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)

Traces the development of the integral and the differential calculus and related theories since ancient times.

Indeed, this is the first serious book-length study of the subject by a philosopher of science.

Philologus, founded in 1846, is one of the oldest and most respected periodicals in the field of Classics. It publishes articles on Greek and Latin literature, historiography, philosophy, history of religion, linguistics, reception, and the history of scholarship. The journal aims to contribute to our understanding of Greco-Roman culture and its lasting influence on European civilization. The journal Philologus, conceived as a forum for discussion among different methodological approaches to the study of ancient texts and their reception, publishes (...) original scholarly contributions with a strong international focus. Publications languages are: German, English, Italian, French. As of 2014 the publication of supplementary volumes will be resumed. The series Philologus. Supplemente / Philologus. Supplementary Volumes will publish monographs and edited volumes pertaining to all aspects of the study of ancient literature and its reception, with a special focus on topics concerning literary theory, philosophy and the history of scholarship from an interdisciplinary perspective. (shrink)

There is a common tradition in European education going back to the Middle Ages which long played a part in providing the curriculum of schools which catered both for the wealthy and for able sons of less well-to-do families. Originally published in 1974, this volume examines the relationship between education and society in the different countries of Europe from which differences in tradition and practice emerge. The countries discussed include: France, Germany, the former Soviet Union, Poland and Sweden.

Originally published in 1972, this book is concerned with education as part of a larger social history. Chapters include: The roots of Anglican supremacy in English education The Board schools of London The use of ecclesiastical records for the history of education Topographical resources: private and secondary education from the sixteenth to the twentieth century.

Originally published in 1971, this volume examines the relationship between the history and sociology of education. History does not stand in isolation, but has much to draw from and contribute to, other disciplines. The methods and concepts of sociology, in particular, are exerting increasing influence on historical studies, especially the history of education. Since education is considered to be part of the social system, historians and sociologists have come to survey similar fields; yet each discipline appears to (...) have its own aims and methodology. (shrink)

Part of the educational system in England has been geared towards the preparation of particular professions, while the identity and status of members of some professions have depended significantly on the general education they have received. Originally published in 1973, this volume explores the interaction between education and the professions. It also looks at the education of the main professions in sixteenth century England and at how twentieth century university teaching is a key profession for the training of new recruits (...) to other professions. (shrink)

Cauchy’s contribution to the foundations of analysis is often viewed through the lens of developments that occurred some decades later, namely the formalisation of analysis on the basis of the epsilon-delta doctrine in the context of an Archimedean continuum. What does one see if one refrains from viewing Cauchy as if he had read Weierstrass already? One sees, with Felix Klein, a parallel thread for the development of analysis, in the context of an infinitesimal-enriched continuum. One sees, with Emile Borel, (...) the seeds of the theory of rates of growth of functions as developed by Paul du Bois-Reymond. One sees, with E. G. Björling, an infinitesimal definition of the criterion of uniform convergence. Cauchy’s foundational stance is hereby reconsidered. (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)

The translation of Newton’s geometrical Propositions in the Principia into the language of the differential calculus in the form developed by Leibniz and his followers has been the subject of many scholarly articles and books. One of the most vexing problems in this translation concerns the transition from the discrete polygonal orbits and force impulses in Prop. 1 to the continuous orbits and forces in Prop. 6. Newton justified this transition by lemma 1 on prime and ultimate ratios which (...) was a concrete formulation of a limit, but it took another century before this concept was established on a rigorous mathematical basis. This difficulty was mirrored in the newly developed calculus which dealt with differentials that vanish in this limit, and therefore were considered to be fictional quantities by some mathematicians. Despite these problems, early practitioners of the differential calculus like Jacob Hermann, Pierre Varignon, and Johann Bernoulli succeeded without apparent difficulties in applying the differential calculus to the solution of the fundamental problem of orbital motion under the action of inverse square central forces. By following their calculations and describing some essential details that have been ignored in the past, I clarify the reason why the lack of rigor in establishing the continuum limit was not a practical problem. (shrink)

The anonimous scholiumOn all forms of syllogism was copied in 1884 from the Paris Codex 2064 by E. Richter. In 1899 M. Wallies published it in the preface to Ammonius' commentary on the Prior Analytics of Aristotle. There appear in that scholium, apart from the complex figure of Galenos, other characteristic forms of inference.Among these forms I found five so-called non-demonstrable stoic syllogisms, three modifications of the law of transposition of which the third is not mentioned by the authors of (...) Princ pia Mathematica, and a modification of the form of inference known as Euclid's law. This form of inference was applied by Euclid in mathematics and by Saccherius in syllogistics; it is mentioned for the first time by Cardan in a treatise of 1570 and later by Clavius in his commentary of 1574 on the Elements of Euclid and in the commentary on Theodosius'Sphaerica of the year 1586.In 1658 Erhard Weigel made the first attempt at refuting the logical law of Euclid as formulated by Cardan and Clavius and in 1686 James Bernoulli tried to prove it. (shrink)

This book is part of a major project undertaken by the Centre for Studies in Civilizations , being one of a total of ninety-six planned volumes. The author is a statistician and computer scientist by training, who has concentrated on historical matters for the last ten years or so. The book has very ambitious aims, proposing an alternative philosophy of mathematics and a deviant history of the calculus. Throughout, there is an emphasis on the need to combine (...) class='Hi'>history and philosophy of mathematics, especially in order to evaluate properly the history of mathematics in India, in particular the history of the calculus.The pivotal goals of the book are to oppose the Eurocentric account of the history of science in general and mathematics in particular; to avoid the usual philosophical idea of the centrality of proof for mathematical knowledge, in favour of the traditional Indian notion of pramāṇa [validation] encompassing empirical elements and emphasizing calculation; to analyze the thousand-year-long development of infinite series in India, starting in the fifth century; and to show ‘how and why the calculus was imported into Europe’ from about 1500. The result is a picture in which inputs from the Indian subcontinent and epistemology are the driving forces of the history of mathematics, as people in the European subcontinent struggle to adopt new calculation techniques from the East in spite of Western philosophico-religious biases . Thus, a ‘first math war’ involved the algorismus de numero indorum, adopted for practical reasons, which forced Europeans to modify their epistemology of number and quantities. A ‘second math war’ revolved around infinite series and the calculus, since the background of Western epistemology created ‘spurious difficulties’ about infinities and infinitesimals, partially resolved with theories of real numbers. And a ‘third math war’ is …. (shrink)

The consensus today is that both Newton and Leibniz created calculus independently. Yet, this was not so clear at the beginning of the eighteenth century. A bitter controversy took place at that time, which came to be known as the ‘calculus war’, probably the greatest clash in the history of science. While it is accepted that the debate started when Fatio de Duillier publicly accused Leibniz of plagiarism in 1699, earlier evidence of its origins can be found (...) in an exchange of letters between Leibniz and Huygens. (shrink)

Traces the origins and development of arithmetic, statistics, geometry, and calculus from the ancient civilizations to the present.

The Pythagorean Life is the most extensive surviving source on Pythagoreanism, and has wider interest as an account of the religious aspirations of late antiquity. "...admirably clear translation and sensible introduction"--The Classical...

This paper is an historical study of Tarski's methodology of deductive sciences (in which a logic S is identified with an operator Cn S, called the consequence operator, on a given set of expressions), from its appearance in 1930 to the end of the 1970s, focusing on the work done in the field by Roberto Magari, Piero Mangani and by some of their pupils between 1965 and 1974, and comparing it with the results achieved by Tarski and the Polish school (...) (Łoś, Suszko, Słupecki, Pogorzelski, Wójcicki). In the last section of the paper we will then compare these works with some recent developments in algebraic logic: this will lead to a better understanding of the results of the methodology of deductive science, but at the same time will show some intrinsic limits to such an approach to logic. Even if Magari's work on diagonizable algebras and universal algebra and Mangani's axiomatization of MV-algebras and results in model theory are rather famous, the articles on closure operators, published in the 1960s, are almost totally unknown outside Italy (mainly because of a linguistic limitation, the papers we analyse having been written and published in Italian). This paper aims to fill the gap in the literature and to enable the international community to get acquainted with this part of Italian logic. The same applies to some works published in Barcelona (in Catalan) at the end of the 1970s, analysed in the last section. (shrink)

In a paper read before the last Congress for the Unity of Science, Dr. Milton Singer distinguishes three main phases in the recent history of logic. The achievement he considers most characteristic of the first period is the development of the class calculus or so-called Boolean algebra. It begins with the work of Boole and DeMorgan and culminates in Schroeder's Algebra of Logic. In a minimum formulation, the results of this first stage can be summed up as, first, (...) simplification and generalization of the traditional syllogistic machinery by means of a rather efficient new symbolism, and, second, adoption of the axiomatic method. To restate the last point: Boolean algebra was, since Leibniz, the first explicit attempt to construct what we call a formal system or calculus. Not realized at that time, however, or at least not adequately expressed, was the importance of the more recent distinction between scientific calculi on the one hand and purely linguistic or logical calculi on the other. As a consequence of this lack of explicit insight, the class calculus appeared to be on a par with scientific calculi, say, for instance, axiomatized geometry, Euclidean or otherwise. Its analytical character, even if it was seen, could not be formulated. (shrink)

In this paper, we endeavour to give a historically accurate presentation of how Leibniz understood his infinitesimals, and how he justified their use. Some authors claim that when Leibniz called them “fictions” in response to the criticisms of the calculus by Rolle and others at the turn of the century, he had in mind a different meaning of “fiction” than in his earlier work, involving a commitment to their existence as non-Archimedean elements of the continuum. Against this, we show (...) that by 1676 Leibniz had already developed an interpretation from which he never wavered, according to which infinitesimals, like infinite wholes, cannot be regarded as existing because their concepts entail contradictions, even though they may be used as if they exist under certain specified conditions—a conception he later characterized as “syncategorematic”. Thus, one cannot infer the existence of infinitesimals from their successful use. By a detailed analysis of Leibniz’s arguments in his De quadratura of 1675–1676, we show that Leibniz had already presented there two strategies for presenting infinitesimalist methods, one in which one uses finite quantities that can be made as small as necessary in order for the error to be smaller than can be assigned, and thus zero; and another “direct” method in which the infinite and infinitely small are introduced by a fiction analogous to imaginary roots in algebra, and to points at infinity in projective geometry. We then show how in his mature papers the latter strategy, now articulated as based on the Law of Continuity, is presented to critics of the calculus as being equally constitutive for the foundations of algebra and geometry and also as being provably rigorous according to the accepted standards in keeping with the Archimedean axiom. (shrink)

A wide-ranging history of the intellectual developments that produced the modern idea of the algorithm. Bringing together the histories of mathematics, computer science, and linguistic thought, Language and the Rise of the Algorithm reveals how recent developments in artificial intelligence are reopening an issue that troubled mathematicians long before the computer age. How do you draw the line between computational rules and the complexities of making systems comprehensible to people? Here Jeffrey M. Binder offers a compelling tour of four (...) visions of universal computation that addressed this issue in very different ways: G. W. Leibniz's calculus ratiocinator; a universal algebra scheme Nicolas de Condorcet designed during the French Revolution; George Boole's nineteenth-century logic system; and the early programming language ALGOL, whose name is short for algorithmic language. These episodes show that symbolic computation has repeatedly become entangled in debates about the nature of communication. To what extent can meaning be controlled by individuals, like the values of a and b in algebra, and to what extent is meaning inevitably social? By attending to this long-neglected question, we come to see that the modern idea of the algorithm is implicated in a long history of attempts to maintain a disciplinary boundary separating technical knowledge from the languages people speak day to day. Machine learning, in its increasing dependence on words, now places this boundary in jeopardy, making its stakes all the more urgent to understand. The idea of the algorithm is a levee holding back the social complexity of language, and it is about to break. This book is about the flood that inspired its construction. (shrink)

How the concept of proof has enabled the creation of mathematical knowledge. The Story of Proof investigates the evolution of the concept of proof--one of the most significant and defining features of mathematical thought--through critical episodes in its history. From the Pythagorean theorem to modern times, and across all major mathematical disciplines, John Stillwell demonstrates that proof is a mathematically vital concept, inspiring innovation and playing a critical role in generating knowledge. Stillwell begins with Euclid and his influence on (...) the development of geometry and its methods of proof, followed by algebra, which began as a self-contained discipline but later came to rival geometry in its mathematical impact. In particular, the infinite processes of calculus were at first viewed as "infinitesimal algebra," and calculus became an arena for algebraic, computational proofs rather than axiomatic proofs in the style of Euclid. Stillwell proceeds to the areas of number theory, non-Euclidean geometry, topology, and logic, and peers into the deep chasm between natural number arithmetic and the real numbers. In its depths, Cantor, Gödel, Turing, and others found that the concept of proof is ultimately part of arithmetic. This startling fact imposes fundamental limits on what theorems can be proved and what problems can be solved. Shedding light on the workings of mathematics at its most fundamental levels, The Story of Proof offers a compelling new perspective on the field's power and progress. (shrink)

As with mathematics, logic is easier to do if its symbols and their rules are better. In a graphic way, the logic symbols introduced in thís paper show their truth-table values, their composite truth-functions, and how to say them as either ?or? or ?if ? then? propositions. Simple rules make the converse, add or remove negations, and resolve propositions.

A three-valued propositional logic is presented, within which the three values are read as ?true?, ?false? and ?nonsense?. A three-valued extended functional calculus, unrestricted by the theory of types, is then developed. Within the latter system, Bochvar analyzes the Russell paradox and the Grelling-Weyl paradox, formally demonstrating the meaninglessness of both.