Physical boundaries and the earliest topologists. Topology has a relatively short history; but its 19th century roots are embedded in philosophical problems about the nature of extended substances and their boundaries which go back to Zeno and Aristotle. Although it seems that there have always been philosophers interested in these matters, questions about the boundaries of three-dimensional objects were closest to center stage during the later medieval and modern periods. Are the boundaries of an object actually existing, less-than-three-dimensional parts of (...) the object—that is, are solids bounded by two-dimensional surfaces, surfaces by one-dimensional “edges” or “physical lines”, edges by dimensionless “simples”? If not, how does a perfectly spherical object manage to touch a perfectly flat object—what part of the sphere is in immediate contact with the plane, if the sphere has no unextended parts? But if such parts be admitted, are we not then saddled with “actual infinities” of simples, lines, and surfaces spread throughout each continuous object—the boundaries of all the object’s internal parts? Does it help to say that these internal boundaries exist only “potentially”? (shrink)

Thinking about space is thinking about spatial things. The table is on the carpet; hence the carpet is under the table. The vase is in the box; hence the box is not in the vase. But what does it mean for an object to be somewhere? How are objects tied to the space they occupy? This book is concerned with these and other fundamental issues in the philosophy of spatial representation. Our starting point is an analysis of the interplay between (...) mereology (the study of part/whole relations), topology (the study of spatial continuity and compactness), and the theory of spatial location proper. This leads to a unified framework for spatial representation understood quite broadly as a theory of the representation of spatial entities. The framework is then tested against some classical metaphysical questions such as: Are parts essential to their wholes? Is spatial colocation a sufficient criterion of identity? What (if anything) distinguishes material objects from events and other spatial entities? The concluding chapters deal with applications to topics as diverse as the logical analysis of movement and the semantics of maps. (shrink)

Process and Reality, Whitehead’s magnum opus, is one of the major philosophical works of the modern world, and an extensive body of secondary literature has developed around it. Yet surely no significant philosophical book has appeared in the last two centuries in nearly so deplorable a condition as has this one, with its many hundreds of errors and with over three hundred discrepancies between the American and the English editions, which appeared in different formats with divergent paginations. The work itself (...) is highly technical and far from easy to understand, and in many passages the errors in those editions were such as to compound the difficulties. The need for a corrected edition has been keenly felt for many decades. (shrink)

Although the relationship of part to whole is one of the most fundamental there is, this is the first full-length study of this key concept. Showing that mereology, or the formal theory of part and whole, is essential to ontology, Simons surveys and critiques previous theories--especially the standard extensional view--and proposes a new account that encompasses both temporal and modal considerations. Simons's revised theory not only allows him to offer fresh solutions to long-standing problems, but also has far-reaching consequences for (...) our understanding of a host of classical philosophical concepts. (shrink)

Of Chisholm’s many signal contributions to analytic metaphysics, perhaps the most important is his treatment of boundaries, a category of entity that has been neglected, to say the least, in the history of ontology. We can gain some preliminary idea of the sorts of problems which the Chisholmian ontology of boundaries is designed to solve, if we consider the following Zeno-inspired thought-experiment.

The paper is a contribution to formal ontology. It seeks to use topological means in order to derive ontological laws pertaining to the boundaries and interiors of wholes, to relations of contact and connectedness, to the concepts of surface, point, neighbourhood, and so on. The basis of the theory is mereology, the formal theory of part and whole, a theory which is shown to have a number of advantages, for ontological purposes, over standard treatments of topology in set-theoretic terms. One (...) central goal of the paper is to provide a rigorous formulation of Brentano's thesis to the effect that a boundary can exist as a matter of necessity only as part of a whole of higher dimension which it is the boundary of. It concludes with a brief survey of current applications of mereotopology in areas such as natural-language analysis, geographic information systems, machine vision, naive physics, and database and knowledge engineering. (shrink)

We can see mereology as a theory of parthood and topology as a theory of wholeness. How can these be combined to obtain a unified theory of parts and wholes? This paper examines various non-equivalent ways of pursuing this task, with specific reference to its relevance to spatio-temporal reasoning. In particular, three main strategies are compared: (i) mereology and topology as two independent (though mutually related) chapters; (ii) mereology as a general theory subsuming topology; (iii) topology as a general theory (...) subsuming mereology. Some more speculative strategies and directions for further research are also considered. (shrink)

This is a revised and extended version of the formal theory of holes outlined in the Appendix to the book "Holes and Other Superficialities". The first part summarizes the basic framework (ontology, mereology, topology, morphology). The second part emphasizes its relevance to spatial reasoning and to the semantics of spatial prepositions in natural language. In particular, I discuss the semantics of ‘in’ and provide an account of such fallacious arguments as “There is a hole in the sheet. The sheet is (...) in the drawer. Ergo *there is a hole in the drawer”. (shrink)

What are the relationships between an entity and the space at which it is located? And between a region of space and the events that take place there? What is the metaphysical structure of localization? What its modal status? This paper addresses some of these questions in an attempt to work out at least the main coordinates of the logical structure of localization. Our task is mostly taxonomic. But we also highlight some of the underlying structural features and we single (...) out the interactions between the notion of localization and nearby notions, such as the notions of part and whole, or of necessity and possibility. A theory of localization—we argue—is needed in order to account for the basic relations between objects and space, and runs afoul a pure part-whole theory. We also provide an axiomatization of the relation of localization and examine cases of localization involving entities different from material objects. (shrink)

There are conflicting intuitions concerning the status of a boundary separating two adjacent entities (or two parts of the same entity). The boundary cannot belong to both things, for adjacency excludes overlap; and it cannot belong to neither, for nothing lies between two adjacent things. Yet how can the dilemma be avoided without assigning the boundary to one thing or the other at random? Some philosophers regard this as a reductio of the very notion of a boundary, which should accordingly (...) be treated a mere façon de parler. In this paper I resist this temptation and examine some ways of taking the puzzle at face value within a realist perspective—treating boundaries as ontologically on a par with (albeit parasitic upon) extended parts. (shrink)

There is a basic distinction, in the realm of spatial boundaries, between bona fide boundaries on the one hand, and fiat boundaries on the other. The former are just the physical boundaries of old. The latter are exemplified especially by boundaries induced through human demarcation, for example in the geographic domain. The classical problems connected with the notions of adjacency, contact, separation and division can be resolved in an intuitive way by recognizing this two-sorted ontology of boundaries. Bona fide boundaries (...) yield a notion of contact that is effectively modeled by classical topology; the analogue of contact involving fiat boundaries calls, however, for a different account, based on the intuition that fiat boundaries do not support the open/closed distinction on which classical topology is based. In the presence of this two-sorted ontology it then transpires that mereotopology—topology erected on a mereological basis—is more than a trivial formal variant of classical point-set topology. (shrink)

Husserls »Logische Untersuchungen« sind eines der folgenreichsten Werke der neueren Philosophiegeschichte. Mit dem ersten Erscheinen in den Jahren 1900 und 1901 (Max Niemeyer Verlag, Halle/Saale) nimmt jene Schule ihren Anfang, deren Name im Untertitel des zweiten Bandes zum ersten Mal sinnfällig wird: die Phänomenologie. Husserl sah damals in diesem Werk »Versuche zur Neubegründung der reinen Logik und Erkenntnistheorie«, die den Grund zu einem größeren Gedankengebäude zu legen imstande waren. Sie wollten freilich kein bloßes Programm sein, sondern »Fundamentalarbeit an den unmittelbar (...) erschauten und ergriffenen Sachen« darstellen. Die weitere Arbeit an der Ausprägung der phänomenologischen Philosophie führte schon nach wenigen Jahren zu neuen Problemen, zu vertieften Erkenntnissen und zu schärferen Formulierungen. Ihnen trägt die zweite, zum Teil tiefgreifend umgearbeitete Auflage Rechnung, deren Text seither in unveränderter Form immer wieder nachgedruckt wurde. (shrink)

Holes are a good example of the sort of entity that down-to-earth philosophers would be inclined to expel from their ontological inventory. In this work we argue instead in favor of their existence and explore the consequences of this liberality—odd as they might appear. We examine the ontology of holes, their geometry, their part-whole relations, their identity and their causal role, the ways we perceive them. We distinguish three basic kinds of holes: blind hollows, perforating tunnels, and internal cavities, treating (...) these uniformly as immaterial bodies. We develop a morphology of holes, focusing on the way a hole can be filled, and then look at the main properties of the resulting conceptual framework: holes are parasitic upon the surfaces of their hosts; holes can move, fuse into each other, split; they can be born, develop, and die. Finally, we examine how some morphological features of holes are represented in perception, including the conditions whereby we have the impression that we see, feel, or even hear a hole. The book has over 150 pictures and is completed by a formal appendix, a section with puzzles and exercises, and a extensive annotated bibliography. (shrink)

We are used to regarding actions and other events, such as Brutus’ stabbing of Caesar or the sinking of the Titanic, as occupying intervals of some underlying linearly ordered temporal dimension. This attitude is so natural and compelling that one is tempted to disregard the obvious difference between time periods and actual happenings in favor of the former: events become mere “intervals cum description”.1 On the other hand, in ordinary circumstances the point of talking about time is to talk about (...) what actually happens or might happen at some time or another. We talk about ‘now’ and ‘then’ in an effort to put some order in our description of what goes on. And since different events seem to overlap in so many different ways, a full account of their temporal relations seems to run afoul of a reductionist strategy. This raises two philosophical questions. The first is whether we can actually go beyond time, as it were, i.e., whether we can take events as bona fide entities and deal with them directly, just as we can deal with spatial entities such as physical bodies or masses without confining ourselves to their spatial representations. This is a controversial issue (though probably not as controversial as it used to be), and ties in with a number of unsettled problems concerning, e.g., the structure of causality or the definition of adequate identity and individuation criteria for events. 2 The second question is whether we can perhaps do without time, i.e., whether we can dispense with time points or intervals as an independent ontological category and focus only on actual or potential happenings, in opposition to the form of reductionism mentioned above—in short, whether we can account for the temporal dimension in terms of suitable relations among events. This is also a highly controversial issue, and relates to the classical dispute concerning relational vs. absolutist conceptions of (space and) time.3 It is this second question that we intend to focus on here.. (shrink)

The concept of niche (setting, context, habitat, environment) has been little studied by ontologists, in spite of its wide application in a variety of disciplines from evolutionary biology to economics. What follows is a first formal theory of this concept, a theory of the relations between objects and their niches. The theory builds upon existing work on mereology, topology, and the theory of spatial location as tools of formal ontology. It will be illustrated above all by means of simple biological (...) examples, but the concept of niche should be understood as being, like concepts such as part, boundary, and location, a structural concept that is applicable in principle to a wide range of different domains. (shrink)

Körner has made an important distinction between dependent and independent particulars, noting that any adequate theory of categories will divide particulars into those that are independent and those that are not. In the present paper, the concept of a spatial boundary is used to illustrate the concept of a dependent particular. It is suggested that, if we follow Brentano and think of such boundaries as ontologically dependent upon the things of which they may be said to be boundaries, then we (...) will be able to throw light upon a number of philosophical questions about the spatial continuum. These questions pertain to such topics as: the distinction between parts and boundaries; the definition of dimensionality; and the nature of contact and physical continuity. (shrink)

Argle. I’ve said it before and I’ll say it again: all things are material. Either holes are somehow material, or else there are no such things. Maybe a hole is the material hole-lining that, as we so misleadingly say, “surrounds” the hole; or else whatever ostensible reference we make to holes is secretly some other sort of language-game altogether, or it’s fictitious reference, or it’s just plain mistaken.

Although Aristotle (Metaphysics, Book IV, Chapter 2) was perhaps the first person to consider the part-whole relationship to be a proper subject matter for philosophic inquiry, the Polish logician Stanislow Lesniewski [15] is generally given credit for the first formal treatment of the subject matter in his Mereology.1 Woodger [30] and Tarski [24] made use of a specific adaptation of Lesniewski's work as a basis for a formal theory of physical things and their parts. The term 'calculus of individuals' was (...) introduced by Leonard and Goodman [14] in their presentation of a system very similar to Tarski's adaptation of Lesniewski's Mereology. Contemporaneously with Lesniewski's development of his Mereology, Whitehead [27] and [28] was developing a theory of extensive abstraction based on the two-place predicate, 'x extends over y\ which is the converse of 'x is a part of y\ This system, according to Russell [22], was to have been the fourth volume of their Pήncipia Mathematica, the never-published volume on geometry. Both Lesniewski [15] and Tarski [25] have recognized the similarities between Whitehead's early work and Lesniewski's Mereology. Between the publication of Whitehead's early work and the publication of Process and Reality [29], Theodore de Laguna [7] published a suggestive alternative basis for Whitehead's theory. This led Whitehead, in Process and Reality, to publish a revised form of his theory based on the two-place predicate, 'x is extensionally connected with y\ It is the purpose of this paper to present a calculus of individuals based on this new Whiteheadian primitive predicate. (shrink)

Qualitative reasoning about space and time - a reasoning at the human level - promises to become a fundamental aspect of future systems that will accompany us in daily activity. The aim of Spatial and Temporal Reasoning is to give a picture of current research in this area focusing on both representational and computational issues. The picture emphasizes some major lines of development in this multifaceted, constantly growing area. The material in the book also shows some common ground and a (...) novel combination of spatial and temporal aspects of qualitative reasoning. Part I presents the overall scene. The chapter by Laure Vieu is on the state of the art in spatial representation and reasoning, and that by Alfonso Gerevini gives a similar survey on research in temporal reasoning. The specific contributions to these areas are then grouped in the two main parts. In Part II, Roberto Casati and Achille Varzi examine the ontological status of spatial entities; Anthony Cohn, Brandon Bennett, John Gooday, and Nicholas Gotts present a detailed theory of reasoning with qualitative relations about regions; Andrew Frank discusses the spatial needs of geographical information systems; and Annette Herskovits focuses on the linguistic expression of spatial relations. In Part III, James Allen and George Ferguson describe an interval temporal logic for the representation of actions and events; Drew McDermott presents an efficient way of predicting the outcome of plan execution; and Erik Sandewall introduces a semantics based on transitions for assessing theories of action and change. In Part IV, Antony Galton's chapter stands clearly between the two areas of space and time and outlines the main coordinates of an integrated approach. (shrink)

This paper expands on the theory of event structures put forward in previous work by further investigating the subtle connections between time and events. Specifically, in the first part we generalize the notion of an event structure to that of a refinement structure, where various degrees of temporal granularity are accommodated. In the second part we investigate how these structures can account for the context-dependence of temporal structures in natural language semantics.

Whitehead, in his famous book Process and Reality, proposed a definition of point assuming the concepts of "region" and "connection relation" as primitive. Several years after and independently Grzegorczyk, in a brief but very interesting paper, proposed another definition of point in a system in which the inclusion relation and the relation of being separated were assumed as primitive. In this paper we compare their definitions and we show that, under rather natural assumptions, they coincide.

The concepts of a spatially continuous substance, of spatial dimension and of spatial boundary are here "analyzed out" of the concepts of individual thing, of constituent and of coincidence. The analysis is based upon the theory of spatial coincidence that was developed by Brentano. Its presuppositions are essentially these: (1) if there are spatial objects of any kind, then there are continuous spatial substances. (2) such substances are possibly such that they are not constituents of any individual thing; and (3) (...) they contain constituents (namely, boundaries) which are necessarily such that they are constituents of spatial substances. (shrink)

This paper describes the topological aspect of a logic-based, artificial intelligence approach to formalising the qualitative description of spatial properties and relations, and reasoning about those properties and relations. This approach, known as RCC theory, has been under development for several years at the University of Leeds. The main rationale for this project is that qualitative descriptions of spatial properties and relationships, and qualitative spatial reasoning, are of fundamental importance in human thinking about the world: even where quantitative spatial data (...) are most important, they must be attached to the components of a perceived spatial structure if we are to make use of them. RCC theory covers other qualitative aspects of spatial description and reasoning, but the topological properties and relations of spatially extended entities are fundamental to our work. The topological formalisms used by mathematicians are, in general, not well suited to the task of formalising the kinds of ‘common-sense’ or ‘everyday’ qualitative spatial description and reasoning which are our primary interest. Nevertheless, we must come to grips with the concepts of topology as practised by mathematicians if we are not to risk constantly ‘reinventing wheels’. (shrink)

Physical boundaries and the earliest topologists. Topology has a relatively short history; but its 19th century roots are embedded in philosophical problems about the nature of extended substances and their boundaries which go back to Zeno and Aristotle. Although it seems that there have always been philosophers interested in these matters, questions about the boundaries of three-dimensional objects were closest to center stage during the later medieval and modern periods. Are the boundaries of an object actually existing, less-than-three-dimensional parts of (...) the object—that is, are solids bounded by two-dimensional surfaces, surfaces by one-dimensional “edges” or “physical lines”, edges by dimensionless “simples”? If not, how does a perfectly spherical object manage to touch a perfectly flat object—what part of the sphere is in immediate contact with the plane, if the sphere has no unextended parts? But if such parts be admitted, are we not then saddled with “actual infinities” of simples, lines, and surfaces spread throughout each continuous object—the boundaries of all the object’s internal parts? Does it help to say that these internal boundaries exist only “potentially”? (shrink)

Contact relations have been studied in the context of qualitative geometry and physics since the early 1920s, and have recently received attention in qualitative spatial reasoning. In this paper, we present a sound and complete proof system in the style of Rasiowa and Sikorski (1963) for relation algebras generated by a contact relation.

This paper presents a calculus for mereotopological reasoning in which two-dimensional spatial regions are treated as primitive entities. A first order predicate language ℒ with a distinguished unary predicate c(x), function-symbols +, · and - and constants 0 and 1 is defined. An interpretation ℜ for ℒ is provided in which polygonal open subsets of the real plane serve as elements of the domain. Under this interpretation the predicate c(x) is read as 'region x is connected' and the function-symbols and (...) constants are given their meaning in terms of a Boolean algebra of polygons. We give an alternative interpretation ������ base on the real closed plane which turns out to be isomorphic to ℜ. A set of axioms and a rule of inference are introduced. We prove the soundness and completeness of the calculus with respect to the given interpretation. (shrink)

Whitehead, in his famous book "Process and Reality", proposed a definition of point assuming the concepts of “region” and “connection relation” as primitive. Several years after and independently Grzegorczyk, in a brief but very interesting paper proposed another definition of point in a system in which the inclusion relation and the relation of being separated were assumed as primitive. In this paper we compare their definitions and we show that, under rather natural assumptions, they coincide.

Die "Paradoxien des Unendlichen" sind ein Klassiker der Philosophie der Mathematik und zugleich eine gute Einführung in das Denken des "Urgroßvaters" der analytischen Philosophie. Das Unendliche - seit jeher ein Faszinosum für die philosophische Reflexion - wurde in der Zeit nach der Grundlegung der Analysis durch Leibniz und Newton in der Mathematik zunächst als Problem betrachtet, das sich nicht vollkommen widerspruchsfrei behandeln lässt. Bernard Bolzano, der heute als "Urgroßvater der analytischen Philosophie" (Michael Dummett) gilt, zeigt in diesem klassisch gewordenen Text (...) jedoch, dass es beim Nachdenken über das Unendliche, sofern man begrifflich differenziert argumentiert und klare Definitionen zugrunde legt, keine echten Widersprüche gibt und sich die "Paradoxien des Unendlichen" auflösen lassen. Bolzano setzt sich zunächst mit den Unendlichkeitsbegriffen der Mathematik und der (idealistisch geprägten) Philosophie seiner Zeit auseinander, bevor er sich Schritt für Schritt die analytischen Grundlagen einer konsistenten Theorie des Unendlichen erarbeitet. Bereits einige Jahrzehnte vor Georg Cantors bis heute maßgeblicher mathematischer Unendlichkeitsdefinition gelingt Bolzano mit diesem Werk, dessen logische Klarheit und dessen Grad an philosophischer Durchdachtheit bis heute Beispielcharakter haben, die Rehabilitation des Unendlichen für die Philosophie der Mathematik. Die "Paradoxien des Unendlichen" wurden zuerst 1851 aus dem Nachlass herausgegeben. Diese Neuausgabe beruht auf dem der Erstausgabe zugrundeliegenden Manuskript. (shrink)

Die in diesem Band vereinten Aufsätze spiegeln Brentanos späte Ansichten wieder; die Einleitung wurde für den allgemein philosophisch interessierten Leser konzipiert und behandelt Brentanos Theorie des Kontinuums, des Ursprungs des ...