, and also the similar pervasiveness of
notions like *continuity,* *integrity, boundary*, *prototypicality*,
etc., we can conjecture that topology will be not merely sufficiently general
to encompass a broad range of cognitive science subject-matters, but also that
it will have the tools to do justice to these subject-matters without imposing
alien features thereon.

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3. See the works by Ore, Hammer, Nöbeling, Netzer in the list below.

6. Brentano 1988, Asenjo 1993, Smith 1987.

9. Ajdukiewicz 1935, Holenstein 1975.

11. Casati 1991, Petitot 1994.

12. See Husserl 1983, pp. 83f, 95, 413, etc. and compare
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13. See Husserl 1983, pp. 275-300, 402-410; Stumpf 1873; Meinong 1903, esp. §2: "Farbengeometrie und Farbenpsychologie", and §5: "Der Farbenraum und seine Dimensionen".

16. See for example Jackendoff 1991, Lakoff 1989, Petitot 1992, 1992a, Wildgen 1982

17. See e.g. Mourelatos 1981, Galton 1984, Hoeksema 1985, Desclès 1989, Brandt 1989.

18. Bozzi 1958, 1959, McCloskey 1983, Smith and Casati 1994.