In recent years, several remarkable results have shown that certain theorems of finite combinatorics are unprovable in certain logical systems. These developments have been instrumental in stimulating research in both areas, with the interface between logic and combinatorics being especially important because of its relation to crucial issues in the foundations of mathematics which were raised by the work of Kurt Godel. Because of the diversity of the lines of research that have begun to shed light on these issues, there (...) was a need for a comprehensive overview which would tie the lines together. This volume fills that need by presenting a balanced mixture of high quality expository and research articles that were presented at the August 1985 AMS-IMS-SIAM Joint Summer Research Conference, held at Humboldt State University in Arcata, California.With an introductory survey to put the works into an appropriate context, the collection consists of papers dealing with various aspects of 'unprovable theorems and fast-growing functions'. Among the topics addressed are: ordinal notations, the dynamical systems approach to Ramsey theory, Hindman's finite sums theorem and related ultrafilters, well quasiordering theory, uncountable combinatorics, nonstandard models of set theory, and a length-of-proof analysis of Godel's incompleteness theorem. Many of the articles bring the reader to the frontiers of research in this area, and most assume familiarity with combinatorics and/or mathematical logic only at the senior undergraduate or first-year graduate level. (shrink)

Almost everyone has an intuitive notion of what a random number is. For example, consider these two series of binary digits: 01010101010101010101 01101100110111100010 The first is obviously constructed according to a simple rule; it consists of the number 01 repeated ten times. If one were asked to speculate on how the series might continue, one could predict with considerable confidence that the next two digits would be 0 and 1. Inspection of the second series of digits yields no such comprehensive (...) pattern. There is no obvious rule governing the formation of the number, and there is no rational way to guess the succeeding digits. The arrangement seems haphazard; in other words, the sequence appears to be a random assortment of 0's and 1's. (shrink)

What could be more certain than the fact that 2 plus 2 equals 4? Since the time of the ancient Greeks mathematicians have believed there is little---if anything---as unequivocal as a proved theorem. In fact, mathematical statements that can be proved true have often been regarded as a more solid foundation for a system of thought than any maxim about morals or even physical objects. The 17th-century German mathematician and philosopher Gottfried Wilhelm Leibniz even envisioned a ``calculus'' of reasoning such (...) that all disputes could one day be settled with the words ``Gentlemen, let us compute!'' By the beginning of this century symbolic logic had progressed to such an extent that the German mathematician David Hilbert declared that all mathematical questions are in principle decidable, and he confidently set out to codify once and for all the methods of mathematical reasoning. (shrink)

For the purposes of this monograph, "by a degree" is meant a degree of recursive unsolvability. A degree [script bold]m is said to be minimal if 0 is the unique degree less than [script bold]m. Each of the six chapters of this self-contained monograph is devoted to the proof of an existence theorem for minimal degrees.

One enduring legacy of the twentieth century will be the slow, certain transformation of the world from insular civilizations to interactive societies enmeshed in global systems of electronic communication, economics, and politics. Financial news from Thailand or Brazil is often more important globally than political events in the old centers of power. Some bemoan the uncertainty and flux of all this. However, the mutual definition of the world’s societies presents an extraordinary opportunity to humanize a situation that all too quickly (...) could degenerate into a Weberian ‘iron cage’ of truly global proportions. What contribution can the world’s great philosophical traditions make toward humanity’s common task of civilizing the rise of globalization? Here the ancient concept of paideia walks upon the stage of the twenty-first century. (shrink)

One enduring legacy of the twentieth century will be the slow, certain transformation of the world from insular civilizations to interactive societies enmeshed in global systems of electronic communication, economics, and politics. Financial news from Thailand or Brazil is often more important globally than political events in the old centers of power. Some bemoan the uncertainty and flux of all this. However, the mutual definition of the world’s societies presents an extraordinary opportunity to humanize a situation that all too quickly (...) could degenerate into a Weberian ‘iron cage’ of truly global proportions. What contribution can the world’s great philosophical traditions make toward humanity’s common task of civilizing the rise of globalization? Here the ancient concept of paideia walks upon the stage of the twenty-first century. (shrink)

One enduring legacy of the twentieth century will be the slow, certain transformation of the world from insular civilizations to interactive societies enmeshed in global systems of electronic communication, economics, and politics. Financial news from Thailand or Brazil is often more important globally than political events in the old centers of power. Some bemoan the uncertainty and flux of all this. However, the mutual definition of the world’s societies presents an extraordinary opportunity to humanize a situation that all too quickly (...) could degenerate into a Weberian ‘iron cage’ of truly global proportions. What contribution can the world’s great philosophical traditions make toward humanity’s common task of civilizing the rise of globalization? Here the ancient concept of paideia walks upon the stage of the twenty-first century. (shrink)

Documents a decade that saw the Association begin negotiations to merge with the Western Philosophical Association that later led to the original organization becoming the Eastern Division of an expanded Association, and a world war that divided friends and colleagues across both geographical and political lines. The addresses, therefore, take on internal and external politics and are often tinged with tragedy. The topics include the problem of transcendence, Bergson and pragmatism, time and the experience of time, the ethics of states, (...) the doctrinaire in a time of crisis, the psychology of punitive justice, art and the democracy, the social significance of education, and the attack on the state. Biographical sketches and photographs are provided for the speakers. Annotation copyrighted by Book News, Inc., Portland, OR. (shrink)

Human beings are strange animals. Although evolutionary theory has brilliantly accounted for the features we share with other creatures—from the genetic code that directs the construction of our bodies to the details of how our muscles and neurons work—we still stand out in countless ways. Our brains are exceptionally large, we alone have truly grammatical language, and we alone compose symphonies, drive cars, eat spaghetti with a fork and wonder about the origins of the universe.

In these Carus Lectures, Annette Baier looks at the relation between individual and shared reasoning, intending, and moral reflection. In each case she emphasizes the interdependence of minds and the role of social practices in setting the norms governing these activities.

This book traces the further development and emergence of American philosophy, particularly Naturalism and Pragmatism, against the backdrop of still-dominant Hegelian philosophy, during the third decade of the 20th century, through the addresses and biographies of the presidents of its oldest and largest philosophical society. Of special interest is the previously unpublished presidential address of Henry Walgrave Stewart, second president of the Pacific Division of The American Philosophical Association. The work contains the biographies, photographs, and addresses of 24 (...) past presidents, and photographs and addresses of others. No other publication has collected the presidential addresses of this, the largest society of philosophers, during its third decade of existence, and in no other publication are the biographies and photographs of all of its presidents available. While some histories of this period in 20th century American philosophy have been written, and while some collections of articles by philosophers exist which include writings of some of these individuals, none has concentrated on a representation of the leadership of. (shrink)

*The views expressed are the author's own. This article should not be construed as representing policies of the American Medical Association.

*Disclaimer: The views expressed are the author's and should not be ascribed to the American Medical Association.

In Culture, Genre, and Literary Vocation, Michael Davitt Bell charts the important and often overlooked connection between literary culture and authors' careers. Bell's influential essays on nineteenth-century American writers—originally written for such landmark projects as The Columbia Literary History of the United States and The Cambridge History of American Literature—are gathered here with a major new essay on Richard Wright. Throughout, Bell revisits issues of genre with an eye toward the unexpected details of authors' lives, and invites us (...) to reconsider the hidden functions that terms such as "romanticism" and "realism" served for authors and their critics. Whether tracing the demands of the market or the expectations of readers, Bell examines the intimate relationship between literary production and culture; each essay closely links the milieu in which American writers worked with the trajectory of their storied careers. (shrink)

The rapidly changing field of genetics affects society through advances in health-care and through implications of genetic research. This study addresses the impacts of new genetic discoveries and technologies on different segments of today's society. The book begins with a chapter on genetic complexity, and subsequent chapters discuss moral and ethical questions arising from today's genetics from the perspectives of health care professionals, the media, the general public, special interest groups and commercial interests.