184 research outputs found
Prototypes, Poles, and Topological Tessellations of Conceptual Spaces
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
Topological Models of Columnar Vagueness
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
Physikalistische Graphologie als Avantgarde der Psychologie oder Physikalismus auf Abwegen
Die Physikalisierung der Psychologie war für Carnap Teil eines Programms, das die Sonderstellung der Psychologie als Wissenschaft des menschlichen Denkens und Fühlens als Illusion entlarven und zeigen sollte, die Psychologie sei ein Teil der Physik wie alle anderen Wissenschaften auch. In etwas anderer Motivation zielte Carnaps Physikalismus ausserdem auf eine Überwindung der Trennung von Geistes–wissenschaften und Naturwissenschaften: Erwiese sich die Psychologie sich als physikalisierbar, wäre das ein wesentlicher Schritt für die Vereinheitlichung der Wissenschaften in Gestalt einer enzyklopädischen „Einheitswissenschaft“ überhaupt.
Carnaps Argument für die Physikalisierbarkeit der Psychologie als ganzer basierte auf der These der Physikalisierbarkeit der Graphologie als zentraler Teildisziplin der Psychologie. Die Graphologie sei der begrifflich am weitesten fortgeschrittene und deshalb am ehesten physikalisierbare Teil der Psychologie. Das verdanke sie in erster Linie den wegweisenden Arbeiten Ludwig Klages’. Erweise sich die Graphologie als physikalisierbar, stehe einer durchgehenden Physikalisierung aller Wissenschaften nichts mehr im Wege.
Als Episode in Carnaps philosophischer Entwicklung ist dem Graphologieprojekt bis heute kaum Aufmerksamkeit geschenkt worden. Das ist ein Versäumnis, manifestiert sich in diesem Projekt doch der allgemeine Stil des Carnapschen Philosophierens besonders deutlich, nämlich von einer sehr abstrakten und idealisierten Vorstellung von Wissenschaft ausgehend weitreichende philosophische Folgerungen zu ziehen
Continuous Lattices and Whiteheadian Theory of Space
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. \u3cbr\u3e\u3cbr\u3e
From Cautious Enthusiasm to Profound Disenchantment - Ernest Nagel and Carnapian Logical Empiricism
The global relation between logical empiricism and American pragmatism is
one of the more difficult problems in history of philosophy. In this paper I’d like to take
a local perspective and concentrate on the details that concern the vicissitudes of a
philosopher who played an important role in the encounter of logical empiricism and
American pragmatism, namely, Ernest Nagel. In this paper, I want to explore some
aspects of Nagel’s changing attitude towards the then „new“ logical-empiricist
philosophy. In the beginning Nagel welcomed logical empiricism whole-heartedly. This
early enthusiasm did not last. At the end of his philosophical career Nagel’s early
positive attitude towards logical empiricism shown in the 1930s had been replaced by a
much more reserved one. Nagel’s growing dissatisfaction with the Carnapian version of
logical empiricist philosophy was clearly expressed in Nagel’s criticism of Carnap’s
inductive logic and more generally in his last book Teleology Revisited and Other
Essays on History and Philosophy of Science. There he critized harshly Carnap’s
philosophy of science in general as ahistoric and non-pragmatist.
One of the distinctive features of Nagel’s philosophy of science is the emphasis that he
put on the role of history of science for philosophy of science. A compelling evidence
for this attitude are his works on the history and philosophy of geometry and algebra
One may say that Carnap and Nagel represented opposed possibilities of how the
profession of a philosopher of science could be understood: Carnap as a „conceptual
engineer“ was engaged in the task of inventing the conceptual tools for a better
theoretical understanding of science, while Nagel was to be considered more as a
„public intellectual“ engaged in the project of realizing a more rational and enlightened
society
Description, Construction and Representation. From Russell and Carnap to Stone
The first aim of this paper is to elucidate Russell’s construction of spatial points, which is to be \u3cbr\u3econsidered as a paradigmatic case of the "logical constructions" that played a central role in his epistemology and theory of science. Comparing it with parallel endeavours carried out by Carnap and Stone it is argued that Russell’s construction is best understood as a structural representation. It is shown that Russell’s and Carnap’s representational constructions may be considered as incomplete and sketchy harbingers of Stone’s representation theorems. The representational program inaugurated by Stone’s theorems was one of the success stories of 20th century’s mathematics. This suggests that representational constructions à la Stone could also be important for epistemology and philosophy of science. More specifically it is argued that the issues proposed by Russellian definite descriptions, logical constructions, and structural representations still have a place on the agenda of contemporary epistemology and philosophy of science. Finally, the representational interpretation of Russell’s logical constructivism is used to shed some new light on the recently vigorously discussed topic of his structural realism
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