Viewed this way, the texts yield striking examples of language and notation that are irreducibly ambiguous and productive because they are ambiguous.

This book draws its inspiration from Hilbert, Wittgenstein, Cavaillès and Lakatos and is designed to reconfigure contemporary philosophy of mathematics by making the growth of knowledge rather than its foundations central to the study of mathematical rationality, and by analyzing the notion of growth in historical as well as logical terms. Not a mere compendium of opinions, it is organised in dialogical forms, with each philosophical thesis answered by one or more historical case studies designed to support, complicate or question (...) it. The first part of the book examines the role of scientific theory and empirical fact in the growth of mathematical knowledge. The second examines the role of abstraction, analysis and axiomatization. The third raises the question of whether the growth of mathematical knowledge constitutes progress, and how progress may be understood. Readership: Students and scholars concerned with the history and philosophy of mathematics and the formal sciences. (shrink)

The Cartesian method, construed as a way of organizing domains of knowledge according to the "order of reasons," was a powerful reductive tool. Descartes made significant strides in mathematics, physics, and metaphysics by relating certain complex items and problems back to more simple elements that served as starting points for his inquiries. But his reductive method also impoverished these domains in important ways, for it tended to restrict geometry to the study of straight line segments, physics to the study of (...) ambiguously constituted bits of matter in motion, and metaphysics to the study of the isolated, incorporeal knower. This book examines in detail the negative and positive impact of Descartes's method on his scientific and philosophical enterprises, exemplified by the Geometry, the Principles, the Treatise of Man, and the Meditations. (shrink)

This book explicates Leibnizian analysis as a search for conditions of intelligibility, and reconsiders his use of principles and methods as well as his account of truth in this way. Via careful reading of well-known, lesser known, and previously unedited texts, it gives a more accurate picture of his philosophical intentions, as well as the relevance of his project to contemporary debate. Two case studies are included, one concerning logic and the other arithmetic; they illustrate a theory of intelligibility that (...) takes as its central notion "possibility for thought", a notion which allows Leibniz to escape certain traps of psychologism, the pseudo-ontology of empiricism, and the empty forms of logicism, and suggests new approaches for contemporary philosophy. "In this remarkable study, Grosholz and Yakira offer a fresh interpretive and conceptual angle on Leibniz's metaphysics. [...] this study deserves high marks for its subtlety, novelty, and creative insight into Leibniz's modes of inquiry as well as for its philosophical acumen." Annals of Science. (shrink)

This book deals with a topic that has been largely neglected by philosophers of science to date: the ability to refer and analyze in tandem. On the basis of a set of philosophical case studies involving both problems in number theory and issues concerning time and cosmology from the era of Galileo, Newton and Leibniz up through the present day, the author argues that scientific knowledge is a combination of accurate reference and analytical interpretation. In order to think well, we (...) must be able to refer successfully, so that we can show publicly and clearly what we are talking about. And we must be able to analyze well, that is, to discover productive and explanatory conditions of intelligibility for the things we are thinking about. The book’s central claim is that the kinds of representations that make successful reference possible and those that make successful analysis possible are not the same, so that significant scientific and mathematical work typically proceeds by means of a heterogeneous discourse that juxtaposes and often superimposes a variety of kinds of representation, including formal and natural languages as well as more iconic modes. It demonstrates the virtues and necessity of heterogeneity in historically central reasoning, thus filling an important gap in the literature and fostering a new, timely discussion on the epistemology of science and mathematics. (shrink)

Most discussions of Leibniz's metaphysics of time and space begin and end with the correspondence between Leibniz and Samuel Clarke, Newton's friend and defender. But Leibniz's ideas about time and space are far richer than this exchange suggests, and Michael Futch shows that the study of those investigations will enhance current discussion among philosophers and cosmologists. Futch's scholarly attention to a wide range of texts is matched by his philosophical acuity. His detailed expositions of texts are not tedious or pedantic (...) because he never fails to bring out their philosophical interest and pertinence. So, too, the book's bibliography is especially valuable.The book opens with a discussion of Newton's substantivalism, which treats time and space as substance-like structures that exist independent of the objects located in them, versus Leibniz's reductionism, which holds time and space to be merely relations, ontologically dependent on the things that exist: space is the order of coexistences, time the order of successions. Futch gives useful historical background that explains why the debate takes the form that it does and what. (shrink)

This chapter treats the reception and assessment of the two English translations of Simone de Beauvoir's Le deuxième sexe, the first by Howard M. Parshley in 1953 and the second by Constance Borde and Sheila Malovany‐Chevallier in 2009. We examine both the criticisms and the appreciations, concluding that the second is superior in many ways to the first. On that basis, we propose a digital edition of the original book and its earlier drafts en face the 2009 English translation, which (...) would also allow as hypertext the addition of curated scholarly notes and textual exegesis, alongside crowd‐sourced discussion. This would both make the book more accessible worldwide, and add the scholarly dimension for which many Beauvoir scholars have long been waiting. (shrink)

W. E. B. Du Bois was one of the most profound and influential African-American intellectuals of the twentieth century. This volume addresses the complexities of Du Bois' legacy, showing how his work gets to the heart of today's theorizing about the color line.

The problematic relation between logic and knowledge has given rise to some of the most important works in the history of philosophy, from Books VIVII of Platos Republic and Aristotles Prior and Posterior Analytics, to Kants Critique of Pure Reason and Mills A System of Logic, Ratiocinative and Inductive. It provides the title of an important collection of papers by Bertrand Russell. However, it has remained an underdeveloped theme in the last century, because logic has been treated as separate from (...) knowledge. This book does not hope to make up for a century-long absence of discussion. Rather, its ambition is to call attention to the theme and stimulating renewed reflection upon it. The book collects essays of leading figures in the field and it addresses the theme as a topic of current debate, or as a historical case study, or when appropriate as both. Each essay is followed by the comments of a younger discussant, in an attempt to transform what might otherwise appear as a monologue into an ongoing dialogue; each section begins with an historical essay and ends with an essay by one of the editors. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Plato and Leibniz against the MaterialistsEmily GrosholzImportant parallels hold between Leibniz’s attitude towards materialism and that of Plato. Both philosophers were interested in and hostile to materialism, and their qualified rejection of materialism became crucial to the systems of their maturity. Leibniz’s attachment to Plato began very early: in a text of 1664 Leibniz quoted the Timaeus, 1 and in another of 1670 he claimed that the Timaeus, along (...) with the Parmenides and the Phaedo, were his favorite dialogues. 2 The influence of the Timaeus is quite apparent in Leibniz’s correspondence with the French Platonist Nicholas Remond (1713–16) and certain essays (particularly “On the Ultimate Origination of Things” [1697] and “Tentamen Anagogicum”), where he appeals to Plato in distinguishing his own position from the materialism of his day, which revived and transformed that of Leucippus and Democritus, Epicurus, and the poet Lucretius.The way in which Leibniz resists and circumvents materialism in these letters and essays has a family resemblance with the way Plato addresses and solves certain metaphysical problems, some of them raised by materialism, in the Timaeus. All the same, the Platonic and Leibnizian strategies also diverge in significant respects. For Plato belongs to the ancient world and Leibniz to the modern world, in which the topic of subjectivity becomes central. Although both philosophers make essential use of abstract mathematical schemata to organize and inform their philosophical positions, Plato tends to give them a constitutive role and Leibniz a regulative role. [End Page 255]What is the point of establishing this analogy, a likeness nuanced by unlikeness, in the metaphysical strategies of Plato and Leibniz? First, it is important to examine philosophical genealogies. In this case the genealogy is attested by Leibniz himself, who feels at once indebted to and impatient with his forebear. Yet it has received much less scholarly recognition than the more obvious link between Aristotle and Leibniz. Second, this kind of examination exhibits the shape and tensile strength of metaphysical systems, as tragedy or fiction exhibits the extent and limits of moral and political systems. I assume that certain metaphysical problems and certain possibilities for metaphysical systems recur through time and that those things that are perennial in philosophy point to what is eternal about it, but I also believe that there is something peculiar about modernity, which imposes a sea-change on philosophy after Descartes. Materialism is a recurrent metaphysical possibility with which contemporary philosophers must still come to terms; the peculiar nature of modern philosophy introduces new ways to address materialism. Surely the study of how Plato and Leibniz took the measure of materialism will be illuminating for philosophers interested in such questions.I. Modern scholars debate over whether Plato knew the writings of Democritus and addressed himself to them. The classical authors were unanimous in the opinion that Plato regarded Democritus as one of his most important opponents. According to an often repeated anecdote, Plato wanted all the writings of Democritus burnt and only gave up this wish when it was clear that the writings were already too widely circulated. 3Commentators have pursued this alleged opposition in various directions. Hammer-Jensen 4 and Natorp 5 have argued that Plato discovered the works of Democritus in the midst of writing the Timaeus and that one can trace their influence in the latter part of the dialogue. Thus they find Plato heavily indebted to Democritus. Taylor, by contrast, rejects the influence of Democritus on Plato altogether. [End Page 256]There is no need to bring in Democritus, and it is perhaps most likely that Plato had never read his works. We cannot prove such a proposition, but we shall at least see that there is not a single sentence in the Timaeus which shows any knowledge of Democritus. Moreover, it is very doubtful whether Plato would have learned anything from Democritus if he had read him. 6However, Taylor’s conviction that Plato would not have been interested in Democritus is based on the assumption, central to his commentary, that the Timaeus must be read as a Pythagorean text, and this in itself is debatable. After all, Plato chose Timaeus the Locrian as the protagonist of... (shrink)

The question of how and why mathematics can be applied to physical reality should be approached through the history of science, as a series of case studies which may reveal both generalizable patterns and salient differences in the grounds and nature of that application from era to era. The present examination of Descartes' Principles of Philosophy Part II, reveals a deep ambiguity in the relation of Euclidean geometry to res extensa, and a tension between geometrical form and 'common motion of (...) parts' as principles of individuation for matter in Cartesian physics. (shrink)

The question of how and why the application of mathematics to physical reality is possible has occupied philosophers for many centuries. In contemporary discussions, Philip Kitcher’s attack on a priorist approaches to the question is particularly interesting, for it suggests that there is no global answer (Kitcher 1983, Chapters 1-4). In this essay, I would like to develop his insight by arguing, first, that the problem of how mathematics relates to physical reality should be addressed by an appeal to the (...) history of mathematics and the sciences in the form of case studies which analyze the peculiar and changing ways in which these domains interact. Generalizations about such interaction should be made carefully, in a kind of local and piecemeal retrospective; I believe that the historical record will show that the grounds of justification and the form of applied mathematics changes dramatically from one historical era to the next. (shrink)

Convinced that logic has a history and that its history always manages to surprise the philosophers, Claude Imbert has devoted much of her work to the study of the Stoic school and of the late-nineteenth-century German logician Gottlob Frege. In the fifth chapter of her book Pour une histoire de la logique, she examines the trajectory of Frege's awareness of what his new logic entails, in particular the way it subverts the project of Kant.

Convinced that logic has a history and that its history always manages to surprise the philosophers, Claude Imbert has devoted much of her work to the study of the Stoic school and of the late-nineteenth-century German logician Gottlob Frege. In the fifth chapter of her book Pour une histoire de la logique, she examines the trajectory of Frege's awareness of what his new logic entails, in particular the way it subverts the project of Kant.

Convinced that logic has a history and that its history always manages to surprise the philosophers, Claude Imbert has devoted much of her work to the study of the Stoic school and of the late-nineteenth-century German logician Gottlob Frege. In the fifth chapter of her book Pour une histoire de la logique, she examines the trajectory of Frege's awareness of what his new logic entails, in particular the way it subverts the project of Kant.

Cartesian method both organizes and impoverishes the domains to which Descartes applies it. It adjusts geometry so that it can be better integrated with algebra, and yet deflects a full-scale investigation of curves. It provides a comprehensive conceptual framework for physics, and yet interferes with the exploitation of its dynamical and temporal aspects. Most significantly, it bars a fuller unification of mathematics and physics, despite Descartes' claims to quantify nature. The work of his contemporaries Galileo and Torricelli, and of his (...) successor Newton, illustrates conceptual possibilities Descartes left aside, due to his attachment to method. (shrink)

The significance of Leibniz’s work as a physical scientist has long been underestimated or misunderstood. This stems in part from the great success of Newton’s physics on the one hand and the influence of Kant’s account of scientific knowledge on the other, both of which tend to obscure Leibniz’s successes and intentions. It is also due to the unavailability or scholarly neglect of key texts which, if properly assessed, illuminate the work of Leibniz in dynamics. In La dynamique de Leibniz, (...) François Duchesneau attempts to give a synoptic view of Leibniz’s accomplishments in this field, to do justice to neglected writings, and to show that the course of Leibniz’s dynamics offers fresh inspiration for current debates in the philosophy of science and epistemology. (shrink)

Mathematics, and mechanics conceived as a formal science, have their own proper subject matters, their own proper unities, which ground the characteristic way of constituting problems and solutions in each domain, the discoveries that expand and integrate domains with each other, and so in particular allow them, in the end, to be connected in a partial way with empirical fact. Criticizing both empiricist and structuralist accounts of mathematics, I argue that only an account of the formal sciences which attributes to (...) them objects as well as structure, proper semantics as well as syntax, can do justice to their intelligibility, heuristic force and explanatory power. (shrink)

In this essay, I consider the use of mathematical statistics in the study of biological systems in the field, using as case studies the work of Ruth Geyer Shaw and her colleagues at the University of Minnesota. To address practical issues, like how to enhance prairie restoration, and how to prepare for (and perhaps prevent) the effect of rapid climate change, she and her colleagues combine mathematical modeling and intensive data collection in the field. Using ANOVA and the more versatile (...) approach of Aster Models, they show the effects of inbreeding (which follows from the fragmentation of prairie) to lower the fitness of plant populations. They study the genetic variance of plants in stable populations, which is important when that population must deal with changing conditions. And they study the consequences of rapid climate change: even good genetic variance and initial fitness may not save a given population from extinction. This leads to a new alliance of science with local, state, federal and global politics. (shrink)

The existence of mathematical objects may be explained in terms of their occurrence in problems. Especially interesting problems arise at the overlap of domains, and the items that intervene in them are hybrids sharing the characteristics of both domains in an ambiguous way. Euclid's geometry, and Leibniz' work at the intersection of geometry, algebra and mechanics in the late seventeenth century, provide instructive examples of such problems and items. The complex and yet still formal unity of these items calls into (...) question certain tenets of Resnik's structuralism, and of the reductive projects of the logicists. (shrink)

The relationship between the objects of mathematics and physics has been a recurrent source of philosophical debate. Rationalist philosophers can minimize the distance between mathematical and physical domains by appealing to transcendental categories, but then are left with the problem of where to locate those categories ontologically. Empiricists can locate their objects in the material realm, but then have difficulty explaining certain peculiar “transcendental” features of mathematics like the timelessness of its objects and the unfalsifiability of (at least some of) (...) its truths. During the past twenty years, the relationship between mathematics and physics has come to seem particularly problematic, in part because of a strong interest in “naturalized epistemology” among American philosophers. The tendency to construe epistemological relations in causal and materialist terms seems to enforce a sharp distinction between mathematical and physical entities, and makes the former seem at best uncomfortably inaccessible and at worst irrelevant. (shrink)

This article describes the debate about space and time in the early modern period focusing on the exchange between Gottfried Leibniz and Isaac Newton. It provides a brief account of Galileo's critique of medieval cosmology, the finite, two-sphere cosmos with fixed places as well as a beginning and an end in time, the related account of motion as finite and in need of an external agent, and the too-limited use of geometry in mechanics. The article reviews in some detail the (...) achievements of Leibniz and Newton in order to make clear the differences in their views about space and time, construed in mathematical, metaphysical, and physical terms as they emerged in the Leibniz/Clarke correspondence. (shrink)