In the present article two possible meanings of the term mathematical structure are discussed: a formal and a nonformal one. It is claimed that contemporary mathematics is structural only in the nonformal sense of the term. Bourbaki's definition of structure is presented as one among several attempts to elucidate the meaning of that nonformal idea by developing a formal theory which allegedly accounts for it. It is shown that Bourbaki's concept of structure was, from a mathematical point of view, a (...) superfluous undertaking. This is done by analyzing the role played by the concept, in the first place, within Bourbaki's own mathematical output. Likewise, the interaction between Bourbaki's work and the first stages of category theory is analyzed, on the basis of both published texts and personal documents. (shrink)

Leo Corry tells the story behind the idea of number, from the early days of the Pythagoreans, up until the turn of the twentieth century. He presents an overview of how numbers were handled and conceived in classical Greek mathematics, in the mathematics of Islam, in European mathematics of the middle ages and the Renaissance, during the scientific revolution, all the way through to the mathematics of the 18th to the early 20th century.

The ArgumentRecent studies in the philosophy of mathematics have increasingly stressed the social and historical dimensions of mathematical practice. Although this new emphasis has fathered interesting new perspectives, it has also blurred the distinction between mathematics and other scientific fields. This distinction can be clarified by examining the special interaction of thebodyandimagesof mathematics.Mathematics has an objective, ever-expanding hard core, the growth of which is conditioned by socially and historically determined images of mathematics. Mathematics also has reflexive capacities unlike those of (...) any other exact science. In no other exact science can the standard methodological framework usedwithinthe discipline also be used to study the nature of the discipline itself.Although it has always been present in mathematical research, reflexive thinking has become increasingly central to mathematics over the past century. Many of the images of the discipline have been dictated by the increase in reflexive thinking which has also determined a great portion of the contemporary philosophy and historiography of mathematics. (shrink)

The ArgumentThe belief in the existence of eternal mathematical truth has been part of this science throughout history. Bourbaki, however, introduced an interesting, and rather innovative twist to it, beginning in the mid-1930s. This group of mathematicians advanced the view that mathematics is a science dealing with structures, and that it attains its results through a systematic application of the modern axiomatic method. Like many other mathematicians, past and contemporary, Bourbaki understood the historical development of mathematics as a series of (...) necessary stages inexorably leading to its current state — meaning by this, the specific perspective that Bourbaki had adopted and were promoting. But unlike anyone else, Bourbaki actively put forward the view that their conception of mathematics was not only illuminating and useful for dealing with the current concerns of mathematics, but that this was in fact the ultimate stage in the evolution of mathematics, bound to remain unchanged by any future development of this science. In this way, they were extending in an unprecedented way the domain of validity of the belief in the eternal character of mathematical truths, from the body to the images of mathematical knowledge.Bourbaki were fond of presenting their insistence on the centrality of the modern axiomatic method as a way to ensure the eternal character of mathematical truth as an offshoot of Hilbert's mathematical heritage. A detailed examination of Hilbert's actual conception of the axiomatic method, however, brings to the fore interesting differences between it and Bourbaki's conception, thus underscoring the historically conditioned character of certain, fundamental mathematical beliefs. (shrink)

This article explores the changing relationships between geometric and arithmetic ideas in medieval Europe mathematics, as reflected via the propositions of Book II of Euclid’s Elements. Of particular interest is the way in which some medieval treatises organically incorporated into the body of arithmetic results that were formulated in Book II and originally conceived in a purely geometric context. Eventually, in the Campanus version of the Elements these results were reincorporated into the arithmetic books of the Euclidean treatise. Thus, while (...) most of the Latin versions of the Elements had duly preserved the purely geometric spirit of Euclid’s original, the specific text that played the most prominent role in the initial passage of the Elements from manuscript to print—i.e., Campanus’ version—followed a different approach. On the one hand, Book II itself continued to appear there as a purely geometric text. On the other hand, the first ten results of Book II could now be seen also as possiblytranslatable into arithmetic, and in many cases even as inseparably associated with their arithmetic representation. (shrink)

The present article discusses the computational tools (both conceptual and material) used in various attempts to deal with individual cases of FLT, as well as the changing historical contexts in which these tools were developed and used, and affected research. It also explores the changing conceptions about the role of computations within the overall disciplinary picture of number theory, how they influenced research on the theorem, and the kinds of general insights thus achieved. After an overview of Kummer’s contributions and (...) its immediate influence, I present work that favored intensive computations of particular cases of FLT as a legitimate, fruitful, and worth-pursuing number-theoretical endeavor, and that were part of a coherent and active, but essentially low-profile tradition within nineteenth century number theory. This work was related to table making activity that was encouraged by institutions and individuals whose motivations came mainly from applied mathematics, astronomy, and engineering, and seldom from number theory proper. A main section of the article is devoted to the fruitful collaboration between Harry S. Vandiver and Emma and Dick Lehmer. I show how their early work led to the hesitant introduction of electronic computers for research related with FLT. Their joint work became a milestone for computer-assisted activity in number theory at large. (shrink)

We mourn the loss of Yehuda Elkana, founding editor of this journal. Setting science in context was a mission of his life. For him this did not mean to relativize and historicize science to the point where it is no longer distinguishable as central to the human quest for knowledge. Rather, an understanding of science as being rooted in social, material, and cultural contexts was for him the key to its central role for solving the problems of humanity with which (...) he was so deeply concerned. (shrink)

The present issue of Science in Context comprises a collection of articles dealing with various, specific aspects of the history of mathematics during the last third of the nineteenth century and the first half of the twentieth. Like the September issue of 2003 of this journal, which was devoted to the history of ancient mathematics, this collection originated in the aftermath of a meeting held in Tel-Aviv and Jerusalem in May 2001, under the title: “History of Mathematics in the Last (...) 25 Years: New Departures, New Questions, New Ideas.” Taken together, these two topical issues are meant as a token of appreciation for the work of Sabetai Unguru and his achievements in the history of mathematics. (shrink)

ArgumentThis article gives the background to a public lecture delivered in Hebrew by Edmund Landau at the opening ceremony of the Hebrew University in Jerusalem in 1925. On the surface, the lecture appears to be a slightly awkward attempt by a distinguished German-Jewish mathematician to popularize a few number-theoretical tidbits. However, quite unexpectedly, what emerges here is Landau's personal blend of Zionism, German nationalism, and the proud ethos of pure, rigorous mathematics – against the backdrop of the situation of Germany (...) after World War I. Landau's Jerusalem lecture thus shows how the Zionist cause was inextricably linked to, and determined by political agendas that were taking place in Europe at that time. The lecture stands in various historical contexts - Landau's biography, the history of Jewish scientists in the German Zionist movement, the founding of the Hebrew University in Jerusalem, and the creation of a modern Hebrew mathematical language. This article provides a broad historical introduction to the English translation, with commentary, of the original Hebrew text. (shrink)

The history of Israeli science and technology offers a unique case study of a young and small nation that has developed an unprecedented love affair with science and technology. Unlike other nineteenth-century ideologies, Zionism was never considered to be founded on science. Nevertheless, from the very start, the Zionist movement perceived the sciences, pure and applied, as central to its program of creating a new Jewish society in the Land of Israel. Modern science was to provide twice for the Jews: (...) a relief from their suffocating religion and the tools needed to recover their ancient land from its ruins. Israel would remain the people of the book, but it would be the Book of Nature, not of God, that would set it free. Sharing the universal knowledge and values of science with mankind, the Jews would finally become both normal and self-determined. Thus, already in the nineteenth century, long before the State of Israel was founded, Zionist visionaries had dreamt of it as a modern version of Francis Bacon's utopian Kingdom of Bensalem, where science and technology would provide health, wealth, and power. (shrink)

This issue of Science in Context presents a collection of historical studies on various aspects of science and its practice as developed in Latin-American contexts. Relatively few scholars working in the history of science, and even in the more general field of “science studies,” have devoted their research to this field. Likewise, relatively little research has been done by scholars of Latin American studies on the cultural, political, and social impact of science, a field that is usually considered to be (...) one of the central, defining aspects of modern, Western civilization. (shrink)

Charles W. Curtis is a prominent mathematician who has made important contributions to the field of representation theory. His textbooks in this field have been classics for a long time. In Pioneers of Representation Theory he has set out to present the historical development of the main ideas of the discipline, from the work of Georg Ferdinand Frobenius in the 1890s up to 1960. In addition to Frobenius, the book focuses mainly on three other “pioneers”: William Burnside, Issai Schur, and (...) Richard Brauer.Curtis states that his interest in the history of representation theory arose gradually, and jointly with that of Reiner, in the process of writing their textbooks. Thus, a main aim of the book is to show how the pioneers reached their important results while using only the mathematics then available to them. In order to do so, Curtis presents successive and thorough accounts of the works in which the new and seminal ideas of the pioneers were introduced and then developed. At the same time, however, he simplifies some of the original arguments by translating them into modern terminology.In an attempt to make the book appealing and accessible to a wider circle of potential readers, some of the necessary mathematics is explained. Thus Chapter 1 presents an introduction to the basic ideas and problems of nineteenth‐century algebra and number theory; later on, as the need arises, additional mathematical sections addressed to nonexperts are included. Of course, even this very clear presentation will be accessible only to readers with at least a graduate‐level knowledge of algebra.Curtis also attempts to provide some historical context. He is particularly concerned to clarify the main problems that the work of each of the mathematicians discussed here was originally meant to resolve. In a book of this kind this is surely one of the major benefits that the reader may expect to obtain.There are biographical sketches of the major figures involved, as well as additional information about other participants. Some illuminating documents are quoted, and Curtis supplies brief accounts of the mathematical traditions within which each of the pioneers was educated and worked. He acknowledges, however, that Pioneers of Representation Theory has only a limited claim as a work of historical research. He thus provides references to more historically oriented work that interested readers can consult. In particular, he mentions several articles by Thomas Hawkins; it should be stressed that much of Hawkins's work has recently been published in book form: Emergence of the Theory of Lie Groups: An Essay in the History of Mathematics, 1869–1926 . Curtis's and Hawkins's books complement each other in many important ways, and each should appeal to readers of the other.Charles Curtis has written an impressive, authoritative, and well‐informed book on a difficult subject. Mathematicians and historians of twentieth‐century mathematics with the relevant background will find it difficult but rewarding reading. Along the way, historiographical questions may arise. To what extent is it possible to translate, as Curtis does, the original proofs of the pioneers into more modern terminology and yet remain faithful to the sources? In most cases, however, it will require the technical expertise of the author himself to answer questions like this one properly. We can only hope that the firm starting point this book provides will lead Curtis to further historical research in this and related fields. (shrink)

This book is a wonderful resource for historians and philosophers of mathematics and physics alike, not just for Hilbert's own work in physics, but also because Corry sets Hilbert in context, bringing out the people with whom Hilbert had contact, describing their work and possible links with Hilbert's work, and describing the activities going on around Hilbert. The historical thesis of this book is that Hilbert worked on a wide range of issues in physics for a period lasting more (...) than two decades, employing and developing his axiomatic approach throughout. One conclusion that follows from this is that Hilbert's 1915–1917 work relating to Einstein's General Theory of Relativity was a natural continuation of Hilbert's pre-existing interests and activities, and not a one-off foray into foreign territory. 1Of especial interest to philosophers of mathematics are two further theses. Corry stresses that for Hilbert geometry is an empirical science, and related to this argues first, that Hilbert intends the axiomatic method to be used in enhancing our understanding of the content of a given theory via relating the results of the axiomatic investigation back to the intuitive content of the axioms; and, second, that to understand Hilbert's axiomatic approach in mathematics we must pay serious attention to his work in physics.Corry also hopes to show ‘the significant and unique contribution of Hilbert to certain important developments in twentieth-century physics’ . 2 In the end, this assessment of Hilbert's contribution to physics is far from clear cut: the two cases where Hilbert goes into the details of a physical theory show him lacking feel for what is important physically with respect to that theory. Nevertheless, philosophers and historians of physics will find a great deal to interest them in the story of Hilbert's involvement in physics, and in the details …. (shrink)

Philosophers often claim that forgiveness is a paradoxical phenomenon. I here examine two of the most widespread ways of dealing with the paradoxical nature of forgiveness. One of these ways, emblematized by Aurel Kolnai, seeks to resolve the paradox by appealing to the idea of repentance. Somehow, if a wrongdoer repents, then forgiving her is no longer paradoxical. I argue that this influential position faces more problems than it solves. The other way to approach the paradox, exemplified here by the (...) work of Jacques Derrida, is just too obscure to be by itself helpful. Yet, I argue that what I take to be its spirit is on the right track. I recommend distinguishing between the definition and the justification of forgiveness, and also between forgiveness understood as a mental phenomenon and an overt, communicative act. These distinctions are not given their due in the specialized literature, and I expose the nefarious consequences of this neglect. By focusing on forgiveness as a mental phenomenon I seek to analyze the root of the talk of paradoxes which surrounds the discussion of forgiveness. Finally, I present an analysis of forgiveness as a pure mental phenomenon, and argue that this analysis is the most important step in understanding forgiveness in any other sense. While my analysis reveals interesting aspects of forgiveness, it reveals, too, that forgiveness is not quite as paradoxical after all. (shrink)

Are consumers of meat morally responsible for harms caused to animals in the produciton of that meat? One common argument for the negative states that in a global market, the decisions of an individual consumer makes little or no difference to whether and how a product is produced, hence the individual consumer cannot be held morally responsible. I argue that this same reasoning would imply that consumers of child-pornography cannot be held morally responsible for the harms done to children in (...) its production. Since society does (rightly) hold consumers of child-pornography responsible for harms done to children depicted, we are being inconsistent if we do not hold consumers in general rsponsible for harms done in the production of that which they consume. (shrink)

The continental philosophy of non-classical logics is a relatively new field that seeks to determine whether any aspects of certain continental philosophers’ thinking can be characterized in terms of non-classical logics. Some of the main figures that have been examined so far are Martin Heidegger, Jacques Derrida, Gilles Deleuze, and François Laruelle. Although many of these studies are grounded in the writings of Graham Priest, who wrote some of the seminal texts in the field, Jc Beall’s work also features prominently (...) in a number of cases. After surveying this field and highlighting Beall’s influence within it, I claim that it can be substantially expanded by making further uses of Beall’s writings, especially ones that introduce non-classical logics in ways that are especially suitable for this particular field of study and also that discuss such topics as negation, subclassical logics, logical pluralism, and glutty futures. (shrink)

The present volume contains articles of well-known representatives of the Meaning-Text Theory (MTT) and other related linguistic theories. Founded by I. Mel'cuk and A. Zholkovsky in the sixties in Moscow, MTT soon became known in the West as a “prominent outsider” theory. The picture changed since then, though. MTT gained importance in several areas of linguistics and computational linguistics. It influenced the design of new grammar formalisms such as Dependency Tree Grammars. Also, specific parts of MTT have been directly overtaken (...) into other theories; consider, for example, the work on integrating Lexical Functions into Pustejovsky's Generative Lexicon. The present volume is a further convincing demonstration of MTT's liveliness and relevance to the field's “burning” issues. The focus of the volume is on semantics, semantic representation and relation of semantics to surface in MTT. Six out of eight articles (Polguère; Escalier & Fournier; Paducheva; St.-Germain; Beck; Bogulavsky) deal with problems related to these topics, while the last two articles of the volume (Sgall and Rambow; Joshi) throw a bridge between MTT, or, more precisely, between dependency-based theories of which MTT is one instantiation, and other linguistic theories. (shrink)

"-- M. J. Silverthorne,The Humanities Association Review Leo Strauss (1899-1973) was the Robert Maynard Hutchins Distinguished Service Professor Emeritus of ...

The difference between cause and effect seems obvious and crucial in ordinary life, yet missing from modern physics. Almost a century ago, Bertrand Russell called the law of causality 'a relic of a bygone age'. In this important collection 13 leading scholars revisit Russell's revolutionary conclusion, discussing one of the most significant and puzzling issues in contemporary thought.

New nursing researchers are faced with a smorgasbord of competing methodologies. Sometimes, they are encouraged to adopt the research paradigms beloved of their senior colleagues. This is a problem if those paradigms are no longer of contemporary methodological relevance. The aim of this paper was to provide clarity about current research paradigms. It seeks to interrogate the continuing viability of positivism as a guiding paradigm for nursing research. It does this by critically analysing the methodological literature. Five major paradigms are (...) addressed: the positivist; the interpretivist/constructivist; the transformative; the realist; and the postpositivist. Acceptance of interpretivist, transformative or realist approaches necessarily entails wholesale rejection of positivism, while acceptance of postpositivism involves its partial rejection. Postpositivism has superseded positivism as the guiding paradigm of the scientific method. The incorporation in randomized controlled trials of postpositivist assumptions indicates that even on the methodological territory that it once claimed as its own, positivism has been rendered redundant as an appropriate paradigm for contemporary nursing research. (shrink)

Jacques Derrida’s and Gilles Deleuze’s philosophies of justice share many similar features. For both, justice involves an overturning of law by extralegal means, made possible by an “undecidability” in the judgment-making process. To distinguish their conceptions of justice, we examine their implicit modes of non-classical reasoning with regard to “otherness,” building from Routley and Routley and Daniel Smith, to conclude that Derrida’s thinking on justice is at least paracomplete (or analetheic) while Deleuze’s is just paraconsistent (or dialetheic).

Enactivism and ecological psychology converge on the relevance of the environment in understanding perception and action. On both views, perceiving organisms are not merely passive receivers of environmental stimuli, but rather form a dynamic relationship with their environments in such a way that shapes how they interact with the world. In this paper, I suggest that while enactivism and ecological psychology enjoy a shared specification of the environment as the cognitive domain, on both accounts, the structure of the environment, itself, (...) is unspecified beyond that of contingent relations with the species-typical sensorimotor capacities of perceiving organisms. This lack of specification creates a considerable gap in theory regarding the organization of organisms as coupled with their environments. I argue that this gap can be filled by drawing from resources in developmental systems theory, namely, specifying the environmental state-space as a developmental niche that shapes and is shaped by individual organisms over developmental and, on a population scale, evolutionary time. Defining the environment as an organism’s developmental niche makes it clearer how and why certain contingencies have arisen, in turn, strengthening a joint appeal to both enactivism and ecological psychology as theories asserting complementarity between organisms and their environments. (shrink)

Jean-Pierre Dupuy argues that our failure to prevent the looming climate catastrophe results from a faulty metaphysics of time: because we believe the present can proceed down one of the many branches that extend into the future, some of which bypass the catastrophe, we do not think it is absolutely urgent to take drastic action now. His solution to this problem of demotivation is “enlightened doomsaying” in “projected time”, which means that we affirm the coming catastrophe as something real in (...) the future rather than being a mere possibility; thus, we regard it seriously enough that we are motivated to take the needed actions to prevent it. One potential obstacle to this proposal is that it requires the forming of consensus and coordination with the powerful players who benefit from our current path and whose apparently near-total grip on this catastrophic future may itself discourage action. We then consider an alternative model based on Gilles Deleuze’s philosophy of the present–future relation. Although it has the branching structure that Dupuy is wary of, it may not suffer from the same problem of demotivation on account of the way it conceives the complex structure of the present event. For this reason, the Deleuzian model may be more suited to motivating action in a world where the future must be fought for rather than unanimously agreed upon. (shrink)

We respond to a phenomenological challenge set forth in Thomas Nagel’s “What Is It Like To Be a Bat?,” namely, to seek a method for obtaining a phenomenological description of non-human animal experience faithful to an animal’s first-person subjective perspective. First, we examine “translational” strategies employing empathy and communication with animals. Then we turn to a “transpositional” strategy from Uexkull’s Umwelt theory in which we objectively determine the components of a non-human animal’s subjective world of experience and then map those (...) coordinates onto our own subjective world. While this method gives us partial access to the animal’s “perception-world” aspect of its Umwelt, it does not inform us about what it is like to live in the interactive, “effect-world” aspect. To better overcome this limitation, we add a “transformational” approach derived from Deleuze’s & Guattari’s notion of “becoming-animal,” in which we take on an animal’s manners and capacities for interacting with the other objects and creatures in its world. (shrink)

To compare Merleau-Ponty’s and Deleuze’s phenomenal bodies, I first examine how for Merleau-Ponty phenomena appear on the basis of three levels of integration: 1) between the parts of the world, 2) between the parts of the body, and 3) between the body and its world. I contest that Deleuze’s attacks on phenomenology can be seen as constructive critiques rather than as being expressions of an anti-phenomenological position. By building from Deleuze’s definition of the phenomenon and from his more phenomenologically relevant (...) writings, we find that phenomena for him are given to the body under exactly the opposite conditions as for Merleau-Ponty, namely that 1) the world’s differences 2) appear to a disordered body that 3) comes into shocking affective contact with its surroundings. I argue that a Deleuzian theory of bodily-given phenomena is better suited than Merleau-Ponty’s model in the task of accounting for the intensity of phenomenal appearings. (shrink)

This paper explores the relationship between the Islamic religion and the level of socially responsible business conduct (SRBC) of Islamic entrepreneurs. The authors find that the common ideas of SRBC correspond with the view of business in Islam, although there are also some notable differences. They also find that Muslim entrepreneurs attach a higher weight to specific elements of SRBC than do non‐Muslims. However, they also find that Muslims are less involved with applying SRBC in practice than non‐Muslim managers.

French philosopher Gilles Deleuze wrote two 'logic' books: Francis Bacon: The Logic of Sensation and The Logic of Sense. However, in neither of these books nor in any other works does Deleuze articulate in a formal way the features of the logic he employs. He certainly does not use classical logic. And the best options for the non-classical logic that he may be implementing are: fuzzy, intuitionist, and many-valued. These are applicable to his concepts of heterogeneous composition and becoming, affirmative (...) synthetic disjunction, and powers of the false. In The Logic of Gilles Deleuze. Volume 1: Basic Principles, Corry Shores examines the applicability of three non-classical logics to Deleuze's philosophy, by building from the philosophical and logical writings of Graham Priest, the world's leading proponent of dialetheism. Through so doing, Shores demonstrates how Deleuze's logic is best understood as a dialetheic, paraconsistent, many-valued logic. (shrink)

A process philosophical interpretation of Deleuze's theories of time encounters problems when formulating an account of Deleuze's portrayal of temporality in The Time-Image, where time is understood as having the structure of instantaneity and simultaneity. I remedy this shortcoming of process philosophical readings by formulating a phenomenological interpretation of Deleuze's second synthesis of time. By employing Deleuze's logic of affirmative synthetic disjunction in combination with his differential calculus interpretation of Spinoza's and Bergson's duration, this phenomenological interpretation portrays time as given (...) to our awareness in immediacy rather than through a continuous process of unfolding. The viability of this alternate approach calls into question the claims that Deleuze is strictly a process philosopher and anti-phenomenologist. (shrink)