This book presents reverse mathematics to a general mathematical audience for the first time. Reverse mathematics is a new field that answers some old questions. In the two thousand years that mathematicians have been deriving theorems from axioms, it has often been asked: which axioms are needed to prove a given theorem? Only in the last two hundred years have some of these questions been answered, and only in the last forty years has a systematic approach been developed. In Reverse (...) Mathematics, John Stillwell gives a representative view of this field, emphasizing basic analysis--finding the "right axioms" to prove fundamental theorems--and giving a novel approach to logic. Stillwell introduces reverse mathematics historically, describing the two developments that made reverse mathematics possible, both involving the idea of arithmetization. The first was the nineteenth-century project of arithmetizing analysis, which aimed to define all concepts of analysis in terms of natural numbers and sets of natural numbers. The second was the twentieth-century arithmetization of logic and computation. Thus arithmetic in some sense underlies analysis, logic, and computation. Reverse mathematics exploits this insight by viewing analysis as arithmetic extended by axioms about the existence of infinite sets. Remarkably, only a small number of axioms are needed for reverse mathematics, and, for each basic theorem of analysis, Stillwell finds the "right axiom" to prove it. By using a minimum of mathematical logic in a well-motivated way, Reverse Mathematics will engage advanced undergraduates and all mathematicians interested in the foundations of mathematics. (shrink)

How the concept of proof has enabled the creation of mathematical knowledge. The Story of Proof investigates the evolution of the concept of proof--one of the most significant and defining features of mathematical thought--through critical episodes in its history. From the Pythagorean theorem to modern times, and across all major mathematical disciplines, John Stillwell demonstrates that proof is a mathematically vital concept, inspiring innovation and playing a critical role in generating knowledge. Stillwell begins with Euclid and his (...) influence on the development of geometry and its methods of proof, followed by algebra, which began as a self-contained discipline but later came to rival geometry in its mathematical impact. In particular, the infinite processes of calculus were at first viewed as "infinitesimal algebra," and calculus became an arena for algebraic, computational proofs rather than axiomatic proofs in the style of Euclid. Stillwell proceeds to the areas of number theory, non-Euclidean geometry, topology, and logic, and peers into the deep chasm between natural number arithmetic and the real numbers. In its depths, Cantor, Gödel, Turing, and others found that the concept of proof is ultimately part of arithmetic. This startling fact imposes fundamental limits on what theorems can be proved and what problems can be solved. Shedding light on the workings of mathematics at its most fundamental levels, The Story of Proof offers a compelling new perspective on the field's power and progress. (shrink)

This paper explores different interpretations of the word ‘deep’ as it is used by mathematicians, with a large number of examples illustrating various criteria for depth. Most of the examples are theorems with ‘historical depth’, in the sense that many generations of mathematicians contributed to their proof. Some also have ‘foundational depth’, in the sense that they support large mathematical theories. Finally, concepts from mathematical logic suggest that it may be possible to order certain theorems or problems according to ‘logical (...) depth’. (shrink)

This Element aims to present an outline of mathematics and its history, with particular emphasis on events that shook up its philosophy. It ranges from the discovery of irrational numbers in ancient Greece to the nineteenth- and twentieth-century discoveries on the nature of infinity and proof. Recurring themes are intuition and logic, meaning and existence, and the discrete and the continuous. These themes have evolved under the influence of new mathematical discoveries and the story of their evolution is, to a (...) large extent, the story of philosophy of mathematics. (shrink)

Hilbert took to using ideal elements in the 1890's, in both algebraic number theory and geometry. His Zahlbericht of 1897 popularized the concept of the ideal introduced by Dedekind in 1871 (which in turn formalized the concept of "ideal number" introduced by Kummer in the 1840's). His geometric work likewise followed a long history of ideal elements, some that originated in geometry and others that originated elsewhere and were applied to geometry. Important examples were:Piero della Francesca's Ideal City.Points at infinity (...) in projective geometry, originating in the 1430s from the "vanishing points" in perspective drawing.Imaginary points in algebraic geometry, originating from the imaginary numbers used by .. (shrink)

StillwellJohn.* * _ A Concise History of Mathematics for Philosophers. _ Cambridge Elements in the Philosophy of Mathematics. Cambridge University Press, 2019. Pp. 69. ISBN: 978-1-108-45623-4, 978-1-108-61012-4. doi.org/10.1017/9781108610124.

Review of John Stillwell, Reverse Mathematics: Proofs from the Inside Out. Princeton, NJ: Princeton University Press, 2018, pp. 200. ISBN 978-0-69-117717-5 (hbk), 978-0-69-119641-1 (pbk), 978-1-40-088903-7 (e-book).

We will next speak of Liberality. Now this is thought to be the mean state, having for its object-matter Wealth: I mean, the Liberal man is praised not in the circumstances of war, nor in those which constitute the character of perfected self-mastery, nor again in judicial decisions, but in respect of giving and receiving Wealth, chiefly the former. By the term Wealth I mean all those things whose worth is measured by money.

Loneliness in medicine is a serious problem not just for patients, for whom illness is intrinsically isolating, but also for physicians in the contemporary condition of medicine. We explore this problem by investigating the ideal physician-patient relationship, whose analogy with friendship has held enduring normative appeal. Drawing from Talbot Brewer and Nir Ben-Moshe, we argue that this appeal lies in a dynamic form of companionship incompatible with static models of friendship-like physician-patient relationships: a mutual refinement of embodied virtue that draws (...) both persons together. The ideal physician-patient relationship has a dialectical character that fosters each member's improvement of phenomenologically recognizing and embodying moral virtues. A key component of this dynamic is a commitment to the common goal of the patient's health, realized through joint interactivities and conversations over time. The physician's presence to the patient's suffering—understood best as an alienating phenomenological condition for the patient—orients and discloses possibilities for virtuous caregiving by structuring the meanings of the goals, conversations, and joint narrative constitutive of their relationship. Presence to suffering, paradoxically, is perhaps an important prerequisite for this dynamic partnership. These activities dialectically build an interpretive horizon of understanding through which moral goods and character refinement—in and for the other—may become revealed for both persons in their shared being-in-the-world. This analysis of suffering, mood, and revealing of (possible) moral goods has implications for addressing the modern problem of loneliness for patients and physicians, who are increasingly inhibited from building flourishing relationships with each other. (shrink)

Many real-world agents recognise that they impose harms by choosing to emit carbon, e.g., by flying. Yet many do so anyway, and then attempt to make things right by offsetting those harms. Such offsetters typically believe that, by offsetting, they change the deontic status of their behaviour, making an otherwise impermissible action permissible. Do they succeed in practice? Some philosophers have argued that they do, since their offsets appear to reverse the adverse effects of their emissions. But we show that (...) they do not. In practice, standard carbon offsetting does not reverse the harms of the original action, nor does it even benefit the same group as was harmed. Standard moral theories hence deny that such offsetting succeeds. Indeed, we show that any moral theory that allows offsetting in this setting faces a dilemma between allowing any wrong to be offset, no matter how grievous, and recognising an implausibly sharp discontinuity between offsettable actions and non-offsettable actions. The most plausible response is to accept that carbon offsetting fails to right our climate wrongs. (shrink)

My critics in this symposium illustrate one principle and three fallacies of disability studies. The principle, which we all share, is that all persons are equal and none are less equal than others. No disability, however slight, nor however severe, implies lesser moral, political or ethical status, worth or value. This is a version of the principle of equality. The three fallacies exhibited by some or all of my critics are the following: Choosing to repair damage or dysfunction or to (...) enhance function, implies either that the previous state is intolerable or that the person in that state is of lesser value or indicates that the individual in that state has a life that is not worthwhile or not thoroughly worth living. None of these implications hold. Exercising choice in reproduction with the aim of producing children who will be either less damaged or diseased, or more healthy, or who will have enhanced capacities, violates the principle or equality. It does not. Disability or impairment must be defined relative either to normalcy, “normal species functioning”, or “species typical functioning”. It is not necessarily so defined. (shrink)

This paper discusses the role of consent in decision making generally and its role in end of life decisions in particular. It outlines a conception of autonomy which explains and justifies the role of consent in decision making and criticises some misapplications of the idea of consent, particular the role of fictitious or “proxy” consents.Where the inevitable outcome of a decision must be that a human individual will die and where that individual is a person who can consent, then that (...) decision is ethical if and only if the individual consents. In very rare and extreme cases such a decision will be ethical in the absence of consent where it would be massively cruel not to end life in order to prevent suffering which is in no other way preventable.Where, however, the human individual is not a person, as is the case with abortion, the death of infants like Mary , or in the very rare and extreme cases of those who have ceased to be persons like Tony Bland, such decisions are governed by the ethics of ending the lives of non-persons. (shrink)

There are already many excellent books on existentialism. Some of them deal with particular problem or particular existentialist writers. Most of those that deal with existentialism as a whole divide their subject-matter according to authors, presenting chapters on Kierkegaard, Heidegger, Sartre, and the rest. Thus I think that there is room for the present book, which attempts a comprehensive examination and evaluation of existentialism, but does so by thematic treatment. That is to say, each chapter deals with a major theme (...) of existentialist philosophy, and these themes are arranged in the order of the existential dialect. Of course, each chapter is illustrated with material from the writings of existentialists, from Kierkegaard to Camus. (shrink)

Cadaver organs should be automatically availableThe shortage of donor organs and tissue for transplantation constitutes an acute emergency which demands radical rethinking of our policies and radical measures. While estimates vary and are difficult to arrive at there is no doubt that the donor organ shortage costs literally hundreds of thousands of lives every year. “In the world as a whole there are an estimated 700 000 patients on dialysis . . .. In India alone 100 000 new patients present (...) with kidney failure each year” . Almost “three million Americans suffer from congestive heart failure . . . deaths related to this condition are estimated at 250 000 each year . . . 27 000 patients die annually from liver disease . . .. In Western Europe as a whole 40 000 patients await a kidney but only . . . 10 000 kidneys”1 become available. Nobody knows how many people fail to make it onto the waiting lists and fail to register in the statistics. “As of 24th November 2002 in the United Kingdom 667 people have donated organs, 2055 people have received transplants, and 5615 people are still awaiting transplants.”2Conscious of the terrible and unnecessary tragedy that figures like these represent I have been advocating for more than 20 years now some radical measures to stem this appalling waste of human life. The measure which is the subject of Hamer and Rivlin’s paper 3 concerns the automatic availability of all cadaver organs—a measure, which I first advocated publicly in 1983.4THE AUTOMATIC AVAILABILITY OF DONOR ORGANSWe need to begin by being clear about just what it is I propose and why. At the moment in the United Kingdom we …. (shrink)

Analysing film. Distinguishing the filmic contribution to meaning -- Examples of filmic "textual organisation" -- Redrawing boundaries -- Organisation of the book -- Semiotics and documents. Semiotics and its relations to film -- The nature of discourse semantics -- The film as cinematographic document -- A combined view: filmic documents for filmic discourse -- Constructing the semiotic mode of film. Semiotic multimodality -- The internal organisation of semiotic strata -- Composing and combining semiotic modes -- Materiality and "epistemological commitment" -- (...) Christian Metz and the grande syntagmatique of the image track. The original model -- Two examples of analysis with the grande syntagmatique -- Revisions and rebuttals -- Foundations for analysis: filmic units. The basic units of film: preliminaries -- Audiovisual iconic representations -- Perception, perceptual realism and reliable measurement -- Multiplicity: from perception to discourse -- Filmic units revisited: discourse-motivated definitions -- The paradigmatic organisation of film. Beyond Metz: towards a grande paradigmatique -- Capturing discourse dependency structures in film -- The paradigmatic dimensions of projection, taxis and plane -- Two examples of paradigmatic analysis -- Summary and conclusions. (shrink)

The reviews of this book which greeted its appearance in America, where it won a Catholic Press Association Religious Book Award, speak for themselves. 'The real core of the book is the question that is raised - the demanding bone-crushing question we all face - alone - at one time - the question of death/life and immortality. In these few pages we set out on a journey - one that winds its way among ancient stories and myths ... one's constant (...) disturbing companion is death - a reality that keeps breaking in upon our imes, our friends, our dreams and hopes and even our changing and becoming. The richness of the book lies in the questioning and some of the answers other men's times have given in attempt to live with this probing. We meet Kazantzakis, Homer, Melville, Hegel, Dante in their humanness in the face of death and in their timeless answers' (National Catholic Reporter). From the Protestant side, Martin Marty commented how John Dunne, 'perhaps the most intuitive and natively talented Catholic around, has cast a new spell', and The Critic remarked: 'As a meditation ought to be, it is simple and profound at once. And it is beautifully done, in some ways the richest of John Dunne's works.' All of which amounts to an invitation to join the author as he poses the timeless dilemma of the human condition and seeks to resolve it through stories of adventures, journeys and voyages inspired by man's encounter with death; stories of childhood, youth, manhood and age; and finally stories of God and of man wrestling with God and the unknown. (shrink)

John Dillon presents an English translation of Alcinous' Handbook of Platonism, accompanied by an introduction and a philosophical commentary which explain the ideas in the work and show their intellectual and historical context. The Handbook purports to be an introduction to the doctrines of Plato, but in fact gives us an excellent survey of Platonist thought in the second century AD.

This brief paper shows that the idea of benefits to the subject compensating for the harms of disability is at best self defeating and at worst sinister. Equally benefits to third parties while real are dubious as compensating factors. This shows that disabilities are just that, a net loss and not a net gain.

Herodotus. Custom is king.--Engels, F. Ethics and law: eternal truths.--Sumner, W. G. Folkways.--Ross, W. D. The meaning of right.--Duncker, K. Ethical relativity?--Herskovits, M. J. Cultural relativism and cultural values.--Kluckhohn, C. Ethical relativity: sic et non.--Taylor, P. W. Social science and ethical relativism.--Ladd, J. The issue of relativism.--Redfield, R. The universally human and the culturally variable.--Bibliography (p. 145-146).

This book provides an intellectual biography of Max Horkheimer during the early and middle phases of his life and analyzes his model of early Critical Theory.

Utilizing a contractualist framework for understanding the basis and limits for the use of force by police, this article offers five limiting principles—respect for status as moral agents, proportionality, minimum force necessary, ends likely to be accomplished, and appropriate motivation—and then discusses uses of force that violate or risk violating those principles. These include, but are not limited to, unseemly invasions, strip searches, perp walks, handcuffing practices, post-chase apprehensions, contempt-of-cop arrests, overuse of intermediate force measures, coerced confessions, profiling, stop and (...) frisk practices, and the administration of street justice. (shrink)

When things go wrong with assisted reproduction we should look at what’s best for everyone in the particular circumstancesA RTBs, as we must now call them, are becoming more and more frequent. In the recent United Kingdom case Mr and Mrs A, a “white” couple, gave birth to twins described as “black”. The mix up apparently occurred because a Mr and Mrs B, a “black” couple, were being treated in the same clinic and Mrs A’s eggs were fertilised with Mr (...) B’s sperm. Mr and Mrs A love the twins and wish to keep them and the facts seem to be that Mrs A is the genetic mother of the twins but her partner is not the genetic father. Under English law, in assisted reproduction, the woman who gives birth to the child is the legal mother and this is true also in cases of surrogacy. I think there is no doubt in this case that English law has taken the right path and that it would be unconscionable to contemplate taking a child away from the woman who had undergone the risks and pains of pregnancy and childbirth and the bonding process that goes along with that when she wishes to keep the child particularly when there is no substantial and pressing evidence that she would be an unsafe parent. I argued this in my book The Value of Life4 and I think it is still true. So in this case justice, morality, and decency are served by confirming Mr and Mrs A as the parents of the children. Since in English law there is some ambiguity over paternity and how that is to be assigned and since Mr A is apparently not the genetic father, I think it is important …. (shrink)

Bu yazı organ nakli ile ilgili şu soruyu ele almaktadır: Doktorlar sağlıklı bir kişinin organlarını alarak organ nakline gereksinim duyan birden fazla kişinin yaşamını kurtarırlarsa ahlaksal açıdan yanlış bir şey yapmış olurlar mı? Her ne kadar sağlıklı bir insanın organlarını alarak onun ölümüne neden olmak kabul edilemez görünse de bunu yapmamanın daha çok sayıda kişinin ölümüne neden olacağı düşünülürse ortada araştırılmaya değer bir soru olduğu görülür.

Open‐ended tasks can be decomposed into the three levels of Newell's Cognitive Band: the Unit‐Task level, the Operation level, and the Deliberate‐Act level. We analyzed the video game Co‐op Space Fortress at these levels, reporting both the match of a cognitive model to subject behavior and the use of electroencephalogram (EEG) to track subject cognition. The Unit Task level in this game involves coordinating with a partner to kill a fortress. At this highest level of the Cognitive Band, there is (...) a good match between subject behavior and the model. The EEG signals were also strong enough to track when Unit Tasks succeeded or failed. The intermediate Operation level in this task involves legs of flight to achieve a kill. The EEG signals associated with these operations are much weaker than the signals associated with the Unit Tasks. Still, it was possible to reconstruct subject play with much better than chance success. There were significant differences in the leg behavior of subjects and models. Model behavior did not provide a good basis for interpreting a subject's behavior at this level. At the lowest Deliberate‐Act level, we observed overlapping key actions, which the model did not display. Such overlapping key actions also frustrated efforts to identify EEG signals of motor actions. We conclude that the Unit‐task level is the appropriate level both for understanding open‐ended tasks and for using EEG to track the performance of open‐ended tasks. (shrink)

What are the purposes of education and what is the relationship between educational research and policy? Using the twin lenses of Visible Learning and educational philosophy these are among the many fascinating topics discussed in extended conversations between John Hattie and Steen Nepper Larsen. This wide-ranging, and informative book offers fundamental propositions about the nature of Education. It maps out in fascinating detail a coming together of Hattie's empirical data and world-famous Visible Learning paradigm with the rich heritage of (...) educational philosophy. As well as exploring the inevitable questions of the purpose of education and the development of students in a learning society. Part clash of cultures, part meeting of minds, always fascinating and illuminating, this intriguing book will inspire teachers, students and parents at all levels of the educational system - from kindergarten through school to university. Conversations include: What is the purpose of education? Does educational data speak for itself? What is the role of the teacher? Is learning a visible phenomenon? Is it important to teach and learn specific subjects? What is the role of Neuroscience research? What is the relationship between educational research and educational politics? What is the role of the state in education? (shrink)

Philosophy and Contemporary Issues contains contemporary readings of philosophical significance designed to introduce philosophy to college and university students. The authors' goal is to demonstrate how philosophy illuminates and helps solve some of the important problems facing contemporary man. This edition contains additional readings.

This paper discusses the provocative views of Skene and Parker as to the role of religious or other ideologically based interest groups in law and policy making. We draw distinctions between doctrine and prejudice and between argument and ideology which we trust take the debate further. Finally we recommend an ethereal, democratic, and populist partial solution.

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work.This work is in the public domain in (...) the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work.As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

In the World Library of Educationalists series, international experts themselves compile career- long collections of what they judge to be their finest pieces-extracts from books, key articles, salient research findings, major theoretical and/practical contributions-so the world can read them in a single manageable volume. Readers will be able to follow the themes and strands of their work and see their contribution to the development of a field. Emeritus Professor John White has spent the last 35 years researching, thinking and (...) writing about some of the key and enduring issues in education. In this book, he brings together 16 key writings in one place. Starting with a specially written Introduction, which gives an overview of John's career and conceptualizes his selection within the development of the field, the chapters cover: · Mind · State and Curriculum · Well-being · Politics · Curriculum subjects. This book not only shows how John's thinking developed during his long and distinguished career; it also gives an insight into the development of the fields to which he contributed. (shrink)