We study Dedekind cuts on ordered Abelian groups. We introduce a monoid structure on them, and we characterise, via a suitable representation theorem, the universal part of the theory of such structures.

Let M be a polynomially bounded, o-minimal structure with archimedean prime model, for example if M is a real closed field. Let C be a convex and unbounded subset of M. We determine the first order theory of the structure M expanded by the set C. We do this also over any given set of parameters from M, which yields a description of all subsets of Mn, definable in the expanded structure.

In this chapter, we will provide an overview of Richard Dedekind's work on continuity, both foundational and mathematical. His seminal contribution to the foundations of analysis is the well-known 1872 booklet Stetigkeit und irrationale Zahlen (Continuity and irrational numbers), which is based on Dedekind's insight into the essence of continuity that he arrived at in the fall of 1858. After analysing the intuitive understanding of the continuity of the geometric line, Dedekind characterized the property of continuity for (...) the real numbers in terms of what are nowadays called 'Dedekind cuts' on the rational numbers. This treatment, which can be characterized as being 'arithmetical' as well as 'axiomatic', will be presented in detail. To better position Dedekind's contributions in their historical context, we will also consider some of his more mathematical treatments of continuity in addition to his foundational work. Of particular interest is the definition of the Riemann surface in his joint work with Heinrich Weber (1882). Moreover, Dedekind's reflections on space and continuity in his unpublished papers 'Allgemeine Sätze über Räume' (General theorems about spaces; before 1870) and 'Beweis und Anwendungen eines allgemeinen Satzes über mehrfach ausgedehnte stetige Gebiete' (Proof and applications of a general theorem about multiply extended continuous domains; 1892) illustrate the wide range and general coherency of his thoughts. By discussing Dedekind's works in which the notion of continuity plays a central role, we will show how Dedekind's approaches became increasingly abstract, while at the same time retaining a common methodology. (shrink)

In 1996, W. Veldman and F. Waaldijk present a constructive (intuitionistic) proof for the homogeneity of the ordered structure of the Cauchy real numbers, and so this result holds in any topos with natural number object. However, it is well known that the real numbers objects obtained by the traditional constructions of Cauchy sequences and Dedekind cuts are not necessarily isomorphic in an arbitrary topos with natural numbers object. Consequently, Veldman and Waaldijk's result does not apply to the ordered (...) structure of Dedekind real numbers in toposes. The main result to be proved in the present paper is that the ordered structure of the Dedekind real numbers object is homogeneous, in any topos with natural numbers object. This result is obtained within the framework of local set theory. (shrink)

This paper uses neo-Fregean-style abstraction principles to develop the integers from the natural numbers (assuming Hume’s principle), the rational numbers from the integers, and the real numbers from the rationals. The first two are first-order abstractions that treat pairs of numbers: (DIF) INT(a,b)=INT(c,d) ≡ (a+d)=(b+c). (QUOT) Q(m,n)=Q(p,q) ≡ (n=0 & q=0) ∨ (n≠0 & q≠0 & m⋅q=n⋅p). The development of the real numbers is an adaption of the Dedekind program involving “cuts” of rational numbers. Let P be a property (...) (of rational numbers) and r a rational number. Say that r is an upper bound of P, written P≤r, if for any rational number s, if Ps then either s<r or s=r. In other words, P≤r if r is greater than or equal to any rational number that P applies to. Consider the Cut Abstraction Principle: (CP) ∀P∀Q(C(P)=C(Q) ≡ ∀r(P≤r ≡ Q≤r)). In other words, the cut of P is identical to the cut of Q if and only if P and Q share all of their upper bounds. The axioms of second-order real analysis can be derived from (CP), just as the axioms of second-order Peano arithmetic can be derived from Hume’s principle. The paper raises some of the philosophical issues connected with the neo-Fregean program, using the above abstraction principles as case studies. (shrink)

It is a commonly held view that Dedekind's construction of the real numbers is impredicative. This naturally raises the question of whether this impredicativity is justified by some kind of Platonism about sets. But when we look more closely at Dedekind's philosophical views, his ontology does not look Platonist at all. So how is his construction justified? There are two aspects of the solution: one is to look more closely at his methodological views, and in particular, the places (...) in which predicativity restrictions ought to be applied; another is to take seriously his remarks about the reals as things created by the cuts, instead of considering them to be the cuts themselves. This can lead us to make finer-grained distinctions about the extent to which impredicative definitions are problematic, since we find that Dedekind's use of impredicative definitions in analysis can be justified by his philosophical views. (shrink)

In the present note, we study a generalization of Dedekind cuts in the context of constructive Zermelo–Fraenkel set theory CZF. For this purpose, we single out an equivalent of CZF's axiom of fullness and show that it is sufficient to derive that the Dedekind cuts in this generalized sense form a set. We also discuss the instance of this equivalent of fullness that is tantamount to the assertion that the class of Dedekind cuts in the rational numbers, (...) in the customary constructive sense including locatedness, is a set. This is to be compared with the situation for the partial reals, a generalization in a different direction: if one drops locatedness from the notion of a Dedekind real, then it becomes inherently impredicative. We make this precise, and further present the curiosity that the irrational Dedekind reals form a set already with exponentiation. (shrink)

Psychopathology is the study of the signs and symptoms of psychiatric disorders - delusions, hallucinations, phobias, depression, for example. This book gives an account of the terms currently in use and attempts an in-depth analysis of the nature of each. The matter is examined both from a philosophical perspective and from the point of view of what is known about the function of the hemispheres of the brain.

Hannah Arendt's early biography of Rahel Varnhagen, an eighteenth-century German-Jew, provides a revolutionary feminist component to her political theory. In it, Arendt grapples with the theoretical constitution of a female subject and relates Jewish alterity, identity, and history to feminist politics. Because she understood the "female condition" of difference as belonging to the political subject rather than an autonomous self, her theory entails a "politics of alterity" with applications for feminist practice.

Hannah Arendt's early biography of Rahel Varnhagen, an eighteenth-century German-Jew, provides a revolutionary feminist component to her political theory. In it, Arendt grapples with the theoretical constitution of a female subject and relates Jewish alterity, identity, and history to feminist politics. Because she understood the "female condition" of difference as belonging to the political subject rather than an autonomous self, her theory entails a "politics of alterity" with applications for feminist practice.

At least 177 scales are available to researchers who want to measure religiosity, but questions exist as to exactly what these scales are measuring and in whom they are measuring it. A review of these scales found a lack items designed to measure ethical action in society or the world as a prophetic response to the experience of the divine. Instead, the vast majority of scales focus on internal experiences and beliefs or institutional relationships. A review of scale norm groups (...) found that norm groups often are not fully described, particularly in the area of race/ethnicity, and when they are described, they reveal an over-reliance on convenience samples of college students and an under-representation of racial/ethnic minority groups. Examples of scales with more fully described and more representative norm groups are given, and recommendations are offered for researchers using and developing religiosity scales. (shrink)

Popular movies present chunk-like events that promote episodic, serial updating of viewers’ representations of the ongoing narrative. Event-indexing theory would suggest that the beginnings of new scenes trigger these updates, which in turn require more cognitive processing. Typically, a new movie event is signaled by an establishing shot, one providing more background information and a longer look than the average shot. Our analysis of 24 films reconfirms this. More important, we show that, when returning to a previously shown location, the (...) re-establishing shot reduces both context and duration while remaining greater than the average shot. In general, location shifts dominate character and time shifts in event segmentation of movies. In addition, over the last 70 years re-establishing shots have become more like the noninitial shots of a scene. Establishing shots have also approached noninitial shot scales, but not their durations. Such results suggest that film form is evolving, perhaps to suit more rapid encoding of narrative events. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Joanne Cutting-Gray HANNAH ARENDT'S RAHEL VARNHAGEN Hannah Arendt fled Nazi Germany in 1933, a year she called the end of Jewish history. She was 27 years old at the time and carried with her a manuscript that was later to become the peculiar biography of an eighteenth-century German-Jewish "pariah," Rahel Varnhagen (1771-1833). The Life of a fewish Woman, subtitle of the biography by Arendt, distills the largely unpublished Varnhagen (...) correspondence and marks the impact of that life upon German-Jewish history.1 Published more than "half a lifetime" later in 1957, it assays the politics of form in a highly unusual manner. By expunging all detail in order to analyze Varnhagen's introspection, Arendt implicitly questions selfhood as the form for identity and as the appropriate form for biography. She reveals that seeking to preserve the self determines the blank dumbness of an unnarratable life. Rahel Levin (Antonie Friederike Robert) Varnhagen, or simply "Rahel " as Arendt calls her, lived a life as complicated as her many names. The tortured relation she had to her identity, her insatiable preoccupation with self, reflects an effort to eradicate herJewishness by baptism and marriage and becomes a paradigm of a story about a woman lost to a sense of others and herself. Neither rich, cultivated, nor beautiful, Rahel lived her unhappiness fervently, with an ecstasy that made her appear to others as startlingly original. This helps to explain how she came to preside over the most important intellectual salon in Germany during the late eighteenth century. Princes, intellectuals, and artists flocked to her attic apartment, attracted by the impenetrable spectacle of her despair. Rahel's intensity pulled others into her orbit—including, one-hundred years later, the intense and self-absorbed Arendt at the beginning Philosophy and Literature, © 1991, 15: 229-245 230Philosophy and Literature of her university education.2 Each woman suffered the private failure of a love affair complicated by the broader, historical issue of racial identity that connected their lives. Prewar Nazi Germany taught Arendt what Varnhagen learned during the period of German-Jewish assimilation in the nineteenth century—"one does not escape Jewishness."3 It is no wonder that, in analogous moments of personal and civic upheaval, Rahel Varnhagen and Hannah Arendt "corresponded," the one by writing letters about her life, the other by giving that life to a world. Like Arendt, Varnhagen was unclassifiably exotic, an unattached, Jewish female seemingly without a history, aligned with no one and nothing. If"Jewishness" meant letting life "rain upon" her, Rahel Varnhagen let it strike her "like a storm without an umbrella." And like Arendt, she refused to hide the traits and talents of her personality under cover of morality and convention. Her letters are not self-congratulatory, for she never wittingly chose notoriety. Indeed, she spent much of her life trying to escape the stigma ofJewishness that marked her destiny. Notwithstanding the milieu of nineteenth-century German romanticism that drew the biographer toward her subject, various events surrounding the text make more compelling narratives than Varnhagen 's life: how the manuscript survived World War II, Arendt's emigration to the United States, and, especially, the public and private influence of Martin Heidegger.4 At the age of 18 (1925), Arendt had discovered Varnhagen during her brief love-affair with the 35-year-old philosophy professor at the University of Marburg. Though Heidegger would later call Arendt the "passion of his life" and inspiration for his work—he was writing Being and Time during that period—his brush with Nazism precipitated their breach. Since the Heidegger-Arendt correspondence remains closed, much about their youthful relationship and life-long intellectual affair remains a mystery open only to speculation. From Arendt's biographer, Elisabeth Young-Bruehl, we know that Varnhagen's story became a consolation to her in her sorrow, a means ofgiving significance to the ambiguities of a private relationship as it was caught up in public history.5 Certainly Heidegger's retreat from political events for the sake of thinking was as strong an influence on Arendt's political development as was Varnhagen's retreat from her Jewish destiny. Nonetheless, it would be a mistake to expect... (shrink)

Pickering & Garrod (P&G) argue that contemporary models of language use are inadequate. This has resulted largely because of an experimental focus on monologue rather than dialogue. I agree with the need for increased experimentation that focuses on the interplay between production and comprehension. However, I have some concerns about the Interactive Alignment model that the authors propose.

Primary, or basic, colors have been discussed for centuries. Over time, three criteria have emerged on their behalf: (a) their physical mixture yielding all other spectral colors, (b) the physiological attunement of receptors or pathways to particular wavelengths, and (c) the etymological history of the color term. These criteria can be applied usefully to taste to clarify issues.

The concepts of invariants and cues are useful, as are those of dorsal and ventral streams, but Norman overgeneralizes when interweaving them. Cues are not confined to identification tasks, invariants not to action, and both can be learned.

Who am I? How do I live a good life? What is reality? Such perennial questions may seem remote from the pleasures of playing videogames for entertainment and fantasy. yet gamers too, in the midst of having fun, are potentially embarked upon a quest for understanding and for meaning. Missions for thoughtful gamers presents a sequence of 40 challenges, ranging from thought experiments to design exercises, each one inviting players to become more creatively curious and self-aware."--Back cover.

At least 177 scales are available to researchers who want to measure religiosity, but questions exist as to exactly what these scales are measuring and in whom they are measuring it. A review of these scales found a lack items designed to measure ethical action in society or the world as a prophetic response to the experience of the divine. Instead, the vast majority of scales focus on internal experiences and beliefs or institutional relationships. A review of scale norm groups (...) found that norm groups often are not fully described, particularly in the area of race/ethnicity, and when they are described, they reveal an over-reliance on convenience samples of college students and an under-representation of racial/ethnic minority groups. Examples of scales with more fully described and more representative norm groups are given, and recommendations are offered for researchers using and developing religiosity scales. (shrink)

Nijhawan argues that neural compensation is necessary to account for couplings of perception and action. Although perhaps true in some cases, computational tolerance for asynchronously arriving continuous information is of more importance. Moreover, some of the everyday venues Nijhawan uses to argue for the relevance of prediction and compensation can be better ascribed to skill.

Hollywood movies can be deeply engaging and easy to understand. To succeed in this manner, feature-length movies employ many editing techniques with strong psychological underpinnings. We explore the origins and development of one of these, the reaction shot. This shot typically shows a single, unspeaking character with modest facial expression in response to an event or to the behavior or speech of another character. In a sample of movies from 1940 to 2010, we show that the prevalence of one type (...) of these shots—which we call the cryptic reaction shot—has grown dramatically. These shots are designed to enhance viewers’ emotional involvement with characters. They depict a facial gesture that reflects a slightly negative and slightly aroused emotional state. Their use at the end of conversations, and typically at the end of scenes, helps to leave viewers in a state of speculation about what the character is thinking and what her thoughts may mean for the ongoing narrative. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Philosophy, Psychiatry, & Psychology 8.4 (2001) 337-338 [Access article in PDF] On Kimura's Écrits de psychopathologie phénomenologique John Cutting This book is a French translation of six articles that the Japanese psychiatrist, Kimura Bin, wrote in the 1970s and 1980s. There is the usual long introduction in such books by the translator. There is also the mandatory explanation of the whole matter as a postface by philosopher Henry Maldiney (...) as if the meaning of anything these days has to be in the form of a sandwich.The first article by Kimura Bin deals with the meaning of certain Japanese words pertaining to self and inter-subjectivity. The critical Japanese word is aida, which means something like interpersonal bonding and is that primordial state of the human being that founds both everyday sociality and the individual self, the latter both in its external and internal composition and orientation: "From time immemorial, the Japanese understood interpersonal social existence, the being between, life in the aida of itself and of others, as the essential trait of an individual human. The interpersonal aida is in some sense the quintessence of known individuality." Schizophrenia, for Kimura, stems from a disturbance in this aida at the most fundamental level, and all its manifestations follow from this.The second article is entitled "Time and Anxiety (Angoisse)." This sets out his major psychopathological thesis—judging from these six articles—which is that schizophrenia and depressive illness differ fundamentally in their temporality. In schizophrenia the subject is geared toward the future: "Subjects with a schizophrenic tendency give sufficient proof in their daily life of a marked inclination to imagine future possibilities or ideas whose feasibility is impossible or implausible." In depressive illness there is a preoccupation with the past: "The lived time of the melancholic type of person has a preponderance towards the past, or, more precisely, towards lived being up to the present."The third article, entitled "Schizophrenic Temporality," amplifies this thesis and introduces his terminology for these, respectively, as the Latin terms, ante festum and post festum. The French translation gives the meaning of the latter as "apres la fête" and "trop tard"—after the holiday, too late. (I shall deal with the potpourri of languages involved below.) This article is the most elaborate, and in it Kimura attempts to derive the core characteristics of the two nosological entities from these two underlying anomalous temporalities.The fourth article is about borderline states, a nosological entity that I have no sympathy for, and I shall not discuss it.The fifth article is entitled "Reflection and Self in Schizophrenia." Again, it concerns a fundamental difference between the mode of reflection in schizophrenia and that in depressive illness, but I cannot say, despite numerous readings, [End Page 337] that I understand it. The reason for this may be my poor French, an inadequate translation, or a confused formulation in the first place, or all three. The gist of it seems to be that in schizophrenia the "other" is a "stranger" projected by the thinking self, whereas in depressive illness the "other" is a "familiar" inhabitant of the sociality adherent to this condition.The sixth article, "Pathology of Immediateness," is an attempt to seek philosophical, theoretical, psychopathological, and even nosological and biological backing for what Kimura has set out earlier, particularly in the third article. Hegel, Lacan, Bergson, Heidegger, Derrida, and Husserl are brought in as philosophical support. Blankenburg and Binswanger provide psychopathological backing. A third prong is the nosological tradition, prevalent in Germany and Japan in the twentieth century, of promoting the concept of an atypical or third type of psychosis with a biological resemblance to epileptic psychosis. All this is then brought together and complicated by a discussion of language and 'mediation' to support the notion that there is a third psychopathological principle—in addition to the ante festum pertaining to schizophrenia and the post festum pertaining to depressive illness—which is that of an intra festum: a preoccupation with the now, characteristic of atypical psychoses with an epileptic basis.These articles tax one's linguistic and conceptual repertoire to the... (shrink)

In three studies, we explored the retention and transfer of tool-making knowledge, learnt from an adult demonstration, to other temporal and task contexts. All studies used a variation of a task in which children had to make a hook tool to retrieve a bucket from a tall transparent tube. Children who failed to innovate the hook tool independently saw a demonstration. In Study 1, we tested children aged 4 to 6 years (N = 53) who had seen the original demonstration (...) 3 months earlier. Performance was excellent at the second time, indicating that children’s knowledge was retained over the 3 month period. In Studies 2 and 3 we explored transfer of the new knowledge to other tasks. In Study 2, children were given two variants of the apparatus that differed in surface characteristics (e.g. shape and colour). Participants generalised their knowledge to these new apparatuses even though the new pipecleaner also differed in size and colour. Five- to 6-year-olds (n = 22) almost always transferred their knowledge to problems where the same tool had to be made. Younger, 3- to 5-year-olds’ (n = 46), performance was more variable. In Study 3, 4- to 7-year-olds (N = 146) saw a demonstration of hook making with a pipecleaner, but then had to make a tool by combining pieces of wooden dowel (or vice versa: original training on dowel, transfer to pipecleaner). Children did not transfer their tool-making knowledge to the new material. Children retained tool-making knowledge over time and transferred their knowledge to new situations in which they needed to make a similar tool from similar materials, but not different materials. We concluded that children’s ability to use tool-making knowledge in novel situations is likely to depend on memory and analogical reasoning, with the latter continuing to develop during middle childhood. (shrink)

The computability of reals was introduced by Alan Turing [20] by means of decimal representations. But the equivalent notion can also be introduced accordingly if the binary expansion, Dedekind cut or Cauchy sequence representations are considered instead. In other words, the computability of reals is independent of their representations. However, as it is shown by Specker [19] and Ko [9], the primitive recursiveness and polynomial time computability of the reals do depend on the representation. In this paper, we explore (...) how the weak computability of reals depends on the representation. To this end, we introduce three notions of weak computability in a way similar to the Ershov's hierarchy of Δ02-sets of natural numbers based on the binary expansion, Dedekind cut and Cauchy sequence, respectively. This leads to a series of classes of reals with different levels of computability. We investigate systematically questions as on which level these notions are equivalent. We also compare them with other known classes of reals like c. e. and d-c. e. reals. (shrink)