Topological Foundations of Cognitive Science

A collection of papers presented at the First International Summer Institute in Cognitive Science, University at Buffalo, July 1994, including the following papers: ** Topological Foundations of Cognitive Science, Barry Smith ** The Bounds of Axiomatisation, Graham White ** Rethinking Boundaries, Wojciech Zelaniec ** Sheaf Mereology and Space Cognition, Jean Petitot ** A Mereotopological Definition of 'Point', Carola Eschenbach ** Discreteness, Finiteness, and the Structure of Topological Spaces, Christopher Habel ** Mass Reference and the Geometry of Solids, Almerindo E. Ojeda ** Defining a 'Doughnut' Made Difficult, N .M. Gotts ** A Theory of Spatial Regions with Indeterminate Boundaries, A.G. Cohn and N.M. Gotts ** Mereotopological Construction of Time from Events, Fabio Pianesi and Achille C. Varzi ** Computational Mereology: A Study of Part-of Relations for Multi-media Indexing, Wlodek Zadrozny and Michelle Ki

“The whole is greater than the part.” Mereology in Euclid's Elements

The present article provides a mereological analysis of Euclid’s planar geometry as presented in the first two books of his Elements. As a standard of comparison, a brief survey of the basic concepts of planar geometry formulated in a set-theoretic framework is given in Section 2. Section 3.2, then, develops the theories of incidence and order (of points on a line) using a blend of mereology and convex geometry. Section 3.3 explains Euclid’s “megethology”, i.e., his theory of magnitudes. In Euclid’s system of geometry, megethology takes over the role played by the theory of congruence in modern accounts of geometry. Mereology and megethology are connected by Euclid’s Axiom 5: “The whole is greater than the part.” Section 4 compares Euclid’s theory of polygonal area, based on his “Whole-Greater-Than-Part” principle, to the account provided by Hilbert in his Grundlagen der Geometrie. An hypothesis is set forth why modern treatments of geometry abandon Euclid’s Axiom 5. Finally, in Section 5, the adequacy of atomistic mereology as a framework for a formal reconstruction of Euclid’s system of geometry is discussed

Model Checking Spatial Logics for Closure Spaces

Spatial aspects of computation are becoming increasingly relevant in Computer Science, especially in the field of collective adaptive systems and when dealing with systems distributed in physical space. Traditional formal verification techniques are well suited to analyse the temporal evolution of programs; however, properties of space are typically not taken into account explicitly. We present a topology-based approach to formal verification of spatial properties depending upon physical space. We define an appropriate logic, stemming from the tradition of topological interpretations of modal logics, dating back to earlier logicians such as Tarski, where modalities describe neighbourhood. We lift the topological definitions to the more general setting of closure spaces, also encompassing discrete, graph-based structures. We extend the framework with a spatial surrounded operator, a propagation operator and with some collective operators. The latter are interpreted over arbitrary sets of points instead of individual points in space. We define efficient model checking procedures, both for the individual and the collective spatial fragments of the logic and provide a proof-of-concept tool

The Introduction of Topology into Analytic Philosophy: Two Movements and a Coda

Both early analytic philosophy and the branch of mathematics now known as topology were gestated and born in the early part of the 20th century. It is not well recognized that there was early interaction between the communities practicing and developing these fields. We trace the history of how topological ideas entered into analytic philosophy through two migrations, an earlier one conceiving of topology geometrically and a later one conceiving of topology algebraically. This allows us to reassess the influence and significance of topological methods for philosophy, including the possible fruitfulness of a third conception of topology as a structure determining similarity

A SPATIAL LOGIC FOR SIMPLICIAL MODELS

Collective Adaptive Systems often consist of many heterogeneous components typically organised in groups. These entities interact with each other by adapting their behaviour to pursue individual or collective goals. In these systems, the distribution of these entities determines a space that can be either physical or logical. The former is defined in terms of a physical relation among components. The latter depends on logical relations, such as being part of the same group. In this context, specification and verification of spatial properties play a fundamental role in supporting the design of systems and predicting their behaviour. For this reason, different tools and techniques have been proposed to specify and verify the properties of space, mainly described as graphs. Therefore, the approaches generally use model spatial relations to describe a form of proximity among pairs of entities. Unfortunately, these graph-based models do not permit considering relations among more than two entities that may arise when one is interested in describing aspects of space by involving interactions among groups of entities. In this work, we propose a spatial logic interpreted on simplicial complexes. These are topological objects, able to represent surfaces and volumes efficiently that generalise graphs with higher-order edges. We discuss how the satisfaction of logical formulas can be verified by a correct and complete model checking algorithm, which is linear to the dimension of the simplicial complex and logical formula. The expressiveness of the proposed logic is studied in terms of the spatial variants of classical bisimulation and branching bisimulation relations defined over simplicial complexes

A Spatial Logic for Simplicial Models

Collective Adaptive Systems often consist of many heterogeneous components typically organised in groups. These entities interact with each other by adapting their behaviour to pursue individual or collective goals. In these systems, the distribution of these entities determines a space that can be either physical or logical. The former is defined in terms of a physical relation among components. The latter depends on logical relations, such as being part of the same group. In this context, specification and verification of spatial properties play a fundamental role in supporting the design of systems and predicting their behaviour. For this reason, different tools and techniques have been proposed to specify and verify the properties of space, mainly described as graphs. Therefore, the approaches generally use model spatial relations to describe a form of proximity among pairs of entities. Unfortunately, these graph-based models do not permit considering relations among more than two entities that may arise when one is interested in describing aspects of space by involving interactions among groups of entities. In this work, we propose a spatial logic interpreted on simplicial complexes. These are topological objects, able to represent surfaces and volumes efficiently that generalise graphs with higher-order edges. We discuss how the satisfaction of logical formulas can be verified by a correct and complete model checking algorithm, which is linear to the dimension of the simplicial complex and logical formula. The expressiveness of the proposed logic is studied in terms of the spatial variants of classical bisimulation and branching bisimulation relations defined over simplicial complexes

Ekonominen unifikaatio filosofisen analyysin metodina

This doctoral dissertation introduces economical unification as a method of analysis and shows how it is applied in dealing with some topics that are central in contemporary philosophy. The method resembles a production line that consists of three successive elements which are interconnected in two stages: Economy > Ontology > Applications In the first stage, an economically unified ontology is explicated by applying the principle of economy, which is an evaluation criterion of alternative ontologies. An economically unified ontology is an empirically sufficient, metaphysically minimal and generally virtuous world-view or a belief system of a human being. In the second stage everything else is dealt with in terms of the ontology. The central argument is that economical unification is a more progressive method than plain conceptual analysis which proceeds in the absence of an economically unified ontology and without the principle of economy. Its progressiveness results from having economy as an unambiguous evaluation criterion, which enables explicating a stable and minimal unified ontology which functions as a common base for all topics, and which enables defining and disambiguating meanings of concepts, thereby facilitating their genuine understanding and resolving problems around them, more efficiently than without an economically unified ontology, and without an unambiguous evaluation criterion that would enable explicating it. The progressiveness of the method is substantiated by applying it in disambiguating some of the central concepts that are dealt with in contemporary philosophy such as time, truth and possibility, and in resolving problems around them. The method works: unification efficiently resolves problems whose central source is disunification itself. In other words, the absence of an economically unified ontology is a central source of problems and ambiguities in contemporary philosophy; in economical unification such problems are resolved by removing their source; their source is removed by replacing the absence of an economically unified ontology by bringing it in the center of the analysis. The holistic method that handles special topics in the top-down order by relying on an understandable world-view, is very different from traditional conceptual analysis that proceeds in the absence of an economically unified ontology, and even in the absence of having it as the goal, i.e., without economy or the degree of virtuousness as the criterion. Moreover, the method was formulated in order to systematically overcome those limitations of plain conceptual analysis which result from their absence. Traditional conceptual analysis proceeds typically by investigating isolated topics and various angles to them, but this does not manage to interconnect the isolated topics and thus does not resolve problems which are due to the isolation itself. It is practically impossible to unify many things by concentrating on one thing only, and the optimal rate of progress in philosophy and in science in general cannot be achieved if the analysis is limited into investigating isolated fragments. In order to achieve the optimal rate of progress, unification is needed in counterbalancing specialization. By looking at many individual pieces together, one can start streamlining them into a functional totality. In this process much is revealed about what kinds of parts are needed in the totality and what are not. The totality consists of interrelated parts, but in economical unification the overall picture of reality guides the development of its parts at least as strongly as the requirements for the parts guide the development of the totality. Economical unification can thus be seen merely as the project getting hold of the natural order where the totality and its parts interact, and whose alternative is to keep on investigating details of isolated parts blindfolded without worrying about their roles in a totality, for all parts that are applicable do have a role in a totality. One can and one should scrutinize any suggested totality and replace it when a better one is available, but not before a better one has been presented. This holds for contemporary paradigmatic theories and for everything that comes after them. This brings the focus to the question of what is the objective meaning of better. The suggested answer is: the more economically unified, the better. The most important starting point in the project of economical unification is the acceptance of the principle of economy or the degree of virtuousness as the evaluation criterion, for without a commonly accepted and acknowledged criterion the path towards consensus is unnecessarily long and painful. The easiest way of accepting economy as the criterion is understanding that its general acceptance would accelerate the progress rate of science, including philosophy: virtuousness as the criterion of theories likely results into more virtuous science, faster than without it. Once we have a common criterion, people no longer have to settle on agreeing to disagree, but people have leaped forward into evaluating which theory is objectively better. Everything can be scrutinised, including economy, but rejecting it without replacing it with a more progressive criterion means that one does not fully appreciate progress.Tämä väitöskirja esittelee ekonomisen unifikaation metodin ja soveltaa sitä filosofian keskeisiin aiheisiin. Metodi muistuttaa tuotantolinjaa mikä koostuu kolmesta peräkkäisestä elementistä jotka liittyvät toisiinsa kahdessa vaiheessa: Ekonomia > Ontologia > Sovellukset Ensimmäisessä vaiheessa eksplikoidaan ekonomisesti unifioitu ontologia soveltamalla ekonomian periaatetta, joka on vaihtoehtoisten ontologioiden arviointikriteeri. Ekonomisesti unifioitu ontologia on empiirisesti riittävä, metafyysisesti minimaalinen ja yleisesti hyveellinen maailmankuva. Toisessa vaiheessa sovellukset käsitellään ontologian avulla. Keskeinen väite on, että ekonominen unifikaatio on edistyksellisempi metodi kuin pelkkä käsitteellinen analyysi mikä etenee ilman ekonomisesti unifioitua ontologiaa, käyttämättä ekonomian periaatetta arviointikriteerinä. Metodin edistyksellisyys juontuu selkeästä ekonomian periaatteesta, mikä mahdollistaa stabiilin, yhtenäisen ja minimaalisen ontologian eksplikoinnin. Ontologia toimii kaikkien sovellusten käsittelyn yhtenäisenä pohjana ja mahdollistaa käsitteiden merkitysten selkeyttämisen ja niiden ympärillä olevien ongelmien ratkaisun, tehokkaammin kuin pelkkä käsitteellinen analyysi ilman yhtenäistä ontologiaa, ja ilman selkeää arviointikriteeriä mikä mahdollistaisi yhtenäisen ontologian eksplikoinnin. Metodin progressiivisuutta alleviivataan soveltamalla sitä. Joitain nykyfilosofian keskeisiä käsiteitä -kuten aika, totuus ja mahdollisuus- selkeytetään ja näitten ympärillä olevia ongelmia ratkaistaan. Metodi toimii: unifikaatio ratkaisee tehokkaasti ongelmia joiden keskinen syy on hajanaisuus. Toisin sanoen, ekonomisesti unifioidun ontologian poissaolo on nykyfilosofian ongelmien ja epämääräisyyksien keskeinen lähde; ekonomisen unifikaation metodissa tällaiset ongelmat ratkaistaan poistamalla niiden lähde; niiden lähde poistetaan korvaamalla ekonomisesti unifioidun ontologian poissaolo tuomalla se analyysin keskelle. Holistinen unifikaation metodi joka käsittelee sovelluksia ylhäältä-alas järjestyksessä, luottamalla intuitiiviseen maailmankuvaan on hyvin erilainen kuin perinteinen käsitteellinen analyysi joka etenee ilman yhtenäistä maailmankuvaa ja jopa tavoittelematta sitä, eli ilman ekonomian periaatetta arviointikriteerinä joka suosii yhtenäistä ja kaikin puolin hyveellistä maailmankuvaa. Ekonominen unifikaatio on luotu jotta näiden puutteesta johtuvista rajoitteista päästäisiin systemaattisesti yli. Perinteinen käsitteellinen analyysi etenee tyypillisesti tutkimalle lukuisia näkökulmia eristyneisiin aiheisiin; tällainen analyysi ei onnistu yhdistämään hajanaisia aiheita eikä täten kykene ratkaisemaan ongelmia joiden syy on hajanaisuus itse. Tieteen ja filosofian optimaalista edistystahtia ei voi saavuttaa rajoittumalla eristyneisiin fragmentteihin. Jotta optimaalinen edistystahti voitaisiin saavuttaa, unifikaatiota tarvitaan spesialisaation vastapainona. Katsomalla monia yksittäisiä osia yhdessä, voi ne saneerata toimivaksi kokonaisuudeksi. Tässä prosessissa selviää minkälaisia osia kokonaisuudessa tarvitaan ja mitä ei. Kokonaisuus koostuu yhteenliittyneistä osista, mutta ekonomisessa unifikaatiossa kokonaismaailmankuva ohjaa osiensa kehitystä vähintään yhtä vahvasti kuin vaatimukset osille ohjaavat kokonaisuuden kehitystä. Ekonomisen unifikaation voi täten nähdä luonnollisen järjestyksen tavoitteluna, missä kokonaisuus ja sen osat ovat interaktiossa, ja jonka vaihtoehto on jatkaa eristyneiden osien yksityiskohtien tutkimista side silmillä, huolehtimatta niiden roolista kokonaisuudessa, vaikka kaikki osat joita voi todellisuudessa käyttää, kuuluvat johonkin kokonaisuuteen. Kaikkia ehdotettuja kokonaisuuksia tulee tarkastella kriittisesti ja korvata ne paremmalla heti kun tällainen on saatavilla, mutta ei ennen kuin tällainen on esitetty. Tämä pätee nykyisiin paradigmaattisiin teorioihin ja kaikkeen mikä tulee niitten jälkeen. Tämä keskittää huomion kysymykseen siitä mikä on sanan parempi objektiivinen merkitys. Ehdotettu vastaus on: mitä enemmän ekonomisesti unifioitu, sitä parempi. Tärkein lähtökohta ekonomisessa unifikaatiossa on luonnollisesti ekonomian periaate tai teoreettisten hyveiden arvostaminen, koska tie konsensukseen on tarpeettoman pitkä ja tuskallinen ilman yleisesti hyväksyttyä ja tiedostettua arviointikriteeriä. Helpoin tapa hyväksyä ekonomian periaate on ymmärtää että sen yleinen hyväksyminen kiihdyttäisi tieteen ja filosofian kehitystahtia: hyveellisyys teorioiden arviointikriteerinä luonnollisesti johtaa hyveellisempään tieteeseen, nopeammin kuin ilman sitä. Yleisen kriteerin nojalla, ihmisten ei tarvitse tyytyä olemaan samaa mieltä siitä että he ovat subjektiivisesti eri mieltä, vaan he ovat harpanneet eteenpäin arvioimaan mikä teoria on objektiivisesti parempi. Ekonomian periaatetta voi kritisoida ja parantaa, mutta sen hylkääminen, korvaamatta sitä edistyksellisemmällä kriteerillä, ei ole edistyksellistä

Infinity

This essay surveys the different types of infinity that occur in pure and applied mathematics, with emphasis on: 1. the contrast between potential infinity and actual infinity; 2. Cantor's distinction between transfinite sets and absolute infinity; 3. the constructivist view of infinite quantifiers and the meaning of constructive proof; 4. the concept of feasibility and the philosophical problems surrounding feasible arithmetic; 5. Zeno's paradoxes and modern paradoxes of physical infinity involving supertasks