Computability theory originated with the seminal work of Gödel, Church, Turing, Kleene and Post in the 1930s. This theory includes a wide spectrum of topics, such as the theory of reducibilities and their degree structures, computably enumerable sets and their automorphisms, and subrecursive hierarchy classifications. Recent work in computability theory has focused on Turing definability and promises to have far-reaching mathematical, scientific, and philosophical consequences. Written by a leading researcher, Computability Theory provides a concise, comprehensive, and authoritative introduction to contemporary (...) computability theory, techniques, and results. The basic concepts and techniques of computability theory are placed in their historical, philosophical and logical context. This presentation is characterized by an unusual breadth of coverage and the inclusion of advanced topics not to be found elsewhere in the literature at this level. The book includes both the standard material for a first course in computability and more advanced looks at degree structures, forcing, priority methods, and determinacy. The final chapter explores a variety of computability applications to mathematics and science. Computability Theory is an invaluable text, reference, and guide to the direction of current research in the field. Nowhere else will you find the techniques and results of this beautiful and basic subject brought alive in such an approachable and lively way. (shrink)

By constructing a maximal incomplete d.r.e. degree, the nondensity of the partial order of the d.r.e. degrees is established. An easy modification yields the nondensity of the n-r.e. degrees and of the ω-r.e. degrees.

We prove the following three theorems on the enumeration degrees of ∑20 sets. Theorem A: There exists a nonzero noncuppable ∑20 enumeration degree. Theorem B: Every nonzero Δ20enumeration degree is cuppable to 0′e by an incomplete total enumeration degree. Theorem C: There exists a nonzero low Δ20 enumeration degree with the anticupping property.

We show that every nonzero $\Delta _{2}^{0}$ e-degree bounds a minimal pair. On the other hand, there exist $\Sigma _{2}^{0}$ e-degrees which bound no minimal pair.

It is shown that there exists a low2 Harrington non-splitting base-that is, a low2 computably enumerable (c.e.) degree a such that for any c.e. degrees x, y, if $0' = x \vee y$ , then either $0' = x \vee a$ or $0' = y \vee a$ . Contrary to prior expectations, the standard Harrington non-splitting construction is incompatible with the $low_{2}-ness$ requirements to be satisfied, and the proof given involves new techniques with potentially wider application.

In this paper we examine a class of pairs of recursively enumerable degrees, which is related to the Slaman-Soare Phenomenon.

A proof is given that 0′ is definable in the structure of the degrees of unsolvability. This answers a long-standing question of Kleene and Post, and has a number of corollaries including the definability of the jump operator.

We say that a computably enumerable (c.e.) degree b is a Lachlan nonsplitting base (LNB), if there is a computably enumerable degree a such that a > b, and for any c.e. degrees w,v ≤ a, if a ≤ w or; v or; b then either a ≤ w or; b or a ≤ v or; b. In this paper we investigate the relationship between bounding and nonbounding of Lachlan nonsplitting bases and the high /low hierarchy. We prove that there (...) is a non-Low2 c.e. degree which bounds no Lachlan nonsplitting base. (shrink)

Turing’s paper on computable numbers has played its role in underpinning different perspectives on the world of information. On the one hand, it encourages a digital ontology, with a perceived flatness of computational structure comprehensively hosting causality at the physical level and beyond. On the other, it can give an insight into the way in which higher order information arises and leads to loss of computational control—while demonstrating how the control can be re-established, in special circumstances, via suitable type reductions. (...) We examine the classical computational framework more closely than is usual, drawing out lessons for the wider application of information–theoretical approaches to characterizing the real world. The problem which arises across a range of contexts is the characterizing of the balance of power between the complexity of informational structure and the means available to bring this information back into the computational fold. We proceed via appropriate mathematical modelling to a more coherent view of the computational structure of information, relevant to a wide spectrum of areas of investigation. (shrink)

The Major Sub-degree Problem of A. H. Lachlan (first posed in 1967) has become a long-standing open question concerning the structure of the computably enumerable (c.e.) degrees. Its solution has important implications for Turing definability and for the ongoing programme of fully characterising the theory of the c.e. Turing degrees. A c.e. degree a is a major subdegree of a c.e. degree b > a if for any c.e. degree x, ${{\bf 0' = b \lor x}}$ if and only if (...) ${{\bf 0' = a \lor x}}$ . In this paper, we show that every c.e. degree b ≠ 0 or 0′ has a major sub-degree, answering Lachlan’s question affirmatively. (shrink)

We show that every nonzero Δ20, e-degree bounds a minimal pair. On the other hand, there exist Σ20, e-degrees which bound no minimal pair.

We show that there exists a properly Σ2 minimal degree b, and moreover that b can be chosen to join with 0′ to 0″ – so that b is a 0″ complement for every degree a such that 0′ ≤ a < 0″.

Computability theory concerns information with a causal–typically algorithmic–structure. As such, it provides a schematic analysis of many naturally occurring situations. Emil Post was the first to focus on the close relationship between information, coded as real numbers, and its algorithmic infrastructure. Having characterised the close connection between the quantifier type of a real and the Turing jump operation, he looked for more subtle ways in which information entails a particular causal context. Specifically, he wanted to find simple relations on reals (...) which produced richness of local computability-theoretic structure. To this extent, he was not just interested in causal structure as an abstraction, but in the way in which this structure emerges in natural contexts. Post’s programme was the genesis of a more far reaching research project. In this article we will firstly review the history of Post’s programme, and look at two interesting developments of Post’s approach. The first of these developments concerns the extension of the core programme, initially restricted to the Turing structure of the computably enumerable sets of natural numbers, to the Ershov hierarchy of sets. The second looks at how new types of information coming from the recent growth of research into randomness, and the revealing of unexpected new computability-theoretic infrastructure. We will conclude by viewing Post’s programme from a more general perspective. We will look at how algorithmic structure does not just emerge mathematically from information, but how that emergent structure can model the emergence of very basic aspects of the real world. (shrink)

We prove that there exists a nonzero recursively enumerable Turing degree possessing a strong minimal cover.

We show that for any computably enumerable set A and any equation image set L, if L is low and equation image, then there is a c.e. splitting equation image such that equation image. In Particular, if L is low and n-c.e., then equation image is n-c.e. and hence there is no low maximal n-c.e. degree.

We prove that there exists a noncappable enumeration degree strictly below 0' e.

We give a corrected proof of an extension of the Robinson Splitting Theorem for the d. c. e. degrees.

We develop an approach to the longstanding conjecture of Kierstead concerning the character of strongly nontrivial automorphisms of computable linear orderings. Our main result is that for any η-like computable linear ordering, such that has no interval of order type η, and such that the order type of is determined by a -limitwise monotonic maximal block function, there exists computable such that has no nontrivial automorphism.

This important new work is a major analysis of the foundation of Eric Voegelin's political science.

Consciousness and Politics begins with an analysis of the problem of the historicity of truth as it was formulated shortly before Voegelin abandoned his eight-volume History of Political Ideas. The analysis then follows a more or less chronological path, discussing the arguments developed in The New Science of Politics, Voegelin's most famous book, the differentiation of consciousness and the problems of myth and nature as presented in the early volumes of Order and History. Starting in the 1960s, Voegelin began a (...) lengthy argument in several volumes that resumed his concern with the philosophy of consciousness, which he had outlined in his early writings, and its connection to what we conventionally call philosophy of history. Voegelin's late and often difficult essays, lectures, and the final volume of Order and History, many scholars have noticed, emphasize the meditative origins of his political science and, more broadly, of philosophy. The concluding chapters analyze this subject-matter and a perennial question that so many of Voegelin's readers have raised: what is the relation of his political science or philosophy to Christianity? (shrink)

Despite a large and multifaceted effort to understand the vast landscape of phenotypic data, their current form inhibits productive data analysis. The lack of a community-wide, consensus-based, human- and machine-interpretable language for describing phenotypes and their genomic and environmental contexts is perhaps the most pressing scientific bottleneck to integration across many key fields in biology, including genomics, systems biology, development, medicine, evolution, ecology, and systematics. Here we survey the current phenomics landscape, including data resources and handling, and the progress that (...) has been made to accurately capture relevant data descriptions for phenotypes. We present an example of the kind of integration across domains that computable phenotypes would enable, and we call upon the broader biology community, publishers, and relevant funding agencies to support efforts to surmount today's data barriers and facilitate analytical reproducibility. (shrink)

In addition to gestalt psychology and phenomenology, merleau-ponty's first book was influenced decisively by hegel. more precisely, it is argued that his critical remarks concerning reflexology as well as idealism and empiricism were inspired in large measure by the interpretation of hegel that merleau-ponty learned from alexandre kojeve.

Although his contributions to philosophy are revered and his writings have been collected, Eric Voegelin’s persona will inevitably fade with the memories of those who knew him. This book preserves the human element of Voegelin by capturing those valuable personal recollections. Barry Cooper and Jodi Bruhn conducted intensive interviews with Voegelin’s wife, his closest friends, and his first-generation students—many of whom have since passed on—in order to bring to print everything important about his life and personality. American scholars (...) will especially appreciate the glimpses provided by Voegelin’s German colleagues into his life in Munich, as well as the thoughts of his students in Vienna. Reflections of people such as Paul Caringella, Bruno Schlesinger, and Heinz Barazon capture Voegelin’s greatness and shortcomings alike and also shed new light on his philosophical quest for truth. By descending progressively further into the past, the book takes readers deeper into the essence of Voegelin as reminiscences become more dramatic. Ranging widely from America back to Germany—with recollections of Gestapo intimidation and eventual emigration—the accounts interweave episodes of pathos, humor, fear, rivalry, and ambition. We witness Voegelin’s persistent and partly self-imposed communication problems and impatience with administrative duties, his respect for prudent political actors and public servants, and his genuine affection not only for his colleagues and best students but also for diligent secretaries and empathetic nurses. Through these recollections, key elements of his personality repeatedly emerge: his intelligence, optimism, and integrity, combined with an acute perception of the significance of his work. This is the most revealing and comprehensive biographical work yet available on a man known to be captivating as a thinker—and now shown to be equally fascinating as a human being. His own publications attest to his mind and methods; _Voegelin Recollected_ provides a deeper understanding of the man himself. (shrink)

*Introduction by Joseph R. Strayer\n(ix-x) Not that Bloch was the greatest French historian of his generation, though he would certainly rank high in any list. Not even that he was the most widely read—others excelled in that art of combining exact knowledge with readability which has distinguished French scholarship for many years. Others have talked about the narrowness of purely political history, the evils of excessive specialization, and the unreality of the conventional periodization of history—without ever leaving their own limited (...) fields. Bloch not only said that history was a whole, that no period and no topic could be understood except in relation to other periods and topics, but he constantly taught [p.x] and wrote in accordance with this belief. …He worked constantly for a “wider, more human history,” for a history which described how and why people live and work together. He saw life as a whole, as a complicated interplay of ideals and realities, of conscious innovation and unconscious conservation. … Throughout the two volumes he sought, above everything else, to understand and explain the state of mind and the habits of life which could produce and support feudal organization.\n(32-33) It is very like the illusion of certain old etymologiests who thought they had said all when they set down the oldest known meaning of a word opposite its present sense, having shown… As if the main problem were not to understand how and why the transition had taken place. As if, above all, the meaning of any word were influenced more by its own past than by the contemporary state of the vocabulary which, in its turn, is determined by the social conditions of the moment. [Y: think about this when I give the meanings of sirhak]\n(64-65) …every historical research supposes that the inquiry has a direction at the very first step. In the beginning, there must be the guiding spirit. Mere passive observation, even supposing such a thing were possible, has never contributed anything productive to any science. \n(71) Every historical book worthy of the name oought to include a chapter, or if one prefers, a series of paragraphs inserted at turning points in the development, which might almost entitled: “How can I know what I am about to say?” I am persuaded that even the lay reader would experience an actual intellectual pleasure in examining these “confessions.”. (shrink)

A comprehensive survey and introduction to the range of Michel Foucault's thought dealing with philosophy in general, sociology, political science, art, literature, and history in particular.

Leo Strauss and Eric Voegelin were political theorists of the first rank whose impact on the study of political science in North America has been profound. A study of their writings is one of the most expeditious ways to explore the core of political science; comparing and contrasting the positions both theorists have taken in assessing that core provides a comprehensive appreciation of the main options of the Western tradition. In fifty-three recently discovered letters, Strauss and Voegelin explore the nature (...) of their similarities and differences, offering trenchant observations about one another's work, about the state of the discipline, and about the influences working on them. The correspondence fleshes out many assumptions made in their published writings, often with a frankness and directness that removes all vestiges of ambiguity. Included with the correspondence are four pivotal re-published essays—"Jerusalem and Athens: Some Preliminary Reflections", "The Gospel and Culture", "Immortality: Experience and Symbol", and "The Mutual Influence of Theology and Philosophy" —and commentaries by James L. Wiser, Hans-Georg Gadamer, Stanley Rosen, Thomas J. J. Altizer, Timothy Fuller, Ellis Sandoz, Thomas L. Pangle, and David Walsh. (shrink)

_Faith and Political Philosophy_ consists of fifty-three letters between Leo Strauss and Eric Voegelin, two of the most important political theorists of the twentieth century. In this correspondence, Strauss and Voegelin explore the nature of their similarities and differences, offering insightful observations about one another's work, about the state of the discipline, and about the influences working on them. The letters shed light on many assumptions made in their published writings, often with an openness that removes all vestiges of uncertainty.

In 1924, not quite two years after receiving his doctorate from the University of Vienna, Eric Voegelin was named a Laura Spelman Rockefeller Memorial Fellow and thus given the opportunity to pursue postdoctoral studies in the United States. For the next twenty-four months, Voegelin worked with some of the most creative scholars in America and at several of the country's great universities, an experience that undoubtedly influenced his scholarly and personal perspectives throughout his life. A more immediate result was the (...) publication in 1928 of _On the Form of the American Mind,_ the young philosopher's first major work, in which his acute perceptions and analyses combine with a conceptual vocabulary struggling to find its own coherence and form. Voegelin begins his inquiry into the form of the American mind with a complex discussion of the concepts of time and existence in European and American philosophy and continues with an extended interpretation of George Santayana, a study of the Puritan mystic Jonathan Edwards, a presentation on Anglo-American jurisprudence, and a consideration of the historian, economist, and political scientist John R. Commons. Although admitting that this diversity of themes seems only loosely connected," Voegelin demonstrates the actual overall unity of these various subjects: each concerns linguistic expressions of a theoretical nature. Analysis of _On the Form of the American Mind_ indicates that Voegelin integrated the approaches of _Lebensphilosophie_ into what Georg Misch called the "philosophical combination of anthropology and history," which characterized contemporary trends within the discourse of the _Geisteswissenschaften_ and finally resulted in a theoretical paradigm of philosophical anthropology. Jürgen Gebhardt and Barry Cooper provide access to this brilliant study with their two-part introduction. The first part considers _On the Form of the American Mind_ in the context of methodological debates ongoing in Germany at the time Voegelin was writing the book; the second describes Voegelin's American experience and compares his work with similar studies written during the post-World War I period. (shrink)

The essays in this collection honor Professor Ellis Sandoz, Hermann Moyse Jr. Distinguished Professor of Political Science, Louisiana State University, and founding director of the Eric Voegelin Institute for American Renaissance Studies, an institute located at Louisiana State University and devoted to research and publication in the fields of political philosophy, constitutional law, and Voegelin studies. Without the tireless leadership—both academic and economic—of Ellis Sandoz, who was one of Eric Voegelin’s early students and his first American doctoral candidate at the (...) University of Munich, the thirty-four volumes of _The Collected Works of Eric Voegelin _would never have neared completion within only twenty years of Voegelin’s death. The authors of the essays, some of the most respected scholars in the field of political philosophy, hail from Austria, Canada, the Czech Republic, Germany, Ireland, and the United States. They have known Sandoz in a variety of ways: as former students, as beneficiaries of his work on constitutionalism, as fellow participants in the publication of _The Collected Works of Eric Voegelin_, as philosophical colleagues, as participating members of the Eric Voegelin Society, and finally as friends. The essays, each written especially for this volume, are grouped into the disciplines of philosophy, literature, and politics and range widely across these subjects, exploring writers as diverse as Xenophon, Natsume Soseki, Freud, and George Santayana and topics ranging from the leadership style of François Mitterrand and the Velvet Revolution to the nature of the Law and of the Political. _Philosophy, Literature, and Politics _is a fitting tribute to a man whose life’s work has been formed by commitment to the philosophical enterprise and by a conviction in Eric Voegelin’s contribution to that enterprise. The seventeen essays in this collection will be of particular interest to scholars of politics, comparative government, Japanese studies, political philosophy, and literary criticism. (shrink)

In 1924, not quite two years after receiving his doctorate from the University of Vienna, Eric Voegelin was named a Laura Spelman Rockefeller Memorial Fellow and thus given the opportunity to pursue postdoctoral studies in the United States. For the next twenty-four months, Voegelin worked with some of the most creative scholars in America and at several of the country's great universities, an experience that undoubtedly influenced his scholarly and personal perspectives throughout his life. A more immediate result was the (...) publication in 1928 of _On the Form of the American Mind,_ the young philosopher's first major work, in which his acute perceptions and analyses combine with a conceptual vocabulary struggling to find its own coherence and form. Voegelin begins his inquiry into the form of the American mind with a complex discussion of the concepts of time and existence in European and American philosophy and continues with an extended interpretation of George Santayana, a study of the Puritan mystic Jonathan Edwards, a presentation on Anglo-American jurisprudence, and a consideration of the historian, economist, and political scientist John R. Commons. Although admitting that this diversity of themes seems only loosely connected," Voegelin demonstrates the actual overall unity of these various subjects: each concerns linguistic expressions of a theoretical nature. Analysis of _On the Form of the American Mind_ indicates that Voegelin integrated the approaches of _Lebensphilosophie_ into what Georg Misch called the "philosophical combination of anthropology and history," which characterized contemporary trends within the discourse of the _Geisteswissenschaften_ and finally resulted in a theoretical paradigm of philosophical anthropology. Jürgen Gebhardt and Barry Cooper provide access to this brilliant study with their two-part introduction. The first part considers _On the Form of the American Mind_ in the context of methodological debates ongoing in Germany at the time Voegelin was writing the book; the second describes Voegelin's American experience and compares his work with similar studies written during the post-World War I period. (shrink)