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 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)

I present a new interpretation of Wittgenstein's later philosophy of logic and mathematics. This interpretation, like others, emphasizes Wittgenstein's attempt to reconcile platonistic and constructivistic approaches. But, unlike other interpretations, mine explains that attempt in terms of Wittgenstein's position about the relations between our concepts of necessity and provability. If what I say here is correct, then we can rescue Wittgenstein from the charge of naive relativism. For his relativism extends only to provability, and not to necessity.

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)

The following three related contributions jointly serve to lift up elements of the thought of the anthropolo­gist Gregory Bateson that can be fruitfully compared with elements of Michael Polanyi’s thought. In a brief introduction, William Stillwell reviews Bateson’s life and developing interests. Stillwell also provides, in a creative dialog form akin to Bateson’s own dialogs, a short review article on Noel Charlton’s Understanding Gregory Bateson: Mind, Beauty and the Sacred Earth. The third piece is Jere Moorman’s short 1991 (...) essay (now out of print) discussing Polanyi’s ideas about tacit knowing and their connection with Bateson’s ideas about the double bind. (shrink)

"Williams's masterful translation satisfies (at last!) a long-standing need. There are lots of good translations of Augustine's great work, but until now we have been forced to choose between those that strive to replicate in English something of the majesty and beauty of Augustine's Latin style and those that opt instead to convey the careful precision of his philosophical terminology and argumentation. Finally, Williams has succeeded in capturing both sides of Augustine's mind in a richly evocative, impeccably reliable, elegantly readable (...) presentation of one of the most impressive achievements in Western thought--Augustine's Confessions." --Scott MacDonald, Professor of Philosophy and Norma K. Regan Professor in Christian Studies, Cornell University. (shrink)

Ikujiro Nonaka, whose formative experience is Japanese, is an established scholar who has written about large business organizations. He sees knowledge at the heart of the organization and its products and aims to develop Michael Polanyi’s conception of tacit knowledge in a practical direction to enhance organizational “knowledge creation.” For Nonaka, what matters is the practice, the doing, the embodiment of knowledge. An organization can amplify and crystallize individuals’ tacit knowledge in a process that allows them to experience deeper understanding. (...) Nonaka holds that it is iimportant to explore the potential that knowledge holds. His spiral process describes disciplined practices that make tacit knowledge independent and available to restructure the organizational knowledge context. (shrink)

The Moral Nexus develops and defends a new interpretation of morality—namely, as a set of requirements that connect agents normatively to other persons in a nexus of moral relations. According to this relational interpretation, moral demands are directed to other individuals, who have claims that the agent comply with these demands. Interpersonal morality, so conceived, is the domain of what we owe to each other, insofar as we are each persons with equal moral standing. The book offers an interpretative argument (...) for the relational approach. Specifically, it highlights neglected advantages of this way of understanding the moral domain; explores important theoretical and practical presuppositions of relational moral duties; and considers the normative implications of understanding morality in relational terms. The book features a novel defense of the relational approach to morality, which emphasizes the special significance that moral requirements have, both for agents who are deliberating about what to do and for those who stand to be affected by their actions. The book argues that relational moral requirements can be understood to link us to all individuals whose interests render them vulnerable to our agency, regardless of whether they stand in any prior relationship to us. It also offers fresh accounts of some of the moral phenomena that have seemed to resist treatment in relational terms, showing that the relational interpretation is a viable framework for understanding our specific moral obligations to other people. (shrink)

This book provides a collection of sources, many of them fragmentary and previously scattered and hard to access, for the development of Peripatetic philosophy in the later Hellenistic period and the early Roman Empire. It also supplies the background against which the first commentator on Aristotle from whom extensive material survives, Alexander of Aphrodisias (fl. c. AD 200), developed his interpretations which continue to be influential even today. Many of the passages are here translated into English for the first time, (...) including the whole of the summary of Peripatetic ethics attributed to 'Arius Didymus'. (shrink)

R.I.G. Hughes presents a series of eight philosophical essays on the theoretical practices of physics. The first two essays examine these practices as they appear in physicists' treatises (e.g. Newton's Principia and Opticks ) and journal articles (by Einstein, Bohm and Pines, Aharonov and Bohm). By treating these publications as texts, Hughes casts the philosopher of science in the role of critic. This premise guides the following 6 essays which deal with various concerns of philosophy of physics such as laws, (...) disunities, models and representation, computer simulation, explanation, and the discourse of physics. (shrink)

We should approach this topic with great fear and respect. It is not a merely “academic” issue. Human rights are trampled upon in the East as in the West, in the North as in the South of our planet. Granting the part of human greed and sheer evil in this universal transgression, could it not also be that Human Rights are not observed because in their present form they do not represent a universal symbol powerful enough to elicit understanding and (...) agreement? (shrink)

Chemical structures are among the trademarks of our profession, as surely chemical as flasks, beakers and distillation columns. When someone sees one of us busily scribbling formulas or structures, he or she has no trouble identifying a chemist. Yet these familiar objects, which accompany our work from start to end, from the initial doodlings (Fig. I) to the final polished artwork in a publication (Fig. II), are deceptively simple. They raise interesting and difficult questions about representation. It is the intent (...) of this article to reflect upon molecular graphics. (shrink)

Ethics and Decision Making in Counseling and Psychotherapy has a distinct and timely focus on counseling as a profession. Chapters address the mental health professions, values in counseling, decision making, ethical principles, ethical standards, technology, ethical climate, and office/administrative practices. The early chapters present a foundation for ethical practice of the profession and provides solid building blocks to the more advanced perspectives in later chapters. Chapters on specialty practice are lively and contemporary overviews of these practice areas in counseling that (...) reflect more recent trends to increased differentiation of these areas of practice. Each specialty has its own chapter to allow for an extended discussion of the unique aspects of the field that provide clarity to understanding the particular ethical issues and standards that apply in each specialty. The specialty areas recognized by CACREP are included: mental health counseling; school counseling; couple, marital, and family counseling; rehabilitation counseling; addictions counseling; career counseling; and group counseling. This book is broad in its coverage of the most practiced specialties in mental health practice, and provides an efficient and effective overview of the broad scope of particular areas addressed in counseling. (shrink)

How do we explain violence? What is so significant of modern forms of violence that it has produced such large-scale destruction in its wake? This volume builds on the political philosophy of Wittgenstein, his notions of peace and violence, to explore how violence in any form is contained within culturally or ideologically formed institutions. Drawing on Wittgenstein's work on language, it explores the link between language and violence, everydayness, culture. It examines everyday instances of micro-violence which we sometimes forget to (...) recall. This book puts forth the claim that any theory of violence will have to touch upon the many myriad - both micro and macro -- political, social and cultural interactions that make up the human condition. The author further comments on the unseen ways in which violence has been instrumentalized in modern history's many stages to create a spectacle of power so as to reinforce authority. The volume will be of great interest to students and scholars of peace and conflict studies, political philosophy, linguistics and modern history. (shrink)

Aesthetics in philosophy of mathematics is too narrowly construed. Beauty is not the only feature in mathematics that is arguably aesthetic. While not the highest aesthetic value, being interesting is a sine qua non for publishability. Of the many ways to be interesting, being explanatory has recently been discussed. The motivational power of what is interesting is important for both directing research and stimulating education. The scientific satisfaction of curiosity and the artistic desire for beautiful results are complementary but both (...) aesthetic. (shrink)

Karl Raimund Poppers (1902-1994) Hauptwerk, die Logik der Forschung (1934), gilt als Grundlagenwerk des kritischen Rationalismus. Der kritische Rationalismus zeigt, warum unser Wissen fehlbar ist und versteht den Erkenntnisfortschritt als Resultat von Hypothesenbildung und -widerlegung. Der Sammelband orientiert sich an der Gliederung der Logik der Forschung. Seine Beiträge kommentieren die jeweiligen Themen nach aktueller Forschungslage.

It is tempting to think that all of normativity, such as our reasons for action, what we ought to do, and the attitudes that it is fitting for us to have, derives from what is valuable. But value-first approaches to normativity have fallen out of favour as the virtues of reasons- and fittingness-first approaches to normativity have become clear. On these views, value is not explanatorily prior to reasons and fit; rather the value of things is understood in terms of (...) the pro-attitudes it is fitting to have, or that we have reasons to have, towards these things. The value-first accounts of reasons and fit that have been proposed have been direct accounts of reasons and fit. On these views, the fittingness of an attitude is explained in terms of the value of having it or the value of the object of that attitude. This chapter details and surveys these problems, explaining which are yet to be overcome, and explaining which problems direct value-first accounts can overcome. This chapter then explores the prospects of an indirect value-first account of reasons and fit. It argues that it avoids the problems that direct value-first accounts face and may be as explanatorily powerful as fittingness- and reasons-first accounts of value and normativity more generally. (shrink)

Is the essay literature or philosophy? A form of art or a form of knowledge? The contemporary essay is torn between its belletrist ancestry and its claim to philosophical legitimacy. The Spanish philosopher Eduardo Nicol captured the genre's uncertain status when he dubbed it “almost literature and almost philosophy” (Nicol 1961:207). The problem is hardly a new one. It goes back to what Plato called the “ancient quarrel” between poetry and philosophy, and more recently to the German Romantic theorist, Friedrich (...) Schlegel, who called for a mode of criticism which would be at once philosophical and poetic. But today, when the status of critical discourse is up for grabs, reflecting the crisis of knowledge in the universities, the question of the essay takes on a new urgency. Now the predominant form of writing in the human sciences, it cannot avoid the challenge to define itself according to the prevailing standards of scientific knowledge and method. (shrink)

The book is written for anyone seriously interested in Plato's thought and in the history of literary theory or of rhetoric. No knowledge of Greek is required. The focus of this account is on how the resources both of persuasive myth and of formal argument, for all that Plato sets them in strong contrast, nevertheless complement and reinforce each other in his philosophy.