This biography of Hypatia, the female philosopher and mathematician in Christian Egypt, provides background on her work and her life as an elite woman at this time. There are many myths about Hypatia, including her research, inventions and the impact of her murder, all based on a handful of contemporary resources. Through presenting the different theories and myths alongside the available evidence, this book will enable the reader to make their own interpretations about her life. Whilst the evidence does leave (...) many questions unanswered, this book provides the evidence as it stands, separating the myth from reality. There is very little published on Hypatia and she forms quite a niche market in the history of ancient Egypt. However, she is an interesting example of how multicultural Alexandria functioned at such an unstable political time, and provides anecdotal evidence of the atrocities that occurred. This book will appeal to scholars, lay people and political and religious researchers, and will show that the history of Egypt does not end at Cleopatra. (shrink)

During the last years of his life, the mathematician Karl Menger worked on a biography of his father, the economist and founder of the Austrian School of Economics, Carl Menger. The younger Menger never finished the work. While working in the Menger collections at Duke University’s David M. Rubenstein Rare Book and Manuscript Library, we discovered draft chapters of the biography, a valuable source of information given that relatively little is known about Carl Menger’s life nearly a hundred years after (...) his death. The unfinished biography covers Carl Menger’s family background and his life through early 1889. In this article, we discuss the biography and the most valuable new insights it provides into Carl Menger’s life, including Carl Menger’s family, his childhood, his student years, his time working as a journalist and newspaper editor, his early scientific career, and his relationship with Crown Prince Rudolf. (shrink)

Henri Poincaré was not just one of the most inventive, versatile, and productive mathematicians of all time--he was also a leading physicist who almost won a Nobel Prize for physics and a prominent philosopher of science whose fresh and surprising essays are still in print a century later. The first in-depth and comprehensive look at his many accomplishments, Henri Poincaré explores all the fields that Poincaré touched, the debates sparked by his original investigations, and how his discoveries still contribute (...) to society today. Math historian Jeremy Gray shows that Poincaré's influence was wide-ranging and permanent. His novel interpretation of non-Euclidean geometry challenged contemporary ideas about space, stirred heated discussion, and led to flourishing research. His work in topology began the modern study of the subject, recently highlighted by the successful resolution of the famous Poincaré conjecture. And Poincaré's reformulation of celestial mechanics and discovery of chaotic motion started the modern theory of dynamical systems. In physics, his insights on the Lorentz group preceded Einstein's, and he was the first to indicate that space and time might be fundamentally atomic. Poincaré the public intellectual did not shy away from scientific controversy, and he defended mathematics against the attacks of logicians such as Bertrand Russell, opposed the views of Catholic apologists, and served as an expert witness in probability for the notorious Dreyfus case that polarized France. Richly informed by letters and documents, Henri Poincaré demonstrates how one man's work revolutionized math, science, and the greater world. (shrink)

In the present paper, I present and discuss some new information about the life of Paul Wittich from Wrocław, Poland (formerly Breslau, Germany), an elusive mathematician and astronomer of the late 16th century. Wittich seems to have played a significant role in the emergence of two important, though short-lived, developments of late 16th-century science: the so-called prosthaphaeresis calculating method, and the geoheliocentric model of the universe usually attributed to Tycho Brahe. His role in both achievements, however, has not been sufficiently (...) explained so far, and it is my belief that one of the main obstacles to that endeavor has been our scarce knowledge about the details of Wittich's life. Thus, I attempt to reconstruct his life with aim of being able to evaluate, eventually, his involvement in the development of mathematics and astronomy towards the end of the 16th century, primarily with the use of whatever scant sources can be found concerning Wittich. The source that I present here—a notebook of Bartholomäus Scultetus, an astronomer from Görlitz—is known and has already been used, but only to a limited extent by the community of historians of science, or by those of astronomy and mathematics. By presenting the information about Wittich from Scultetus's notebook, I aim to bring this source to the attention of those communities, clarify some hitherto unclear details from Wittich's life, and set out new paths for further investigation into his biography. (shrink)

ArgumentThe article describes a special type of scientific and philosophical “non-conformism” as exemplified in the versatile work of Richard von Mises. While the historical impact of von Mises' practical and organizational work in applied mathematics is beyond doubt, it is shown that von Mises' insistence on cognitive connectibility of various scientific domains was not, in the end, successful although it stimulated the theoretical discussion considerably. Von Mises developed a principally critical attitude towards what he considered “one-sided” in several streams of (...) modernity and tended to be “anti-modern” even in those of his activities which belonged to “modernism.” Reasons can be found in his biographical experiences, in particular in his unorthodox academic education and repeated emigrations. Thus biographical research has considerable potential for historical explanation. The paper delves into the existing premises and sources for a scientific biography of Richard von Mises which is yet to be written, and publishes in the appendix a revealing biographical sketch authored by von Mises' widow, the applied mathematician Hilda Geiringer. (shrink)

Vladimir Arnold, an eminent mathematician of our time, is known both for his mathematical results, which are many and prominent, and for his strong opinions, often expressed in an uncompromising and provoking manner. His dictum that "Mathematics is a part of physics where experiments are cheap" is well known. This book consists of two parts: selected articles by and an interview with Vladimir Arnold, and a collection of articles about him written by his friends, colleagues, and students. The book is (...) generously illustrated by a large collection of photographs, some never before published. The book presents many a facet of this extraordinary mathematician and man, from his mathematical discoveries to his daredevil outdoor adventures. (shrink)

Photographs of the philosopher, his family, and associates provide an intimate look at a private and self-effacing man whose work has had a lasting impact on twentieth-century thought.

Dirk van Dalen’s biography studies the fascinating life of the famous Dutch mathematician and philosopher Luitzen Egbertus Jan Brouwer. Brouwer belonged to a special class of genius; complex and often controversial and gifted with a deep intuition, he had an unparalleled access to the secrets and intricacies of mathematics. Most mathematicians remember L.E.J. Brouwer from his scientific breakthroughs in the young subject of topology and for the famous Brouwer fixed point theorem. Brouwer’s main interest, however, was in the foundation (...) of mathematics which led him to introduce, and then consolidate, constructive methods under the name ‘intuitionism’. This made him one of the main protagonists in the ‘foundation crisis’ of mathematics. As a confirmed internationalist, he also got entangled in the interbellum struggle for the ending of the boycott of German and Austrian scientists. This time during the twentieth century was turbulent; nationalist resentment and friction between formalism and intuitionism led to the Mathematische Annalen conflict. It was here that Brouwer played a pivotal role. The present biography is an updated revision of the earlier two volume biography in one single book. It appeals to mathematicians and anybody interested in the history of mathematics in the first half of the twentieth century. (shrink)

Frank Ramsey was a brilliant Cambridge philosopher, mathematician, and economist who died in 1930 at 26 having made landmark contributions to decision theory, game theory, mathematics, logic, semantics, philosophy of science, and the theory of truth. This rich biography tells the story of his extraordinary life and intellectual achievement.

Two recently published scholarly biographies of leading nineteenth-century British mathematicians, pioneers in abstract algebra and personal friends, provide an informed comparison of their lives and work.

A biography of the seventeenth-century French mathematician and philosopher whose revolutionary methods of reasoning greatly influenced future scientific thinking.

The important and influential mathematician Felix Hausdorff was also the author of several remarkable philosophical books and essays. About his unusual biography between philosophy and topology we have only little knowledge. We know only few personal documents and letters. Among Hausdorff’s letters that have been preserved (about 130), the letters and postcards to the librarian, translator and artist Franz Meyer in Jena (10) give an important insight into Hausdorff’s biography and his philosophical development. They also inform about his way from (...) a philosopher, who liked to play with Nietzsche’s thoughts, to a pure mathematician. (shrink)

In his biography of Emil Artin, Richard Brauer describes the years from 1931–1941 as a time when “Artin spoke through his students and through the members of his mathematical circle” rather than through written publications. This paper explores these seemingly quiet years when Artin immigrated to America and disseminated ideas about algebraic number theory during this time in his collaboration with George Whaples, a young American mathematician who had just completed his Ph.D. at the University of Wisconsin. The main result (...) of their work is the use of the product formula for valuations to come up with an axiomatic characterization of both algebraic number fields and algebraic function fields with a finite field of constants. These two families of fields are exactly the fields for which class field theory is known to hold. We situate their mathematical work in the broader context of algebraic number theory and their lives within the broader historical context. (shrink)

This is a biography of the works of Richard Price, 1723-1791, one of the leading radical intellectuals of the late-18th century. By profession a dissenting minister, he was also a mathematician, a political pamphleteer, particularly on the American and French Revolutions, and a moral philosopher.

Charles Sanders Peirce was born in September 1839 and died five months before the guns of August 1914. He is perhaps the most important mind the United States has ever produced. He made significant contributions throughout his life as a mathematician, astronomer, chemist, geodesist, surveyor, cartographer, metrologist, engineer, and inventor. He was a psychologist, a philologist, a lexicographer, a historian of science, a lifelong student of medicine, and, above all, a philosopher, whose special fields were logic and semiotics. He is (...) widely credited with being the founder of pragmatism. In terms of his importance as a philosopher and a scientist, he has been compared to Plato and Aristotle. He himself intended "to make a philosophy like that of Aristotle." Peirce was also a tormented and in many ways tragic figure. He suffered throughout his life from various ailments, including a painful facial neuralgia, and had wide swings of mood which frequently left him depressed to the state of inertia, and other times found him explosively violent. Despite his consistent belief that ideas could find meaning only if they "worked" in the world, he himself found it almost impossible to make satisfactory economic and social arrangements for himself. This brilliant scientist, this great philosopher, this astounding polymath was never able, throughout his long life, to find an academic post that would allow him to pursue his major interest, the study of logic, and thus also fulfill his destiny as America's greatest philosopher. Much of his work remained unpublished in his own time, and is only now finding publication in a coherent, chronologically organized edition. Even more astounding is that,despite many monographic studies, there has been no biography until now, almost eighty years after his death. Brent has studied the Peirce papers in detail and enriches his account with numerous quotations from letters by Peirce and by his friends. This is a fascinating account of a p. (shrink)

This volume celebrates the various facets of Alan Turing (1912–1954), the British mathematician and computing pioneer, widely considered as the father of computer science. It is aimed at the general reader, with additional notes and references for those who wish to explore the life and work of Turing more deeply. -/- The book is divided into eight parts, covering different aspects of Turing’s life and work. -/- Part I presents various biographical aspects of Turing, some from a personal point of (...) view. -/- Part II presents Turing’s universal machine (now known as a Turing machine), which provides a theoretical framework for reasoning about computation. His 1936 paper on this subject is widely seen as providing the starting point for the field of theoretical computer science. -/- Part III presents Turing’s working on codebreaking during World War II. While the War was a disastrous interlude for many, for Turing it provided a nationally important outlet for his creative genius. It is not an overstatement to say that without Turing, the War would probably have lasted longer, and may even have been lost by the Allies. The sensitive nature of Turning’s wartime work meant that much of this has been revealed only relatively recently. -/- Part IV presents Turing’s post-War work on computing, both at the National Physical Laboratory and at the University of Manchester. He made contributions to both hardware design, through the ACE computer at the NPL, and software, especially at Manchester. Part V covers Turing’s contribution to machine intelligence (now known as Artificial Intelligence or AI). Although Turing did not coin the term, he can be considered a founder of this field which is still active today, authoring a seminal paper in 1950. -/- Part VI covers morphogenesis, Turing’s last major scientific contribution, on the generation of seemingly random patterns in biology and on the mathematics behind such patterns. Interest in this area has increased rapidly in recent times in the field of bioinformatics, with Turing’s 1952 paper on this subject being frequently cited. -/- Part VII presents some of Turing’s mathematical influences and achievements. Turing was remarkably free of external influences, with few co-authors – Max Newman was an exception and acted as a mathematical mentor in both Cambridge and Manchester. -/- Part VIII considers Turing in a wider context, including his influence and legacy to science and in the public consciousness. -/- Reflecting Turing’s wide influence, the book includes contributions by authors from a wide variety of backgrounds. Contemporaries provide reminiscences, while there are perspectives by philosophers, mathematicians, computer scientists, historians of science, and museum curators. Some of the contributors gave presentations at Turing Centenary meetings in 2012 in Bletchley Park, King’s College Cambridge, and Oxford University, and several of the chapters in this volume are based on those presentations – some through transcription of the original talks, especially for Turing’s contemporaries, now aged in their 90s. Sadly, some contributors died before the publication of this book, hence its dedication to them. -/- For those interested in personal recollections, Chapters 2, 3, 11, 12, 16, 17, and 36 will be of interest. For philosophical aspects of Turing’s work, see Chapters 6, 7, 26–31, and 41. Mathematical perspectives can be found in Chapters 35 and 37–39. Historical perspectives can be found in Chapters 4, 8, 9, 10, 13–15, 18, 19, 21–25, 34, and 40. With respect to Turing’s body of work, the treatment in Parts II–VI is broadly chronological. We have attempted to be comprehensive with respect to all the important aspects of Turing’s achievements, and the book can be read cover to cover, or the chapters can be tackled individually if desired. There are cross-references between chapters where appropriate, and some chapters will inevitably overlap. -/- We hope that you enjoy this volume as part of your library and that you will dip into it whenever you wish to enter the multifaceted world of Alan Turing. (shrink)

In 1916 Bertrand Russell was prosecuted and fined for publishing (in defence of a conscientious objector) 'statements likely to prejudice the recruiting and discipline of His Majesty's forces.' He was almost immediately afterwards dismissed from his Lectureship at Trinity College, Cambridge, by the College Council. This expulsion provoked a storm of protest and the true facts of the case became obscured by misconceptions, prejudices and uninformed gossip, to the discredit of the College. In 1942, therefore G. H. Hardy the mathematician (...) printed for private circulation to another generation of Fellows at Trinity a full account of the incident in an attempt to explain what really happened. This is now made public. Besides provoking an authoritative record of a celebrated but misinterpreted episode in Russell's eventful academic career, this document contains interesting evidence about attitudes to pacifism in the First World War and in particular about the sympathies of such distinguished colleagues and contemporaries of Russell as Cornford, Housman, McTaggart and Whitehead. (shrink)

Paradoxes of the Infinite presents one of the most insightful, yet strangely unacknowledged, mathematical treatises of the 19 th century: Dr Bernard Bolzano’s Paradoxien . This volume contains an adept translation of the work itself by Donald A. Steele S.J., and in addition an historical introduction, which includes a brief biography as well as an evaluation of Bolzano the mathematician, logician and physicist.

_Paradoxes of the Infinite_ presents one of the most insightful, yet strangely unacknowledged, mathematical treatises of the 19 th century: Dr Bernard Bolzano’s _Paradoxien_. This volume contains an adept translation of the work itself by Donald A. Steele S.J., and in addition an historical introduction, which includes a brief biography as well as an evaluation of Bolzano the mathematician, logician and physicist.

_Paradoxes of the Infinite_ presents one of the most insightful, yet strangely unacknowledged, mathematical treatises of the 19 th century: Dr Bernard Bolzano’s _Paradoxien_. This volume contains an adept translation of the work itself by Donald A. Steele S.J., and in addition an historical introduction to the masterpiece, which includes a brief biography as well as an evaluation of Bolzano the mathematician, logician and physicist.

The life and work of Hermann Günther Graßmann (1809–1877) attract not only ever again the attention of mathematicians, mathematical historians and those interested in the history of mathematics, they constitute also a challenge for the methodology of historiographical research. This challenge persists since Friedrich Engel’s biography of 1911; there, two sources were presented and interpreted in a not legitimate manner which even mislead since then various scholars. This paper faces the intricate task to unravel not only the methodological shortcomings (...) of Engel’s biography, but also to re-assess the misinterpretations – particularly of alleged influences on Graßmann’s approaches – induced by Engel’s misleading claims. Based upon broader historically contextualised analyses of Graßmann’s innovative elements in his theory of extension and upon new primary sources for his student times at the Gymnasium and Berlin University, the paper presents a novel assessment of Graßmann’s approaches. (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)

Mathematical correspondence offers a rich heritage for the history of mathematics and science, as well as cultural history and other areas. It naturally covers a vast range of topics, and not only of a scientific nature; it includes letters between mathematicians, but also between mathematicians and politicians, publishers, and men or women of culture. Wallis, Leibniz, the Bernoullis, D'Alembert, Condorcet, Lagrange, Gauss, Hermite, Betti, Cremona, Poincaré and van der Waerden are undoubtedly authors of great interest and their letters (...) are valuable documents, but the correspondence of less well-known authors, too, can often make an equally important contribution to our understanding of developments in the history of science. Mathematical correspondences also play an important role in the editions of collected works, contributing to the reconstruction of scientific biographies, as well as the genesis of scientific ideas, and in the correct dating and interpretation of scientific writings. This volume is based on the symposium “Mathematical Correspondences and Critical Editions,” held at the 6th International Conference of the ESHS in Lisbon, Portugal in 2014. In the context of the more than fifteen major and minor editions of mathematical correspondences and collected works presented in detail, the volume discusses issues such as • History and prospects of past and ongoing edition projects, • Critical aspects of past editions, • The complementary role of printed and digital editions, • Integral and partial editions of correspondence, • Reproduction techniques for manuscripts, images and formulae, and the editorial challenges and opportunities presented by digital technology. (shrink)

This study suggests that news obituaries have a role to play in educating practical reason using The New York Times’ Overlooked project to illustrate. The argument draws from virtue ethicist Alasdair MacIntyre’s book Ethics in the Conflicts of Modernity. A close reading of Overlooked’s15 initial obituaries used the biographies in MacIntyre’s book as templates. The analysis concluded that the articles on LGBTQ activist Marsha P. Johnson and novelist Charlotte Brontë illustrated lives that were happy in an Aristotelian sense despite misfortune. (...) The articles on journalist Ida B. Wells and mathematician Ada Lovelace illustrated incomplete lives despite overall directedness. The articles on novelist Nella Larsen and Brooklyn Bridge supervisor Emily Warren Roebling illustrated the challenge of reconciling different aspects of one’s life to achieve unity of character. This finding was closely tied to how structural injustice prevents or distorts the kinds of relationships MacIntyre says agents need to flourish. (shrink)

Girolamo Saccheri (1667--1733) was an Italian Jesuit priest, scholastic philosopher, and mathematician. He earned a permanent place in the history of mathematics by discovering and rigorously deducing an elaborate chain of consequences of an axiom-set for what is now known as hyperbolic (or Lobachevskian) plane geometry. Reviewer's remarks: (1) On two pages of this book Saccheri refers to his previous and equally original book Logica demonstrativa (Turin, 1697) to which 14 of the 16 pages of the editor's "Introduction" are devoted. (...) At the time of the first edition, 1920, the editor was apparently not acquainted with the secondary literature on Logica demonstrativa which continued to grow in the period preceding the second edition \ref[see D. J. Struik, in Dictionary of scientific biography, Vol. 12, 55--57, Scribner's, New York, 1975]. Of special interest in this connection is a series of three articles by A. F. Emch [Scripta Math. 3 (1935), 51--60; Zbl 10, 386; ibid. 3 (1935), 143--152; Zbl 11, 193; ibid. 3 (1935), 221--333; Zbl 12, 98]. (2) It seems curious that modern writers believe that demonstration of the "nondeducibility" of the parallel postulate vindicates Euclid whereas at first Saccheri seems to have thought that demonstration of its "deducibility" is what would vindicate Euclid. Saccheri is perfectly clear in his commitment to the ancient (and now discredited) view that it is wrong to take as an "axiom" a proposition which is not a "primal verity", which is not "known through itself". So it would seem that Saccheri should think that he was convicting Euclid of error by deducing the parallel postulate. The resolution of this confusion is that Saccheri thought that he had proved, not merely that the parallel postulate was true, but that it was a "primal verity" and, thus, that Euclid was correct in taking it as an "axiom". As implausible as this claim about Saccheri may seem, the passage on p. 237, lines 3--15, seems to admit of no other interpretation. Indeed, Emch takes it this way. (3) As has been noted by many others, Saccheri was fascinated, if not obsessed, by what may be called "reflexive indirect deductions", indirect deductions which show that a conclusion follows from given premises by a chain of reasoning beginning with the given premises augmented by the denial of the desired conclusion and ending with the conclusion itself. It is obvious, of course, that this is simply a species of ordinary indirect deduction; a conclusion follows from given premises if a contradiction is deducible from those given premises augmented by the denial of the conclusion---and it is immaterial whether the contradiction involves one of the premises, the denial of the conclusion, or even, as often happens, intermediate propositions distinct from the given premises and the denial of the conclusion. Saccheri seemed to think that a proposition proved in this way was deduced from its own denial and, thus, that its denial was self-contradictory (p. 207). Inference from this mistake to the idea that propositions proved in this way are "primal verities" would involve yet another confusion. The reviewer gratefully acknowledges extensive communication with his former doctoral students J. Gasser and M. Scanlan. ADDED 14 March 14, 2015: (1) Wikipedia reports that many of Saccheri's ideas have a precedent in the 11th Century Persian polymath Omar Khayyám's Discussion of Difficulties in Euclid, a fact ignored in most Western sources until recently. It is unclear whether Saccheri had access to this work in translation, or developed his ideas independently. (2) This book is another exemplification of the huge difference between indirect deduction and indirect reduction. Indirect deduction requires making an assumption that is inconsistent with the premises previously adopted. This means that the reasoner must perform a certain mental act of assuming a certain proposition. It case the premises are all known truths, indirect deduction—which would then be indirect proof—requires the reasoner to assume a falsehood. This fact has been noted by several prominent mathematicians including Hardy, Hilbert, and Tarski. Indirect reduction requires no new assumption. Indirect reduction is simply a transformation of an argument in one form into another argument in a different form. In an indirect reduction one proposition in the old premise set is replaced by the contradictory opposite of the old conclusion and the new conclusion becomes the contradictory opposite of the replaced premise. Roughly and schematically, P,Q/R becomes P,~R/~Q or ~R, Q/~P. Saccheri’s work involved indirect deduction not indirect reduction. (3) The distinction between indirect deduction and indirect reduction has largely slipped through the cracks, the cracks between medieval-oriented logic and modern-oriented logic. The medievalists have a heavy investment in reduction and, though they have heard of deduction, they think that deduction is a form of reduction, or vice versa, or in some cases they think that the word ‘deduction’ is the modern way of referring to reduction. The modernists have no interest in reduction, i.e. in the process of transforming one argument into another having exactly the same number of premises. Modern logicians, like Aristotle, are concerned with deducing a single proposition from a set of propositions. Some focus on deducing a single proposition from the null set—something difficult to relate to reduction. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Ideas 63.4 (2002) 555-575 [Access article in PDF] Measurer of All Things:John Greaves (1602-1652), the Great Pyramid, and Early Modern Metrology Zur Shalev [Figures]Writing from Istanbul to Peter Turner, one of his colleagues at Merton College, Oxford, John Greaves was deeply worried: Onley I wonder that in so long time since I left England I should neither have received my brasse quadrant which I (...) left to be finished for my journey thither, nor any notice of it [...]. I agreed with mr. Allen upon price and the time that he should finish it, if he hath failed me he hath done me the greatest injury that can be. 1A great injury indeed, because Greaves's journey to Italy and the Levant was all about measuring—luckily the instrument did reach him at some later stage. The thirty-six-year-old Professor of Geometry at Gresham College was taking the measurements of countless monuments and objects in the locations he visited. In Rome he measured, among many other ancient structures, Cestius' Pyramid and St. Peter's basilica. In Lucca, deeply impressed, he counted his paces around the beautiful city walls. In Siena he observed together with a "Mathematical Professor" one of the Sidera Medicea using "a glass." In Egypt he even hurt his eyes gazing at the sun, looking for sunspots and measuring its diameter. 2 His measuring mission, however, culminated in the fixing of the [End Page 555] latitudes of Istanbul, Rhodes, and Alexandria. As Bishop Juxton wrote before the trip to Greaves's employers at Gresham College (apparently using Greaves's own promotional language): This worke I find by the best astronomers, especially by Ticho Brache [ sic ] and Kepler, hath beene much desired as tending to the advancement of that science, and I hope it wil be an honour to that nation and prove ours if we first observe it. 3A mathematician-Orientalist, commanding the ancient and modern astronomical and geographical literature of Latin, Greek, Hebrew, Arabic, and Persian authors, Greaves was arguably the best qualified European at the period to perform the task. That he miscalculated the latitude of Rhodes is of less consequence for his present-day readers. 4 However, one of the most obscure and therefore telling of his measurement activities was the survey of the Pyramids of Giza, which resulted in the Pyramidographia (1646). 5 This remarkable learned treatise and travel account hybrid, which is at the focus of the present study, gives us a glimpse into the rich and complex world of scientific antiquarians.Greaves is most conveniently remembered today as an Orientalist. While we must be thankful to Edward Said for broadening the meaning of Orient-alism—from an academic discipline, accumulating objective knowledge of the East, into a much wider cultural discourse, his emphasis on the nineteenth and twentieth centuries and on the European colonial mindset is less useful for making historical sense of what early modern Orientalists were doing—physically and culturally. We can understand Greaves's "Oriental" enterprise in its [End Page 556] depth and variety only within long established European traditions of scholarship and keeping the intellectual concerns of his time in mind. 6But beyond articulating for us the nature of early Orientalism, the Pyramidographia, as well as Greaves's entire scientific career, situated in their wider European context, provide a fine entry point into a foreign world of learned practices and methods. It teaches us how well into the seventeenth century astronomy and philology, observation and bookishness, could coalesce in one figure, in one enterprise. 7 We still lack a modern biography of Greaves, a complex protagonist of a complex period, and even a full evaluation of his intellectual work. While this paper surely cannot compensate for that, I do attempt here a brief exposition of Greaves's Pyramidographia. 8Modern historians of archaeology and Egyptology, preoccupied mostly with the disciplines' progress and development of scientific standards, have noted the Pyramidographia in passing and praised its precise language and rigorous research... (shrink)

The Companion Encyclopedia is the first comprehensive work to cover all the principal lines and themes of the history and philosophy of mathematics from ancient times up to the twentieth century. In 176 articles contributed by 160 authors of 18 nationalities, the work describes and analyzes the variety of theories, proofs, techniques, and cultural and practical applications of mathematics. The work's aim is to recover our mathematical heritage and show the importance of mathematics today by treating its interactions with the (...) related disciplines of physics, astronomy, engineering and philosophy. It also covers the history of higher education in mathematics and the growth of institutions and organizations connected with the development of the subject. Part 1 deals with mathematics in various ancient and non-Western cultures from antiquity up to medieval and Renaissance times. Part 2 treats developments in all the main areas of mathematics during the medieval and Renaissance periods up to and including the early 17th century. Parts 3-10 are divided into the main branches into which mathematics developed from the early 17th century onwards: calculus and mathematical analysis, logic and foundations, algebras, geometries, mechanics, mathematical physics and engineering, and probability and statistics. Parts 11-13 review the history of mathematics from an international perspective. The teaching of mathematics in higher education is examined in various countries, and mathematics in culture, art and society is covered. The Companion Encyclopedia features annotated bibliographies of both classic and contemporary sources; black and white illustrations, line figures and equations; biographies of major mathematicians and historians and philosophers of mathematics; a chronological table of main events in the developments of mathematics; and a fully integrated index of people, events and topics. (shrink)

Grown up as versatile people, Ottoman intellectuals had holistic views towards science, art and literature, and wrote in a variety of disciplines. It was not uncommon for a mathematician to write in philosophy, for a ḥadīth (report of the words and deeds of the Prophet) scholar to write history books, for a statesman to be busy with calligraphy or for a Shaykh al-Islām (the highest ranking Islamic legal authority) to have a “Dīwān” (a collection of poems). However, possibly due to (...) specialization-based scientific understanding of modern researchers, one side of these people generally came to the fore and some capabilities of them remained unknown. Grown up in Istanbul, capital town of Ottoman Empire, in late 16th and early 17th centuries, Okchuzāde Mehmed Shāhī was also one of these personalities called “hezār-fan” (polymath). Mehmed Shāhī served in top level offices of the State, was busy with art and literature, and became known for inshāʾ (composition) kind of works written with elaborate prose. Mehmed Shāhī earned a deserved fame with these capabilities and he was also a poet. Indeed, due to poems he wrote in his early ages, he was noticed by some biography writers about poets and his poems were mentioned in some anthologies. In addition to poetry anthologies, he had some classical works, and some couplets, quatrains and verses included in his prose works. In the present study, a collection and evaluation of poems by Mehmed Shāhī were presented.Summary: Ottoman statesman Okchuzāde Mehmed Shāhī was born in 1562 AD (970 AH) in Istanbul. He was son of Okchuzāde Mehmed Pasha (d. 995 AH/1587 AD) and served as Kātib (scribe), Defterdār (treasurer), Reīs-ul-kuttāb (chief of the scribes), Governor General of Cyprus and Aleppo. His real name was Mehmed, but he was mentioned in different sources by names such as Mehmed Bey, Mehmed Shāh, Mehmed Shāhī, Mir Mehmed Shāhī and Okchuzāde Mehmed Shāhī. Starting at a young age, he was well educated in madrasah (college). With his vibrant character and superior genius, he came to the forefront among his peers. He was educated under the supervision of Māʿlūlzāde Nakīb Efendi (d. 993 AH/1585 AD) and graduated in 988 AH/1580 AD at the age of 18. When Nakīb Efendi became Shaykh al-Islām in the same year, he started his official duty as Dīwān-ı Humāyūn (imperial council) secretary. With the duty of Executive Assistant, he became the secretary of Damad İbrahim Pasha who started to his office in 1004 AH/1596 AD. He worked under the patronage of the Pasha until Pasha’s death in 1010 AH/1601 AD. Mehmed Shāhī succeeded Musa Chelebi as Reīs-ul-kuttāb and Basdefterdar (chief treasurer) in 1005 AH/1596 AD, and was appointed as title deed and cadastre director in 1006 AH/1597 AD. However, he was dismissed from this duty and was included among the officials to join military campaign. He was prosecuted and tried for imprisonment for the accusation of coming to Istanbul without permission. He succeeded Hamza Pasha as Nishanci (high-ranking bureaucrat) in 1007 AH/1599 AD and was assigned to military campaign. Mehmed Shāhī was dismissed from his duty in 1008 AH/1600 AD. Although he was reappointed to his duty in the same year, he was dismissed again in 1010 AH/1601 AD. While Hamza Pasha was reappointed to his office, Mehmed Shāhī was not assigned to any duty. He succeeded Murad Efendi as the Defterdar of Egypt in 1013 AH/1605 AD. After this duty, one of the vassal Sanjak (administrative district) Beyliks (principalities) in Egypt was given to him. When vassal Sanjak Beyliks were abolished by Kara Mehmed Pasha, Beylerbeyi (governor) of Egypt, in 1016 AH/1608 AD, he returned to Istanbul. He was not appointed to an office for a long time. In 1029 AH/1620 AD, he was appointed as title deed and cadastre director in 1029 AH/1620 AD, and became Nishanci again in 1030 AH/1621 AD. He served in military campaign to Poland, but was dismissed from this duty during the campaign. He was appointed as Nishanci for the fifth time during the second reign of Sultan Mustafa (1622-1623). He was dismissed in 1033 AH/1624 AD and retired. In early 1039 AH/1630 AD, he died at the age of 69 in Istanbul. His tomb is near Ekmekchioghlu tomb in the vicinity of Sunbul Efendi.What is known about his family is limited to his son named Vakʿi Ahmed Efendi (d. 1060 AH/1650 AD). Ahmet Efendi learned Naskh, Thuluth and tevkīʿi types of calligraphy styles from his father. He served as anthology writer for Shaykh al-Islām Bahāī Efendi. He died in 1060 AH/1650 AD when he was serving as teacher in Sulaymaniyah Madrasah. His tomb is around Shaykh Vefa tomb.In addition to his administrative duties, Mehmed Shāhī was also interested in poetry, prose, calligraphy, mysticism, ḥadīth and tafsīr (exegesis). Possibly also due to his dismissal of duties many times in his life, he produced many works. Among them are Aḥsan al-ḥadīth, an-Naẓm al-mubīn fī āyāt al-erbaʿīn, al-Maqām al-Maḥmūd, Tarjuma-i Tuḥfa as-ṣalavāt, Jāmiʿ al-ghāyāt, Mukhtār al-akhyār, Qānūn-i Jadīd-i Arāḍi, Munsheat ve Dīwān. Famed for his prose works, the author was also a poet. With his couplets which he wrote under pseudonym of Zaynī, he managed to get on the anthology of Kınalızāde at his twenties. He was celebrated with the expression of “Arrow of word from his bow of soul has reached the target of approval from masters; he is at a young age (at his twenties)”. Subsequently adopting pseudonym of Shāhī, the poet had some poems published within the works of Beyānī, Nevʿizāde and Mustaqimzāde as well as within his own prose works. Mustaqimzāde even mentioned that Mehmed Shāhī had a dīwān. Nevertheless, this work of him has not been found yet. A total of 30 poetry works of Mehmed Shāhī were found in various sources, except for his translations of 40 ḥadīth and 40 Qur’ān verses written in poetry. Of his 30 poetry works, 26 were written in Turkish and 4 in Persian. Various sections of ʿarūḍ (meter) were used in these works written as couplets, quatrains and verses. It could be stated that his poems rich in Persian and Arabic words had a rich rhyme frame. In his poetry works in which religious, literary, historical, legendary and mysticism themes were worked, concepts and figures such as Allah, the Lord, Messiah, Messenger, Risālet (messengership), intercession, Kāfūr al-Ikhshīdī (966-968), Mustaʿsim (d. 656 AH/1258 AD), Ibn Alkamī (d. 656 AH/1258 AD), Naṣīr-i Tūsī (d. 672 AH/1274 AD), Jamshīd, Keyumars, Uftade, Hudāyī, Majūsī, Kizilbash, Bayramī and muftī (jurist capable of giving a formal legal opinion). His poems were about intercession demand from the Messenger (PBUH), sin of serving to Persian, importance of an invention, praise of Uftade Sultan (d. 988 AH/1580 AD), effect of prayer, fate of administrator favoring his relatives, futility of envying the rich, importance of fighting with Zoroastrian, benefits of exhibition of sorrow, value of skillful people, resembling of the world to a bride with a thousand grooms, beauty of eloquence, ignominy of leaving chastity for lust, asking for forgiveness for sins, talent of understanding the words, not giving quarter to enemies, praise of Hz.ʿUmar, humbleness, value of lack of worldly things, sufferance due to the beauty of the beautiful, needing not to wonder why the beloved one is far away, evil of protecting the snot, richness of Arabic language, deeds of Anatolian fāḍils (virtuous), importance of endeavor in succeeding the closeness to God and respectability of ʿAzīz Maḥmūd Hudāyī (d. 1038 AH/1628 AD). Okchuzāde Mehmed Shāhī was an Ottoman statesman who also earned a well-deserved fame for his works written in inshāʾ (composition) genre of elaborate prose. In addition, he also had poetry works. Praised for the poems which he wrote in his early ages by anthology writers who included his poems in their anthologies, the poet also had some couplets and quatrains scattered among his own prose works. His poems in the form of couplets and quatrains within abovementioned works mostly were about religious, literary, legendary, historical and mysticism themes.Mehmed Shāhī was knowledgeable in tafsīr, ḥadīth, mysticism and calligraphy, and had good levels of Arabic and Persian. Writing poems in Persian, Mehmed Shāhī was decorated his prose works with Arabic expressions such as Qur’ān verses, ḥadīths and proverbs. It could be stated that such decorations made the language of his poems more difficult. Nevertheless, his poems are plainer and easier compared to his prose works. The poet who also had lyrical poems generally preferred ḥikamī (sapiential) style and wrote genres such as munacaat (prayer to God), shafaatnāme (intercession epistles), historical quatrain, medhiye (panegyric) and hicviye (epigram). His poems in anthologies were lyrical, those in Aḥsan al-ḥadīth and an-Naẓm al-mubīn were didactical and those in Munsheat were epic. Based on an evaluation of these poems, it could be stated that he was a poet with a style, though not a poet of first class. (shrink)

This book is dedicated to the life and work of logician Janusz Czelakowski on the topic of logical consequence. It consists of three parts – a biography, a survey and research sections. The volume begins with an autobiographic chapter by Janusz Czelakowski followed by a historical chapter written by Jacek Malinowski. The survey section forms the backbone of the volume with each chapter covering one of Janusz Czelakowski’s results. They focus on his results in the area of logical consequence, demonstrate (...) how his results influenced following research, and presents potential future results, problems and applications. This volume is of interest to logicians and mathematicians. (shrink)

This book is about the relationship between necessary reasoning and visual experience in Charles S. Peirce’s mathematical philosophy. It presents mathematics as a science that presupposes a special imaginative connection between our responsiveness to reasons and our most fundamental perceptual intuitions about space and time. Central to this view on the nature of mathematics is Peirce’s idea of diagrammatic reasoning. In practicing this kind of reasoning, one treats diagrams not simply as external auxiliary tools, but rather as immediate visualizations of (...) the very process of the reasoning itself. Thus conceived, one's capacity to diagram their thought reveals a set of characteristics common to ordinary language, visual perception, and necessary mathematical reasoning. The book offers an original synthetic approach that allows tracing the roots of Peirce’s conception of a diagram in certain patterns of interrelation between his semiotics, his pragmaticist philosophy, his logical and mathematical ideas, bits and pieces of his biography, his personal intellectual predispositions, and his scientific practice as an applied mathematician. (shrink)

The present paper is aimed at showing some correlations between Charles Peirce’s life, his intellectual habits as a logician and mathematician, his semiotic theory and his practice as a geodesist. For this purpose, it makes use of Peirce’s ideas about the nature of visual experience, some facts of his intellectual biography, and his definitions of sign and the term “virtual.” It appears that Peirce’s mature pragmatist and semiotic ideas find some support in his early practice as a scientist and a (...) mathematician, thus providing an interesting example of the intersection of scientific practice and philosophical speculation. (shrink)

The mathematical and epistemological origins of Hermann Graßmann’s innovative work have always attracted the interest of mathematicians and historians. Since Friedrich Engel’s biography, a favourite source for these interpretations has been two curriculum vitae, which were, however, only known from several excerpts. The complete texts are edited here for the first time. They are presented and commented on in their respective contexts, namely the examinations required for a career as Gymnasium teacher and as Protestant pastor. Graßmann’s relations to Schleiermacher’s (...) theology and to his philosophy, as well as Graßmann’s philosophical and mathematical motivations, are discussed in the light of these two documents. (shrink)

Jakob Friedrich Fries (1773-1843): A Philosophy of the Exact Sciences -/- Shortened version of the article of the same name in: Tabula Rasa. Jenenser magazine for critical thinking. 6th of November 1994 edition -/- 1. Biography -/- Jakob Friedrich Fries was born on the 23rd of August, 1773 in Barby on the Elbe. Because Fries' father had little time, on account of his journeying, he gave up both his sons, of whom Jakob Friedrich was the elder, to the Herrnhut Teaching (...) Institution in Niesky in 1778. Fries attended the theological seminar in Niesky in autumn 1792, which lasted for three years. There he (secretly) began to study Kant. The reading of Kant's works led Fries, for the first time, to a deep philosophical satisfaction. His enthusiasm for Kant is to be understood against the background that a considerable measure of Kant's philosophy is based on a firm foundation of what happens in an analogous and similar manner in mathematics. -/- During this period he also read Heinrich Jacobi's novels, as well as works of the awakening classic German literature; in particular Friedrich Schiller's works. In 1795, Fries arrived at Leipzig University to study law. During his time in Leipzig he became acquainted with Fichte's philosophy. In autumn of the same year he moved to Jena to hear Fichte at first hand, but was soon disappointed. -/- During his first sojourn in Jenaer (1796), Fries got to know the chemist A. N. Scherer who was very influenced by the work of the chemist A. L. Lavoisier. Fries discovered, at Scherer's suggestion, the law of stoichiometric composition. Because he felt that his work still need some time before completion, he withdrew as a private tutor to Zofingen (in Switzerland). There Fries worked on his main critical work, and studied Newton's "Philosophiae naturalis principia mathematica". He remained a lifelong admirer of Newton, whom he praised as a perfectionist of astronomy. Fries saw the final aim of his mathematical natural philosophy in the union of Newton's Principia with Kant's philosophy. -/- With the aim of qualifying as a lecturer, he returned to Jena in 1800. Now Fries was known from his independent writings, such as "Reinhold, Fichte and Schelling" (1st edition in 1803), and "Systems of Philosophy as an Evident Science" (1804). The relationship between G. W. F. Hegel and Fries did not develop favourably. Hegel speaks of "the leader of the superficial army", and at other places he expresses: "he is an extremely narrow-minded bragger". On the other hand, Fries also has an unfavourable take on Hegel. He writes of the "Redundancy of the Hegelistic dialectic" (1828). In his History of Philosophy (1837/40) he writes of Hegel, amongst other things: "Your way of philosophising seems just to give expression to nonsense in the shortest possible way". In this work, Fries appears to argue with Hegel in an objective manner, and expresses a positive attitude to his work. -/- In 1805, Fries was appointed professor for philosophy in Heidelberg. In his time spent in Heidelberg, he married Caroline Erdmann. He also sealed his friendships with W. M. L. de Wette and F. H. Jacobi. Jacobi was amongst the contemporaries who most impressed Fries during this period. In Heidelberg, Fries wrote, amongst other things, his three-volume main work New Critique of Reason (1807). -/- In 1816 Fries returned to Jena. When in 1817 the Wartburg festival took place, Fries was among the guests, and made a small speech. 1819 was the so-called "Great Year" for Fries: His wife Caroline died, and Karl Sand, a member of a student fraternity, and one of Fries' former students stabbed the author August von Kotzebue to death. Fries was punished with a philosophy teaching ban but still received a professorship for physics and mathematics. Only after a period of years, and under restrictions, he was again allowed to read philosophy. From now on, Fries was excluded from political influence. The rest of his life he devoted himself once again to philosophical and natural studies. During this period, he wrote "Mathematical Natural Philosophy" (1822) and the "History of Philosophy" (1837/40). -/- Fries suffered from a stroke on New Year's Day 1843, and a second stroke, on the 10th of August 1843 ended his life. -/- 2. Fries' Work Fries left an extensive body of work. A look at the subject areas he worked on makes us aware of the universality of his thinking. Amongst these subjects are: Psychic anthropology, psychology, pure philosophy, logic, metaphysics, ethics, politics, religious philosophy, aesthetics, natural philosophy, mathematics, physics and medical subjects, to which, e.g., the text "Regarding the optical centre in the eye together with general remarks about the theory of seeing" (1839) bear witness. With popular philosophical writings like the novel "Julius and Evagoras" (1822), or the arabesque "Longing, and a Trip to the Middle of Nowhere" (1820), he tried to make his philosophy accessible to a broader public. Anthropological considerations are shown in the methodical basis of his philosophy, and to this end, he provides the following didactic instruction for the study of his work: "If somebody wishes to study philosophy on the basis of this guide, I would recommend that after studying natural philosophy, a strict study of logic should follow in order to peruse metaphysics and its applied teachings more rapidly, followed by a strict study of criticism, followed once again by a return to an even closer study of metaphysics and its applied teachings." -/- 3. Continuation of Fries' work through the Friesian School -/- Fries' ideas found general acceptance amongst scientists and mathematicians. A large part of the followers of the "Fries School of Thought" had a scientific or mathematical background. Amongst them were biologist Matthias Jakob Schleiden, mathematics and science specialist philosopher Ernst Friedrich Apelt, the zoologist Oscar Schmidt, and the mathematician Oscar Xavier Schlömilch. Between the years 1847 and 1849, the treatises of the "Fries School of Thought", with which the publishers aimed to pursue philosophy according to the model of the natural sciences appeared. In the Kant-Fries philosophy, they saw the realisation of this ideal. The history of the "New Fries School of Thought" began in 1903. It was in this year that the philosopher Leonard Nelson gathered together a small discussion circle in Goettingen. Amongst the founding members of this circle were: A. Rüstow, C. Brinkmann and H. Goesch. In 1904 L. Nelson, A. Rüstow, H. Goesch and the student W. Mecklenburg travelled to Thuringia to find the missing Fries writings. In the same year, G. Hessenberg, K. Kaiser and Nelson published the first pamphlet from their first volume of the "Treatises of the Fries School of Thought, New Edition". -/- The school set out with the aim of searching for the missing Fries' texts, and re-publishing them with a view to re-opening discussion of Fries' brand of philosophy. The members of the circle met regularly for discussions. Additionally, larger conferences took place, mostly during the holidays. Featuring as speakers were: Otto Apelt, Otto Berg, Paul Bernays, G. Fraenkel, K. Grelling, G. Hessenberg, A. Kronfeld, O. Meyerhof, L. Nelson and R. Otto. On the 1st of March 1913, the Jakob-Friedrich-Fries society was founded. Whilst the Fries' school of thought dealt in continuum with the advancement of the Kant-Fries philosophy, the members of the Jakob-Friedrich-Fries society's main task was the dissemination of the Fries' school publications. In May/June, 1914, the organisations took part in their last common conference before the gulf created by the outbreak of the First World War. Several members died during the war. Others returned disabled. The next conference took place in 1919. A second conference followed in 1921. Nevertheless, such intensive work as had been undertaken between 1903 and 1914 was no longer possible. -/- Leonard Nelson died in October 1927. In the 1930's, the 6th and final volume of "Treatises of the Fries School of Thought, New Edition" was published. Franz Oppenheimer, Otto Meyerhof, Minna Specht and Grete Hermann were involved in their publication. -/- 4. About Mathematical Natural Philosophy -/- In 1822, Fries' "Mathematical Natural Philosophy" appeared. Fries rejects the speculative natural philosophy of his time - above all Schelling's natural philosophy. A natural study, founded on speculative philosophy, ceases with its collection, arrangement and order of well-known facts. Only a mathematical natural philosophy can deliver the necessary explanatory reasoning. The basic dictum of his mathematical natural philosophy is: "All natural theories must be definable using purely mathematically determinable reasons of explanation." Fries is of the opinion that science can attain completeness only by the subordination of the empirical facts to the metaphysical categories and mathematical laws. -/- The crux of Fries' natural philosophy is the thought that mathematics must be made fertile for use by the natural sciences. However, pure mathematics displays solely empty abstraction. To be able to apply them to the sensory world, an intermediatory connection is required. Mathematics must be connected to metaphysics. The pure mechanics, consisting of three parts are these: a) A study of geometrical movement, which considers solely the direction of the movement, b) A study of kinematics, which considers velocity in Addition, c) A study of dynamic movement, which also incorporates mass and power, as well as direction and velocity. -/- Of great interest is Fries' natural philosophy in view of its methodology, particularly with regard to the doctrine "leading maxims". Fries calls these "leading maxims" "heuristic", "because they are principal rules for scientific invention". -/- Fries' philosophy found great recognition with Carl Friedrich Gauss, amongst others. Fries asked for Gauss's opinion on his work "An Attempt at a Criticism based on the Principles of the Probability Calculus" (1842). Gauss also provided his opinions on "Mathematical Natural Philosophy" (1822) and on Fries' "History of Philosophy". Gauss acknowledged Fries' philosophy and wrote in a letter to Fries: "I have always had a great predilection for philosophical speculation, and now I am all the more happy to have a reliable teacher in you in the study of the destinies of science, from the most ancient up to the latest times, as I have not always found the desired satisfaction in my own reading of the writings of some of the philosophers. In particular, the writings of several famous (maybe better, so-called famous) philosophers who have appeared since Kant have reminded me of the sieve of a goat-milker, or to use a modern image instead of an old-fashioned one, of Münchhausen's plait, with which he pulled himself from out of the water. These amateurs would not dare make such a confession before their Masters; it would not happen were they were to consider the case upon its merits. I have often regretted not living in your locality, so as to be able to glean much pleasurable entertainment from philosophical verbal discourse." -/- The starting point of the new adoption of Fries was Nelson's article "The critical method and the relation of psychology to philosophy" (1904). Nelson dedicates special attention to Fries' re-interpretation of Kant's deduction concept. Fries awards Kant's criticism the rationale of anthropological idiom, in that he is guided by the idea that one can examine in a psychological way which knowledge we have "a priori", and how this is created, so that we can therefore recognise our own knowledge "a priori" in an empirical way. Fries understands deduction to mean an "awareness residing darkly in us is, and only open to basic metaphysical principles through conscious reflection.". -/- Nelson has pointed to an analogy between Fries' deduction and modern metamathematics. In the same manner, as with the anthropological deduction of the content of the critical investigation into the metaphysical object show, the content of mathematics become, in David Hilbert's view, the object of metamathematics. -/-. (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)

Hello Mr John Lucas, I go to school in Perth in Western Australia. In the subject mathematics at my school, we were given a project to research a given mathematician and write a report on them. I was given you. I have to incorporate some information about the mathematical times in which you live and to attempt to include details of the contribution that you made to the field of mathematics. I also have to include a short biography of your (...) life. If it is alright with you, would you be able to give me a brief biography of your life and a very brief summary of the contributions you have made to the field of mathematics. If it is possible, even the mathematical times in which we live. I hope this is not too much trouble and if you could, would you be able to e-mail it to me as soon as possible please since we don't have long to finish the project. Please e-mail it back to me on my dad's address above. I hope this is not a bother to you. Remember, I am only a high school student and I might not understand all that you hopefully send. Thankyou very much. (shrink)

This essay explores ethical foundations for meeting the digital challenge via a case study of the work, life, and virtues of the greatest mathematician and natural scientist of the eighteenth century, Leonhard Euler. By biography and history one can learn of the gifts of human strength, practices, good will, dependence on others, and friendships which made possible Euler’s own astonishing corpus of work and that of many other scientists, mathematicians, engineers, and technologists. Digital technology results from a combination of (...) science (demonstrable knowledge and method), technology, engineering, and art (forms and artifacts of making and expressing), and mathematics (abstract numerical, algebraic, geometrical, formal, and digital concepts, rules, representations, and logics). Joint reflection on the biographical, historical, and natural sources of mathematics and the digital is essential for any humanization or ascesis in response to the perils and promises of digital technology for human thriving. As ethics enable and embody an ethos, so technologies are means and manifestations of a telos. Thus, thought and action for thriving through the digital needs to contemplate and conciliate the ends of humans and of the digital. (shrink)

This article offers a critical edition of a handwritten and unpublished letter by the philosopher and mathematician Lazarus Bendavid. Created around 1805, the autograph is of great interest to the intellectual biography of the GermanJewish author as well as for the historiography of philosophy at the end of the 18th and the beginning of the 19th century. The text describes and evaluates the situation of philosophy in Germany after Kant's death in great detail, mentioning the names of no less than (...) 30 philosophers. The following edition includes a reconstruction of the communicative situation that led up to the letter, especially the identification of its recipient, and a brief characterization of its content, the rendition of the transcribed text in a text critical form, a thorough comment on the mentioned authors and works, a methodologically oriented editorial addendum and an interpretative commentary in the spirit of constellation research. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Hypatia’s Daughters: Fifteen Hundred Years of Women Philosophers ed. by Linda Lopez McAllisterSue M. WeinbergLinda Lopez McAllister, editor. Hypatia’s Daughters: Fifteen Hundred Years of Women Philosophers. Bloomington and Indianapolis: Indiana University Press, 1996. Pp. xiv + 345. Cloth, $49.95. Paper, $22.50.Hypatia: born in the fourth century A.D.: philosopher, mathematician, astronomer, teacher; brutally murdered in Alexandria in 415 A.D—whether for holding religious views regarded as heretical or because she (...) was a woman who dared to be a scholar and teacher remains a matter of speculation. Today, she gives her name to the journal for feminist philosophy. She is a symbol for those women who, over the centuries, have struggled to be part of the philosophical life of their time.Until very recently, the centuries that separated Hypatia from the present were assumed to have been a philosophical desert for women. But times have changed. Hypatia’s Daughters is a sign of this change. It contains eighteen articles on seventeen philosophers: Hypatia; Hildegard of Bingen, Heloise, and Christine de Pisan from the Middle Ages; Elizabeth of Bohemia, Sor Juana de la Cruz, Anne Conway, Damaris Masham, Catharine Trotter, Belle van Zuylen and Mary Wollstonecraft from the seventeenth and eighteenth centuries; Anne Doyle Wheeler and Harriet Taylor Mill from the nineteenth; Charlotte Perkins Oilman, Edith Stein, Hannah Arendt, and Angela Davis from the late nineteenth century into contemporary times. Few are known as well as they deserve, but many names are familiar even to non-specialists. Their ideas, however, are another matter.The collection in no way constitutes a definitive selection of women philosophers; those discussed are not presented as the most significant women philosophers in the history of philosophy. (A different kind of study would be necessary for evaluations of that kind.) The women here studied have little in common as a group; they wrote on many topics of philosophical concern; their views and their supporting reasoning differ. Further, as the subjects of the articles differ in their ideas, so do the contributors’ approaches and their methods of studying those ideas differ. These differences are the strengths of a collection that is stimulating, enlightening, provocative. The following examples indicate their range.Andrea Nye examines the correspondence of Heloise and Abelard and, in a second article, of Elizabeth of Bohemia and Descartes. Heloise and Elizabeth are usually regarded as passive followers or, at best, no more than serious questioners seeking answers of their correspondents. Nye’s fresh look at the Heloise-Abelard correspondence finds not only important differences between two ways of thinking, but also sees already underlying their disagreement what she calls the “macroquestion” of philosophy’s future: philosophy as “a professional discipline independent of political or social concerns or... an ongoing critical and cultural discourse.” (25) Elizabeth’s persistent questions to Descartes reveal ideas that challenge his views, especially on moral issues and physiological-psychological [End Page 164] phenomena; these are views that have a greater complexity than the basic issue of mind-body interaction that is usually identified as central to their exchange.Writing on Harriet Taylor Mill, Jo Ellen Jacobs examines changing attitudes towards her subject, beginning with views of Taylor Mill’s contemporaries and moving through the decades into our time. She finds parallels between criticisms of Taylor Mill and attitudes towards the so-called intellectual woman. In times encouraging of women, Taylor Mill is regarded as having made important contributions to the development of J. S. Mill’s thinking, in addition to writing and speaking about her own ideas and positions that did not always follow those of J. S. Mill. In less encouraging times, her significance tends to be minimized.Hannah Arendt is not usually thought of as a feminist. Joanne Cutting-Gray, through her discussion of Arendt’s early book on Rahel Varnhagen, develops implications of that study for Arendt as a “feminist thinker,” in particular through her focus on “the politics of alterity.” Cutting-Gray finds in the Varnhagen biography “an analogy between Jewish and female alterity [that] can stretch our understanding of women and politics.” (281)And there is more, much more: Martha Brandt Bolton’s scholarly examination of the philosophical works... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Was Isaac Newton an Arian?Thomas C. PfizenmaierHistorians of Newton's thought have been wide ranging in their assessment of his conception of the trinity. David Brewster, in his The Life of Sir Isaac Newton (1831), was fully convinced that Newton was an orthodox trinitarian, although he recognized that "a traditionary belief has long prevailed that Newton was an Arian."1 Two reasons were used to defend his conclusion that Newton was (...) orthodox. The first was a letter from John Craig, a friend of Newton, written shortly after Newton's death to John Conduitt, the husband of Newton's niece. In this letter Craig remarked that Newton's theological opinions "were sometimes different from those which are commonly received" but that he hoped Conduitt would publish Newton's theological papers, "that the world may see that Sir Isaac Newton was as good a Christian as he was a mathematician and philosopher."2 The second reason with which Brewster defended his conclusion was his acknowledgment that the doctrine of the trinity itself had variations.I had no hesitation when writing the Life of Sir Isaac Newton in 1830, in coming to the conclusion that he was a believer in the Trinity; and in giving this opinion on the creed of so great a man, and so indefatigable a student of scripture, I was well aware that there are various forms of Trinitarian truth, and various modes of expressing it, which have been received as orthodox in the purest societies of the Christian Church.3L. T. More in his biography Isaac Newton gently chided Brewster for not publishing some of the crucial manuscripts, which eventually became [End Page 57] part of the Keynes collection.4 From these additional manuscripts More reached the conclusion that Newton was not orthodox but an Arian. He proceeded to say that Newton was not only an Arian but, because of the manner in which he understood Jesus' role as prophet, a Unitarian.5More recently the famous British economist John M. Keynes, who considered it a great "impiety" that the Portsmouth family's collection of Newton's non-scientific manuscripts were dispersed at the Sotheby's auction in 1936 and who recovered about half of them creating the "Keynes" collection at Cambridge, provided yet a third interpretation of Newton's thought.Very early in life Newton abandoned orthodox belief in the Trinity. At this time the Socinians were an important Arian sect amongst intellectual circles. It may be that Newton fell under Socinian influences, but I think not. He was rather a Judaic monotheist of the school of Maimonides. He arrived at this conclusion, not on so-to-speak rational or skeptical grounds, but entirely on the interpretation of ancient authority. He was persuaded that the revealed documents give no support to the Trinitarian doctrines which were due to late falsifications. The revealed God was one God.6Among contemporary scholars, the consensus is that Newton was an Arian rather than a Socinian.7 This view is clearly expressed by Richard [End Page 58] Westfall who writes, "Well before 1675, Newton had become an Arian in the original sense of the term. He recognized Christ as a divine mediator between God and man, who was subordinate to the Father who created him."8 Yet Frank Manuel, in his The Religion of Isaac Newton (1974), advises that we not try to push Newton into a theological categorization too quickly. He reminds us that Newton was, if nothing else, an original thinker.It is an error to seize upon his antitrinitrianism in order to pigeonhole him in one of the recognized categories of heresy—Arian, Socinian, Unitarian, or Deist.9So how did Isaac Newton conceive of the trinity? In exploring his writing on the subject we will be guided by two principles mentioned above. The first is Brewster's insight that the trinitarian doctrine has in fact been understood with a certain degree of variety within the sphere of orthodoxy. The second is from Manuel's reminder, that we must resist the temptation to make Newton "fit" some predetermined category or school of trinitarian thought. We will begin by looking at the resources available for studying... (shrink)

During the fifteenth century, the Scientific Revolution signaled a major shift in the way people viewed the natural world. Today, René Descartes is perhaps best known as the father of modern Western philosophy, but he also played an important role in the development of a rational approach toward scientific questions. He was a gifted mathematician and his examinations of the natural world led him to develop theories about light, the formation of the universe, and how the human mind works. This (...) biography shows how Descartes's rational method inspired his own discoveries and continues to resonate today. (shrink)

William Burnside was one of the three most important algebraists who were involved in the transformation of group theory from its nineteenth-century origins to a deeper twentieth-century subject. Building on work of earlier mathematicians, they were able to develop sophisticated tools for solving difficult problems. His works are of enormous historical importance; they remain also a source of inspiration and information. The works of his contemporaries, such as Klein, Frobenius, Schur, have been published as also have the works of (...) his immediate successors such as Phillip Hall. All of Burnside's papers are reproduced here, organized chronologically and with a detailed bibliography. Walter Feit has contributed a foreword, and a collection of introductory essays are included to provide a commentary on Burnside's work and set it in perspective along with a modern biography that draws on archive material. This is the first reference volume of Burnside's collected papers, enhanced by a series of critical essays on his work, and is important for group theorists and historians and philosophers of mathematics and fills the gap in this area of literature. (shrink)

This paper compares and contrasts the main metaphysical and religious ideas of the eighteenth-century political theorist and chemist Joseph Priestley (1733–1804) with those of his correspondent Ruder Boscovic (1711–1787), astronomer, poet, mathematician, diplomat and Jesuit priest. It points out the theological differences between the two thinkers resulting from the divergent ontological and metaphysical implications of the theory of point atomism that they shared. This theory they placed in a wider context, considering both its limits and its value as a contribution (...) to ongoing speculations concerning the nature of space and time, theological conundrums such as free will and the mind-body problem. Where they differed, however, was that Priestley, a chemist rather than a mathematician, used point atomism mainly to support his campaign to further materialism and discredit Christianity. Boscovic, on the other hand, carefully distinguished the truths of faith and reason and was therefore indignant at Priestley's misuse of his speculations. Boscovic's protest was probably motivated in part by the materialism of atheists such as the Baron d’Holbach (1723–1789). Priestley was no atheist, however. Interestingly, François Arouet de Voltaire (1698–1778), unaware of the works of either Boscovic or Priestley, devised a theistic materialism in his last years that in many respects resembled Priestley's. I begin with brief biographies of the two thinkers, and outline their intermittent relationship. (shrink)