This paper intends to further the understanding of the formal properties of (higher-order) vagueness by connecting theories of (higher-order) vagueness with more recent work in topology. First, we provide a “translation” of Bobzien's account of columnar higher-order vagueness into the logic of topological spaces. Since columnar vagueness is an essential ingredient of her solution to the Sorites paradox, a central problem of any theory of vagueness comes into contact with the modern mathematical theory of topology. Second, Rumfitt’s recent topological reconstruction (...) of Sainsbury’s theory of prototypically defined concepts is shown to lead to the same class of spaces that characterize Bobzien’s account of columnar vagueness, namely, weakly scattered spaces. Rumfitt calls these spaces polar spaces. They turn out to be closely related to Gärdenfors’ conceptual spaces, which have come to play an ever more important role in cognitive science and related disciplines. Finally, Williamson’s “logic of clarity” is explicated in terms of a generalized topology (“locology”) that can be considered an alternative to standard topology. Arguably, locology has some conceptual advantages over topology with respect to the conceptualization of a boundary and a borderline. Moreover, in Williamson’s logic of clarity, vague concepts with respect to a notion of a locologically inspired notion of a “slim boundary” are (stably) columnar. Thus, Williamson’s logic of clarity also exhibits a certain affinity for columnar vagueness. In sum, a topological perspective is useful for a conceptual elucidation and unification of central aspects of a variety of contemporary accounts of vagueness. (shrink)

We examine some of Connes’ criticisms of Robinson’s infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes’ own earlier work in functional analysis. Connes described the hyperreals as both a “virtual theory” and a “chimera”, yet acknowledged that his argument relies on the transfer principle. We analyze Connes’ “dart-throwing” thought experiment, but reach an opposite conclusion. In S , all definable sets of reals are (...) Lebesgue measurable, suggesting that Connes views a theory as being “virtual” if it is not definable in a suitable model of ZFC. If so, Connes’ claim that a theory of the hyperreals is “virtual” is refuted by the existence of a definable model of the hyperreal field due to Kanovei and Shelah. Free ultrafilters aren’t definable, yet Connes exploited such ultrafilters both in his own earlier work on the classification of factors in the 1970s and 80s, and in Noncommutative Geometry, raising the question whether the latter may not be vulnerable to Connes’ criticism of virtuality. We analyze the philosophical underpinnings of Connes’ argument based on Gödel’s incompleteness theorem, and detect an apparent circularity in Connes’ logic. We document the reliance on non-constructive foundational material, and specifically on the Dixmier trace −∫ (featured on the front cover of Connes’ magnum opus) and the Hahn–Banach theorem, in Connes’ own framework. We also note an inaccuracy in Machover’s critique of infinitesimal-based pedagogy. (shrink)

We seek to elucidate the philosophical context in which one of the most important conceptual transformations of modern mathematics took place, namely the so-called revolution in rigor in infinitesimal calculus and mathematical analysis. Some of the protagonists of the said revolution were Cauchy, Cantor, Dedekind,and Weierstrass. The dominant current of philosophy in Germany at the time was neo-Kantianism. Among its various currents, the Marburg school (Cohen, Natorp, Cassirer, and others) was the one most interested in matters scientific and mathematical. Our (...) main thesis is that Marburg neo-Kantian philosophy formulated a sophisticated position towards the problems raised by the concepts of limits and infinitesimals. The Marburg school neither clung to the traditional approach of logically and metaphysically dubious infinitesimals, nor whiggishly subscribed to the new orthodoxy of the “great triumvirate” of Cantor, Dedekind, and Weierstrass that declared infinitesimals conceptus nongrati in mathematical discourse. Rather, following Cohen’s lead, the Marburg philosophers sought to clarify Leibniz’s principle of continuity, and to exploit it in making sense of infinitesimals and related concepts. (shrink)

Abstract. In Dynamics of Reason Michael Friedman proposes a kind of synthesis between the neokantianism of Ernst Cassirer, the logical empiricism of Rudolf Carnap, and the historicism of Thomas Kuhn. Cassirer and Carnap are to take care of the Kantian legacy of modern philosophy of science, encapsulated in the concept of a relativized a priori and the globally rational or continuous evolution of scientific knowledge,while Kuhn´s role is to ensure that the historicist character of scientific knowledge is taken seriously. More (...) precisely, Carnapian linguistic frameworks, guarantee that the evolution of science procedes in a rational manner locally,while Cassirer’s concept of an internally defined conceptual convergence of empirical theories provides the means to maintain the global continuity of scientific reason. In this paper it is argued that Friedman’s neokantian account of scientific reason based on the concept of the relativized a priori underestimates the pragmatic aspects of the dynamics of scientific reason. To overcome this short-coming, I propose to reconsider C.I. Lewis’s account of a pragmatic the priori, recently modernized and elaborated by Hasok Chang. This may be<br><br><br><br><br><br><br><br><br><br&g t;<br><br><br><br><br><br>Keywords: Dynamics of reason, Paradigms, Logical Empiricism,Neokantianism, Pragmatism, Mathematics, Communicative Rationality. (shrink)

Abstract. The aim of this paper is to present a topological method for constructing discretizations (tessellations) of conceptual spaces. The method works for a class of topological spaces that the Russian mathematician Pavel Alexandroff defined more than 80 years ago. Alexandroff spaces, as they are called today, have many interesting properties that distinguish them from other topological spaces. In particular, they exhibit a 1-1 correspondence between their specialization orders and their topological structures. Recently, a special type of Alexandroff spaces was (...) used by Ian Rumfitt to elucidate the logic of vague concepts in a new way. According to his approach, conceptual spaces such as the color spectrum give rise to classical systems of concepts that have the structure of atomic Boolean algebras. More precisely, concepts are represented as regular open regions of an underlying conceptual space endowed with a topological structure. Something is subsumed under a concept iff it is represented by an element of the conceptual space that is maximally close to the prototypical element p that defines that concept. This topological representation of concepts comes along with a representation of the familiar logical connectives of Aristotelian syllogistics in terms of natural settheoretical operations that characterize regular open interpretations of classical Boolean propositional logic. In the last 20 years, conceptual spaces have become a popular tool of dealing with a variety of problems in the fields of cognitive psychology, artificial intelligence, linguistics and philosophy, mainly due to the work of Peter Gärdenfors and his collaborators. By using prototypes and metrics of similarity spaces, one obtains geometrical discretizations of conceptual spaces by so-called Voronoi tessellations. These tessellations are extensionally equivalent to topological tessellations that can be constructed for Alexandroff spaces. Thereby, Rumfitt’s and Gärdenfors’s constructions turn out to be special cases of an approach that works for a more general class of spaces, namely, for weakly scattered Alexandroff spaces. This class of spaces provides a convenient framework for conceptual spaces as used in epistemology and related disciplines in general. Alexandroff spaces are useful for elucidating problems related to the logic of vague concepts, in particular they offer a solution of the Sorites paradox (Rumfitt). Further, they provide a semantics for the logic of clearness (Bobzien) that overcomes certain problems of the concept of higher2 order vagueness. Moreover, these spaces help find a natural place for classical syllogistics in the framework of conceptual spaces. The crucial role of order theory for Alexandroff spaces can be used to refine the all-or-nothing distinction between prototypical and nonprototypical stimuli in favor of a more fine-grained gradual distinction between more-orless prototypical elements of conceptual spaces. The greater conceptual flexibility of the topological approach helps avoid some inherent inadequacies of the geometrical approach, for instance, the so-called “thickness problem” (Douven et al.) and problems of selecting a unique metric for similarity spaces. Finally, it is shown that only the Alexandroff account can deal with an issue that is gaining more and more importance for the theory of conceptual spaces, namely, the role that digital conceptual spaces play in the area of artificial intelligence, computer science and related disciplines. Keywords: Conceptual Spaces, Polar Spaces, Alexandroff Spaces, Prototypes, Topological Tessellations, Voronoi Tessellations, Digital Topology. (shrink)

The aim of this paper is to present a topological method for constructing discretizations of topological conceptual spaces. The method works for a class of topological spaces that the Russian mathematician Pavel Alexandroff defined more than 80 years ago. The aim of this paper is to show that Alexandroff spaces, as they are called today, have many interesting properties that can be used to explicate and clarify a variety of problems in philosophy, cognitive science, and related disciplines. For instance, recently, (...) Ian Rumfitt used a special type of Alexandroff spaces to elucidate the logic of vague concepts in a new way. Moreover, Rumfitt’s class of Alexandroff spaces can be shown to provide a natural topological semantics for Susanne Bobzien’s “logic of clearness”. Mainly due to the work of Peter Gärdenfors and his collaborators, conceptual spaces have become an increasingly popular tool of dealing with a variety of problems in the fields of cognitive psychology, artificial intelligence, linguistics and philosophy. For Gärdenfors’s conceptual spaces, geometrically defined discretizations play an essential role. These tessellations can be shown to be extensionally equivalent to topological tessellations that can be constructed for Alexandroff spaces in general. Thereby, Rumfitt’s and Gärdenfors’s constructions turn out to be special cases of an approach that works for a more general class of spaces, namely, for weakly scattered Alexandroff spaces. The main aim of this paper is to show that this class of spaces provides a convenient framework for conceptual spaces as used in epistemology and related disciplines in general. Weakly scattered Alexandroff spaces are useful for elucidating problems related to the logic of vague concepts, in particular they offer a solution of the Sorites paradox. Further, they provide a semantics for the logic of clearness that overcomes certain problems of the concept of higher-order vagueness. Moreover, these spaces help find a natural place for classical syllogistics in the framework of conceptual spaces. The specialization order of Alexandroff spaces can be used to refine the all-or-nothing distinction between prototypical and non-prototypical stimuli in favor of a gradual distinction between more or less prototypical elements of conceptual spaces. The greater conceptual flexibility of the topological approach helps avoid some inherent inadequacies of the geometrical approach, for instance, the so-called “thickness problem”. Finally, it is shown that the Alexandroff spaces offer an appropriate framework to deal with digital conceptual spaces that are gaining more and more importance in the areas of artificial intelligence, computer science and related disciplines. (shrink)

Nach gängiger Auffassung nahm das Thema „Werte” in Carnaps Philosophie nur einen geringen Stellenwert ein. In dieser Arbeit soll gezeigt werden, daß diese Einschätzung der Korrektur bedarf: So wird der im „Aufbau“ vorgetragene Entwurf eines Konstitutionssystems mit Werten als der höchsten Schicht des Konstitutionssystems abgeschlossen. Auch die Quasianalyse als allgemeine Konstitutionsmethode steht in enger Beziehung zur Unterscheidung zwischen „Sein“ und „Gelten“, die für den werttheoretisch orientierten Neukantianismus der südwest-deutschen Schule charakteristisch war. Allgemein erlaubt die Wertthematik einen Blick auf Unterströmungen des (...) carnapschen Denkens, die in den offiziellen logisch-empiristischen Darstellungen seines Denkens meist vernachlässigt werden. Die Wertthematik, so die These dieser Arbeit, eröffnet überdies eine neue Perspektive auf die spezifisch Jenaer Konstellation von Neukantianismus und Lebensphilosophie, die Carnaps Philosophie wesentlich geprägt hat. (shrink)

David Lewis famously argued against structural universals since they allegedly required what he called a composition “sui generis” that differed from standard mereological com¬position. In this paper it is shown that, although traditional Boolean mereology does not describe parthood and composition in its full generality, a better and more comprehensive theory is provided by the foundational theory of categories. In this category-theoretical framework a theory of structural universals can be formulated that overcomes the conceptual difficulties that Lewis and his followers (...) regarded as unsurmountable. As a concrete example of structural universals groups are considered in some detail. (shrink)

Abstract. Traditional epistemology of knowledge and belief can be succinctly characterized as JTB-epistemology, i.e., it is characterized by the thesis that knowledge is justified true belief. Since Gettier’s trail-blazing paper of 1963 this account has become under heavy attack. The aim of is paper is to study the Gettier problem and related issues in the framework of topological epistemic logic. It is shown that in the framework of topological epistemic logic Gettier situations necessarily occur for most topological models of knowledge (...) and belief. On the other hand, there exists a special class of topological models (based on so called nodec spaces) for which traditional JTB-epistemology is valid. Further, it is shown that for each topological model of Stalnaker’s combined logic KB of knowledge and belief a canonical JTB-model (its JTB-doppelganger) can be constructed that shares many structural properties with the original model but is free of Gettier situations. The topological model and its JTB-doppelganger both share the same justified belief operator and have very similar knowledge operators. Seen from a somewhat different perspective, the JTB-account of epistemology amounts to a simplification of a more general epistemological account of knowledge and belief that assumes that these two concepts may differ in some cases. The JTB-account of knowledge and belief assumes that the epistemic agent’s cognitive powers are rather large. Thereby in the JTB-epistemology Gettier cases do not occur. Eventually, it is shown that for all topological models of Stalnaker’s KB-logic Gettier situations are topologically characterized as nowhere dense situations. This entails that Gettier situations are epistemologically invisible in the sense that they can neither be known nor believed with justification with respect to the knowledge operator and the belief operator of the models involved. (shrink)

In this paper I want to show that topology has a bearing on the theory of tropes. More precisely, I propose a topological ontology of tropes. This is to be understood as follows: trope ontology is a „one-category”-ontology countenancing only one kind of basic entities, to wit, tropes. 1 Hence, individuals, properties, relations, etc. are to be constructed from tropes.

Since antiquity well into the beginnings of the 20th century geometry was a central topic for philosophy. Since then, however, most philosophers of science, if they took notice of topology at all, considered it as an abstruse subdiscipline of mathematics lacking philosophical interest. Here it is argued that this neglect of topology by philosophy may be conceived of as the sign of a conceptual sea-change in philosophy of science that expelled geometry, and, more generally, mathematics, from the central position it (...) used to have in philosophy of science and placed logic at center stage in the 20th century philosophy of science. Only in recent decades logic has begun to loose its monopoly and geometry and topology received a new chance to find a place in philosophy of science. (shrink)

The notion of idealization has received considerable attention in contemporary philosophy of science but less in philosophy of mathematics. An exception was the ‘critical idealism’ of the neo-Kantian philosopher Ernst Cassirer. According to Cassirer the methodology of idealization plays a central role for mathematics and empirical science. In this paper it is argued that Cassirer's contributions in this area still deserve to be taken into account in the current debates in philosophy of mathematics. For extremely useful criticisms on earlier versions (...) I am grateful to B.P. Larvor and another anonymous journal referee. CiteULike Connotea Del.icio.us What's this? (shrink)

The main thesis of this paper is that Pap’s The Functional A Priori of Physical Theory (Pap 1946, henceforth FAP) and Cassirer’s Determinism and Indeterminism in Modern Physics (Cassirer 1937, henceforth DI) may be conceived as two kindred accounts of a late Neo-Kantian philosophy of science. They elucidate and clarify each other mutually by elaborating conceptual possibilities and pointing out affinities of neo-Kantian ideas with other currents of 20th century’s philosophy of science, namely, pragmatism, conventionalism, and logical empiricism. Taking into (...) account these facts, it seems not too far fetched to conjecture that under more favorable circumstances Pap could have served as a mediator between the “analytic” and “continental” tradition thereby overcoming the dogmatic dualism of these two philosophical currents that has characterized philosophy in the second half the 20th century. (shrink)

The aim of this paper is to show that a comprehensive account of the role of representations in science should reconsider some neglected theses of the classical philosophy of science proposed in the first decades of the 20th century. More precisely, it is argued that the accounts of Helmholtz and Hertz may be taken as prototypes of representational accounts in which structure preservation plays an essential role. Following Reichenbach, structure-preserving representations provide a useful device for formulating an up-to-date version of (...) a (relativized) Kantian a priori. An essential feature of modern scientific representations is their mathematical character. That is, representations can be conceived as (partially) structure-preserving maps or functions. This observation suggests an interesting but neglected perspective on the history and philosophy of this concept, namely, that structure-preserving representations are closely related to a priori elements of scientific knowledge. Reichenbach’s early theory of a relativized constitutive but non-apodictic a priori component of scientific knowledge provides a further elaboration of Kantian aspects of scientific representation. To cope with the dynamic aspects of the evolution of scientific knowledge, Cassirer proposed a re-interpretation of the concept of representation that conceived of a particular representation as only one phase in a continuous process determined by pragmatic considerations. Pragmatic aspects of representations are further elaborated in the classical account of C.I. Lewis and the more modern of Hasok Chang. (shrink)

Einführung in die Philosophie Rudolf Carnaps.

Carnap's quasi-analysis is usually considered as an ingenious but definitively flawed approach in epistemology and philosophy of science. In this paper it is argued that this assessment is mistaken. Quasi-analysis can be reconstructed as a representational theory of constitution of structures that has applications in many realms of epistemology and philosophy of science. First, existence and uniqueness theorems for quasi-analytical representations are proved. These theorems defuse the classical objections against the quasi-analytical approach launched forward by Goodman and others. Secondly, the (...) constitution of various kinds of structures is treated in detail: order structures, comparative similarity structures, mereological, mereotopological, and topological structures are considered. In particular, it is pointed out that there exist interesting relations between quasi-analysis and modern theories of pointless topology. (shrink)

Value judgments are meaningless. This thesis was one of the notorious tenets of Carnap's mature logical empiricism. Less well known is the fact that in the Aufbau values were considered as philosophically respectable entities that could be constituted from value experiences. About 1930, however, values and value judgments were banished to the realm of meaningless metaphysics, and Carnap came to endorse a strict emotivism. The aim of this paper is to shed light on the question why Carnap abandoned his originally (...) positive attitude concerning values. It is argued that his non-cognitivist attitude was the symptom of a deep-rooted and never properly dissolved tension between conflicting inclinations towards Neokantianism and Lebensphilosophie. In America Carnap's non-cognitivism became a major obstacle for a closer collaboration between logical empiricists and American pragmatists. Carnap's persisting adherence to the dualism of practical life and theoretical science was the ultimate reason why he could not accept Morris's and Kaplan's pragmatist theses that cognitivism might well be compatible with a logical and empiricist scientific philosophy. (shrink)

The aim of this paper is to show that (elementary) topology may be useful for dealing with problems of epistemology and metaphysics. More precisely, I want to show that the introduction of topological structures may elucidate the role of the spatial structures (in a broad sense) that underly logic and cognition. In some detail I’ll deal with “Cassirer’s problem” that may be characterized as an early forrunner of Goodman’s “grue-bleen” problem. On a larger scale, topology turns out to be useful (...) in elaborating the approach of conceptual spaces that in the last twenty years or so has found quite a few applications in cognitive science, psychology, and linguistics. In particular, topology may help distinguish “natural” from “not-so-natural” concepts. This classical problem that up to now has withstood all efforts to solve (or dissolve) it by purely logical methods. Finally, in order to show that a topological perspective may also offer a fresh look on classical metaphysical problems, it is shown that Leibniz’s famous principle of the identity of indiscernibles is closely related to some well-known topological separation axioms. More precisely, the topological perspective gives rise in a natural way to some novel variations of Leibniz’s principle. (shrink)

Felix Klein and Abraham Fraenkel each formulated a criterion for a theory of infinitesimals to be successful, in terms of the feasibility of implementation of the Mean Value Theorem. We explore the evolution of the idea over the past century, and the role of Abraham Robinson's framework therein.

Husserl's mathematical philosophy of science can be considered an anticipation of the contemporary postpositivistic semantic approach, which regards mathematics and not logic as the appropriate tool for the exact philosophical reconstruction of scientific theories. According to Husserl, an essential part of a theory's reconstruction is the mathematical description of its domain, that is, the world (or the part of the world) the theory intends to talk about. Contrary to the traditional micrological approach favored by the members of the Vienna Circle, (...) Husserl, inspired by modern geometry and set theory, aims at a macrological analysis of scientific theories that takes into account the global structures of theories as structured wholes. This is set in the complementary theories of manifolds and theory forms considered by Husserl himself as the culmination of his formal theory of science. (shrink)

According to general wisdom, Carnap's quasianalysis is an ingenious but definitively flawed approach to epistemology and philosophy of science. I argue that this assessment is mistaken. Rather, Carnapian quasianalysis can be reconstructed as a special case of a general theory of structural representation. This enables us to exploit some interesting analogies of quasianalysis with the representational theory of measurement. It is shown how Goodman's well-known objections against the quasianalytical approach may be defused in the new framework. As an application, I (...) sketch how the thesis of empirical underdetermination of theories may be elucidated in the framework of quasianalysis. (shrink)

The aim of this paper is to discuss the “Austro-American” logical empiricism proposed by physicist and philosopher Philipp Frank, particularly his interpretation of Carnap’s Aufbau, which he considered the charter of logical empiricism as a scientific world conception. According to Frank, the Aufbau was to be read as an integration of the ideas of Mach and Poincaré, leading eventually to a pragmatism quite similar to that of the American pragmatist William James. Relying on this peculiar interpretation, Frank intended to bring (...) about a rapprochement between the logical empiricism of the Vienna Circle in exile and American pragmatism. In the course of this project, in the last years of his career, Frank outlined a comprehensive, socially engaged philosophy of science that could serve as a “link between science and philosophy”. (shrink)

Abstract. Value judgments are meaningless. This thesis was one of the notorious tenets of Carnap’s mature logical empiricism. Less well known is the fact that in the Aufbau values were con-sidered as philosophically respectable entities that could be constituted from value experiences. About 1930, however, values were banished to the realm of meaning-less me-taphysics, and Carnap came to endorse a strict emotivism. The aim of this paper is to shed new light on the question why Carnap abandoned his originally positive (...) attitude concerning values. It is argued that Carnap’s non-cognitivist attitude was the symptom of a deep-rooted and never properly dissolved tension be-tween his conflicting inclinations towards Neokantianism and Lebensphilosophie. In America Carnap’s non-cognitivism became a major obstacle for a closer collaboration between lo-gical empiricists and American pragmatists. Carnap’s persisting ad---herence to the dualism of practical life and theoretical science was the ultimate reason why he could not accept Morris’s and Kaplan’s pragmatist the-ses that cognitivism might well are compatible with a logical and empiricist scientific philosophy. (shrink)

In this paper some classical representational ideas of Hertz and Duhem are used to show how the dichotomy between representation and intervention can be overcome. More precisely, scientific theories are reconstructed as complex networks of intervening representations (or representational interventions). The formal apparatus developed is applied to elucidate various theoretical and practical aspects of the in vivo/in vitro problem of biochemistry. Moreover, adjoint situations (Galois connections) are used to explain the relation berween empirical facts and theoretical laws in a new (...) way. (shrink)

Abstract: One of the institutional highlights of the encounter between Austrian “wissen¬schaftliche Philosophie” and French “philosophie scientifique” in the first half of the 20th century was the “First International Congress for Unity of Science” that took place 1935 in Paris. In my contribution I deal with an episode of the philosophical mega-event whose protagonist was the American philosopher and semiotician Charles William Morris. At the Paris congress he presented his programme of a comprehensive, practice-oriented scientific philosophy and, in a more (...) elaborated version he published it two years later in Logical Positivism, Pragmatism and Scientific Empiricism (Morris 1937). Morris aimed at a synthesis of formalism, pragmatism, and traditional empiricism that combined the virtues of these accounts while avoided their shortocmings. The core of approach was a comprehensive theory of the concept of meaning. Through an analysis of the concept of meaning he sought to sort out the existing differences and the options for a possible future rapprochment between logical empiricism and pragmatism. Against the overly narrow logical empiricist understanding of philosophy as the syntax of the language of science Morris argued for a “scientific pragmatism” that comprised four levels: (1) Philosophy as Logic of Science, (2) Philosophy as Clarification of Meaning (Peirce), (3) Philosophy as Empirical Axiology (Dewey), and (4) Philosophy as Empirical Cosmology (Whitehead). (shrink)

In the framework of set theory we cannot distinguish between natural and non-natural predicates. To avoid this shortcoming one can use mathematical structures as conceptual spaces such that natural predicates are characterized as structurally nice subsets. In this paper topological and related structures are used for this purpose. We shall discuss several examples taken from conceptual spaces of quantum mechanics (orthoframes), and the geometric logic of refutative and affirmable assertions. In particular we deal with the problem of structurally distinguishing between (...) natural colour predicates and Goodmanian predicates like grue and bleen. Moreover the problem of characterizing natural predicates is reformulated in such a way that its connection with the classical problem of geometric conventionalism becomes manifest. This can be used to shed some new light on Goodman's remarks on the relative entrenchment of predicates as a criterion of projectibility. (shrink)

Ernst Cassirer, 2011, Symbolische Prägnanz, Ausdrucksphänomen und „Wiener Kreis“, Nachgelassene Manuskripte und Texte, vol. 4, ed. Christian Möckel, 478pp., Hamburg, Felix Meiner Verlag.

In this paper a solution of Whitehead’s problem is presented: Starting with a purely mereological system of regions a topological space is constructed such that the class of regions is isomorphic to the Boolean lattice of regular open sets of that space. This construction may be considered as a generalized completion in analogy to the well-known Dedekind completion of the rational numbers yielding the real numbers . The argument of the paper relies on the theories of continuous lattices and “pointless” (...) topology.

. (shrink)

The aim of this paper is make a contribution to the ongoing search for an adequate concept of the a priori element in scientific knowledge. The point of departure is C.I. Lewis’s account of a pragmatic a priori put forward in his "Mind and the World Order" (1929). Recently, Hasok Chang in "Contingent Transcendental Arguments for Metaphysical Principles" (2008) reconsidered Lewis’s pragmatic a priori and proposed to conceive it as the basic ingredient of the dynamics of an embodied scientific reason. (...) The present paper intends to further elaborate Chang’s account by relating it with some conceptual tools from cognitive semantics and certain ideas that first emerged in the context of the category-theoretical foundations of mathematics. (shrink)

The aim of this paper is to elucidate the mereological structure of complex states of affairs without relying on the problematic notion of structural universals. For this task tools from graph theory, lattice theory, and the theory of relational systems are employed. Our starting point is the mereology of similarity structures. Since similarity structures are structured sets, their mereology can be considered as a generalization of the mereology of sets..

In the philosophy of the analytical tradition, set theory and formal logic are familiar formal tools. I think there is no deep reason why the philosopher’s tool kit should be restricted to just these theories. It might well be the case—to generalize a dictum of Suppes concerning philosophy of science—that the appropriate formal device for doing philosophy is mathematics in general; it may be set theory, algebra, topology, or any other realm of mathematics. In this paper I want to employ (...) elementary topological considerations to shed new light on the intricate problem of the relation of qualities and similarity. Thereby I want to make plausible the general thesis that topology might be a useful device for matters epistemological. (shrink)

A possible world is a junky world if and only if each thing in it is a proper part. The possibility of junky worlds contradicts the principle of general fusion. Bohn (2009) argues for the possibility of junky worlds, Watson (2010) suggests that Bohn‘s arguments are flawed. This paper shows that the arguments of both authors leave much to be desired. First, relying on the classical results of Cantor, Zermelo, Fraenkel, and von Neumann, this paper proves the possibility of junky (...) worlds for certain weak set theories. Second, the paradox of Burali-Forti shows that according to the Zermelo-Fraenkel set theory ZF, junky worlds are possible. Finally, it is shown that set theories are not the only sources for designing plausible models of junky worlds: Topology (and possibly other "algebraic" mathematical theories) may be used to construct models of junky worlds. In sum, junkyness is a relatively widespread feature among possible worlds. (shrink)

In this paper it is shown that Heyting and Co-Heyting mereological systems provide a convenient conceptual framework for spatial reasoning, in which spatial concepts such as connectedness, interior parts, (exterior) contact, and boundary can be defined in a natural and intuitively appealing way. This fact refutes the wide-spread contention that mereology cannot deal with the more advanced aspects of spatial reasoning and therefore has to be enhanced by further non-mereological concepts to overcome its congenital limitations. The allegedly unmereological concept of (...) boundary is treated in detail and shown to be essentially affected by mereological considerations. More precisely, the concept of boundary turns out to be realizable in a variety of different mereologically grounded versions. In particular, every part K of a Heyting algebra H gives rise to a well-behaved K-relative boundary operator. (shrink)

The notion of the a priori is an important legacy of Kant for modern philosophy of science. In the course of the 20th century, a variety of proposals was put forward all of which claimed to overcome the inadequacies of Kant’s original proposal of a synthetic a priori for modern science.