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Since the birth of philosophy, space has
been a topic of investigation which has posed
puzzles and challenges. Today, spatial representation
is receiving even wider attention. From philosophers
to computer scientists, passing through logicians,
linguists, and cognitive scientists, many
minds with different backgrounds are looking
at ways to formally represent space and then
reason with and about spatial structures.
The book under review is the joint effort
of two philosophers who have already shown
their nifty-ness in explaining the nature
of some spatial phenomena in Holes and Other Superficialities (Casati
and Varzi, 1994).
The present book is not the continuation
of Holes, rather it takes a broader perspective,
using holes when necessary to explain spatial
peculiarities, but providing a lot of different
examples too. To put it differently, if one
has enjoyed Holes, one will appreciate Parts and Places even more, if instead one has to make a
choice between the two books from Casati
and Varzi, it is strongly recommended to
start with Parts and Places.
Already in the choice of the title, Casati
and Varzi make the first smart move: rather
than resortingto the usual, but somehow cryptic,
name of the field they are mainly dealing
– that is Mereotopology– they use two much
simpler synonyms – Parts and Places. (The
etymology of the word mereology comes from the Greek meros = part, and that of topology also from Greek topos = common place.)The word Parts accounts for a fundamental property of spatial
entities: they are are built of simpler constituents
and/or together with other entities build
more complex wholes. The word Places tells us that dealing with space is dealing
with locations and located entities.
The book brings mereotopology to a wider
public, by using simple words next to more
technical ones and by providing several examples
from our everyday experience of space next
to formal theories of space. Furthermore,
it provides a vast survey of the field, ranging
from the seminal work of Whitehead to the
most recent publications, which makes it
an excellent starting point for the graduate
student and researcher interested in qualitative
formal representations of space. The reader
interested in implementing spatial related
systems might consider the prevalent philosophical
view of the field to be a weakness. She/hewill
sometimes get lost in the philosophical subtleties
of the more abstract concepts and, at times,
wonder about possible implications of the
conclusions reached by the authors on implementations.
The starting point of the investigation is
the identification of appropriate theories
of space. Of course, different tasks need
different approaches. Geologists may resort
to geometry for “measuring the earth,” physicists
to vector spaces to compute the trajectory
of a satellite and engineers to tensor spaces
for building a skyscraper. What then is appropriate
at both the philosophical and cognitive level?
Mereology and topology seem adequate, going
– when necessary – into geometry. The first
four chapters of the volume are dedicated
to presentation of various theories with
particular emphasison the advantages and
drawbacks of each one of them. The underlying
conjecture of Casati and Varziis that the
best way to go is to start with mereology
and then extend it with topology (and other
theory fragments). The following five chapters
each deeply analyzes one different peculiar
property of space; while the following two
deal with applications to specific space-related
issues, such as events in space and maps.
The volume ends with a chapter of open questions,
notes (all grouped at the end of the volume,
which is a bit unhandy), references and a
well compiled index of terms, names and symbols.
Chapter 2 introduces the reader to the spatial
entities that will accompany her/him through
the book: glasses, tables, chairs, donuts,
and, why not, bikinis. But what are the unifying
elements of these sorts of objects? All of
them are composed of hierarchically organized
parts, made of matter and may be one piece
or not. It is the last distinction which
already provides for a surprise.
Mereology, often presented as a theory of
parts and wholes, is actually not enough
to capture the three main features of spatial
entities. Wholeness is a monadic property,
while parthood is a dyadic one which cannot
account for the former. Perhaps, topology
can do the trick. Topology can indeed be
seen as the theory of connection, so self-connectedness
can be expressed, in addition the notion
of set inclusion could work to account for
parthood. This time, it is a very peculiar
spatial phenomenon, that of holes, to impose
a deeper reflection. Topology cannot account
for superficial holes and it has to resort
tothe complement of an object to describe
the hole, i. e., it has to explicitly reference
the immaterial object. The lesson here is
that one needs a formal mereological theory
together with a topological one, the two
theories should be independent, while mutually
related. To put it in the words of Casati
and Varzi: “Mereology alone is too weak;
topology alone is too strong.”
The deficiency of mereology with respect
to expressing wholeness can only be explained
within a formal theory of mereology, and
Chapter 3 provides various such theories.
Actually, after an historical overview of
mereology, the authors present a full hierarchy.
The basic one, called ground mereology M,
has only three simple axioms for the parthood
predicate: reflexivity, antisymmetry and
transitivity. No surprise here, something
is part of itself, if two items are mutually
proper part of each other then they are the
same, and finally, a part of something which
is part of a bigger whole, is also part of
the bigger whole. Extensions of M go into
various directions depending on which extra
axioms are added: supplementation, extensionality,
closure principles, and fusion axioms all
provide for acceptable theories of parts.
The top of the hierarchy is what the authors
call general extensional mereology GEM, and
they also point out that GEM has some spatial
flavor: it behaves, as proven by Tarski in
1935, like set inclusion.
Chapter 4 is another technical piece of work;
equipped with mereological theories, the
authors present topological extensions, again
in the form of a hierarchy. In such extended
theories – i. e., in mereotopology – one
can define predicates for Tangential Part,
Internal Overlap, Internal Proper Part, just
to name a few. A remark, which also applies
to the previous chapter, is that emphasis
is put on the axiomatization of the theories
and on the differences in expressive power
among them, but little is said or shown about
the underlying models and model theoretic
issues.
Boundaries (Chapter 5) are ineludible when
dealing with spatial objects. When there
is some-thing, then there is a boundary.
It may be fuzzy or not, but it is there for
sure. Or is it not? Boundaries are absolutely
spatial entities, but at the same time they
do not take up place. Furthermore, when an
object is divided into several pieces, which
one gets the boundary? Spatial, but also
temporal, boundaries are a source of puzzles
that have to be dealt with when devising
a theory of space. Theories can treat boundaries
in different ways, including disregarding
them from the start, but an ontological commitment
must be taken. The authors suggest, though,
that boundaryless theories are blind of important
distinctions. In a boundaryless theory, there
is no difference between the notion of continuity
and that of contiguity. Another important
characteristic is the impossibility of having
a boundary without having an object: “boundaries
are ontologically dependent entities.” Another
class of puzzling spatial entities which
parasitically live in spatial objects is
that of holes (Chapter 8).Holes, like boundaries,
exist when there is an object they live in:
there is no way of isolating thehole of the
donut while removing the donut. Holes, as
any other parasite, need to be understood
andclassified. In doing so, the authors can
use topology in most cases, but in some other
cannot (forexample topology is blind for
hallows), whence the need for a separate
theory of holes to integratein a full mereotopological
spatial framework.
different way of thinking is required for
counterparts and spatial essentialism (Chapters
6 and9, respectively), because these are
modal notions. Casati and Varzi show how
a theory that accountsfor counterparts and
potential objects can be kept inside the
bounds of first order logic (by includingpossible
worlds in the domain of quantification),
but a first order theory must be extended
by meansof modal operators for the mereological
essentialism. So far, spatially extended
objects and regions have been used as synonyms,
but actually the firstare entities that occupy
the latter. An object has a region as “address,”
and this address can be of manysorts
(permanent vs. temporary, minimal vs. broad,
structured vs. unstructured). The authors
proposea theory to distinguish objects from
the regions they occupy, a theory of location
(Chapter 7). Detailson the interrelation
of a location predicate with the ones for
connection and parthood are spelled out.
The last two chapters, which go towards applications,
are somehow controversial. Chapter 10 isabout
temporal events and their relation to space,
an issue of sure interest. At times, the
assumptionseems to be explicitly made that
time can be treated with the same tools as
space. Probably the drivingforce here is
in the linguistic similarities between spatial
and temporal prepositions, but still it mayturn
out to be a dangerous line of thinking (to
get an impression of some purely spatial
behaviors, seeLemon and Pratt, 1997). Nevertheless,
the integration of time into a spatial framework
brings newinteresting formalizations into
the picture, e. g., those concerning movement.
The following chapteris also concerned with
applications. The chapter on maps (Chapter
11) is intriguing for a numberof reasons:
first, the application is an important one
(think for example of geographical informationsystems,
global positioning applications and even
wearable computing devices), second, the
mainidea of Casati and Varzi is quite clever:
between the real world we populate and the
paper map wehave in our hands while traveling
there is a semantic layer: that of formal
maps. Formal maps arerelated to the spatial
entities of the real world by an isomorphism,
thus the task of the authors is todefine
a semantics for the formal maps. The attempt
is successful, a problem though resides in
thedifficulties to then take the next step:
from formal maps to real world ordinary maps.
The authorssuggest that “this is analogous
to the project of providing a formal semantics
for formal languages. It will then be possible
to ask whether some features of this semantics
can be used to describe thesemantic structure
of ordinary maps. The aim is to eventually
be in a position to move from thesemantics
of formal maps to the formal semantics of
maps.” There is an obvious worry here: if
onthe one hand, it is true that an isomorphism
can be established between the space to be
representedand a formal map, it is not at
all clear how to establish an isomorphism
(or any other kind of precisemapping) between
formal maps and ordinary maps. Consider a
river, it is a connected spatial entity,
in a formal map it would be represented as
a self-connected entity, but then, in an
ordinary map, itwould not. In fact, road
bridges, train bridges, and other entities
are drawn above the river and makeit a scattered
region in an ordinary map. Hence, even though
the idea that formal maps exist and thatthe
laws of mereotopology govern them is very
interesting in principle, it does not seem
to decreasethe gap between formal theories
of maps and actually implementable systems.
All in all, Parts and Places is a very well
thought and carefully written book. It is
clearly intendedfor the philosopher and the
cognitive scientist, but it is also recommend
to the computer scientist. The latter is
likely to be carried away by Casati and Varzi’s
way of explaining the matter, first posingquestions
(that most often one would not have thought
of) and then answering them. The volume is
rich of mind opening puzzles that borrow
from everyday life embedded in space and
time, which is bound to capture and hold
the reader’s attention throughout the book.
References
Casati, R. and Varzi, A., 1994, Holes and
Other Superficialities, Cambridge, MA: MIT
Press.
Lemon, O. and Pratt, I., 1997, “Spatial logic
and the complexity of diagrammatic reasoning,
”Machine Graphics and Vision 6, 89–108. Marco
Aiello Institute for Logic,
Language and Computation, and Intelligent
Sensory and Information Systems University
of Amsterdam Plantage Muidergracht 241018
TV Amsterdam The NetherlandsE-mail: aiellom@ieee.org
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Evans Experientialism 
Parts and Places