ReferencesLadyman, J., Linnebo, Ø., and Bigaj, T. (2013). Entanglement and non-factorizability. Studies in History and Philosophy of Modern Physics, 44:215–221.Leinaas, J. and Myrheim, J. (1977). On the theory of identical particles. Il Nuovo Cimento B Series 11, 37(1):1–23.
I will briefly describe three such systems, each of which will be based on the individual notion of region and the relational notion
of being part of. Additionaly, every system will have its own specific notion:
- of half-plane, in case of the system of Aleksander Śniatycki from 
- of oval, in case of the system of Giangiancomo Gerla and Rafał Gruszczynski from 
- of sphere, in case of the system of Alfred Tarski from .
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.
 R. Gruszczynski, A. Pietruszczak, Full development of geometry of solids, Bulletin of Symbolic Logic, 14(4), 481-540, 2008.
 A. Śniatycki, An axiomatics of non-desarguean geometry based on the half-plane as the primitive notion, Dissertationes Mathematicae, no. LIX, PWN, Warszawa, 1968.
 A. Tarski Les fondements de la géometrié de corps, Księga Pamiatkowa Pierwszego Polskiego Zjazdu Matematycznego, suplement to Annales de la Societé Polonaise de Mathématique, Kraków, 1929, pp. 29-33.
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.
b) the level of ideas – general object, species, genera, hierarchy of general objects, species difference, property of idea and property given in a content stratum of idea,
c) the level of concepts – concept, the structure of concepts, content of concept, positive and negative content, extension of concept.Definitions of terms and notions in question and some theorems I gave in set-theoretical language. So, now the problem is: is it possible to collect some set of ontological notions defined in topological 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 is topology on X; this attempt will allow us to compare the individual with a concept of object given by Twardowski in his Zur Lehre vom Inhalt und Gegenstand der Vorstellungen; open sets of TX we can interpret as first order ingredients of an individual and elements of open sets as second order ingredients; both first and second order ingredients are properties of 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; every curve from <x, y, z, t> to <x’, y’, z’, t’> we interpret as a real object (individual); with each point <x, y, z, t> of an object o we will join a set of properties P; next we will define essential properties, obtained and lost properties, and using different structures of time we will give semantically described temporal logic.
Complaining that „[v]on der Geometria Situs… wissen und haben wir nach anderthalbhundert Jahren noch nicht viel mehr wie nichts“, Carl Friedrich Gauss, as a clear example what can be done to change this unfortunate state of the matter, gave in 1833 a derivation of the so called linking number formula for two intertwined curves. For Gauss his result provided also a link between geometria (or, occasionally, analysis) situs (topology) and geometria magnitudinis (analytical geometry). Nearly half a century later James Clerk Maxwell realized its connections with electromagnetic theory explicitly referring to its topological origins. In this way topology entered physics.
Using the above and other examples (Dirac and t’Hooft-Polyakov monopoles, Aharonov-Bohm effect, dislocations in cristals) I will try to show the role played by topology in physical theories. I will argue that relatively ‘innocent” concept of continuity on which topology is based, imposes concrete restrictions on properties of existing and non-existing (Dirac monopole) physical quantities, providing, on the other hand, new tools (topological quantum computing).
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.
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.
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 . (Brough, J.B).
E. Husserl, Formal and Transcendental Logic, 1969 , 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)
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.
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.