In recent philosophy of mathematics avariety of writers have presented ``structuralist''views and arguments. There are, however, a number ofsubstantive differences in what their proponents take``structuralism'' to be. In this paper we make explicitthese differences, as well as some underlyingsimilarities and common roots. We thus identifysystematically and in detail, several main variants ofstructuralism, including some not often recognized assuch. As a result the relations between thesevariants, and between the respective problems theyface, become manifest. Throughout our focus is onsemantic and metaphysical issues, (...) including what is orcould be meant by ``structure'' in this connection. (shrink)

This paper considers philosophical approaches that are relevant to the intertwinement of logic, metaphysics, and psychology proposed by the Aquinas commentator Tommaso de Vio Cardinal Cajetan, the humanist Petrus Ramus, the pure Aristotelian Cornelius Martini, the Semi-Ramist Bartholomaeus Keckermann, and the lexicographer Rudolf Goclenius. Mostly, however, it is about Ramus and his followers, the Ramists, because of the role they played in exacerbating a discussion on the constitution of objectivity during the Renaissance that was to have an impact on Cartesian (...) andpost-Cartesian theories of subjectivity. Finally, keeping in mind that Kant was familiar with the secunda Petri, i.e., with the second part of Ramus's logic, namely the theory of judgment, some common ground is recognizable between Ramus and Kant as well. (shrink)

Historians of science find it difficult to pinpoint to an exact period in which symbolic algebra came into existence. This can be explained partly because the historical process leading to this breakthrough in mathematics has been a complex and diffuse one. On the other hand, it might also be the case that in the early twentieth century, historians of mathematics over emphasized the achievements in algebraic procedures and underestimated the conceptual changes leading to symbolic algebra. This paper attempts to provide (...) a more precise setting for the historical context in which this decisive step to symbolic reasoning took place. For that purpose we will consider algebraic problem solving as model-based reasoning and symbolic representation as a model. This allows us to characterize the emergence of symbolic algebra as a shift from a geometrical to a symbolic mode of representation. The use of the symbolic as a model will be situated in the context of mercantilism where merchant activity of exchange has led to reciprocal relations between money and wealth. (shrink)

Important study focuses on the revival and assimilation of ancient Greek mathematics in the 13th–16th centuries, via Arabic science, and the 16th-century development of symbolic algebra. This brought about the crucial change in the concept of number that made possible modern science — in which the symbolic "form" of a mathematical statement is completely inseparable from its "content" of physical meaning. Includes a translation of Vieta's Introduction to the Analytical Art. 1968 edition. Bibliography.

The Renaissance has long been recognized as a brilliant moment in the development of Western civilization. Little attention has been devoted, however, to the distinct contribution of philosophy to Renaissance culture. This volume introduces the reader to the philosophy written, read, taught, and debated during the period traditionally credited with the "revival of learning." Beginning with original sources still largely inaccessible to most readers, and drawing on a wide range of secondary studies, the author examines the relation of Renaissance philosophy (...) to humanism and the universities, the impact of rediscovered ancient sources, the recovery of Plato and the Neoplatonists, and the evolving ascendancy of Aristotle. Renaissance Philosophy also explores the original contributions of major figures including Bruni, Valla, Ficino, Pico della Mirandola, Pomponazzi, Machiavelli, More, Vitoria, Montaigne, Bruno, and Camapanella. (shrink)

Lorenzo Valla's rhetorical reform of logic resulted in important changes in sixteenth-century mathematical sciences, and not only in mathematical education and in the use of mathematics in other sciences, but also in mathematical theory itself. Logic came to be identified with dialectic, syllogisms with enthymemes and necessary truth with the limit case of probable truth. Two main ancient authorities mediated between logical and mathematical concerns: Cicero and Proclus. Cicero's 'common notions' were identified with Euclid's axioms, so that mathematics could be (...) viewed as core knowledge shared by all humankind. Proclus' interpretation of Euclid's axioms gave rise to the idea of a universal human natural light of reasoning and of a mathesis universalis as a basic mathematics common to both arithmetic and geometry and as an art of thinking interpretable as algebra. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Symbolic Mathematics and the Intellect Militant: On Modern Philosophy’s Revolutionary SpiritCarl PageWhat makes modern philosophy different? My question presupposes the legitimacy of calling part of philosophy “modern.” That presupposition is in turn open to question as regards its meaning, its warrant, and the conditions of its applicability. 1 Importance notwithstanding, such further inquiries all start out from the phenomenon upon which everyone agrees: philosophy running through Plato and Aristotle (...) looks significantly different from philosophy running from Descartes to Kant.My concern in this essay is with the phenomenon of the difference itself, rather than with the second-order questions associated with how properly to assign it historical meaning. I take the difference between ancient and modern philosophy to be as significant as differences in philosophy’s history can be: modern philosophy rests on a new interpretation of the nature and fulfillment of human reason, and disputes about the nature of human reason are the ultimate battles of philosophy. But the general thesis is not my main point. 2 The focus of this essay falls on what may be called the integrity of the phenomenon, on the specific interpretation of human reason that lends modern philosophy its peculiar face. [End Page 233]Modern philosophy’s strikingly revolutionary spirit is my point of departure. When Descartes writes in the first of his Meditations that “it was necessary, once in the course of my life, to demolish everything completely and start again right from the foundations if I wanted to establish anything at all in the sciences that was stable and likely to last,” 3 he reveals the same enthusiasm for total reform later found in Kant: “This attempt to alter the procedure which has hitherto prevailed in metaphysics, by completely revolutionizing it in accordance with the example set by the geometers and physicists, forms indeed the main purpose of this critique of pure speculative reason.” 4 How exactly did philosophy become so convinced that its central tradition—a sprawling, disorganized, ugly city, as Descartes has it in the Discourse (I:116)—needed razing to the ground in the interest of some rational town-planning? Moreover, the calls for revolution have not abated, despite contemporary disillusionment with both Cartesian rationalism and Enlightenment philosophy in general; they have grown more shrill. The confidence with which rationalism, foundationalism, universalism, logocentrism, Platonism, and so on are currently set at naught for the sake of contingency, particularity, and difference reveals the same revolutionary and totalizing spirit that marks the earlier phase of philosophy’s modernity. Such enthusiasm is reason’s freedom taken to an extreme. 5 What inspires this march of the intellect militant? What, if anything, justifies its hubristic self-assertions in the domain of philosophy? These are the questions I address.Descartes is commonly identified as the father of modern philosophy. While the full story of modern philosophy’s parentage is more complicated than this, it is fair to say that in Descartes self-consciousness of a new mode of doing philosophy emerges with a focus and revolutionary sense of purpose that caught philosophical imagination in his own time and continues to do so in ours. 6 Motifs of modern philosophy may be found in many places—Machiavelli, Hobbes, Francis Bacon, Nicholas of Cusa, Giordano Bruno, Jakob Boehme—nonetheless, Descartes’s intensely single-minded, even jealous advocacy commends itself to all but the most stubborn antiquarian mentality as modernity’s almost perfect philosophical representative. [End Page 234]That Descartes stands on a remarkable philosophical cusp is apparent in the contrast between the title and the subject matter of his most influential philosophical work: Meditationes de Prima Philosophiae (1641). To that point prima philosophia or First Philosophy had been construed as the metaphysics. It was not concerned with the critical question of how metaphysical sciences are possible and was not directly related to any doctrine of the human soul—except perhaps on the one point of the divinity of nous, the human soul’s highest part. In Descartes’s Meditations, on the other hand, the landscape has altered. Doubt, certainty, knowledge, the ego cogitans and its stream of representations are the new subject matter. What is first in philosophy is no longer what... (shrink)

In the seventeenth century the microscope opened up a new world of observation, and, according to Catherine Wilson, profoundly revised the thinking of scientists and philosophers alike. The interior of nature, once closed off to both sympathetic intuition and direct perception, was now accessible with the help of optical instruments. The microscope led to a conception of science as an objective, procedure-driven mode of inquiry and renewed interest in atomism and mechanism. Focusing on the earliest forays into microscopical research, from (...) 1620 to 1720, this book provides us with both a compelling technological history and a lively assessment of the new knowledge that helped launch philosophy into the modern era. Wilson argues that the discovery of the microworld--and the apparent role of living animalcula in generation, contagion, and disease--presented metaphysicians with the task of reconciling the ubiquity of life with human-centered theological systems. It was also a source of problems for philosophers concerned with essences, qualities, and the limits of human knowledge, whose positions are echoed in current debates about realism and instrument-mediated knowledge. Covering the contributions of pioneering microscopists (Leeuwenhoek, Swammerdam, Malpighi, Grew, and Hooke) and the work of philosophers interested in the microworld (Bacon, Descartes, Leibniz, Malebranche, Locke, and Berkeley), she challenges historians who view the abstract sciences as the sole catalyst of the Scientific Revolution as she stresses the importance of observational and experimental science to the modern intellect. (shrink)

The Ethics of Geometry is a study of the relationship between philosophy and mathematics. Essential differences in the ethos of mathematics, for example, the customary ways of undertaking and understanding mathematical procedures and their objects, provide insight into the fundamental issues in the quarrel of moderns with ancients. Two signal features of the modern ethos are the priority of problem-solving over theorem-proving, and the claim that constructability by human minds or instruments establishes the existence of relevant entities. These figures are (...) combined in the emblematic statement of Salomon Maimon, "In mathematical construction we are, as it were, gods." Construction is the mark of modernity. The disciplines of classical philology, literary interpretation and the history of philosophy and of mathematics are woven together in this volume. (shrink)

This book offers a balanced and comprehensive account of philosophical thought from the middle of the fourteenth century to the emergence of modern philosophy at the turn of the seventeenth century.

In sixteenth century Paris a circle of mathematicians produced texts of the most advanced kind of algebra. This "French algebraic tradition" will be the context for Viete's symbolic algebra. Comparing French algebraic texts with Italian and German, and examining the publishing context , I establish a periodization in two phases. ;Jacques Peletier stands for the introduction of the abacus tradition and algebra at the court. Peletier's algebraic program is connected to his theory of rhetoric. He establishes a genre of texts (...) devoted to algebra in vernacular, promoting French as a scientific language. Rhetorical criteria order L'Algebre, emphasizing structure and theory . He innovates on Cardano's and Stifel's treatments of equations in several unknowns. ;Phase two changes style and content; Guillaume Gosselin is representative. Manuals shift focus from problems and questions to equations and classification. Solution of equations becomes the purpose of algebra. Problems are no longer addressed as such, but conceived in their most general form as corresponding equations. Algebrists connect with a milieu of jurists important in politics and scholarship. This erudite milieu favored the recovery of Diophantus' Arithmetic, which provides a set of theoretical problems on numbers. Diophantus solved problems by transforming them into particular cases; Gosselin reaches a general solution by transforming them into equations. Gosselin's notation makes the difference. The algebrists create an illustrious genealogy for algebra, deriving it from Greece rather than abacus schools. ;Three features of this tradition can be traced back to Italian humanism: Cardano's and Tartaglia's algebra, imitatio , and the construction of a history for the discipline. The French algebrists radically transformed all three. Their translatio authorized them to abandon links to the medieval tradition and to build a new discipline that they could see as national. Preparing the adoption of this discipline by the legal elite was the rhetorical interpretation of logic developed in Paris at the time. This provided a theoretical frame in which generalized algebraic problems were seen as Cicero's quaestiones infinitae, i.e. scientific questions. (shrink)