From ancient times to the present, the discovery and presentation of new proofs of previously established theorems has been a salient feature of mathematical practice. Why? What purposes are served by such endeavors? And how do mathematicians judge whether two proofs of the same theorem are essentially different? Consideration of such questions illuminates the roles that proofs play in the validation and communication of mathematical knowledge and raises issues that have yet to be resolved by mathematical logicians. The Appendix, in (...) which several proofs of the Fundamental Theorem of Arithmetic are compared, provides a miniature case study. (shrink)

Die von Karl Menger und seinen Mitarbeitern (darunter Kurt Gödel) herausgegebenen "Ergebnisse eines Mathematischen Kolloquiums" zählen zu den wichtigsten Quellenwerken der Wissenschafts- und Geistesgeschichte der Zwischenkriegszeit, mit bahnbrechenden Beiträgen von Menger, Gödel, Tarski, Wald, John von Neumann und vielen anderen. In diesem Band liegt der Inhalt erstmals gesammelt vor. Der Nobelpreisträger Gerard Debreu schrieb die Einleitung, die Kommentare wurden vom Logiker und Gödel-Biographen John Dawson jr., dem Topologen Ryszard Engelking und dem Wirtschaftstheoretiker Werner Hildenbrand verfasst. Außerdem enthält (...) der Band einen biographischen Aufsatz über Karl Menger sowie einen von Menger verfassten Überblick über die wichtigsten topologischen und geometrischen Arbeiten des Kolloquiums. (shrink)

Leading legal scholar John Witte, Jr. explores the role religion played in the development of rights in the Western legal tradition and traces the complex interplay between human rights and religious freedom norms in modern domestic and international law. He examines how US courts are moving towards greater religious freedom, while recent decisions of the pan-European courts in Strasbourg and Luxembourg have harmed new religious minorities and threatened old religious traditions in Europe. Witte argues that the robust promotion and (...) protection of religious freedom is the best way to protect many other fundamental rights today, even though religious freedom and other fundamental rights sometimes clash and need judicious balancing. He also responds to various modern critics who see human rights as a betrayal of Christianity and religious freedom as a betrayal of human rights. (shrink)

As initially envisioned, Gödel's Collected Works were to include transcriptions of material from his mathematical workbooks. In the end that material, as well as some other manuscript items from Gödel's Nachlass, had to be left out. This note describes some of the unpublished items in the Nachlass that are likely to attract the notice of scholars and surveys the extent of shorthand transcription efforts undertaken hitherto. Some examples of sources outside Gödel's Nachlass that may be of interest to Gödel scholars (...) are also indicated. (shrink)

Though regarded today as one of the most important results in logic, the compactness theorem was largely ignored until nearly two decades after its discovery. This paper describes the vicissitudes of its evolution and transformation during the period 1930-1970, with special attention to the roles of Kurt Gödel, A. I. Maltsev, Leon Henkin, Abraham Robinson, and Alfred Tarski.

In September of 2002, without fanfare, Ivor Grattan-Guinness retired from the faculty of Middlesex University. His scholarly activity, however, has continued unabated, his time now no longer fetter...

This article provides an English translation of a historic discussion on the foundations of mathematics, during which Kurt GÖdel first announced his incompleteness theorem to the mathematical world. The text of the discussion is preceded by brief background remarks and commentary.

The word “euthanasia” is hopelessly overloaded with emotional connotations. It means so many things to many different people. The implications of euthanasia associated with the Second World War have often rendered the term unsuitable for discussions of a rational manner. As far as I am concerned, what happened in Germany under Hitler had nothing to do with the classic meaning of a gentle and easy death but was rather simply a policy of mass murder.

Religious considerations in medical decision-making have enjoyed newfound attention in recent years, challenging the assumption that the domains of biological and spiritual flourishing can be cleanly separated in clinical practice. A surprising majority of patients desire their physicians to engage their religious and spiritual concerns, yet most never receive such attention, particularly in cases near the end of life where such attention seems most warranted.1–3 As physicians Aparna Sajja and Christina Puchalski recently wrote in the AMA Journal of Ethics theme (...) issue on ‘Religion and Spirituality in Healthcare Practice’, modern medical training is ‘falling short of preparing physicians to help patients with the metaphysical needs of their illness’.4 It would seem then that both patients and physicians might challenge the conclusions proffered by philosophers Jake Greenblum and Ryan Hubbard in ‘Responding to Religious Patients: Why Physicians Have No Business Doing Theology’, in which they claim that physicians ought not to engage religious reasoning for medical decisions. Their conclusion relies on two core arguments: a ‘public reason’ argument and a ‘fiduciary argument’, both of which contend that theology and religious considerations have no place in medical reasoning. We find that two central errors undergird these arguments. First is the fraught notion of an epistemically and ethically normative public reason, and second is a fundamental mischaracterisation of the fiduciary nature of the patient–physician relationship and the nature of medicine itself. In response, we argue that all physicians essentially ‘do theology’ and that, rather than divorcing theology from medical practice, clinicians should make it their business to do theology well. To support the exclusion of religious reasoning from medical decision-making, the authors open with a ‘public reason argument’, suggesting that physicians serve in roles closely resembling those of public officials. As such, the physician as public official ought to limit himself …. (shrink)

Legal Validity Legal validity governs the enforceability of law, and the standard of legal validity enhances or restricts the ability of the political ruler to enforce his will through legal coercion. Western law adopts three competing standards of legal validity. Each standard emphasizes a different dimension of law (Berman 1988, p. 779), and each has […].

Despite recent advances in HIV prevention and treatment, high HIV incidence persists among people who inject drugs. Difficult legal and political environments and lack of services for PWID likely contribute to high HIV incidence. Some advocates question whether any HIV prevention research is ethically justified in settings where healthcare system fails to provide basic services to PWID and where implementation of research findings is fraught with political barriers. Ethical challenges in research with PWID include concern about whether research evidence will (...) be translated into practice; concerns that research might exacerbate background risks; and ethical challenges regarding the standard of HIV prevention in research. While these questions arise in other research settings, for research with PWID, these questions are especially controversial. This paper analyses four ethical questions in determining whether research could be ethically acceptable: Can researchers ensure that research does not add to the burden of social harms and poor health experienced by PWID? Should research be conducted in settings where it is uncertain whether research findings will be translated into practice? When best practices in prevention and care are not locally available, what standard of care and prevention is ethically appropriate? Does the conduct of research in settings with oppressive policies constitute complicity? We outline specific criteria to address these four ethical challenges. We also urge researchers to join the call to action for policy change to provide proven safe and effective HIV prevention and harm reduction interventions for PWID around the world. (shrink)

According to several commentators, Kurt Godel's incompleteness discoveries were assimilated promptly and almost without objection by his contemporaries - - a circumstance remarkable enough to call for explanation. Careful examination reveals, however, that there were doubters and critics, as well as defenders and rival claimants to priority. In particular, the reactions of Carnap, Bernays, Zermelo, Post, Finsler, and Russell, among others, are considered in detail. Documentary sources include unpublished correspondence from Godel's Nachlass.

We develop a modified system of standard logic, Augmented Standard Logic (ASL), and we employ ASL in an effort to show that, contrary to prevailing opinion, both Aristotle and Diodorus presented impressive arguments, having valid structures and highly plausible premisses, in their famous fatalism debate. We argue that ASL, which contains standard logic and a full system of modal and temporal logic emanating from a modicum of primitives, should not only enable one to appreciate the sophisticated philosophizing which characterized this (...) ancient debate, but should prove to be quite useful in application to contemporary issues. (shrink)

If stem cell-based therapies are developed, we will likely confront a difficult problem of justice: for biological reasons alone, the new therapies might benefit only a limited range of patients. In fact, they might benefit primarily white Americans, thereby exacerbating long-standing differences in health and health care.

I attempt to describe and defend an alternative definition of insanity. I claim that my definition follows from an adequate general understanding of legal excuse and that it describes correctly the question that jurors in the recent Andrea Yates case and others like it ought to be faced with. My essay has three parts. In the first, I briefly criticize M'Naghten- and Durham-inspired insanity statutes. In the second, I sketch and defend a general understanding of legal excuse and try to (...) show how an adequate insanity statute follows from it. Finally, in the third section, I describe and defend a rough version of the statute itself. In the course of doing so, I try to explain why I believe that statute is superior to the American Law Institute's various model insanity statutes. (shrink)

We shall get to the crux of the issue if we ask, "What is it that reason is after, and hopes to achieve?" According to Blanshard, it is the grasp of necessity; it is the knowledge not simply of what is, but also and at the same time of what in the strictest sense must be. For reason seeks a final answer to the question, "Why?"; and only when one's knowledge is necessary is a further "Why?" found to be entirely (...) superfluous. Now perfect necessity may be grasped, thinks Blanshard, only in a context of total knowledge. The final goal of reason is thus a knowledge of the world which is perfectly necessary, and by virtue of this fact perfectly complete. (shrink)

What were seen widely in Hume's time as "the obvious dictates of reason" are rarely if ever seen as such today. One reads now that the table does not seem to diminish as one removes oneself from it; instead it appears roughly the same in size all the while. And, what if it did seem to diminish? This would not prove that the existent of which one is visually aware is diminishing, and is therefore but the image of the table (...) rather than the table itself. The appearance of an object is not another object; it is just the original object appearing. Quite simply, things may seem otherwise than they are. (shrink)