The European birth of modern science: an exercise in macro and comparative history Content Type Journal Article Category Essay Review Pages 1-9 DOI 10.1007/s11016-012-9645-6 Authors John A. Schuster, Unit for History and Philosophy of Science and Sydney Centre for the Foundations of Science, University of Sydney, Sydney, NSW 2006, Australia Journal Metascience Online ISSN 1467-9981 Print ISSN 0815-0796.
There is growing interest in clinical ethics. However, we still have sparse knowledge about what is actually going on in the everyday practice of clinical ethics consultations. This paper introduces a descriptive evaluation tool to present, discuss and compare how clinical ethics case consultations are actually carried out. The tool does not aim to define ‘best practice’. Rather, it facilitates concrete comparisons and evaluative discussions of the role, function, procedures and ideals inherent in clinical ethics case consultation practices. The tool (...) was developed during meetings of the European Clinical Ethics Network. Based on written reports and participation in the network meetings, the development and the content of the tool and the results of its application in presenting and discussing 10 case consultations are summarized. The tool facilitated understanding of the details of clinical ethics case consultations across individuals and institutions with various experiences and cultures, and comparison between various practices. (shrink)
The most comprehensive collection of essays on Descartes' scientific writings ever published, this volume offers a detailed reassessment of Descartes' scientific work and its bearing on his philosophy. The 35 essays, written by some of the world's leading scholars, cover topics as diverse as optics, cosmology and medicine, and will be of vital interest to all historians of philosophy or science.
Working in the weakening of constructive Zermelo-Fraenkel set theory in which the subset collection scheme is omitted, we show that the binary refinement principle implies all the instances of the exponentiation axiom in which the basis is a discrete set. In particular binary refinement implies that the class of detachable subsets of a set form a set. Binary refinement was originally extracted from the fullness axiom, an equivalent of subset collection, as a principle that was sufficient to prove that the (...) Dedekind reals form a set. Here we show that the Cauchy reals also form a set. More generally, binary refinement ensures that one remains in the realm of sets when one starts from discrete sets and one applies the operations of exponentiation and binary product a finite number of times. (shrink)
In the early decades of the seventeenth century, various attempts were made to develop a dynamical vocabulary on the basis of work in the practical mathematical disciplines, particularly statics and hydrostatics. The paper contrasts the Mechanica and Archimedean approaches, and within the latter compares conceptions of statics and hydrostatics and their possible extensions in the work of Stevin, Beeckman and Descartes. Descartes’ approach to hydrostatics, a discussion of which forms the core of the paper, is shown to be quite different (...) from that of his contemporaries, above all in its attempt to provide a natural-philosophical grounding for hydrostatics while at the same time using it to develop a range of concepts, approaches and ways of thinking through problems that would shape Descartes’ mature work in optics and cosmology.Author Keywords: Scientific revolution; René Descartes; Isaac Beeckman; Simon Stevin; History of mechanics; History of statics; History of optics. (shrink)
The Dedekind cuts in an ordered set form a set in the sense of constructive Zermelo—Fraenkel set theory. We deduce this statement from the principle of refinement, which we distill before from the axiom of fullness. Together with exponentiation, refinement is equivalent to fullness. None of the defining properties of an ordering is needed, and only refinement for two—element coverings is used. In particular, the Dedekind reals form a set; whence we have also refined an earlier result by Aczel and (...) Rathjen, who invoked the full form of fullness. To further generalise this, we look at Richman's method to complete an arbitrary metric space without sequences, which he designed to avoid countable choice. The completion of a separable metric space turns out to be a set even if the original space is a proper class; in particular, every complete separable metric space automatically is a set. (shrink)
It is folklore that if a continuous function on a complete metric space has approximate roots and in a uniform manner at most one root, then it actually has a root, which of course is uniquely determined. Also in Bishop's constructive mathematics with countable choice, the general setting of the present note, there is a simple method to validate this heuristic principle. The unique solution even becomes a continuous function in the parameters by a mild modification of the uniqueness hypothesis. (...) Moreover, Brouwer's fan theorem for decidable bars turns out to be equivalent to the statement that, for uniformly continuous functions on a compact metric space, the crucial uniform “at most one” condition follows from its non-uniform counterpart. This classification in the spirit of the constructive reverse mathematics, as propagated by Ishihara and others, sharpens an earlier result obtained jointly with Berger and Bridges. (shrink)
How are the various classically equivalent definitions of compactness for metric spaces constructively interrelated? This question is addressed with Bishop-style constructive mathematics as the basic system – that is, the underlying logic is the intuitionistic one enriched with the principle of dependent choices. Besides surveying today's knowledge, the consequences and equivalents of several sequential notions of compactness are investigated. For instance, we establish the perhaps unexpected constructive implication that every sequentially compact separable metric space is totally bounded. As a by-product, (...) the fan theorem for detachable bars of the complete binary fan proves to be necessary for the unit interval possessing the Heine-Borel property for coverings by countably many possibly empty open balls. (shrink)
We extend the concept of apartness spaces to the concept of quasi-apartness spaces. We show that there is an adjunction between the category of quasi-apartness spaces and the category of neighbourhood spaces, which indicates that quasi-apartness is a more natural concept than apartness. We also show that there is an adjoint equivalence between the category of apartness spaces and the category of Grayson’s separated spaces.
Summary Descartes' two treatises of corpuscular-mechanical natural philosophy?Le Monde (1633) and the Principia philosophiae (1644/1647)?differ in many respects. Some historians of science have studied their significantly different theories of matter and elements. Others have routinely noted that the Principia cites much evidence regarding magnetism, sunspots, novae and variable stars which is absent from Le Monde. We argue that far from being unrelated or even opposed intellectual practices inside the Principles, Descartes' moves in matter and element theory and his adoption of (...) wide swathes of novel matters of fact, were two sides of the same coin?that coin being his strategies for improving the systematic power, scope and consistency of the natural philosophy presented in the Principia. We find that Descartes' systematising strategy centred upon weaving ranges of novel matters of fact into explanatory and descriptive narratives with cosmic sweep and radical realist Copernican intent. Gambits of this type have recently been labelled as ?cosmographical? (the natural philosophical relating of heavens and earth in contemporary usage). Realist Copernican natural philosophers, from Copernicus himself, through Bruno, Gilbert and Galileo did this to varying degrees; but, we suggest, Descartes presented in Books III and IV of the Principia the most elaborate and strategically planned version of it, underneath the ostensible textbook style of the work. (shrink)
The existence and uniqueness of a maximum point for a continuous real—valued function on a metric space are investigated constructively. In particular, it is shown, in the spirit of reverse mathematics, that a natural unique existence theorem is equivalent to the fan theorem.
A form of Kripke's schema turns out to be equivalent to each of the following two statements from metric topology: every open subspace of a separable metric space is separable; every open subset of a separable metric space is a countable union of open balls. Thus Kripke's schema serves as a point of reference for classifying theorems of classical mathematics within Bishop-style constructive reverse mathematics.
Dini's theorem says that compactness of the domain, a metric space, ensures the uniform convergence of every simply convergent monotone sequence of real-valued continuous functions whose limit is continuous. By showing that Dini's theorem is equivalent to Brouwer's fan theorem for detachable bars, we provide Dini's theorem with a classification in the recently established constructive reverse mathematics propagated by Ishihara. As a complement, Dini's theorem is proved to be equivalent to the analogue of the fan theorem, weak König's lemma, in (...) the original classical setting of reverse mathematics started by Friedman and Simpson. (shrink)
The topic of this article is the formal topology abstracted from the Zariski spectrum of a commutative ring. After recollecting the fundamental concepts of a basic open and a covering relation, we study some candidates for positivity. In particular, we present a coinductively generated positivity relation. We further show that, constructively, the formal Zariski topology cannot have enough points.
The theory of apartness spaces, and their relation to topological spaces (in the point–set case) and uniform spaces (in the set–set case), is sketched. New notions of local decomposability and regularity are investigated, and the latter is used to produce an example of a classically metrisable apartness on R that cannot be induced constructively by a uniform structure.
One of the chief concerns of the young Descartes was with what he, and others, termed “physico-mathematics”. This signalled a questioning of the Scholastic Aristotelian view of the mixed mathematical sciences as subordinate to natural philosophy, non explanatory, and merely instrumental. Somehow, the mixed mathematical disciplines were now to become intimately related to natural philosophical issues of matter and cause. That is, they were to become more ’physicalised’, more closely intertwined with natural philosophising, regardless of which species of natural philosophy (...) one advocated. A curious, short-lived yet portentous epistemological conceit lay at the core of Descartes’ physico-mathematics—the belief that solid geometrical results in the mixed mathematical sciences literally offered windows into the realm of natural philosophical causation—that in such cases one could literally “see the causes”. Optics took pride of place within Descartes’ physico-mathematics project, because he believed it offered unique possibilities for the successful vision of causes. This paper traces Descartes’ early physico-mathematical program in optics, its origins, pitfalls and its successes, which were crucial in providing Descartes resources for his later work in systematic natural philosophy. It explores how Descartes exploited his discovery of the law of refraction of light—an achievement well within the bounds of traditional mixed mathematical optics—in order to derive—in the manner of physico-mathematics—causal knowledge about light, and indeed insight about the principles of a “dynamics” that would provide the laws of corpuscular motion and tendency to motion in his natural philosophical system. (shrink)
We investigate how nonstandard reals can be established constructively as arbitrary infinite sequences of rationals, following the classical approach due to Schmieden and Laugwitz. In particular, a total standard part map into Richman's generalised Dedekind reals is constructed without countable choice.
We try to recast in modern terms a choice principle conceived by Beppo Levi. who called it the Approximation Principle (AP). Up to now. there was almost no discussion about Levi's contribution. due to the quite obscure formulation of AP the author has chosen. After briefly reviewing the historical and philosophical surroundings of Levi's proposal. we undertake our own attempt at interpreting AP. The idea underlying the principle. as well as the supposed faithfulness of our version to Levi's original intention. (...) are then discussed. Finally. an application of AP to a property of metric spaces is presented. with the aim of showing how AP may work in contexts where other forms of choice are commonly at use. (shrink)
This paper analyses the relationship between religion and the field of medicine and health care in light of other recent studies. Generally, religion and spirituality have a positive impact on disease. For patients diagnosed with malignancies and chronic diseases, religion is an important dimension of healing. From ancient times, God has been considered an inspiration for the physician's knowledge and healing resources. Some authors have proposed a brief history of spiritual and religious states that the doctor can apply to his (...) patient. Religiosity and spirituality allow patients to receive better social support and to benefit greatly from resources provided by religious organizations (cultural activities, jobs, and health care counseling). The two terms "religion" and "spirituality" have different meanings but are always in connection. Many studies emphasize that people with greater religiosity and spirituality have a lower prevalence of depression and suicide, better quality of life, and greater survival. Additionally the article discusses the complementary health care benefits of religious fasting. Caloric and protein restrictions promoted by religious fasting were associated with improvement in control or prophylaxis of many diseases and with longevity. (shrink)