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

Factors influencing parasite-related suppression of mating behavior in the isopod Caecidotea intermedius

Parasites with indirect life cycles often facilitate changes in their intermediate hosts in ways that increase the likelihood of transmission to their definitive hosts. Acanthocephalan infections typically correlate with altered pigmentation, antipredatory behavior, and changes in mating behavior in arthropod intermediate hosts that increase risks of predation by definitive vertebrate hosts. Additionally, these changes have been shown to associate with the developmental stage of the parasite which facilitates the likelihood of survival in the final host. These changes have been proposed to due to direct manipulation by the parasite, host counteradaptation to minimize the costs of infection, or are an indirect byproduct of pathology. The acanthocephalan parasite, Acanthocephalus dirus, infects the stream-dwelling isopod Caecidotea intermedius as an intermediate host and one of several freshwater fishes as a definitive host. Inside the isopod, A. dirus develops from the early non-infective acanthor and acanthella (immature) stages to the late infective cystacanth stage (mature, capable of transmission to the final host). Developmental stage of A. dirus also correlates with changes in isopod color, antipredatory behavior, and mating dynamics. C. intermedius infected with late-stage parasites have been shown to have reduced pairing success in nature. Additionally, it has been shown that male mating responsiveness (e.g. willingness to mate) is reversible (from no mating attempts to positive mating attempts). However, little is known about the potential ultimate and proximate mechanisms underlying these relationships. Additionally, the potential role of host counteradaptation (compensation) during early stages of infection has not been examined. To examine isopod mating behavior in early-stage infections, I used field-based experiments to assess if host compensation was occurring in male C. intermedius. I found that infected isopods did not increase their mating effort compared to uninfected males. Thus, I concluded that male isopods do not compensate for a future reproductive loss. To assess factors that influence male mating responsiveness in late stages of parasite development, I used a combination of field and lab-based experiments. Since chemical cues have been shown to be important in aquatic environments and because predation is necessary for completion of the parasite life cycle, I examined if predator cues could influence male mating responsiveness using a lab-based experiment. I found that predator cues alone do not appear to be influencing mating response. However, I did find that reversibility of mating response can be maintained in a laboratory-setting. I also examined if mating responsiveness is flexible and reversible in nature using a field-based experiment. I found that male mating responsiveness is flexible in nature towards the end of C. intermedius life cycle. I also found that reversibility of mating response occurs within 200 minutes of removal from a natural setting. Thus, it is unlikely that mating responsiveness could be due to an indirect effect of pathology. The ultimate mechanisms I have studied indicate that parasite manipulation is the most likely cause of mating behavior in C. intermedius. Early-stage parasites can not survive transmission to the definitive host. Thus, manipulation of male mating behavior is not beneficial to the parasite at this life stage. Additionally, since male mating behavior is flexible and reversible in nature, it is plausible that parasites can manipulate this behavior to conserve energy (absence of predators or towards end of the breeding season) and increase the likelihood of survival into the definitive host. I examined if neuromodulation could be a proximate mechanism controlling mating behavior. Dopamine and serotonin levels were assayed for infected and uninfected isopods with suppressed mate guarding behavior. I found no difference between infection status and either dopamine or serotonin levels. Thus, these neurohormone levels did not appear to be influencing mating behavior in male C. intermedius