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)

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)

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)

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)

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)

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)

As the last of the Jedi, Luke must now pass on what he has learned of the Force, presumably to restart the Jedi Order. In the imaginations of many, Luke simply must have continued his rise, becoming one of the most powerful living beings in the universe. Deleuze draws his notion of the forces of becoming partly from German philosopher Friedrich Nietzsche, who says that the world is “a monster of energy, without beginning, without end” that “only transforms itself” as (...) “a play of forces and waves of forces, a sea of forces flowing and rushing together, eternally changing, with an ebb and a flood of its forms.” Deleuze's interpretation is that all bodies, all physical things, are “nothing but forces” defined solely by their activities of affecting – and being affected by – other things with which they interact. (shrink)

In this classic analysis, Leo Strauss pinpoints what is original and innovative in the political philosophy of Thomas Hobbes.

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.

Recent feminist philosophy of language has highlighted the ways that the speech of women can be unjustly impeded, because of the way their gender affects the uptake their speech receives. In this chapter, I explore how similar processes can undermine the speech of a different sort of speaker: Indigenous communities. This involves focusing on Indigeneity rather than gender as the salient social identity, and looking at the ways that group speech, rather than only individual speech, can be unjustly impeded. To (...) do this, I make use of the notion of ‘discursive injustice’ that has been developed by Quill Kukla, and, with reference to three case studies, show how discursive injustice can effectively derail the speech of Indigenous communities. (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)

Public participation in scientific research has gained prominence in many scientific fields, but the theory of participatory research is still limited. In this paper, we suggest that the divergence of values and goals between academic researchers and public participants in research is key to analyzing the different forms this research takes. We examine two existing characterizations of participatory research: one in terms of public participants' role in the research, the other in terms of the virtues of the research. In our (...) view, each of these captures an important feature of participatory research but is, on its own, limited in what features it takes into account. We introduce an expanded conception of norms of collaboration that extends to both academic researchers and public participants. We suggest that satisfying these norms requires consideration of the two groups' possibly divergent values and goals, and that a broad characterization of participatory research that starts from participants' values and goals can motivate both public participants’ role in the research and the virtues of the research. The resulting framework clarifies the similarities and differences among participatory projects and can help guide the responsible design of such projects. (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)

The Leo Strauss transcript project -- Editor's introduction: Strauss, Nietzsche, and the history of political philosophy -- Editorial headnote -- Introduction: Nietzsche's philosophy, existentialism, and the problem of our age -- Restoring nature as ethical principle: Zarathustra, prologue -- The creative self: Zarathustra, part 1, 1-8 -- The true individual as the highest goal: Zarathustra, part 1, 9-15 -- Postulated nature and final truth: Zarathustra, part 1, 16-22 -- Truth, interpretation, and intelligibility: Zarathustra, part 2, 1-12 -- Will to power (...) and self-overcoming: Zarathustra, part 2, 15-20 -- Summary and review: fusing Plato and the creative self -- Greek philosophy and the Bible; nature and history: Zarathustra, part 2, 20-22 -- Eternal recurrence: Zarathustra, part 2, 21; part 3, 1-13 -- Survey: Nietzsche and political philosophy -- The goodness of the whole, Socratic and Heideggerian critiques: Zarathustra, part 4-12 -- Creative contemplation: Zarathustra, part 3, 13 -- Restoring the sacred and the final question: Zarathustra, part 4. (shrink)

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)

Leo Strauss's introductions to ten writings of Moses Mendelssohn -- Preliminary remark by Alexander Altmann -- Introduction to Pope a metaphysician! -- Introduction to "Epistle to Mr. Lessing in Leipzig" -- Introduction to Commentary on Moses Maimonides' "Logical terms" -- Introduction to Treatise on evidence in metaphysical sciences -- Introduction to Phädon -- Introduction to Treatise on the incorporeality of the human soul -- Introduction to "On a handwritten essay of Mr. de Luc's" -- Introduction to The soul -- Introduction (...) to Morning hours and to the friends of Lessing -- Introduction to God's cause, or providence vindicated. (shrink)

This collection of Leo Abse's last essays are writings that he was working on from 2006 up to and during his final illness.

For Deleuze, the creation and conveyance of meaning requires not a strict fidelity to an original idea, message or image but rather its deformation. The forces causing such disfigurations operate in gesture, vocalisation and text, with one level sometimes disrupting the others. Among them, gesture plays an especially important role, given Deleuze’s attention to bodily experience. He locates it in theatre, painting and cinema, particularly in the works of Carmelo Bene, Francis Bacon and Jerry Lewis. In these cases, instead of (...) conveying a recognisable message, a deformative gesture sends a shock wave that scrambles normal meaning assignments and codings. Yet, although gesture is a main channel for these communication disruptive forces, their origin may lie elsewhere, namely, in a ‘pick-up’ of influences resulting from an inter-affective encounter between heterogeneous bodies. (shrink)

Deleuze never explicitly formulates his philosophy of logical truth‐values. It thus remains an open question as to the number and types he held there to be. Despite his explicit comments on these matters, additional textual evidence suggests that in his thinking on the event, he favored a third truth‐value, holding either the analetheic view that some truth‐bearers can be truth‐valueless or the dialetheic view that some truth‐bearers can be both true and false. I first argue that taking a logical approach (...) to Deleuze's thinking is feasible, despite his and others' claims that might suggest otherwise. Next, I examine his explicit statements to show that they cannot be taken at face value and that, rather, we need to transpose his claims into contemporary terminology in order to accurately assess them. I lastly turn to his Leibniz‐inspired philosophy of time to argue that the affirmations involved in this conception strongly suggest a dialetheic tendency in his thinking. (shrink)