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

The Case for Corporate Climate Ratings: Nudging Financial Markets

Capital markets are cast as both villain and hero in the climate playbill. The trillions of dollars required to combat climate change leave ample room for heroics from the financial sector. For the time being, however, capital continues to flow readily toward fossil fuels and other carbon-intensive industries. Drawing on the results of an empirical study, this Article posits that ratings of corporate climate risk and governance can help overcome pervasive information asymmetries and nudge investors toward more climate-conscious investment choices with welfare-enhancing effects.In the absence of a meaningful price on carbon, three private ordering initiatives are trying to mobilize capital markets as a force for good in the war on carbon. But shareholder climate activism, calls for better climate-related financial disclosures, and the divestment movement have yet to usher in the paradigm shift toward low-carbon capitalism.Corporate climate ratings overcome existing information asymmetries to nudge investors toward more carbon-conscious allocation of their assets. Every year, rating agencies like Standard & Poor’s, Moody’s, and Fitch pass judgment on over one hundred trillion dollars’ worth of securities. Modeled after these well-established ratings of creditworthiness, independent ratings of companies’ climate risk and governance can redirect the flow of capital away from high-carbon assets toward more climate-friendly options—without the need for government authorization or other market-distorting interventions.A series of survey experiments with over fifteen hundred participants test, and demonstrate, the capacity of corporate climate ratings to promote low-carbon investment. Inclusion of climate ratings among the performance metrics commonly considered by investors significantly increases investment in the stock of companies with favorable climate ratings, even when other stocks boast a stronger return profile. Variations in the ratings’ framing and format, informed by insights from behavioral economics and finance, facilitate recommendations for best practices

Detecting directional coupling in the human epileptic brain: Limitations and potential pitfalls

We study directional relationships—in the driver-responder sense—in networks of coupled nonlinear oscillators using a phase modeling approach. Specifically, we focus on the identification of drivers in clusters with varying levels of synchrony, mimicking dynamical interactions between the seizure generating region (epileptic focus) and other brain structures. We demonstrate numerically that such an identification is not always possible in a reliable manner. Using the same analysis techniques as in model systems, we study multichannel electroencephalographic recordings from two patients suffering from focal epilepsy. Our findings demonstrate that—depending on the degree of intracluster synchrony—certain subsystems can spuriously appear to be driving others, which should be taken into account when analyzing field data with unknown underlying dynamics

What Makes a Theory of Infinitesimals Useful? A View by Klein and Fraenkel

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

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

Requirements for a Renewables Revolution

This Article identifies and analyzes the obstacles presently barring the rise of renewables, evaluates the role of the current policy favorite emission pricing, and offers design recommendations for a comprehensive U.S. renewables policy. Successful climate change mitigation requires a timely shift to renewable sources of energy, such as sunlight, wind or tides, to decarbonize today’s high-carbon electricity sector. But market pull alone is not strong enough. This Article discusses the most widely cited economic barriers and identifies and evaluates additional obstacles related to the electricity sector’s regulatory framework. Emission pricing is largely considered the most efficient policy to drive the timely transition to a renewables-based electricity sector. This Article argues that, for political, conceptual, and, most of all, regulatory reasons, emission pricing will not fuel the rise of renewables at the speed necessary for successful climate change mitigation. Rather, a comprehensive renewables policy is required to address each and every one of the existing barriers. Drawing on the policy experience of other sectors and nations, I offer recommendations for the design of a comprehensive U.S. renewables policy. Many of the proposed policy recommendations aim at non-economic barriers, which can be overcome through regulatory intervention. Once these barriers have been removed, policy support for renewables can focus on the remaining economic barriers and, hence, becomes far less costly. In light of the plethora of obstacles to a timely transition to renewables, this Article calls for concerted policy action by scientists, engineers, economists, lawyers, marketers, and educators to fuel the renewables revolution