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)

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.

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.

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)

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)

Dieser Titel aus dem De Gruyter-Verlagsarchiv ist digitalisiert worden, um ihn der wissenschaftlichen Forschung zugänglich zu machen. Da der Titel erstmals im Nationalsozialismus publiziert wurde, ist er in besonderem Maße in seinem historischen Kontext zu betrachten. Mehr erfahren Sie.

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)

Things and Their Place in Theories Our talk of external things, our very notion of things, is just a conceptual apparatus that helps us to foresee and ...

This volume honours the life and work of Solomon Feferman, one of the most prominent mathematical logicians of the latter half of the 20th century. In the collection of essays presented here, researchers examine Feferman’s work on mathematical as well as specific methodological and philosophical issues that tie into mathematics. Feferman’s work was largely based in mathematical logic, but also branched out into methodological and philosophical issues, making it well known beyond the borders of the mathematics community. With regard to (...) methodological issues, Feferman supported concrete projects. On the one hand, these projects calibrate the proof theoretic strength of subsystems of analysis and set theory and provide ways of overcoming the limitations imposed by Gödel’s incompleteness theorems through appropriate conceptual expansions. On the other, they seek to identify novel axiomatic foundations for mathematical practice, truth theories, and category theory. In his philosophical research, Feferman explored questions such as “What is logic?” and proposed particular positions regarding the foundations of mathematics including, for example, his “conceptual structuralism.” The contributing authors of the volume examine all of the above issues. Their papers are accompanied by an autobiography presented by Feferman that reflects on the evolution and intellectual contexts of his work. The contributing authors critically examine Feferman’s work and, in part, actively expand on his concrete mathematical projects. The volume illuminates Feferman’s distinctive work and, in the process, provides an enlightening perspective on the foundations of mathematics and logic. (shrink)

Wilfred Sieg. Mechanical Procedures and Mathematical Experience.

A complete translation of Aristotle's classic that is both faithful and readable, along with an introduction that provides the modern reader with a means of understanding this seminal work and its impact on our culture. In this volume, Joe Sachs (translator of Aristotle's _Physics, Metaphysics,_ and the _Nicomachean Ethics _)also supplements his excellent translation with well-chosen notes and glossary of important terms. Focus Philosophical Library translations are close to and are non-interpretative of the original text, with the notes and a (...) glossary intending to provide the reader with some sense of the terms and the concepts as they were understood by Aristotle’s immediate audience. (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)

Hilbert's finitist program was not created at the beginning of the twenties solely to counteract Brouwer's intuitionism, but rather emerged out of broad philosophical reflections on the foundations of mathematics and out of detailed logical work; that is evident from notes of lecture courses that were given by Hilbert and prepared in collaboration with Bernays during the period from 1917 to 1922. These notes reveal a dialectic progression from a critical logicism through a radical constructivism toward finitism; the progression has (...) to be seen against the background of the stunning presentation of mathematical logic in the lectures given during the winter term 1917/18. In this paper, I sketch the connection of Hilbert's considerations to issues in the foundations of mathematics during the second half of the 19th century, describe the work that laid the basis of modern mathematical logic, and analyze the first steps in the new subject of proof theory. A revision of the standard view of Hilbert's and Bernays's contributions to the foundational discussion in our century has long been overdue. It is almost scandalous that their carefully worked out notes have not been used yet to understand more accurately the evolution of modern logic in general and of Hilbert's Program in particular. One conclusion will be obvious: the dogmatic formalist Hilbert is a figment of historical (de)construction! Indeed, the study and analysis of these lectures reveal a depth of mathematical-logical achievement and of philosophical reflection that is remarkable. In the course of my presentation many questions are raised and many more can be explored; thus, I hope this paper will stimulate interest for new historical and systematic work. (shrink)

David Hilbert was one of the great mathematicians who expounded the centrality of their subject in human thought. In this collection of essays, Wilfried Sieg frames Hilbert's foundational work, from 1890 to 1939, in a comprehensive way and integrates it with modern proof theoretic investigations.

Explores the nature and types of mystical experience and discusses the value of mysticism for humanity.

It is a remarkable fact that Hilbert's programmatic papers from the 1920s still shape, almost exclusively, the standard contemporary perspective of his views concerning (the foundations of) mathematics; even his own, quite different work on the foundations of geometry and arithmetic from the late 1890s is often understood from that vantage point. My essay pursues one main goal, namely, to contrast Hilbert's formal axiomatic method from the early 1920s with his existential axiomatic approach from the 1890s. Such a contrast illuminates (...) the circuitous beginnings of the finitist consistency program and connects the complex emergence of existential axiomatics with transformations in mathematics and philosophy during the 19th century; the sheer complexity and methodological difficulties of the latter development are partially reflected in the well known, but not well understood correspondence between Frege and Hilbert. Taking seriously the goal of formalizing mathematics in an effective logical framework leads also to contemporary tasks, not just historical and systematic insights; those are briefly described as “one direction” for fascinating work. (shrink)

Alonzo Church's mathematical work on computability and undecidability is well-known indeed, and we seem to have an excellent understanding of the context in which it arose. The approach Church took to the underlying conceptual issues, by contrast, is less well understood. Why, for example, was "Church's Thesis" put forward publicly only in April 1935, when it had been formulated already in February/March 1934? Why did Church choose to formulate it then in terms of Gödel's general recursiveness, not his own λ (...) -definability as he had done in 1934? A number of letters were exchanged between Church and Paul Bernays during the period from December 1934 to August 1937; they throw light on critical developments in Princeton during that period and reveal novel aspects of Church's distinctive contribution to the analysis of the informal notion of effective calculability. In particular, they allow me to give informed, though still tentative answers to the questions I raised; the character of my answers is reflected by an alternative title for this paper, Why Church needed Gödel's recursiveness for his Thesis. In Section 5, I contrast Church's analysis with that of Alan Turing and explore, in the very last section, an analogy with Dedekind's investigation of continuity. (shrink)

Wilfried Sieg. Calculations by Man and Machine: Conceptual Analysis.

W. V. Quine is one of the most eminent philosophers alive today. Now in his mid-eighties he has produced a sharp, sprightly book that encapsulates the whole of his philosophical enterprise, including his thinking on all the key components of his epistemological stance--especially the value of logic and mathematics. New readers of Quine may have to go slowly, fathoming for themselves the richness that past readers already know lies between these elegant lines. For the faithful there is much to ponder. (...) In this short book, based on lectures delivered in Spain in 1990, Quine begins by locating his work historically. He provides a lightning tour of the history of philosophy (particularly the history of epistemology), beginning with Plato and culminating in an appreciative sketch of Carnap's philosophical ambitions and achievements. This leads, in the second chapter, to an introduction to Quine's attempt to naturalize epistemology, which emphasizes his continuities with Carnap rather than the differences between them. The next chapters develop the naturalistic story of the development of science to take account of how our conceptual apparatus is enhanced so that we can view the world as containing re-identifiable objects. Having explained the role of observation sentences in providing a checkpoint for assessing scientific theories, and having despaired of constructing an empirical criterion to determine which sentences are meaningful, Quine in the remaining chapters takes up a variety of important issues about knowledge. He concludes with an extended treatment of his views about reference and meaning and his attitudes toward psychological and modal notions. The presentation is distinctive, and the many small refinements of detail and formulation will fascinate all who know Quine's philosophy. (shrink)

"More than any thing else technology creates our world. It creates our wealth, our economy, our very way of being," says W. Brian Arthur. Yet, until now the major questions of technology have gone unanswered. Where do new technologies come from -- how exactly does invention work? What constitutes innovation, and how is it achieved? Why are certain regions -- Cambridge, England, in the 1920s and Silicon Valley today -- hotbeds of innovation, while others languish? Does technology, like biological life, (...) evolve? How do new industries, and the economy itself, emerge from technologies? In this groundbreaking work, pioneering technology thinker and economist W. Brian Arthur sets forth a boldly original way of thinking about technology that gives answers to these questions. The Nature of Technology is an elegant and powerful theory of technology's origins and evolution. It achieves for the progress of technology what Thomas Kuhn's The Structure of Scientific Revolutions did for scientific progress. Arthur explains how transformative new technologies arise and how innovation really works. Conventional thinking ascribes the invention of technologies to "thinking outside the box," or vaguely to genius or creativity, but Arthur shows that such explanations are inadequate. Rather, technologies are put together from pieces -- themselves technologies -- that already exist. Technologies therefore share common ancestries and combine, morph, and combine again to create further technologies. Technology evolves much as a coral reef builds itself from activities of small organisms -- it creates itself from itself; all technologies are descended from earlier technologies. Drawing on a wealth of examples, from historical inventions to the high-tech wonders of today, and writing in wonder fully engaging and clear prose, Arthur takes us on a mind-opening journey that will change the way we think about technology and how it structures our lives. (shrink)

Herbrand's Theorem, in the form of $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\exists } $$ -inversion lemmata for finitary and infinitary sequent calculi, is the crucial tool for the determination of the provably total function(al)s of a variety of theories. The theories are (second order extensions of) fragments of classical arithmetic; the classes of provably total functions include the elements of the Polynomial Hierarchy, the Grzegorczyk Hierarchy, and the extended Grzegorczyk Hierarchy $\mathfrak{E}^\alpha $ , α < ε0. A subsidiary aim of the paper is to show (...) that the proof theoretic methods used here are distinguished by technical elegance, conceptual clarity, and wide-ranging applicability. (shrink)

Wilfred Sieg. Relative Consistency and Accesible Domains.

Through clear explanation of its characteristic concepts and doctrines, and paying close attention to the published works of its philosophers, the volume ...

Wilfried Sieg and John Byrnes. AnModel for Parallel Computation: Gandy's Thesis.

Church's and Turing's theses dogmatically assert that an informal notion of effective calculability is adequately captured by a particular mathematical concept of computability. I present an analysis of calculability that is embedded in a rich historical and philosophical context, leads to precise concepts, but dispenses with theses. To investigate effective calculability is to analyze symbolic processes that can in principle be carried out by calculators. This is a philosophical lesson we owe to Turing. Drawing on that lesson and recasting work (...) of Gandy, I formulate boundedness and locality conditions for two types of calculators, namely, human computing agents and mechanical computing devices (discrete machines). The distinctive feature of the latter is that they can carry out parallel computations. The analysis leads to axioms for discrete dynamical systems (representing human and machine computations) and allows the reduction of models of these axioms to Turing machines. Cellular automata and a variety of artificial neural nets can be shown to satisfy the axioms for machine computations. (shrink)