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5th International Ontological Workshop
Topological Philosophy
The Conference is organized by International Center for FormalOntology, Warsaw University of Technology.
Edited by:
Krzysztof Siemie«czukBartªomiej Skowron
©International Center for Formal Ontology
First published 2016
Publisher: International Center for Formal Ontology, Warsaw Uni-versity of Technology, Warsaw
http://www.icfo.ans.pw.edu.pl/
Contents
1 Committees 3
2 Graduate School on Topological Philosophy 4
3 Conference Schedule 5
4 Abstracts 8
Tomasz Bigaj & John Dougherty On the Topological Ap-proach to the Metaphysical Problem of Indistinguish-able Quantum Particles . . . . . . . . . . . . . . . . . 8
Romain Dufêtre Holism, Porosity and Topological Philos-ophy. A Set-Theoretical Introduction . . . . . . . . . 10
Samuel Fletcher Topological Structure on Scienti�c Theories 10Benjamin Feintzeig Topological Considerations in the Con-
struction of Quantum Theories . . . . . . . . . . . . 11Rafaª Gruszczy«skiHalf-Planes, Ovals and Spheres. Point-
Free Systems of A�ne and Euclidean Geometry . . . 11Laurenz Hudetz Representing Points as Classes of Mereotopo-
logically Structured Basic Entities . . . . . . . . . . . 12Janusz KaczmarekMathematical Tools in Ontology of Ideas,
Concepts and Individuals . . . . . . . . . . . . . . . . 14Marek Ku± What Topology Can O�er Physics. Topological
Constraints and Predictions in Classical and Quan-tum Worlds. . . . . . . . . . . . . . . . . . . . . . . . 15
Andrzej Leder In What Kind of Space Do Husserl's Lec-tures on the Phenomenology of the Consciousness ofInternal Time Take Place? . . . . . . . . . . . . . . . 16
Nasim Mahoozi Can Topology Justify �Vague Existence�? . 17Nikolay Milkov Wittgenstein's Ways . . . . . . . . . . . . . 18Thomas Mormann (De)Constructing Points: From Topol-
ogy to Mereology and Back . . . . . . . . . . . . . . . 19Stefano Papa Time Granularity and the Formal Ontology
of Time-Awareness. A Husserlian Argument for aTopology of Temporal Information . . . . . . . . . . . 19
Marek Piwowarczyk Problems with Extension . . . . . . . 22
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Tomasz Placek Bifurcating Universes without BifurcatingPaths . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Ian Pratt-Hartmann A Skeptical Look at Region-based The-ories of Space . . . . . . . . . . . . . . . . . . . . . . 23
Peter Simons Connectedness and Ontological Unity . . . . 24Bartªomiej Skowron Perzanowski's Combination Ontologic
in Hilbert's Cube . . . . . . . . . . . . . . . . . . . . 24Achille Varzi The Boundaries of Things: Where Topology
Meets Metaphysics . . . . . . . . . . . . . . . . . . . 25
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1 Committees
Organising Committee:
� Mirosªaw Szatkowski (International Center for Formal Ontol-ogy, Warsaw University of Technology)
� Bartªomiej Skowron (International Center for Formal Ontol-ogy, Warsaw University of Technology)
Scienti�c Committee:
� Thomas Mormann (University of the Basque Country, Spain)
� Tomasz Placek (Jagiellonian University, Poland)
� Ian Pratt-Hartmann (University of Manchester, UK)
� Peter Simons (Trinity College Dublin, Ireland)
� Mirosªaw Szatkowski (LMU, Munich, Germany)
� Achille Varzi (Columbia University, USA)
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2 Graduate School on Topological
Philosophy
International Center for Formal Ontology
6-7 February 2016
Okopowa 55, Warsaw, Poland
School Schedule
Saturday, February 6
10.00�11.30 Basic Topology I (Roland Zarzycki)
11.30�12.00 Co�ee
12.00�13.30 Basic Topology II (Roland Zarzycki)
13.30�15.00 Dinner
15.00�16.30 Basic Topology III (Roland Zarzycki)
16.30�17.00 Co�ee
17.00�18.30 Basic Topology IV (Roland Zarzycki)
Sunday, February 7
10.00�11.30 Topological Philosophy I (Thomas Mormann)
11.30�12.00 Co�ee
12.00�13.30 Topological Philosophy II (Thomas Mormann)
13.30�15.00 Dinner
15.00�16.30 Parts and Boundaries (Achille Varzi)
16.30�17.00 Co�ee
17.00�18.30 On the Logic and Metaphysics of the Concept ofDiscernibility (Tomasz Bigaj)
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3 Conference Schedule
Monday, February 8
9.00 Opening Address�Józef Lubacz (Chairman of the ICFO Pro-gram Council, Director of the Institute of Telecommunica-tions, Warsaw University of Technology, Poland)
Chair: Peter Simons
9.15�10.00 Thomas Mormann (University of the Basque Coun-try)(De)Constructing Points: From Topology to Mereology andBack
10.00�10.45 Ian Pratt-Hartmann (University of Manchester, UK)A Skeptical Look at Region-Based Theories of Space
10.45�11.30 Rafaª Gruszczy«ski (Nicolaus Copernicus Univer-sity in Toru«, Poland)Half-Planes, Ovals and Spheres. Point-Free Systems of A�neand Euclidean Geometry
11.30�12.00 Co�ee
Chair: Achille Varzi
12.00�12.45 Peter Simons (Trinity College Dublin, Ireland)Connectedness and Ontological Unity
12.45�13.30 Janusz Kaczmarek (University of �ód¹, Poland)Mathematical Tools in Ontology of Ideas, Concepts and Indi-viduals
13.30�15.00 Lunch
Chair: Tomasz Bigaj
15.00�15.30 Laurenz Hudetz (University of Salzburg, Austria)Representing Points as Classes of Mereotopologically Struc-tured Basic Entities
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15.30�16.00 Bartªomiej Skowron (International Center for For-mal Ontology, Warsaw University of Technology; Ponti�calUniversity of John Paul II in Kraków, Poland)Perzanowski's Combination Ontologic in Hilbert's Cube
16.00�16.30 Co�ee
Chair: Samuel Fletcher
16.30�17.00 Romain Dufêtre (Université Paris I Panthéon � Sor-bonne, France)Holism, Porosity and Topological Philosophy. A Set-TheoreticalIntroduction
17.30�18.00 Nasim Mahoozi (University of Barcelona, Spain)Can Topology Justify �Vague Existence�?
18.00�18.30 Benjamin Feintzeig (University of California, USA)Topological Considerations in the Construction of QuantumTheories
19.30 Gala Dinner
Tuesday, February 9
Chair: Thomas Mormann
9.00�9.45 Achille Varzi (Columbia University, USA)The Boundaries of Things: Where Topology Meets Metaphysics
9.45�10.30 Tomasz Placek (Jagiellonian University, Poland)Bifurcating Universes Without Bifurcating Paths
10.30�11.15 Tomasz Bigaj (University of Warsaw, Poland; Uni-versity of Bristol, UK) & John Dougherty (University of Cal-ifornia, USA)On the Topological Approach to the Metaphysical Problem ofIndistinguishable Quantum Particles
11.15�11.45 Co�ee
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Chair: Alexander Zuevsky
11.45�12.30 Marek Ku± (Center for Theoretical Physics, PolishAcademy of Sciences, Poland)What Topology Can O�er Physics. Topological Constraintsand Predictions in Classical and Quantum Worlds.
12.30�13.15 Samuel Fletcher (Munich Center for MathematicalPhilosophy, Germany & University of Minnesota, Twin Cities,USA)Topological Structure on Scienti�c Theories
13.30�15.00 Lunch
Chair: Janusz Kaczmarek
15.00�15.45 Nikolay Milkov (University of Paderborn, Germany)Wittgenstein's Ways
15.45�16.30 Andrzej Leder (Polish Academy of Sciences, Poland)In What Kind of Space Husserl's Lectures on the Phenomenol-ogy of the Consciousness of Internal Time Take Place?
16.30�17.15 Stefano Papa (University of Vienna, Austria)Time Granularity and the Formal Ontology of Time-Awareness.A Husserlian Argument for a Topology of Temporal Informa-tion
17.15�17.30 Closing Address
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4 Abstracts
On the Topological Approach to the MetaphysicalProblem of Indistinguishable Quantum Particles
Tomasz Bigaj & John Dougherty
Tomasz Bigaj
University of Warsaw, Poland
University of California, San Diego, USA
John Dougherty
University of California, USA
[email protected], [email protected]
The standard quantum theory of many particles imposes animportant restriction on the available states of particles of the sametype. This restriction takes on the form of the symmetrizationpostulate, according to which the state of a system of �indistin-guishable� particles has to be either symmetric (for bosons) or anti-symmetric (fermions) with respect to the permutation of individualparticles. The symmetrization postulate is applied to reduce thenumber of accessible states that can be identi�ed in a full tensorproduct of N individual (labeled) Hilbert spaces. However, thereare some alternative ways of representing mathematically statesof indistinguishable particles. In the topological approach, thecon�guration space is obtained by identifying all the elements ofthe full tensor product of individual spaces that di�er only withrespect to the permutation of the elements (this procedure is knownas �quotienting out�). The resulting con�guration space appears tohave new interesting topological properties due to the existence ofsingularities at points where two or more particles possess the samestate. In particular, it can be shown (Leinaas & Myrheim 1977)that the di�erence in global topology between con�guration spacesfor distinguishable and indistinguishable particles naturally leadsto the symmetry constraints on the states of particles of the sametype. In this article the topological approach will be compared to
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yet another method of representing the states of indistinguishableparticles recently suggested by Ladyman et al. (2013). In this latestproposal the states of N particles of the same type are constructedwith the help of the symmetric or antisymmetric �wedge� productrather than the full tensor product. The wedge product of twovectors is de�ned as the equivalence class that contains all andonly vectors of the ordinary tensor product for which the operationof symmetrization (antisymmetrization) gives the same result aswhen applied to the direct product of the initial vectors. Boththe topological approach and the wedge formalism will be analyzedwith respect to their ability to shed new light on the metaphys-ical problem of indistinguishable particles, which is the questionwhether quantum particles of the same type can be discerned byany meaningful physical properties or relations, and whether theycan thus achieve the status of individual objects equipped with theirown unique identities. Another question addressed in the paper willbe the problem of the redundancy of some parts of the mathematicalformalism used in the description of physical reality (the problemof �surplus structure� in Michael Redhead's terminology). It turnsout that both the topological approach and the wedge formalismpresent us with their own unique ways of eliminating such surplusstructures in the case of the quantum theory of many particles.
References
[1] Ladyman, J., Linnebo, Ø., and Bigaj, T. (2013). Entanglementand non-factorizability. Studies in History and Philosophy of Mod-ern Physics, 44:215�221.
[2] Leinaas, J. and Myrheim, J. (1977). On the theory of identicalparticles. Il Nuovo Cimento B Series 11, 37(1):1�23.
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Holism, Porosity and Topological Philosophy.A Set-Theoretical Introduction
Romain Dufêtre
Université Paris I Panthéon � Sorbonne, France
This paper aims to bring out an introduction to the study ofholism and more speci�cally holistic relation. A holistic relationshall be considered as a maximally porous relation and two de�ni-tions of porosity shall be provided. The �rst, abstract, the de�nitioninvolves entails that inclusion is not a porous relation that willbe brie�y applied to the comprehension of classical holistic philo-sophical systems. Another de�nition of porosity shall be presentedinvolving topological open ball in a metric space and the second partwill provide an introduction to the tools used by mathematiciansin the topological study of the taxonomy of holehood.
Topological Structure on Scienti�c Theories
Samuel Fletcher
Munich Center for Mathematical Philosophy, Germany
University of Minnesota, Twin Cities, USA
I review and amplify on some of the many uses of representinga scienti�c theory in a particular context not just as a collectionof models, but as a topological space. Topological structure ona set encodes among that set's elements a notion of similarity,which proves fruitful in the analysis of a variety of issues central tothe philosophy of science. These include intertheoretic reduction,idealization and approximation, emergence, the epistemic connec-tion between modeling and knowledge, and modality in science.The morals are twofold: �rst, the further adoption of topological
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(and related) methods has the potential to aid decisive progress inphilosophy of science; and second, the selection and justi�cationof a topology is not a matter of technical convenience, but ratheroften involves great conceptual and philosophical subtlety.
Topological Considerations in the Constructionof Quantum Theories
Benjamin Feintzeig
University of California, USA
It is well known that the process of quantization, constructing aquantum theory out of a classical theory, is not always a well-de�nedprocedure in physics. There are many inequivalent methods thatlead to di�erent choices for what to use as our quantum theory.In this paper, I show that paying close attention to topological in-formation in classical physics, which encodes manifestly physicallysigni�cant notions of approximation, can help us choose betweencompeting quantization procedures. I show that by requiring thetopological information about approximation to line up betweenclassical and quantum physics, we constrain and inform the quan-tum theories that we end up with.
Half-Planes, Ovals and Spheres. Point-FreeSystems of A�ne and Euclidean Geometry
Rafaª Gruszczy«ski
Nicolaus Copernicus University in Toru«, Poland
The task of point-free geometry is to construct a system ofgeometry in which the notion of point is not assumed as basic.We will brie�y present three such systems, each of which will bebased on the individual notion of region and the relational notion ofbeing part of . Additionaly, every system will have its own speci�cnotion:
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� of half-plane, in case of the system of Aleksander �niatyckifrom [3]
� of oval, in case of the system of Giangiancomo Gerla andRafaª Gruszczy«ski from [2]
� of sphere, in case of the system of Alfred Tarski from [4].
We will describe the axioms of the aforementioned theories anddisplay pre-theoretical (spatial) intuitions behind them. Explana-tion how elementary geometrical notions of point , betweenness andequidistance are de�ned within the appropriate systems will also beincluded.
References
[1] G. Gerla, R. Gruszczy«ski Point-free geometry, ovals and half-
planes, submitted
[2] R. Gruszczy«ski, A. Pietruszczak, Full development of geometry of
solids, Bulletin of Symbolic Logic, 14(4), 481-540, 2008.
[3] A. �niatycki, An axiomatics of non-desarguean geometry based on
the half-plane as the primitive notion, Dissertationes Mathemati-cae, no. LIX, PWN, Warszawa, 1968.
[4] A. Tarski Les fondements de la géometrié de corps, Ksi¦ga Pami¡t-kowa Pierwszego Polskiego Zjazdu Matematycznego, suplement toAnnales de la Societé Polonaise de Mathématique, Kraków, 1929,pp. 29�33.
Representing Points as Classesof Mereotopologically Structured Basic Entities
Laurenz Hudetz
University of Salzburg, Austria
It has been suggested by a number of authors (most promi-nently Whitehead and Russell) that spacetime points should be
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identi�ed with classes of mereotopologically structured basic en-tities. These suggestions are mainly motivated by either of thetwo following views: (a) the empiricist or relationist view thatstatements about spacetime points should be reduced to statementsdescribing relations between epistemologically or metaphysicallypreferable entities such as processes and events; (b) the view thattalk about spacetime points should be meaningful even in the casethat the mereological structure of spacetime regions is atomless(given substantivalism about spacetime regions).
In order to evaluate the feasibility of such suggestions, twomain questions need to be answered: (Q1) Under which condi-tions is a point representation method�i.e. a method of iden-tifying points with classes of mereotopologically structured basicentities�generally adequate? (Q2) Are there any adequate pointrepresentation methods in that sense? My talk addresses exactlythese two questions.
If we want to treat question Q1 in a systematic and rigorousway, we �rst of all need a uni�ed formal framework for analysingand evaluating di�erent point representation methods. I proposea uni�ed framework, in which I explicate a general notion of pointrepresentations and the notion of general adequacy of point repre-sentations. Thereby, we obtain an answer to Q1 and transform theinformal question Q2 into a precise, mathematical question.
I then examine important point representation methods withinthe proposed framework and present the main results I have achievedso far. It can be proven in a rigorous manner that the method whichidenti�es points with limited maximal round �lters�as suggestedby Roeper (1997) and Mormann (2010)�is generally adequate. Soquestion Q2 has a positive answer. Other salient methods suchas the method employing ultra�lters as points (along the lines ofStone's representation theorem) and the method using completelyprime �lters (as usual in point-free topology) can be proven to benot generally adequate and we can pinpoint the reason for theirinadequacy.
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Mathematical Tools in Ontology of Ideas,Concepts and Individuals
Janusz Kaczmarek
University of �ód¹, Poland
kaczmarek@�lozof.uni.lodz.pl
The topic of our conference is: topological philosophy. I under-stand it as application and utilizing topology in philosophy and itsproblems�but not conversely. Due to the fact that I deal with on-tology primarily I will understand the topic as a question: to whatextent can topology contribute ontology or if topological conceptsand tools allow to interpret concepts and problem of ontology. Inmy monograph Individuals. Ideas. Concepts. . . I proposed somecollection of terms and notions that are important to ontologicalinvestigations. I pointed out the following levels and relevant terms:
a) the level of individuals�(terms) individual, property, essen-tial and attributive property, positive and negative property,complete object, extension of idea,
b) the level of ideas�general object, species, genera, hierarchyof general objects, species di�erence, property of idea andproperty given in a content stratum of idea,
c) the level of concepts�concept, the structure of concepts, con-tent of concept, positive and negative content, extension ofconcept.
De�nitions of terms and notions in question and some theoremsI gave in set-theoretical language. So, now the problem is: isit possible to collect some set of ontological notions de�ned intopological language?
At the conference I will propose two small and modest ideas:
1) every individual (object) o is understood as a pair (X,TX),where X is interpreted as non-empty set of properties and TX
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is topology on X; this attempt will allow us to compare theindividual with a concept of object given by Twardowski inhis Zur Lehre vom Inhalt und Gegenstand der Vorstellungen;open sets of TX we can interpret as �rst order ingredients ofan individual and elements of open sets as second order ingre-dients; both �rst and second order ingredients are propertiesof an individual,
2) let X be a 4-dimensional connected space, 〈x, y, z, t〉 ∈ X,〈x, y, z〉 are space coordinates and t is time coordinate; everycurve from 〈x, y, z, t〉 to 〈x′, y′, z′, t′〉 we interpret as a realobject (individual); with each point 〈x, y, z, t〉 of an objecto we will join a set of properties P ; next we will de�neessential properties, obtained and lost properties, and usingdi�erent structures of time we will give semantically describedtemporal logic.
What Topology Can O�er Physics. TopologicalConstraints and Predictions in Classical and
Quantum Worlds.
Marek Ku±
Center for Theoretical Physics, Polish Academy of Sciences, Poland
Complaining that �[v]on der Geometria Situs... wissen undhaben wir nach anderthalbhundert Jahren noch nicht viel mehrwie nichts�, Carl Friedrich Gauss, as a clear example what can bedone to change this unfortunate state of the matter, gave in 1833a derivation of the so called linking number formula for two inter-twined curves. For Gauss his result provided also a link betweengeometria (or, occasionally, analysis) situs (topology) and geome-tria magnitudinis (analytical geometry). Nearly half a century laterJames Clerk Maxwell realized its connections with electromagnetictheory explicitly referring to its topological origins. In this waytopology entered physics.
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Using the above and other examples (Dirac and t'Hooft-Polyakovmonopoles, Aharonov-Bohm e�ect, dislocations in cristals) I will tryto show the role played by topology in physical theories. I will arguethat relatively `innocent� concept of continuity on which topologyis based, imposes concrete restrictions on properties of existing andnon-existing (Dirac monopole) physical quantities, providing, onthe other hand, new tools (topological quantum computing).
In What Kind of Space Do Husserl's Lectureson the Phenomenology of the Consciousness
of Internal Time Take Place?
Andrzej Leder
Polish Academy of Sciences, Poland
aleder@i�span.waw.pl
Husserl's sentence: �The manner in which variation of curva-ture makes the various sorts of space-like manifolds pass into oneanother, gives the philosopher who has familiarized himself with theelements of the Riemannian-Helmholtzian theory a certain pictureof the manner of the mutual legal connection among pure formsof theory of determinately distinct types.��introduces a certainstrong philosophical assumption. Namely, it assumes that the �pureforms of theory�, being the conditions of theoretical thinking, maybe �pictured� or represented by certain �multiplicities� or manifoldsdescribable by concepts deriving from topology. The question ofspatial structures in which thinking takes place and the questionof how various spaces di�er from one another become a matter ofimportance, perhaps even of a fundamental theoretical importance.
Hence, a question arises about the limits of applicability ofthis strategy. Is it applicable to deductive theory only, or to anyacademic theory, or perhaps to all rational thinking? Finally, wouldit be possible, perhaps, to extend this form of representing thinkingconditions to any act of consciousness, i.e. to what we call thinkingin general? This would mean, however, that that multiplicity we areasking about, as well as its topology, are equivalent to a certain level
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of�or perhaps to the whole of?�the structure of consciousnesswhich is studied by a phenomenologist when he turns away fromthe intentional object itself and towards the act.
Can Topology Justify �Vague Existence�?
Nasim Mahoozi
University of Barcelona, Spain
The goal of the paper is to justify why topology provides a per-spicuous tool to describe admissible precisi�cations for existence,where precisi�cation is the meaning of precise expression.
In the literature on vagueness, it is almost a common belief thatexistence cannot be vague. The proponents of semantic vagueness,according to which vagueness is due to our language, for example,claim that although vagueness is due to semantic indecision there ispart of language, namely logic that is precise and so existence whichis a language counterpart of existential quanti�er cannot be vague.In this paper, I will accept the relationship between existence andunrestricted existential quanti�er for the sake of argument. What Iwould like to question is the claim that existential quanti�er cannotbe vague. The outline of the paper is as follows. After a shortintroduction, I will explain the main sources of vagueness and whyit is a common belief that existence cannot be vague. Section 3 isdevoted to Sider's explanation of why existential quanti�ers, whichare unrestricted, are precise. I would scrutinize his argument andtry to come up with a new perspective to defend vague existence.Finally, in section four, the ultimate aim is to de�ne precisi�cationsas topological operators. I hope I can bring up convincing reasonsto show that topology is an appropriate tool to be used in the case ofvagueness. Applying topology to vagueness is not new. I will give ashort literature review on works related to topology and vagueness.
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Wittgenstein's Ways
Nikolay Milkov
University of Paderborn, Germany
Aristotle �rst investigated di�erent modes, or ways of being.Unfortunately, in the modern literature the discussion of this con-cept has been largely neglected. Only recently, the interest towardsthe concept of ways increased. Usually, it is explored in connectionwith the existence of universals and particulars. Some authors claimthat universals, the shape of my house, for example, are simplydi�erent ways in which its building blocks (the particulars) areordered. Another group of metaphysicians focuses attention on therelation between universals and particulars. Well-known conceptionis that it is a relation of supervenience that eschews reductivism.In the last few years the relation between universals and particularswas also explored with the help of the concept of grounding.
The approach we are going to follow in this paper is di�erent. Itdiscusses Wittgenstein's Tractarian conception of higher ontologicallevels as ways of arranging elements of lower ontological levels. Inthe Tractatus, Wittgenstein developed his ontology of ways (Artund Weise) in six steps: (i) Constructing states of a�airs out ofobjects; (ii) Constructing propositions out of states of a�airs; (iii)Constructing propositional signs; (iv) Constructing thoughts withthe help of propositional signs; (v) Constructing truth / falsity; (vi)Constructing works of art.
Wittgenstein's Tractarian ontology remained ontology of oneworld, in opposition to the ontology of many subordinated worldsof his teachers Frege (the author of the conception of three worlds)and Russell (the author of the Theory of Types): the transitionfrom one ontological level into another, higher level doesn't mean atransition from one world into another. This is the main advantageof Wittgenstein's constructivist ontology of ways: it makes thebelief in emergence of new worlds redundant. Another its advantageis that it suggests a tangible, topological, solution to the problem.
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(De)Constructing Points:From Topology to Mereology and Back
Thomas Mormann
University of the Basque Country
Points are considered as fundamental ingredients of topologyspaces. For instance, a topological space is de�ned as a set X of�points� endowed with some �topological structure� encapsulated inthe set OX of open subsets of X, OX being a subset of the powerset PX ofX. The set OX of open sets of a topological space has thelattice-theoretical structure of a complete Heyting algebra. As iswell known, many basic concepts of topology can actually expressedwithout points, but using only the lattice-theoretical structure ofOX only, for instance continuity and connectedness. This has led towhat has been described as �pointfree topology�. Indeed, pointfreetopology may be characterized as a kind of non-classical mereologybased on systems of regions exhibiting the structure of completeHeyting algebras instead of Boolean algebras as is the case for classi-cal mereology. On the other hand, given an appropriate (pointfree)Heyting mereological algebras H, it is possible to construct forH a set of ersatz points pt(H). This set pt(H) may be endowedwith a canonical topological structure O(pt(H)) isomorphic to H.In this way, under some mild restrictions, topological spaces andmereological systems may be considered as equivalent.
Time Granularity and the Formal Ontologyof Time-Awareness. A Husserlian Argumentfor a Topology of Temporal Information
Stefano Papa
University of Vienna, Austria
The Phenomenological Analysis of Inner or Immanent Time-Consciousness is meant to make explicit structural dependencies in
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the �eld of predication and imagination. Reasoning about tempo-ral information (which is central in modern systems of temporallogic) is touched upon only insofar, as an explication of abstrac-tive re�ection itself can be made available. Husserl proposes afourfold sense of �absence� in his analysis of temporal awareness:�representation�; what is here absent, is the object denoted by apredication. Representation is dependent upon imagination. Anobject absent in an act of imagining is �vacant� in the sense thatthe item itself is not to be retrieved by imagining it. Vacancy itselfis dependent upon a third more radical absence, the absence ofTime-Consciousness. Absence in this third sense is to be furtherexplicated as awareness of time as immanent duration and innersequence of given items. The methodical points of view whichare operational in the interpretaion of temporal awarenes, and areto be applied to the experience of linear ordering of informationitems (for example, a musical gestalt); are the ones of the formalontology developed in the Logical Investigations. The expressions�stream in a stream�, or �transcendence in the immanence� (inCartesian Meditations, with an eye to intersubjectivity), or else�living present� are metaphorical. They point to the fourth senseof absence.
From the point of view of Phenomenology, the formal meth-ods used in contemporary logics to model temporal reasoning areidealizations and abstractive formalizations, onesidedly founded inthe above mentioned structures of dependency. As an example ofsuch idealizations, one could mention Kamp's theorem: the de�n-ability of all temporal operators in terms of �since� and �until�, isbound to the condition that �time� is interpreted as a continuouslinear ordering. Independently from the completeness-result, thistheorem is important also because it poses the issue of referringto the same language to describe a situation with respect to dif-ferent temporal scalings (granularity).Time granularity is linked tosemantic properties of representation systems. In contemporarylogics it is formally treated by de�ning an algebra for granularities(set-theoretical approach) and by combination of simple temporallogics into a system for time granularity (logical approach). At thispoint, however, phenomenological considerations could motivatea shift in the study of time granularity: since, as stated above,
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time-awareness is an eidetic layer to be abstracted from the pre-sentation of some sequential items (melody), frames (algebras) asformalizations of �nite operations have a shortcome, because theydon't contain a formalization of what they might be about (butthis might be at the root of the problems posed by granularity, asillustrated by the synchronization problem). This situation canbe changed by enriching a given frame (operations) with a setof points (observations), and a subset of their cartesian product.This construction is called a topological system. The introductionof a topology has thus been motivated by phenomenological con-siderations on time-awareness and its structural role in systemsof formal representation. That is, the aim of this paper is tovindicate a systematic function of phenomenological re�ections forthe representation of formal categories in a given formalized system.
References
[1] E. Husserl, Logical Investigations (J. Findlay). 1970.
[2] E. Husserl, On the Phenomenology of the Consciousness of Inter-
nal Time (1893�1917), 1990 [1928]. (Brough, J. B).
[3] E. Husserl, Formal and Transcendental Logic, 1969 [1929], Cairns,D., trans. The Hague: Nijho�.
[4] H. Kamp. Formal Properties of `now'. Theoria, 37:227�273, 1971.
[5] H.Kamp. Events, Instantsand Temporal Reference. In: R. Bäuerle,U. Egli, and A. von Stechow, editors, Semantics from Di�erentPoints of View, pages 376�417. Springer-Verlag, 1979.
[6] Jerome Euzenat, Angelo Montanari. Time granularity. In: MichaelFisher, Dov Gabbay, Lluis Vila. Handbook of temporal reasoningin arti�cial intelligence, Elsevier, pp. 59�118, 2005, Foundations ofarti�cial intelligence.
[7] S. Vickers, Topology via Logic. Cambridge 1989 (1996).
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Problems with Extension
Marek Piwowarczyk
John Paul II Catholic University of Lublin, Poland
In my talk I want to examine the ontological status of extension,analyze the problem of in�nite regress entangled in the conceptof extended object and take a look at possible solutions to theproblem.
Extension is an essential structural property consisting in havingparts which lie outside each other. Each part is also extensive sowe have a vicious regress. In order to avoid it we can: (1) postulatenon-extensive beings which compose extensive ones; (2) say that theregress is not vicious (gunk theory); (3) postulate objects composedof parts dependent with respect to their extension on wholes (Icall them �primordially extended objects�); (4) postulate extendedsimples. I try to show that no solution is satisfactory.
Bifurcating Universes without Bifurcating Paths
Tomasz Placek
Jagiellonian University, Poland
This talk is a part of a larger argument for studying possibleevolutions of individual objects rather than possible evolutions ofan entire universe in investigations concerning determinism. Thetalk draws one's attention to an odd phenomenon in the initial valueproblem of general relativity: each object (massive or mass-less) hasa unique and well-de�ned possible evolution (geodesic), whereas agiven 3-dimensional space has more than one possible development(i.e., Lorentzian 4-dimensional spacetime satisfying Einstein �eldequations). Moreover, the union of these non-isomorphic devel-opments forms a non-Hausdor� manifold; the manifold does not
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permit bifurcating geodesics, however. These features have conse-quences for a project of de�ning possible histories in topologicalterms, for instance, as maximal Hausdor� sub-manifolds of somemaster manifold that captures all historical possibilities.
A Skeptical Lookat Region-based Theories of Space
Ian Pratt-Hartmann
University of Manchester, UK
One of the many achievements of coordinate geometry has beento provide a conceptually elegant and unifying account of spatialentities. According to this account, the primitive constituents ofspace are points, and all other spatial entities�lines curves, surfacesand bodies�are nothing other than the sets of those points whichlie on them. The success of this reduction is so great that theidenti�cation of all spatial objects with sets of points has come toseem almost axiomatic.
For most of the previous century, however, a small but tenaciousband of authors has suggested that more parsimonious and concep-tually satisfying representations of space are obtained if we adoptan ontology in which regions, not points, are the primitive spatialentities. These, and other, considerations have prompted the de-velopment of formal languages whose variables range over certainsubsets (not points) of speci�ed classes of geometrical structures.We call the study of such languages `mereogeometry'.
In the past two decades, the Computer Science community inparticular has produced a steady �ow of new technical results inmereogeometry, especially concerning the computational complex-ity of region-based topological formalisms with limited expressivepower. The purpose of this talk is to provide a conceptual frame-work for assessing the philosophical signi�cance of this work. Asusual, to grasp the philosophy, one �rst needs to master the math-ematics.
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Connectedness and Ontological Unity
Peter Simons
Trinity College Dublin, Ireland
A topological space is path connected when any two pointsare connected by a line. This de�nition, more intuitive than thestandard de�nition of connectedness, goes over neatly to graphtheory. It is argued that it is the concept we need to make sense ofthe notion of a uni�ed or single entity. Some mereological theoriesassume that any collection of objects comprise a whole. This letsin cross-categorical and gerrymandered entities and it is arguedthat, contrary to many ontologists' views, it renders mereology non-innocent. To retain a useful, more restrictive but still very generalnotion of a natural or integrated whole, generalised connectednessis precisely the notion we need.
Perzanowski's Combination Ontologicin Hilbert's Cube
Bartªomiej Skowron
International Center for Formal Ontology, Warsaw University of Technology,
Poland
Perzanowski's combination ontologic is the ontology of elementsand their combinations. Perzanowski, after Leibniz, concluded thatcombinations are de�ned by the internal features of the elementswhich de�ne the structure of connections between the elements,i.e. whether one element is connected with another depends on theelements' insides. We present the view that the structure of thecombination depends �rstly on structural, topological and a prioriforms and then it can be determined by the features of elements.
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We suggest enriching the structure of the ontological universe byits topologisation. The examples of ontological worlds presentedhere prove that this is sensible and necessary. By modelling theontological universe with the use of Hilbert's cube we show therelations between the notion of dimension and the notions of situ-ation and a possible world. We also put forth a new interpretationof ontological rationalism.
The Boundaries of Things: Where TopologyMeets Metaphysics
Achille Varzi
Columbia University, USA
Philosophical re�ections on the topological notion of bound-ary tend to focus on the opposition between boundaries as basic,lower-dimensional entities and boundaries as derived, higher-orderabstractions. This opposition re�ects two fundamentally di�erentways of understanding the structure of space and time and hasimportant ontological consequences when it comes to the structureand nature of those entities that may be said to be located in spaceand time, such as objects and events. There is, in addition, a secondimportant distinction that may be drawn, and whose ontologicalrami�cations extend even further�the distinction between natural(or bona �de) and arti�cial (or �at) boundaries. The former arejust the natural boundaries of old, as grounded in some factualdiscontinuity or qualitative heterogeneity between an entity and itssurroundings; the latter are exempli�ed especially by boundaries in-duced through human cognition and social practices and lie skew toany objective di�erentiations in the underlying wordly material (aswith the contours in a Seurat painting, or the borders of Wyoming).The distinction bites deeply, for it can be drawn across the board:not merely in the domain of boundaries but also in relation tothose entities that may be said to have boundaries. If a certainobject or event enjoys a natural boundary, its identity and survival
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conditions do not depend on us; it is a bona �de, mind-independententity of its own. By contrast, if its boundary is of the arti�cialsort, then the entity itself is to some degree a �at entity, a productof our worldmaking. Now, we may disagree on which entities are ofwhich sort, and any such disagreement will re�ect a correspondingdisagreement in matters of metaphysical realism. Indeed, it canbe argued that the question of realism is, in an important sense,the question of what are the bona �de boundaries, the boundariesthat �carve at the joints�. Here I am especially interested in limitcase: What if there aren't any? What if all boundaries�hence allentities�were on closer look and to some extent the product ofsome cognitive or social �at?
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