Abstracts

• of half-plane, in case of the system of Aleksander Śniatycki from [3]
• of oval, in case of the system of Giangiancomo Gerla and Rafał Gruszczynski from [2]
• of sphere, in case of the system of Alfred Tarski from [4].

I will analyze the axioms of the aforementioned theories and display pre-theoretical (spatial) intuitions behind them.
Explanation how elementary geometrical notions of point, betweenness and equidistance are defined within
the appropriate systems will also be included.

It has been suggested by a number of authors (most prominently Whitehead and Russell) that spacetime points should be identified with classes of mereotopologically structured basic entities. These suggestions are mainly motivated by either of the two following views: (a) the empiricist or relationist view that statements about spacetime points should be reduced to statements describing relations between epistemologically or metaphysically preferable entities such as processes and events; (b) the view that talk about spacetime points should be meaningful even in the case that the mereological structure of spacetime regions is atomless (given substantivalism about spacetime regions).

In order to evaluate the feasibility of such suggestions, two main questions need to be answered: (Q1) Under which conditions is a point representation method-i.e. a method of identifying points with classes of mereotopologically structured basic entities—generally adequate? (Q2) Are there any adequate point representation methods in that sense? My talk addresses exactly these two questions.
If we want to treat question Q1 in a systematic and rigorous way, we first of all need a unified formal framework for analysing and evaluating different point representation methods. I propose a unified framework, in which I explicate a general notion of point representations and the notion of general adequacy of point representations. Thereby, we obtain an answer to Q1 and transform the informal question Q2 into a precise, mathematical question.

I then examine important point representation methods within the proposed framework and present the main results I have achieved so far. It can be proven in a rigorous manner that the method which identifies points with limited maximal round filters-as suggested by Roeper (1997) and Mormann (2010)-is generally adequate. So question Q2 has a positive answer. Other salient methods such as the method employing ultrafilters as points (along the lines of Stone’s representation theorem) and the method using completely prime filters (as usual in point-free topology) can be proven to be not generally adequate and we can pinpoint the reason for their inadequacy.

Husserl’s sentence: “The manner in which variation of curvature makes the various sorts of space-like manifolds pass into one another, gives the philosopher who has familiarized himself with the elements of the Riemannian-Helmholtzian theory a certain picture of the manner of the mutual legal connection among pure forms of theory of determinately distinct types.” – introduces a certain strong philosophical assumption. Namely, it assumes that the “pure forms of theory”, being the conditions of theoretical thinking, may be “pictured” or represented by certain “multiplicities” or manifolds describable by concepts deriving from topology. The question of spatial structures in which thinking takes place and the question of how various spaces differ from one another become a matter of importance, perhaps even of a fundamental theoretical importance.

Hence, a question arises about the limits of applicability of this strategy. Is it applicable to deductive theory only, or to any academic theory, or perhaps to all rational thinking? Finally, would it be possible, perhaps, to extend this form of representing thinking conditions to any act of consciousness, i.e. to what we call thinking in general? This would mean, however, that that multiplicity we are asking about, as well as its topology, are equivalent to a certain level of – or perhaps to the whole of? – the structure of consciousness which is studied by a phenomenologist when he turns away from the intentional object itself and towards the act.

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Nikolay Milkov

Wittgenstein’s Ways

Aristotle first investigated different modes, or ways of being. Unfortunately, in the modern literature the discussion of this concept has been largely neglected. Only recently, the interest towards the concept of ways increased. Usually, it is explored in connection with the existence of universals and particulars. Some authors claim that universals, the shape of my house, for example, are simply different ways in which its building blocks (the particulars) are ordered. Another group of metaphysicians focuses attention on the relation between universals and particulars. Well-known conception is that it is a relation of supervenience that eschews reductivism. In the last few years the relation between universals and particulars was also explored with the help of the concept of grounding.The approach we are going to follow in this paper is different. It discusses Wittgenstein’s Tractarian conception of higher ontological levels as ways of arranging elements of lower ontological levels. In the Tractatus, Wittgenstein developed his ontology of ways (Art und Weise) in six steps: (i) Constructing states of affairs out of objects; (ii) Constructing propositions out of states of affairs; (iii) Constructing propositional signs; (iv) Constructing thoughts with the help of propositional signs; (v) Constructing truth / falsity; (vi) Constructing works of art.Wittgenstein’s Tractarian ontology remained ontology of one world, in opposition to the ontology of many subordinated worlds of his teachers Frege (the author of the conception of three worlds) and Russell (the author of the Theory of Types): the transition from one ontological level into another, higher level doesn’t mean a transition from one world into another. This is the main advantage of Wittgenstein’s constructivist ontology of ways: it makes the belief in emergence of new worlds redundant. Another its advantage is that it suggests a tangible, topological, solution of the problem.

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Thomas Mormann

(De)constructing Points: From Topology to Mereology and Back

Points are considered as fundamental ingredients of topology spaces. For instance, a
topological space is defined as a set X of “points” endowed with some “topological structure” encapsulated in the set OX of open subsets of X, OX being a subset of the power set PX of X. The set OX of open sets of a topological space has the lattice-theoretical structure of a complete Heyting algebra. As is well known, many basic concepts of topology can actually expressed without points, but using only the lattice-theoretical structure of OX only, for instance continuity and connectedness. This has led to what has been described as “pointfree topology”. Indeed, pointfree topology may be characterized as a kind of non-classical mereology based on systems of regions exhibiting the structure of complete Heyting algebras instead of Boolean algebras as is the case for classical mereology. On the other hand, given an appropriate (pointfree) Heyting mereological algebras H, it is possible to construct for H a set of ersatz points pt(H). This set pt(H) may be endowed with a canonical topological structure O(pt(H)) isomorphic to H. In this way, under some mild restrictions, topological spaces and mereological systems may be considered as equivalent.

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Stefano Papa

Time Granularity and the Formal Ontology of Time-Awareness. A Husserlian Argument for a Topology of Temporal Information

The Phenomenological Analysis of Inner or Immanent Time-Consciousness is meant to make explicit structural dependencies in the field of predication and imagination. Reasoning about temporal information (which is central in modern systems of temporal logic) is touched upon only insofar, as an explication of abstractive reflection itself can be made available. Husserl proposes a fourfold sense of „absence“ in his analysis of temporal awareness: .“representation“; what is here absent, is the object denoted by a predication. Representation is dependent upon imagination. An object absent in an act of imagining is „vacant“ in the sense that the item itself is not to be retrieved by imagining it. Vacancy itself is dependent upon a third more radical absence, the absence of Time-Consciousness. Absence in this third sense is to be further explicated as awareness of time as immanent duration and inner sequence of given items. The methodical points of view which are operational in the interpretaion of temporal awarenes, and are to be applied to the experience of linear ordering of information items (for example, a musical gestalt); are the ones of the formal ontology developed in the Logical Investigations. The expressions „stream in a stream“, or „transcendence in the immanence“ (in Cartesian Meditations, with an eye to intersubjectivity), or else „living present“ are metaphorical. They point to the fourth sense of absence.
From the point of view of Phenomenology, the formal methods used in contemporary logics to model temporal reasoning are idealizations and abstractive formalizations, onesidedly founded in the above mentioned structures of dependency. As an example of such idealizations, one could mention Kamp’s theorem: the definability of all temporal operators in terms of „since“ and „until“, is bound to the condition that „time“ is interpreted as a continuous linear ordering. Indipendently from the completeness-result, this theorem is important also because it poses the issue of referring to the same language to describe a situation with respect to different temporal scalings (granularity).Time granularity is linked to semantic properties of representation systems. In contemporary logics it is formally treated by defining an algebra for granularities (set-theoretical approach) and by combination of simple temporal logics into a system for time granularity (logical approach). At this point, however, phenomenological considerations could motivate a shift in the study of time granularity: Since, as stated above, time-awareness is an eidetic layer to be abstracted from the presentation of some sequential items (melody), frames (algebras) as formalizations of finite operations have a shortcome, because they don’t contain a formalization of what they might be about (but this might be at the root of the problems posed by granularity, as illustrated by the synchronization problem). This situation can be changed by enriching a given frame (operations) with a set of points (observations), and a subset of their cartesian product. This construction is called a topological system. The introduction of a topology has thus been motivated by phenomenological considerations on time-awareness and its structural role in systems of formal representation. That is, the aim of this paper is to vindicate a systematic function of phenomenological reflections for the representation of formal categories in a given formalized system.E. Husserl, Logical Investigations (J. Findlay). 1970
E. Husserl, On the Phenomenology of the Consciousness of Internal Time (1893–1917), 1990 [1928]. (Brough, J.B).
E. Husserl, Formal and Transcendental Logic, 1969 [1929], Cairns, D., trans. The Hague: Nijhoff.
H. Kamp. Formal Properties of ‘now’. Theoria, 37:227–273, 1971.
H.Kamp. Events, Instantsand Temporal Reference. In: R.Bäuerle, U.Egli, and A.von Stechow, editors, Semantics from Different Points of View, pages 376–417. Springer-Verlag, 1979.
Jerome Euzenat, Angelo Montanari. Time granularity. In: Michael Fisher, Dov Gabbay, Lluis Vila. Handbook of temporal reasoning in artificial intelligence, Elsevier, pp.59-118, 2005, Foundations of artificial intelligence
S. Vickers, Topology via Logic. Cambridge 1989 (1996)

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Marek Piwowarczyk

Problems with extension

In my talk I want to examine the ontological status of extension, analyze the problem of infinite regress entangled in the concept of extended object and take a look at possible solutions to the problem.

Extension is an essential structural property consisting in having parts which lie outside each other. Each part is also extensive so we have a vicious regress. In order to avoid it we can: (1) postulate non-extensive beings which compose extensive ones; (2) say that the regress is not vicious (gunk theory); (3) postulate objects composed of parts dependent with respect to their extension on wholes (I call them “primordially extended objects”); (4) postulate extended simples. I try to show that no solution is satisfactory.

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Tomasz Placek

Bifurcating universes without bifurcating paths

This talk is a part of a larger argument for studying possible evolutions of individual objects rather than possible evolutions of an entire universe in investigations concerning determinism. The talk draws one’s attention to an odd phenomenon in the initial value problem of general relativity: each object (massive or mass-less) has a unique and well-defined possible evolution (geodesic), whereas a given 3-dimensional space has more than one possible development (i.e., Lorentzian 4-dimensional spacetime satisfying Einstein field equations). Moreover, the union of these non-isomorphic developments forms a non-Hausdorff manifold; the manifold does not permit bifurcating geodesics, however. These features have consequences for a project of defining possible histories in topological terms, for instance, as maximal Hausdorff sub-manifolds of some master manifold that captures all historical possibilities.

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Ian Pratt-Hartmann

A Skeptical Look at Region-based Theories of Space

One of the many achievements of coordinate geometry has been to provide a conceptually elegant and unifying account of spatial entities. According to this account, the primitive constituents of space are points, and all other spatial entities—lines curves, surfaces and bodies—are nothing other than the sets of those points which lie on them. The success of this reduction is so great that the identification of all spatial objects with sets of points has come to seem almost axiomatic.

For most of the previous century, however, a small but tenacious band of authors has suggested that more parsimonious and conceptually satisfying representations of space are obtained if we adopt an ontology in which regions, not points, are the primitive spatial entities. These, and other, considerations have prompted the development of formal languages whose variables range over certain subsets (not points) of specified classes of geometrical structures. We call the study of such languages `mereogeometry’.

In the past two decades, the Computer Science community in particular has produced a steady flow of new technical results in mereogeometry, especially concerning the computational complexity of region-based topological formalisms with limited expressive power. The purpose of this talk is to provide a conceptual framework for assessing the philosophical significance of this work. As usual, to grasp the philosophy, one first needs to master the mathematics.

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Peter Simons

Connectedness and Ontological Unity

A topological space is path connected when any two points are connected by a line. This definition, more intuitive than the standard definition of connectedness, goes over neatly to graph theory. It is argued that it is the concept we need to make sense of the notion of a unified or single entity. Some mereological theories assume that any collection of objects comprise a whole. This lets in cross-categorical and gerrymandered entities and it is argued that, contrary to many ontologists’ views, it renders mereology non-innocent. To retain a useful, more restrictive but still very general notion of a natural or integrated whole, generalised connectedness is precisely the notion we need.

Perzanowski’s combination ontologic is the ontology of elements and their combinations. Perzanowski, after Leibniz, concluded that combinations are defined by the internal features of the elements which define the structure of connections between the elements, i.e. whether one element is connected with another depends on the elements’ insides. We present the view that the structure of the combination depends firstly on structural, topological and a priori forms and then it can be determined by the features of elements. We suggest enriching the structure of the ontological universe by its topologisation. The examples of ontological worlds presented here prove that this is sensible and necessary. By modelling the ontological universe with the use of Hilbert’s cube we show the relations between the notion of dimension and the notions of situation and a possible world. We also put forth a new interpretation of ontological rationalism.