This book gives a detailed treatment of functional interpretations of arithmetic, analysis, and set theory. The subject goes back to Gödel's Dialectica interpretation of Heyting arithmetic which replaces nested quantification by higher type operations and thus reduces the consistency problem for arithmetic to the problem of computability of primitive recursive functionals of finite types. Regular functional interpretations, i.e. Dialectica and Diller-Nahm interpretation as well as Kreisel's modified realization, together with their Troelstra-style hybrids, are applied to constructive as well as (...) classical systems of arithmetic, analysis, and set theory. They yield relative consistency and conservativity results and closure under relevant rules of the theories in question as well as axiomatic characterizations of the functional translations. Prerequisites are: familiarity with classical and intuitionistic predicate logic, basics of computability theory, Gödel's incompleteness theorems. (shrink)

Gödel interpreted Heyting arithmetic HA in a “logic-free” fragment T 0 of his theory T of primitive recursive functionals of finite types by his famous Dialectica-translation D . This works because the logic of HA is extremely simple. If the logic of the interpreted system is different—in particular more complicated—, it forces us to look for different and more complicated functional translations. We discuss the arising logical problems for arithmetical and set theoretical systems from HA to CZF . We want (...) to test the thesis: While the functionals take care of the proof theoretic strength of the interpreted system, it is the functional translation that has to cope with the logical complexities of the system. (shrink)

Gödel's dialectica interpretation of Heyting arithmetic HA may be seen as expressing a lack of confidence in our understanding of unbounded quantification. Instead of formally proving an implication with an existential consequent or with a universal antecedent, the dialectica interpretation asks, under suitable conditions, for explicit 'interpreting' instances that make the implication valid. For proofs in constructive set theory CZF-, it may not always be possible to find just one such instance, but it must suffice to explicitly name a set (...) consisting of such interpreting instances. The aim of eliminating unbounded quantification in favor of appropriate constructive functionals will still be obtained, as our ∧-interpretation theorem for constructive set theory in all finite types CZFω- shows. By changing to a hybrid interpretation ∧q, we show closure of CZFω- under rules that – in stronger forms – have already been studied in the context of Heyting arithmetic. In a similar spirit, we briefly survey modified realizability of CZFω- and its hybrids. Central results of this paper have been proved by Burr 2000a and Schulte 2006, however, for different translations. We use a simplified interpretation that goes back to Diller and Nahm 1974. A novel element is a lemma on absorption of bounds which is essential for the smooth operation of our translation. (shrink)

Gödel's dialectica interpretation of Heyting arithmetic HA may be seen as expressing a lack of confidence in our understanding of unbounded quantification. Instead of formally proving an implication with an existential consequent or with a universal antecedent, the dialectica interpretation asks, under suitable conditions, for explicit ‘interpreting’ instances that make the implication valid. For proofs in constructive set theory CZF‐, it may not always be possible to find just one such instance, but it must suffice to explicitly name a set (...) consisting of such interpreting instances. The aim of eliminating unbounded quantification in favor of appropriate constructive functionals will still be obtained, as our ∧‐interpretation theorem for constructive set theory in all finite types CZFω‐ shows. By changing to a hybrid interpretation ∧q, we show closure of CZFω‐ under rules that – in stronger forms – have already been studied in the context of Heyting arithmetic. In a similar spirit, we briefly survey modified realizability of CZFω‐ and its hybrids. Central results of this paper have been proved by Burr 2000a and Schulte 2006, however, for different translations. We use a simplified interpretation that goes back to Diller and Nahm 1974. A novel element is a lemma on absorption of bounds which is essential for the smooth operation of our translation. (shrink)

La Constance est le seul traité philosophique de Juste Lipse. La constance y est comprise comme endurance face aux malheurs publics, fondée sur une théorie métaphysique de la nécessité de l'ordre du monde crée par un dieu bon et provident. Il s'agit d'une oeuvre majeure du courant néostoïcien.

Because of its elegant simplicity, the Arabidopsis root has become a model for studying plant organogenesis. In this review we focus on recent results indicating the importance of signaling in root development. A role for positional information in root cell specification has been demonstrated by ablation analyses. Through mutational analysis, genes have been identified that play a role in radial pattern formation. The embryonic phenotypes of these mutants raised the possibility that division patterns in post‐embryonic roots are dependent on signaling (...) that originates during embryonic development. Analysis of expression of the SCARECROW gene indicates that it may play a role in this ‘top‐down’ signaling process. Characterization of root epidermis development has led to the identification of negative regulators of root‐hair formation. These appear to set up a prepattern which is reinforced by signaling by plant hormones. (shrink)

Karl Barth and Alvin Plantinga are not thought of as theological allies. Barth is famous for his opposition to philosophy's role in theology, while Plantinga is famous for his emphasis on warranted belief. Kevin Diller argues that they actually offer a unified response to the central epistemological dilemma in theology.

This chapter contains section titled: Introduction Emergence of the Stability‐Diversity—Complexity Debate Mathematization of Ecological Stability The End of the Consensus Contextualization and Classification of Ecological Stability Measures of Ecological Diversity and Complexity Evaluating Stability‐Diversity—Complexity Relationships Acknowledgments References Further Reading.

Rudolf Carnap is increasingly regarded as one of the most important philosophers of the twentieth century. He was one of the leading figures of the logical empiricist movement associated with the Vienna Circle and a central figure in the analytic tradition more generally. He made major contributions to philosophy of science and philosophy of logic, and, perhaps most importantly, to our understanding of the nature of philosophy as a discipline. In this volume a team of contributors explores the major themes (...) of his philosophy and discusses his relationship with the Vienna Circle and with philosophers such as Frege, Husserl, Russell, and Quine. New readers will find this the most convenient and accessible guide to Carnap currently available. Advanced students and specialists will find a conspectus of recent developments in the interpretation of Carnap. (shrink)

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work.This work is in the public domain in (...) the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work.As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

It is commonly held that Karl Barth emphatically rejected the usefulness of philosophy for theology. In this essay I explore the implications of Barth's theological epistemology for the relationship and proper boundaries between philosophy and theology, given its origin in Barth's theology of revelation. I seek to clarify Barth's position with respect to philosophy by distinguishing the contingency of its offence from any necessary incompatibility. Barth does not reject philosophy per se, but the way in which philosophy is typically conducted. (...) This is made explicit through an analysis of Barth's censure of the uncritical acceptance in theology of modernist philosophical presuppositions. I nuance Barth's response to a collection of philosophical assumptions that are rarely distinguished in theological literature. Finally, I highlight a representative instance of Barth's reflections on philosophy in relationship to theology, to demonstrate that the criterion for evaluating the usefulness of philosophical assumptions and methods in the service of theology is the same criterion by which theology is itself evaluated. (shrink)

In 17th-century intellectual life, the ideas of the Renaissance humanist Justus Lipsius (1547-1606) were omnipresent. The publication of his 'Politica' in 1589 had made Lipsius's name as an original and controversial political thinker. The sequel, the 'Monita et exempla politica' ('Admonitions'), published in 1605, was meant as an illustration of Lipsius's political thought as expounded in the 'Politica'. Its aim was to offer concrete models of behavior for rulers against the background of Habsburg politics. Lipsius's later political treatise also (...) forms an indispensable key to interpret the place and function of the 'Politica' in Lipsius's political discourse and in early modern political thought. The 'Political admonitions and examples' - widely read, edited and translated in the 17th and 18th centuries - show Lipsius's pivotal role in the genesis of modern political philosophy. (shrink)

The paper presents a series of German-Jewish readings of Lessing's "Nathan the Wise" stretching from the Enlightenment to the early post-1945 period. Already the first Jewish reader, Moses Mendelssohn, did not focus his interpretation of this drama on the so-called "parabel of the rings," where Nathan is commonly said to preach religious tolerance. Rather, Mendelssohn concentrates on act IV, scene 7, which expounds Lessing's concept of the relationship between Judaism and Christianity and Nathan's experience of Christian persecution. With the upsurge (...) of German anti-Semitism in the late 19th and 20th century, this scene served first as a sign of German-Christian empathy for Jewish suffering, and thus of hope, then as a reminder of recent prosecutions. It seemed to foreshadow, and eventually became overshadowed by, the Shoah. (shrink)

Gottfried Keller's “A Village Romeo and Juliet” contains an astonishing number of neuter terms that are of key importance to this novella. This observation is in line with Keller's predilection for miniaturized worlds and diminutives, played out in this narrative through two basic traits: the reduction of the Shakespearean tragedy to a novella that is a “village story” and the nicknames given to its female protagonist. Around one pole of neutrality the story assembles a cluster of terms that convey things (...) in an abstract, generalized manner. Other neutral nouns make deeply personal concerns look as if they were governed by some overriding, a-personal force. In this vein, the text tends to de-personalize human actions by expressing them in the grammatical form of nominalized infinitives. Inconspicuous idiomatic phrases seem to hide, as in the Freudian Id, the menacing power of the drives. Precisely by revealing that the narrative's sphere is constructed of neutral elementary particles, the text transforms and transubstantiates what it represents into the neutrality of the script. It does so solely, it seems, to make palpable that this literature is charged with the sensuality of the genders. (shrink)

ZusammenfassungObwohl Sportinteraktionen selten ohne einen Unparteiischen ablaufen, ist dieser Umstand soziologisch kaum erforscht. Die Spielleitung im Fußball, dem hier das Hauptinteresse gehört, ist in der Regel an einen neutralen Dritten delegiert, der das Spiel situativ beeinflusst, wie z.B. Elfmeterentscheidungen zuweilen drastisch vor Augen führen. Über einen Vergleich von Spielen ohne Schiedsrichter mit dem Kreisligafußball frage ich nach den strukturellen Unterschieden, die die Anwesenheit eines Schiedsrichters erzeugt. Dabei zeigt sich, dass ihre Präsenz die lautere und unlautere Konkurrenz im Spiel erhöht und (...) dass sich Spieler und Schiedsrichter in einer strategischen Interaktion befinden. Schließlich beschreibe ich die dramaturgischen Herausforderungen für Schiedsrichter, wie etwa ihre Darstellung von Neutralität und sicheren Entscheidungen. (shrink)