The identification of an informal concept of ‘effective calculability’ with a rigorous mathematical notion like ‘recursiveness’ or ‘Turing computability’ is still viewed as problematic, and I think rightly so. I analyze three different and conflicting perspectives Gödel articulated in the three decades from 1934 to 1964. The significant shifts in Gödel's position underline the difficulties of the methodological issues surrounding the Church-Turing Thesis.

In his “Kant and Finitism” Tait attempts to connect his analysis of finitist arithmetic with Kant’s perspective on arithmetic. The examination of this attempt is the basis for a distinctive view on the dramatic methodological shift from Kant to Dedekind and Hilbert. Dedekind’s 1888 essay “Was sind und was sollen die Zahlen?” gives a logical analysis of arithmetic, whereas Hilbert’s 1899 book “Grundlagen der Geometrie” presents such an analysis of geometry or, as Hilbert puts it, of our spatial intuition. This (...) shift in the late ninteenth century required a radical expansion of logic: first by the inclusion of principles for “systems” and “mappings”, but second by a structuralist broadening of axioms and inferential principles. The interaction of mathematics and logic in mathematical logic opened, around 1920, fields of investigation with enormous impact on the philosophy of mathematics, promoting a deeper integration of mathematical practice and philosophical reflection. (shrink)

Dedekind’s structuralism is a crucial source for the structuralism of mathematical practice—with its focus on abstract concepts like groups and fields. It plays an equally central role for the structuralism of philosophical analysis—with its focus on particular mathematical objects like natural and real numbers. Tensions between these structuralisms are palpable in Dedekind’s work, but are resolved in his essay Was sind und was sollen die Zahlen? In a radical shift, Dedekind extends his mathematical approach to “the” natural numbers. He creates (...) the abstract concept of a simply infinite system, proves the existence of a “model”, insists on the stepwise derivation of theorems, and defines structure-preserving mappings between different systems that fall under the abstract concept. Crucial parts of these considerations were added, however, only to the penultimate manuscript, for example, the very concept of a simply infinite system. The methodological consequences of this radical shift are elucidated by an analysis of Dedekind’s metamathematics. Our analysis provides a deeper understanding of the essay and, in addition, illuminates its impact on the evolution of the axiomatic method and of “semantics” before Tarski. This understanding allows us to make connections to contemporary issues in the philosophy of mathematics and science. (shrink)

J. Gorczyca doziert Ethik an der Gregorianischen Universität in Rom. In seinem Entwurf einer christlichen Ethik geht er über die traditionelle Auffassung einer philosophischen Ethik hinaus, einer Ethik, die sich der deduktiven Methode bedient, um mit Kraft der Vernunft normative Prinzipien für das menschliche Handeln systematisch aufzubauen. Dagegen betrachtet Gorczyca die philosophische Ethik aus der anthropologischen Perspektive, d.h. aus der Perspektive des Subjektes des ethischen Denkens. Der objektive Hauptinhalt seiner Ethik bekommt dadurch eine neue Dimension.

J. Gorczyca doziert Ethik an der Gregorianischen Universität in Rom. In seinem Entwurf einer christlichen Ethik geht er über die traditionelle Auffassung einer philosophischen Ethik hinaus, einer Ethik, die sich der deduktiven Methode bedient, um mit Kraft der Vernunft normative Prinzipien für das menschliche Handeln systematisch aufzubauen. Dagegen betrachtet Gorczyca die philosophische Ethik aus der anthropologischen Perspektive, d.h. aus der Perspektive des Subjektes des ethischen Denkens. Der objektive Hauptinhalt seiner Ethik bekommt dadurch eine neue Dimension.

This paper reexamines the long-standing problem of the nature and magnitude of the catastrophic Hellenic expedition to Egypt c. 460-454. An uneasy scholarly consensus posits that many fewer than the 200 triremes implied by Thucydides were involved in the momentous defeat, yet the arguments employed by proponents and detractors of this hypothesis have not been decisive. This paper attempts to develop a better understanding of the final stages of the campaign in order to settle the question of losses. Thucydides offers (...) the most reliable narrative of the events in Egypt , but the compressed nature of the pentekontaetia has left us with a brief, lacunary text. Examination of the verbs poliorkein and kataklēiein and the noun poliorkia in appropriate contexts throughout Thucydides' history reveals that the words connote a tight blockade that seeks to deny all supplies to the besieged; the terms do not normally convey less stringent varieties of military harassment. Application of this understanding to the passages in question shows that the 200 triremes initially mentioned by Thucydides could not possibly have been engaged in Egypt when the siege of Prosopitis island began: a force of such size under a tight blockade could never have held out for 18 months. This conclusion is supported by an economic and demographic survey of the fifth-century B.C. Egyptian Delta, which suggests that resources would not have been plentiful in the region. A much smaller Greek force, perhaps 40 to 50 triremes, must have been involved in the final siege. (shrink)

Connaître la date du premier siège de Nisibe est important car celle-ci permet de connaître la date de la mort de l'évêque le plus connu de cette époque : Jacques de Nisibe. L'A. pense à l'année 337. Jacques a du mourir pendant le siège et non après comme l'attestent certaines sources. Il est mort le 15 juillet 337 et a été enterré dans les murs de la cité.

Lewis and Charny have come under siege for suggesting remote questioning to decide appropriate medical care. While the criticisms are theoretically valid, the idea is so important practically that Lewis and Charny should be supported and their approach investigated as a way of making medical treatment at least more open and possibly more fair.

This paper offers an intellectual discourse for destination managers by exploring alternative branding approaches used during the colonial period in Kenya, now that the image is under siege both internally through socio-economic instability and unprecedented levels of poaching, and externally through travel warnings, outright trafficking in big game trophies, the constant threat of terror attacks, and poor global rankings in the Travel and Tourism Competitive Index. The paper conforms to the mission of thought and practice by identifying practical ways of (...) promoting tourist destination Kenya through an in-depth analysis of historical experiences. (shrink)

The knots of images in Aeschylus’ Septem with their exuberant and powerful vocabulary give the play the aura that prompted Gorgias and Aristophanes to proclaim it “full of Ares”. The ferocity of the ancient siege is brought to life in the destruction of the city the chorus imagines and in the duels at the seven gates that achieve epic proportions through the dueling speeches of Eteocles and his scout. The play’s transparent dependence on language for its emotive effect readily invites (...) close examination. Aeschylus’ Septem abounds with contemporary cosmological and mathematical ideas and images. The poet uses them consistently throughout the play, I suggest, to make of Thebes a seven-sided, shield-shaped cosmos, ruled over by the seven-month lord of sevens (heptamagetes), Apollo, and the completion of whose circumference seals the unified fate of Polynices and Eteocles. (shrink)

In the Transcendental Aesthetic (TA), Kant is typically held to make negative assertations about “things in themselves,” namely that they are not spatial or temporal. These negative assertions stand behind the “neglected alternative” problem for Kant’s transcendental idealism. According to this problem, Kant may be entitled to assert that spatio-temporality is a subjective element of our cognition, but he cannot rule out that it may also be a feature of the objective world. In this paper, I show in a new (...) way how Kant’s view (focusing on his conclusions about space) is not subject to this objection, by showing that he does not make the denial about mind-independent reality that he is typically held to make. The argument develops consequences of a new reading of Kant’s expression “an sich selbst” (‘in itself’; ‘in themselves’). I argue that “an sich selbst” or “per se” has a special, judgment-level role, so that this expression does not form new noun-terms adjectivally. It follows that the conceptual unit of Kant’s “Conclusions” in the TA is simply “things” (Dinge), since “things in themselves” is not a nominal expression; Kant adopts the Wolffian ontological use of “thing” as the basic kind-term for any existent. The arguments that things per se are not in space are arguments that space cannot be a necessary property or relations of things as a kind. I show that this does not involve the positive claim about mind-independent reality that inspires the neglected alternative objection. (shrink)

In this paper we elaborate on the question how to bridge the gap between contemporary (New) music and the tradition of the past, often called ‘classical’ music. First we analyze the notion of tradition (in classical music) as being distinct from traditional music, nationalism and traditionalism. A central role in this paper is dedicated to the role of counterpoint education following J.J. Fux’s Gradus ad Parnassum in the development of Central-European classical music between the late Renaissance and late Romantic periods. (...) The developments in the 20th century New Music reveal several important trend breaks. The controversies raised during the early Renaissance, regarding the practice of polyphonic singing, are discussed with respect to their impact on music development in later ages. From these controversies and the re-discovery of Nicholas de Cusa’s view on mysticism, the experience of the non-experience in polyphonic music is elucidated. Herewith, an illuminative heuristic is found in the enfolding-unfolding paradigm both in music and pictural arts, from the Renaissance till the present. (shrink)

Asperó and Schindler have completely solved the Axiom [Formula: see text] vs. [Formula: see text] problem. They have proved that if [Formula: see text] holds then Axiom [Formula: see text] holds, with no additional assumptions. The key question now concerns the relationship between [Formula: see text] and Axiom [Formula: see text]. This is because the foundational issues raised by the problem of Axiom [Formula: see text] vs. [Formula: see text] arguably persist in the problem of Axiom [Formula: see text] vs. (...) [Formula: see text]. The first of our two main theorems is that Axiom [Formula: see text] is equivalent to Axiom [Formula: see text], and as a corollary we show that Axiom [Formula: see text] fails in all the known models of [Formula: see text]. This suggests that [Formula: see text] actually refutes Axiom [Formula: see text]. Our second main theorem is that the [Formula: see text] Conjecture holds assuming [Formula: see text]. This is the strongest partial result known on this conjecture which is one of the central open problems of [Formula: see text]-theory and [Formula: see text]-logic. These results identify a fundamental asymmetry between the Continuum Hypothesis and any axiom which is both [Formula: see text]-expressible and which implies [Formula: see text], on the basis of generic absoluteness for the simplest of the nontrivial sentences of Third-Order Number Theory. These are the [Formula: see text]-sentences with no parameters. Such sentences are those which simply assert the existence of a set [Formula: see text] for which some property involving only quantification over [Formula: see text] holds. (shrink)

The seventh edition of this pragmatic guide to determining right and wrong in the workplace is updated with new case studies and ancillary materials to combine stakeholder perspectives with a deep dive on workplace ethics issues. Using a unique stakeholder-based approach, this book takes business ethics out of the theory realm and provides practical ways to analyze any business decision. Including dozens of cases, Joseph Weiss looks beyond the impacts of ethical lapses on share price and profit to focus on (...) relationships between stakeholders, including individuals, groups, corporations, and governments. The new seventh edition includes seven new cases and ten new Point/Counterpoint exercises, which challenge students to argue both sides of a contemporary issue. It incorporates the latest research as well as updated cases dealing with recent scandals at organizations such as Boeing, Facebook, and Uber. Professors will have access to online support, including teaching notes, PowerPoint slides, and sample tests. The book is equipped with a step-by-step guide to implementing stakeholder analysis, tips to help students apply ethical principles in their personal and professional lives, and comprehensive coverage of employee workplace issues. Addressing relevant topics such as risk management, preferential hiring, and sexual harassment in conjunction with the digital divide, bribery, the environment, and more, Business Ethics, seventh edition explores ethical implications in today's increasingly interconnected global business environment. (shrink)

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. This volume, the fifteenth publication in the Lecture Notes in Logic series, collects papers presented at the symposium 'Reflections on the Foundations of Mathematics' held in celebration of Solomon Feferman's 70th birthday (The 'Feferfest') at Stanford University, California in 1988. (...) Feferman has shaped the field of foundational research for nearly half a century. These papers reflect his broad interests as well as his approach to foundational research, which emphasizes the solution of mathematical and philosophical problems. There are four sections, covering proof theoretic analysis, logic and computation, applicative and self-applicative theories, and philosophy of modern mathematical and logic thought. (shrink)

This masterful work on Kant's Religion within the Boundaries of Mere Reason explores Kant's treatment of the Idea of God, his views concerning evil, and the moral grounds for faith in God. Kant and Religion works to deepen our understanding of religion's place and meaning within the history of human culture, touching on Kant's philosophical stance regarding theoretical, moral, political, and religious matters. Wood's breadth of knowledge of Kant's corpus, philosophical sharpness, and depth of reflection sheds light not only on (...) Kant, but also on the fate of religion and its relation to philosophy in the modern world. (shrink)

While the idea of philosophy as conceptual analysis has attracted many adherents and undergone a number of variations, in general it suffers from an authority problem with two dimensions. First, it is unclear why the analysis of a concept should have objective authority: why explicating what we mean should express how things are. Second, conceptual analysis seems to lack intersubjective authority: why philosophical analysis should apply to more than a parochial group of individuals. I argue that Hegel’s conception of social (...) ontology, focused on his concept of “ethical life” (Sittlichkeit), helps to explain how concepts (and their explication) have both objective and intersubjective authority in the social domain. Hegel claims that modern institutions are the product of self-conscious purposes, so that they are conceptually constituted. Concepts do not just represent these objects and so depend on a contingent relation to them. As many contemporary social ontologists agree, this means that our concepts of these institutions are uniquely epistemically “transparent.” They have objective authority. Concepts have intersubjective authority in the modern world as well, according to Hegel. However, I show that this feature of Hegel’s account does not rely on the solution to a philosophical problem. Rather, since the concepts of modern ethical life are “essentially contested,” their content depends on the practical and political agreement of modern subjects. This means that concepts can only have objective authority if some prior intersubjective agreement has been reached. The role of philosophy as conceptual analysis is thus importantly dependent on political developments. (shrink)

Inleidend studieboek op universitair niveau over de Nederlandse rechtswetenschap.

De Interpretatione is among Aristotle's most influential and widely read writings; C. W. A. Whitaker presents the first systematic study of this work, and offers a radical new view of its aims, its structure, and its place in Aristotle's system. He shows that De Interpretatione is not a disjointed essay on ill-connected subjects, as traditionally thought, but a highly organized and systematic treatise on logic, argument, and dialectic.

Wilfred Sieg. Mechanical Procedures and Mathematical Experience.

This book explores Kant's views on the concept of God and on the attempt to demonstrate God's existence as a means of understanding Kant's work as a whole and of achieving a proper appreciation of the contents of Kant's moral faith.

The latest volume in the Cambridge Histories of Philosophy series, The Cambridge History of Philosophy in the Nineteenth Century brings together twenty-nine leading experts in the field and covers the years 1790-1870. Their twenty-seven chapters provide a comprehensive survey of the period, organizing the material topically. After a brief editor's introduction, it begins with three chapters surveying the background of nineteenth century philosophy: followed by two on logic and mathematics, two on nature and natural science, five on mind and language, (...) including psychology, the human sciences and aesthetics, four on ethics, three on religion, seven on society, including chapters on the French Revolution, the decline of natural right, political economy, and social discontent, and three on history, dealing with historical method, speculative theories of history and the history of philosophy. The essays are framed by an editor's introduction and a bibliography. (shrink)

Machine learning algorithms pervade contemporary society. They are integral to social institutions, inform processes of governance, and animate the mundane technologies of daily life. Consistently, the outcomes of machine learning reflect, reproduce, and amplify structural inequalities. The field of fair machine learning has emerged in response, developing mathematical techniques that increase fairness based on anti-classification, classification parity, and calibration standards. In practice, these computational correctives invariably fall short, operating from an algorithmic idealism that does not, and cannot, address systemic, Intersectional (...) stratifications. Taking present fair machine learning methods as our point of departure, we suggest instead the notion and practice of algorithmic reparation. Rooted in theories of Intersectionality, reparative algorithms name, unmask, and undo allocative and representational harms as they materialize in sociotechnical form. We propose algorithmic reparation as a foundation for building, evaluating, adjusting, and when necessary, omitting and eradicating machine learning systems. (shrink)

Wilfred Sieg and Dirk Schlimm. Dedekind's Analysis of Number: Systems and Axioms.

We establish by elementary proof-theoretic means the conservativeness of two subsystems of analysis over primitive recursive arithmetic. The one subsystem was introduced by Friedman [6], the other is a strengthened version of a theory of Minc [14]; each has been shown to be of considerable interest for both mathematical practice and metamathematical investigations. The foundational significance of such conservation results is clear: they provide a direct finitist justification of the part of mathematical practice formalizable in these subsystems. The results are (...) generalized to relate a hierarchy of subsystems, all contained in the theory of arithmetic properties, to a corresponding hierarchy of fragments of arithmetic. The proof theoretic tools employed there are used to re-establish in a uniform, elementary way relationships between various fragments of arithmetic due to Parsons, Paris and Kirby, and Friedman. (shrink)