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)
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 Documentary sources include unpublished correspondence from Godel's Nachlass.
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 This note describes some of the unpublished items in the Nachlass that are likely to attract the notice of scholars and surveys the extent of shorthand transcription efforts undertaken hitherto.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.
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.
Summary Background Self-binding directives instruct clinicians to overrule treatment refusal during future severe episodes of illness. These directives are promoted as having potential to increase autonomy for individuals with severe episodic mental illness. Although lived experience is central to their creation, service users’ views on self-binding directives have not been investigated substantially. This study aimed to explore whether reasons for endorsement, ambivalence, or rejection given by service users with bipolar disorder can address concerns regarding self-binding directives, decision-making capacity, and human (...) rights. Methods This study used qualitative data from an internet-based survey distributed to the mailing list of the UK charity Bipolar UK, which contained multiple closed and open questions on advance decision-making in bipolar disorder. Quantitative analysis of a closed question about self-binding directives had already demonstrated endorsement amongst a very high proportion of participants with bipolar disorder who completed the survey. We conducted thematic analysis of responses from those participants who answered a subsequent open question about reasons for their view. Research was co-produced within a multi-disciplinary team, with clinical, legal, and ethical expertise, and lived experience of bipolar disorder. Ideas and methodologies associated with all these areas of expertise were used in the analysis of these reasons and to gain insight into the thoughts of individuals with bipolar disorder about self-binding directives and associated issues. Findings Between Oct 23 and Dec 5, 2017, 932 individuals with a self-reported clinical diagnosis of bipolar disorder completed the internet survey, with 565 (154 men; 400 women; 11 transgender or other) providing free text answers to the open question. A large majority of respondents endorsed self-binding directives, nearly all describing a determinate shift to types of distorted thinking and decision-making when unwell as their key justification. Responses indicating ambivalence were dominated by logistical concerns about the drafting and implementation of self-binding directives, while those who rejected self-binding directives also cited logistical concerns, validity of their thinking when unwell, and potential contravention of human rights. Interpretation This study is, to our knowledge, the first large study of reasons why mental health service users might endorse or reject the use of self-binding directives. The findings provide empirical support for introducing self-binding directives into mental health advance decision-making practice and policy and may help to address enduring ethical concerns surrounding possible implementation of the directive while a person retains decision-making capacity. The opinions expressed here in responses given by multiple service users with bipolar disorder challenge a prominent view within international disability rights debates that involuntary treatment and recognition of impaired mental capacity constitute inherent human rights violations. Funding The Wellcome Trust . (shrink)