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)

History and Philosophy of Modern Mathematics was first published in 1988. Minnesota Archive Editions uses digital technology to make long-unavailable books once again accessible, and are published unaltered from the original University of Minnesota Press editions. The fourteen essays in this volume build on the pioneering effort of Garrett Birkhoff, professor of mathematics at Harvard University, who in 1974 organized a conference of mathematicians and historians of modern mathematics to examine how the two disciplines (...) approach the history of mathematics. In History and Philosophy of Modern Mathematics, William Aspray and Philip Kitcher bring together distinguished scholars from mathematics, history, and philosophy to assess the current state of the field. Their essays, which grow out of a 1985 conference at the University of Minnesota, develop the basic premise that mathematical thought needs to be studied from an interdisciplinary perspective. The opening essays study issues arising within logic and the foundations of mathematics, a traditional area of interest to historians and philosophers. The second section examines issues in the history of mathematics within the framework of established historical periods and questions. Next come case studies that illustrate the power of an interdisciplinary approach to the study of mathematics. The collection closes with a look at mathematics from a sociohistorical perspective, including the way institutions affect what constitutes mathematical knowledge. (shrink)

This edited volume, aimed at both students and researchers in philosophy, mathematics and history of science, highlights leading developments in the overlapping areas of philosophy and the history of modern mathematics. It is a coherent, wide ranging account of how a number of topics in the philosophy of mathematics must be reconsidered in the light of the latest historical research and how a number of historical accounts can be deepened by embracing philosophical questions.

Politics and Modernity provides a critical review of the key interface of contemporary political theory and social theory about the questions of modernity and postmodernity. Review essays offer a broad-ranging assessment of the issues at stake in current debates. Among the works reviewed are those of William Connolly, Anthony Giddens, J[um]urgen Habermas, Alasdair MacIntyre, Richard Rorty, Charles Taylor and Roy Bhaskar. As well as reviewing the contemporary literature, the contributors assess the historical roots of current problems in the works of (...) Nietzsche and Max Weber. (shrink)

A History of Mathematics: From Mesopotamia to Modernity covers the evolution of mathematics through time and across the major Eastern and Western civilizations. It begins in Babylon, then describes the trials and tribulations of the Greek mathematicians. The important, and often neglected, influence of both Chinese and Islamic mathematics is covered in detail, placing the description of early Western mathematics in a global context. The book concludes with modern mathematics, covering recent developments such (...) as the advent of the computer, chaos theory, topology, mathematical physics, and the solution of Fermat's Last Theorem.Containing more than 100 illustrations and figures, this text, aimed at advanced undergraduates and postgraduates, addresses the methods and challenges associated with studying the history of mathematics. The reader is introduced to the leading figures in the history of mathematics and their fields. An extensive bibliography with cross-references to key texts will provide invaluable resource to students and exercises will stretch the more advanced reader. (shrink)

A History of Mathematics: From Mesopotamia to Modernity covers the evolution of mathematics through time and across the major Eastern and Western civilizations. It begins in Babylon, then describes the trials and tribulations of the Greek mathematicians. The important, and often neglected, influence of both Chinese and Islamic mathematics is covered in detail, placing the description of early Western mathematics in a global context. The book concludes with modern mathematics, covering recent developments such (...) as the advent of the computer, chaos theory, topology, mathematical physics, and the solution of Fermat's Last Theorem. Containing more than 100 illustrations and figures, this text, aimed at advanced undergraduates and postgraduates, addresses the methods and challenges associated with studying the history of mathematics. The reader is introduced to the leading figures in the history of mathematics and their fields. An extensive bibliography with cross-references to key texts will provide invaluable resource to students and exercises will stretch the more advanced reader. (shrink)

The book offers a collection of essays on various aspects of Leibniz’s scientific thought, written by historians of science and world-leading experts on Leibniz. The essays deal with a vast array of topics on the exact sciences: Leibniz’s logic, mereology, the notion of infinity and cardinality, the foundations of geometry, the theory of curves and differential geometry, and finally dynamics and general epistemology. Several chapters attempt a reading of Leibniz’s scientific works through modern mathematical tools, and compare Leibniz’s results (...) in these fields with 19th- and 20th-Century conceptions of them. All of them have special care in framing Leibniz’s work in historical context, and sometimes offer wider historical perspectives that go much beyond Leibniz’s researches. A special emphasis is given to effective mathematical practice rather than purely epistemological thought. The book is addressed to all scholars of the exact sciences who have an interest in historical research and Leibniz in particular, and may be useful to historians of mathematics, physics, and epistemology, mathematicians with historical interests, and philosophers of science at large. (shrink)

The book offers a collection of essays on various aspects of Leibniz’s scientific thought, written by historians of science and world-leading experts on Leibniz. The essays deal with a vast array of topics on the exact sciences: Leibniz’s logic, mereology, the notion of infinity and cardinality, the foundations of geometry, the theory of curves and differential geometry, and finally dynamics and general epistemology. Several chapters attempt a reading of Leibniz’s scientific works through modern mathematical tools, and compare Leibniz’s results (...) in these fields with 19th- and 20th-Century conceptions of them. All of them have special care in framing Leibniz’s work in historical context, and sometimes offer wider historical perspectives that go much beyond Leibniz’s researches. A special emphasis is given to effective mathematical practice rather than purely epistemological thought. The book is addressed to all scholars of the exact sciences who have an interest in historical research and Leibniz in particular, and may be useful to historians of mathematics, physics, and epistemology, mathematicians with historical interests, and philosophers of science at large. (shrink)

Review by A. Kanamori, Boston University (author of The Higher Infinite), review in The Bulletin of Symbolic Logic: “Notwithstanding and braving the daunting complexities of this labyrinth, José Ferreirós has written a magisterial account of the history of set theory which is panoramic, balanced and engaging. Not only does this book synthesize much previous work and provide fresh insights and points of view, but it also features a major innovation, a full-fledged treatment of the emergence of the set-theoretic approach (...) in mathematics from the early nineteenth century. Much of history is in the details, and much of good history is in tone, emphasis, and the drawing together of different strands. Without being Whiggish, Ferreirós is never far from major synthetic themes, themes that in the main work toward balance and pull back from excesses of emphasis of some recent histories. The main criticism of the book would have to be on matters of broad interpretation. Yet, Ferreirós brings a considerable mathematical acuity and expertise to his enterprise, and this makes his viewpoints the more compelling.” . (shrink)

François Viète is often regarded as the first modern mathematician on the grounds that he was the first to develop the literal notation, that is, the use of two sorts of letters, one for the unknown and the other for the known parameters of a problem. The fact that he achieved neither a modern conception of quantity nor a modern understanding of curves, both of which are explicit in Descartes’ Geometry, is to be explained on this view (...) “by an incomplete symbolization rather than by any obstacle intrinsic in the system.” Descartes’ Geometry provides only a “clearer expression” of themes already sounded in Viète’s work, one that perfects Viète’s literal calculus and gives it “its modern form”; it merely continues the “‘new’ and ‘pure’ algebra which Viète first established as the ‘general analytic art’.” It can seem, furthermore, that this must be right, that had there been some obstacle intrinsic to Viète’s system that barred the way to a modern conception of quantity and a modern understanding of curves, then Descartes’ Geometry would have had to have taken a very different form than it did. As it was, Descartes had only to improve Viète’s symbolism, free himself of the last vestiges of the ancient view of geometrical and arithmetical objects, and apply the new symbolism to the study of curves in order to achieve what Viète did not but could have. But what, really, is the status of this “could have”; what would it actually have taken for Viète to achieve Descartes’ results in the Geometry? The interest of the question lies in its potential to better our understanding of the nature of modern mathematics. (shrink)

This edited volume explores the previously underacknowledged 'pre-history' of mathematical structuralism, showing that structuralism has deep roots in the history of modern mathematics. The contributors explore this history along two distinct but interconnected dimensions. First, they reconsider the methodological contributions of major figures in the history of mathematics. Second, they re-examine a range of philosophical reflections from mathematically-inclinded philosophers like Russell, Carnap, and Quine, whose work led to profound conclusions about logical, epistemological, and (...) metaphysic. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Abraham, Planter of Mathematics":Histories of Mathematics and Astrology in Early Modern EuropeNicholas PopperFrancis Bacon's 1605 Advancement of Learning proposed to dedicatee James I a massive reorganization of the institutions, goals, and methods of generating and transmitting knowledge. The numerous defects crippling the contemporary educational regime, Bacon claimed, should be addressed by strengthening emphasis on philosophy and natural knowledge. To that end, university positions were to be (...) created devoted to "Artes and Sciences at large," rather than to the professions. High salaries would render lecturers "able and sufficient," undistracted from their task. Most famously, he argued that teaching of the "operatiue studie of many Scyences" should involve sophisticated technical education. The study of natural philosophy demanded not only books, but globes, astrolabes, and other "instrumentals." Most significantly, yielding reliable and meaningful knowledge from experiential gleanings required a rigorous system of deductive reasoning.The legacy of this colossal proposal has earned Bacon honored status as devisor of the scientific method.1 But Bacon's educational reform extended [End Page 87] beyond the methods of producing and transmitting knowledge. To facilitate more efficient "vse and administration" of the knowledge produced by his system, he also demanded a searching examination of the history of learning. This history would provide a mirror enabling his contemporaries to deploy the fruits of his method, by considering how learning in the past had been used successfully or ill-advisedly. Producing this history was the ambition of the Advancement, for he explained: "no man hath propounded to himselfe the generall state of learning to bee described and represented from age to age, as many have done the works of Nature, & the State civile and Ecclesiastical." He continued:And yet I am not ignorant that in divers particular sciences, as of the Iurisconsults, the Mathematicians, the Rhetoricians, the Philosophers, there are set down some smal memorials of the Schooles, Authors, and Bookes: and so likewise some barren relations touching the Invention of Arts, or usages. But a iust story of learning, containing the Antiquities and Originalls of Knowledges, & their Sects; their Inventions, their Traditions; their diverse Administrations and Managings; their Flourishings, their Oppositions, Decayes, Depressions, Oblivions, Removes, with the causes, and occasions of them, and all other events concerning learning, throughout the ages of the world; I may truly affirme to be wanting.2Bacon thus positioned himself not only as a Father of Modern Science, but also a Father of the History of Science.Following Bacon's suggestion, I will examine the "small memorials" of the history of mathematics—and particularly the mathematical art of astrology—in the sixteenth and seventeenth centuries. My conclusion, however, will not bear out Bacon's claims. Despite his frustration, Bacon was only one of many early modern scholars appraising the role of mathematics within history. And he devoted less energy than others to mapping its origins and tracing its transmissions between communities. In fact, Bacon's 1605 proposal for a history of mathematics was already out-of-date. Discussions [End Page 88] of the history of mathematics had been rife on the continent and in England throughout the previous century.3The evidence that Renaissance scholars inherited was sprawling and inconclusive. Several genealogies for mathematics could be found within classical Greek, Latin, and patristic references. One lineage claimed mathematics began in ancient Assyria, where the priestly caste, the Chaldeans, practiced a form of mathematics that seemed a corrupt admixture of philosophy, medicine, and religion, and relied heavily on observation of the heavens. Other scholars traced the origins to Egypt, claiming that the field developed to survey lands frequently flooded by the Nile. The real problem for ancient, late antique, and medieval scholars had not been the origins of mathematics, but what exactly mathematics were. For some, the term strictly referred to the quadrivium: geometry, arithmetic, astronomy, and music. But for others, mathematics' origins amongst the Egyptians or Chaldeans inextricably linked it to forms of astrological divination, augury, and necromancy that were unsavory to both Latin and Christian traditions. Mathematici were included alongside ghastly lists of Magi, Brahmins, Aruspices, Genethliaci, and other diabolic practitioners of idolatrous magic. These might or might not be distinguished from mathematici such as Pythagoras... (shrink)

Abraham Robinson’s framework for modern infinitesimals was developed half a century ago. It enables a re-evaluation of the procedures of the pioneers of mathematical analysis. Their procedures have been often viewed through the lens of the success of the Weierstrassian foundations. We propose a view without passing through the lens, by means of proxies for such procedures in the modern theory of infinitesimals. The real accomplishments of calculus and analysis had been based primarily on the elaboration of novel (...) techniques for solving problems rather than a quest for ultimate foundations. It may be hopeless to interpret historical foundations in terms of a punctiform continuum, but arguably it is possible to interpret historical techniques and procedures in terms of modern ones. Our proposed formalisations do not mean that Fermat, Gregory, Leibniz, Euler, and Cauchy were pre-Robinsonians, but rather indicate that Robinson’s framework is more helpful in understanding their procedures than a Weierstrassian framework. (shrink)

ABSTRACT For some years now, the history of modern mathematics and the history of modern science have developed independently. A step toward a reunification that would benefit both disciplines could come about through a revived appreciation of mathematical practice. Detailed studies of what mathematicians actually do, whether local or broadly based, have often led in recent work to examinations of the social, cultural, and national contexts, and more can be done. Another recent approach toward a (...) historical understanding of the abstractness of modern mathematics has been to see it as a species of modernism, and this thesis will be tested by the raft of works on the history of modern applied mathematics currently under way. (shrink)

It can not be our purpose to give here a complete account of the phenomenological history of mathematical doctrine. It will be enough to refer to the battle of opinions in mathematical theory which was waged within the last eighty-five years, since Riemann's inaugural lecture on Non-Euclidean Geometry. Furthermore, the revolution which Einstein's theory of general relativity created is indicative of the complete absence of any general awareness that mathematics as a science has any relation to social reality. (...) If we are going to establish such relations, we are conscious of the fact that they would be hotly contended by many mathematicians. (shrink)

AbstractŁukasiewicz introduced a new methodological approach to the history of logic. It consists of the use of modern formal logic in the research of the history of logic. Although he was not the first to use formal logic in his historical research, Łukasiewicz was the first who used it consistently and formulated it as a requirement for a historian of logic. The aim of this paper is to present Łukasiewicz's contribution and the history of its formulation. (...) In addition, the paper will present its reception. There are two main themes in Łukasiewicz’s work on the history of logic. He discussed Stoic logic and Aristotle’s syllogistic. The second part of the paper focuses on current reception of Łukasiewicz’s contribution and on the question of whether there is a difference between knowledge about his contribution to Stoic logic and to Aristotle’s syllogistic, given that it has been hypothesised that Łukasiewicz’s contribution is currently more known among Aristotelian scholars than among those who focus on Stoic logic. Based on the papers that have appeared in the last five years at Web of Science and Scopus this hypothesis has not been confirmed. (shrink)

By taking the work and life of the historian of mathematics Heinrich Wieleitner as an example, this study aims to highlight the many interrelations between the historiography of mathematics, mathematics education, and science communication in mathematics.By integrating aspects of the history of media, this case study also explores mathematical public relations work in the 20th century and draws attention to the important persons, institutions and contents. The focus is on the Weimar period, in which the (...) self-understanding of mathematics was challenged in different ways by far-reaching cultural debates. The article demonstrates that as a consequence of a changing media landscape, Weimar culture turned out to be a suitable environment for the successful self-presentation of mathematics. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Histories of Science in Early Modern Europe:IntroductionRobert GouldingIn 1713, Pierre Rémond de Montmort wrote to the mathematician Nicolas Bernoulli:It would be desirable if someone wanted to take the trouble to instruct how and in what order the discoveries in mathematics have come about.... The histories of painting, of music, of medicine have been written. A good history of mathematics, especially of geometry, would be a (...) much more interesting and useful work.... Such a work, if done well, could be regarded to some extent as a history of the human mind, since it is in this science, more than in anything else, that man makes known that gift of intelligence that God has given him to rise above all other creatures.1Such a history of mathematics was attempted by Jean-Etienne Montucla in his Histoire des mathématiques (first printed in 1758, and reissued in a [End Page 33] greatly expanded form in 1799).2 Montucla's great work is generally acknowledged as the first genuine history of mathematics. According to some modern historians, previous attempts at such a history had amounted to little more than collections of anecdotes, biographies or exhaustive bibliographies: "jumbles of names, dates and titles," as one writer in the Dictionary of Scientific Biography characterized them.3 Montucla, on the other hand, was thoroughly animated by the Enlightenment project expressed in Rémond's letter, and his Histoire was a philosophical history of the "development of the human mind," as he himself described it.4 It was precisely Montucla's vision of what mathematics meant and his conviction that mathematics itself must change in order to reflect the historical elevation of the human intellect, that allowed him to transform the scattered dates and anecdotes of his predecessors into a genuine history. All subsequent histories of mathematics—until the most recent social histories5 —have been in a sense "footnotes to Montucla."The papers here argue that there were indeed histories of mathematics before Montucla which are worthy of scholarly attention, and that the dismissal of Renaissance histories of science as nothing more than a "cloud of fine adjectives and metaphors" is unfair (although it contains a germ of truth). Anthony Grafton has argued in an article on the historical writings of Cardano, Rheticus, and Kepler, that the purpose of such texts was not so much to trace what actually happened, as to justify the study of a subject often derided—by both humanist and scholastic writers—as obscure, useless and undignified.6 The following four papers go further in arguing that writings in this genre (or, at least, some writings in this genre) were not limited to this characteristically humanist objective. Authors did not just use their histories to persuade others. They themselves relied on historical narratives in order to think about their discipline, define its parts, distinguish among its acceptable and unacceptable forms and prescribe its content [End Page 34] and method of teaching. By placing their discipline into a historical context shared by other, more mainstream humanistic arts, moreover, they could draw upon the large, narrative structures which Renaissance humanists had adapted to understand human intellectual and cultural development, origins, progress and decline.James Byrne's article highlights a disparity of vision between those humanists who treated the origin of the arts as just another subject for humanistic display; and those (like Regiomontanus) who were invested in the arts and wrote their histories from an insider's perspective. Alongside Regiomontanus, one could include several other authors treated in this collection, such as Thomas Vaughan, Peter Ramus, and Henry Savile. In each case, history provided a means for working through problems of the legitimacy and nature of their disciplines. While their writings are not "histories of the human mind," their interests went beyond mere rehearsal of anecdotes. Ramus's histories, for instance, written 200 years before Montucla's Histoire, had the kind of unifying vision which de Montmort demanded, and shared Montucla's concern to find a meaning in the development of mathematics. Ramus, and the supporters and detractors he spawned, were convinced that constructing the history of this science was crucial to understanding how and why human beings have... (shrink)

ArgumentTwo simultaneous episodes in late nineteenth-century mathematical research, one by Karl Hermann Brunn and another by Hermann Minkowski, have been described as the origin of the theory of convex bodies. This article aims to understand and explain how and why the concept of such bodies emerged in these two trajectories of mathematical research; and why Minkowski's – and not Brunn's – strand of thought led to the development of a theory of convexity. Concrete pieces of Brunn's and Minkowski's mathematical work (...) in the two episodes will, from the perspective of the above questions, be presented and analyzed with the use of the methodological framework of epistemic objects, techniques, and configurations as adapted from Hans-Jörg Rheinberger's work on empirical sciences to the historiography of mathematics by Moritz Epple. Based on detailed descriptions and a comparison of the objects and techniques that Brunn and Minkowski studied and used in these pieces it will be concluded that Brunn and Minkowski worked in different epistemic configurations, and it will be argued that this had a significant influence on the mathematics they developed for those bodies, which can provide answers to the two research questions listed above. (shrink)

One influential interpretation of Newton's formulation of his calculus has regarded his work as an organized, cohesive presentation, shaped primarily by technical issues and implicitly motivated by a knowledge of the form which a "finished" calculus should take. Offered as an alternative to this view is a less systematic and more realistic picture, in which both philosophical and technical considerations played a part in influencing the structure and interpretation of the calculus throughout Newton's mathematical career. This analysis sees the development (...) of Newton's calculus not principally as a calculated movement toward "rigorous" justification (in the sense of Cauchy, Weierstrass, and their 19th century contemporaries) and refinement of techniques, but as an evolution in the light of his involvement in debates over philosophical and scientific issues central to the rise of modern philosophy and science in the 17th and early 18th centuries. (shrink)

In a series of papers read to the Cambridge Philosophical Society through the 1820s, the Cambridge mathematician George Peacock laid the foundation for a natural history of arithmetic that would tell a story of human progress from counting to modern arithmetic. The trajectory of that history, Peacock argued, established algebraic analysis as a form of universal reasoning that used empirically warranted operations of mind to think with symbols on paper. The science of counting would suggest arithmetic, arithmetic (...) would suggest arithmetical algebra, and, finally, arithmetical algebra would suggest symbolic algebra. This philosophy of suggestion provided the foundation for Peacock's “principle of equivalent forms,” which justified the practice of nineteenth-century English symbolic algebra. Peacock's philosophy of suggestion owed a considerable debt to the early Cambridge Philosophical Society culture of natural history. The aim of this essay is to show how that culture of natural history was constitutively significant to the practice of nineteenth-century English algebra. (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)

Logic has attained in our century a development incomparably greater than in any past age of its long history, and this has led to such an enrichment and proliferation of its aspects, that the problem of some kind of unified recom prehension of this discipline seems nowadays unavoidable. This splitting into several subdomains is the natural consequence of the fact that Logic has intended to adopt in our century the status of a science. This always implies that the general (...) optics, under which a certain set of problems used to be con sidered, breaks into a lot of specialized sectors of inquiry, each of them being characterized by the introduction of specific viewpoints and of technical tools of its own. The first impression, that often accompanies the creation of one of such specialized branches in a diSCipline, is that one has succeeded in isolating the 'scientific core' of it, by restricting the somehow vague and redundant generality of its original 'philosophical' configuration. But, after a while, it appears that some of the discarded aspects are indeed important and a new specialized domain of investigation is created to explore them. By follOwing this procedure, one finally finds himself confronted with such a variety of independent fields of research, that one wonders whether the fact of labelling them under a common denomination be nothing but the contingent effect of a pure historical tradition. (shrink)