In considering the appropriate use of the terms “science” and “scientific instrument,” tracing the history of “mathematical instruments” in the early modern period is offered as an illuminating alternative to the historian's natural instinct to follow the guiding lights of originality and innovation, even if the trail transgresses contemporary boundaries. The mathematical instrument was a well-defined category, shared across the academic, artisanal, and commercial aspects of instrumentation, and its narrative from the sixteenth to the eighteenth century was largely (...) independent from other classes of device, in a period when a “scientific” instrument was unheard of. (shrink)

The leaders of the Scientific Revolution were not Baconian in temperament, in trying to build up theories from data. Their project was that same as in Aristotle's Posterior Analytics: they hoped to find necessary principles that would show why the observations must be as they are. Their use of mathematics to do so expanded the Aristotelian project beyond the qualitative methods used by Aristotle and the scholastics. In many cases they succeeded.

This book brings together papers of the conference on 'Space, Geometry and the Imagination from Antiquity to the Modern Age' held in Berlin, Germany, 27-29 August 2012. Focusing on the interconnections between the history of geometry and the philosophy of space in the pre-Modern and Early Modern Age, the essays in this volume are particularly directed toward elucidating the complex epistemological revolution that transformed the classical geometry of figures into the modern geometry of space. Contributors: (...) Graciela De Pierris Franco Farinelli Michael Friedman Daniel Garber Jeremy Gray Gary Hatfield Andrew Janiak Douglas Jesseph Alexander Jones Henry Mendell David Rabouin. (shrink)

In this article, modern standards of early years mathematics education are criticized and a proposal for change is presented. Today's early years mathematics education standards rest on a view of mathematics that became obsolete already at the end of the 19th century while the spirit of children's mathematics is precisely the spirit of modern mathematics. The proposal for change is not a return to the “new mathematics” movement, but something different.

This Encyclopedia offers a fresh, integrated and creative perspective on the formation and foundations of philosophy and science in European modernity. Combining careful contextual reconstruction with arguments from traditional philosophy, the book examines methodological dimensions, breaks down traditional oppositions such as rationalism vs. empiricism, calls attention to gender issues, to ‘insiders and outsiders’, minor figures in philosophy, and underground movements, among many other topics. In addition, and in line with important recent transformations in the fields of history of science and (...) early modern philosophy, the volume recognizes the specificity and significance of early modern science and discusses important developments including issues of historiography (such as historical epistemology), the interplay between the material culture and modes of knowledge, expert knowledge and craft knowledge. This book stands at the crossroads of different disciplines and combines their approaches – particularly the history of science, the history of philosophy, contemporary philosophy of science, and intellectual and cultural history. It brings together over 100 philosophers, historians of science, historians of mathematics, and medicine offering a comprehensive view of early modern philosophy and the sciences. It combines and discusses recent results from two very active fields: early modern philosophy and the history of (early modern) science. (shrink)

In Studies on Early Modern Aristotelianism Paul Richard Blum shows that Aristotle’s thought remained the touchstone of modern philosophy; for it was the philosophy taught at universities. The concept of philosophy at Jesuit schools forms the first part of this book. Their impact on the sciences and mathematics in combination with Renaissance ideas of nature is the topic of the second part. The transformation of Aristotelian metaphysics and theology under the influence of the Renaissance is the (...) third area of this book. Surprising continuity from the late Middle Ages into modernity and the radical difference of subject centered modern philosophy from ‘teachable’ school philosophy are innovative in these studies. (shrink)

In the course of the eighteenth and nineteenth centuries the paradigm of a new science, political economy, was established. It was a science distinct from the Aristotelian sub-disciplines of practical philosophy named oikonomía and politiké, and emphasis on its character of science not unlike the natural sciences – still called ‘natural philosophy’ – mirrored precisely a willingness to stress its autonomy from two other sub-disciplines of practical philosophy, that is, ethics and politics. However, the new science resulted from a transformation (...) of part of traditional practical philosophy, allowing the inclusion of bodies of knowledge accumulated by experts of commerce and public finance. Such bodies of knowledge were unified by the (true or alleged) discovery of regularities, mechanisms, causal connections making for a new partial order within the overall social order. How far this paved the way to a science similar to mathematics rather left a normative discipline as alive as ever was a recurrent question for at least a century, until the marginalist revolution opened the way for a sharp division, leaving ‘economics’ as a science of causes and effects facing ‘economic policy’ as a discourse on ends. (shrink)

This book on medieval and early modern corpuscular matter theories presents the research results of nineteen scholars, who show that his modern model of matter has some of its roots in physical, medical, mathematical, alchemical, and theological conceptions developed in the Middle Ages.

The article investigates the role of symbolic means of knowledge representation in concept development using the historical example of medieval diagrams of change employed in early modern work on the motion of fall. The parallel cases of Galileo Galilei, Thomas Harriot, and René Descartes and Isaac Beeckman are discussed. It is argued that the similarities concerning the achievements as well as the shortcomings of their respective work on the motion of fall can to a large extent be attributed (...) to their shared use of means of knowledge representation handed down from antiquity and the Middle Ages. While the interpretation of medieval diagrams was unproblematic in the scholastic context from which they arose, in the early modern context, which was characterized by the confluence of natural philosophy and practical mathematics, it became ambiguous. It was the early modern mathematicians’ work within this contradictory framework that brought about a new conceptualization of motion which, in particular, eventually led to an infinitesimal concept of velocity. In this process, the diagrams themselves remained largely unchanged and thus functioned as a catalyst for concept development. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Philosophy 41.2 (2003) 273-274 [Access article in PDF] Christoph Lüthy, John E. Murdoch, and William R. Newman, editors. Late Medieval and Early Modern Corpuscular Matter Theories. Leiden: Brill, 2001. Pp. viii + 610. Cloth, $186.00. The nineteen papers of this weighty (handsomely produced, but expensive) volume are mostly devoted to the views of one thinker or group of persons on "corpuscularism" (see (...) 17ff.), in either physics or mathematics, with few synthetic or diachronic studies. Only a quarter of the volume is devoted to the medieval period: Danielle Jacquart on minima in the Salerno school of medicine (twelfth century); the late George Molland on Roger Bacon's and Grosseteste's "corpuscular tendencies"; Charles Lohr on Ramon Lull; John Murdoch on minima naturalia in the medieval and Renaissance periods; and the greater part of William Newman's study of corpuscularian conceptions in "Aristotelian alchemy" from Geber to Sennert. The early-modern part includes studies on mostly expected figures: Francesco Patrici, Giordano Bruno, the Northumberland circle, Francis Bacon, David Gorlaeus (a little-known Dutch theology student; 1591-1612, sic), Daniel Sennert, Honoré Fabri, Galileo, Gassendi, Charleton, Boyle, Locke, and Homberg. Alan Gabbey more generally distinguishes different patterns of explanations in mechanical philosophy.As far as I can judge, the authors are knowledgeable and competent, and the studies make important contributions. But just because this is a high-quality volume produced with the ambition of making a step toward rewriting the history of a major chapter in the history of science, its two significant and principled deficiencies must be pointed out.The volume follows the intention of "show[ing] the debt of early-modern matter theory to previous traditions and thereby [explaining] its bewildering heterogeneity" (back cover). Now, the study of the corpuscular traditions in the pre-modern period is certainly an important desideratum (although this volume again attends chiefly to the early modern corpuscularians). But the motivation and perspective here announced—the medieval period [End Page 273] is to be studied because early modern science is indebted to it—are astoundingly Whiggish. Worse: the study of the entire Arabic-Islamic tradition is dismissed, on the grounds that "Islamic atomism does not seem to have been of any influence upon Christian fourteenth-century atomism" (8): this is not only Whiggism, but also frank and explicit euro- and Christian-centrism, which in the beginning of the twenty-first century is simply stunning. The method amusingly consists in affirming the consequent: presupposing that "Islamic atomism" did not influence the West and early-modern science, no scholar was asked to examine whether or not this was indeed the case, with the congenial result that the volume seemingly confirms the initial presupposition. Q.E.D. Similarly and no less regrettably, the fact that (ps.-) Geber's Summa Perfectionis is not, as Lasswitz had thought, the translation of an Arabic work is here used as grounds on which to dismiss altogether his claim concerning the importance of the Arabic input (13, but see also 293). This way, incomprehension between civilizations is reinforced instead of being reduced, and I believe we did not need September 11 to know this.The thusly dismissed Mu'tazilite school of Islamic theology had important repercussions in Judaism, so that referring to it as "Islamic atomism" is misleading (see notably Hagai Ben-Shammai, "Studies in Karaite Atomism," Jerusalem Studies in Arabic and Islam 6 [1985]: 243-93). The volume's forty-page cumulative bibliography, for which the editors are to be congratulated, reflects the volume's parochial viewpoint: H. A. Wolfson's important The Philosophy of the Kalam (Cambridge, MA: Harvard University Press, 1976), with some 150 pages on atomism, is not included; nor is the (English version of the) landmark study by the late S. Pines, Studies in Islamic Atomism (Jerusalem: The Magnes Press, 1997; the original 1936 German version is mentioned). Wolfson's study, incidentally, shows that the editors' presupposition (8) that the Kalâm tradition could have influenced the West only via the Latin translation of Maimonides' Guide of the Perplexed is unwarranted, for Isaac... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Gothicism and Early Modern Historical EthnographyKristoffer NevilleGothicism: Problems and PossibilitiesEarly-modern Gothicism, or self-identification with the Gothic peoples described by classical authors, has usually been considered a Scandinavian, and particularly Swedish, affair. Particularly in the sixteenth and seventeenth centuries, the Swedish court and universities insisted militantly that the kingdom was the Gothic homeland, and this has fostered an assumption that Gothicism represents a kind of embryonic nationalism. (...) This interpretation was almost inevitable given the circumstances of modern interest in the phenomenon. Scandinavian scholars were the first to pick up the Gothic thread in the earlier twentieth century, and Swedes in particular have dominated the literature on Gothicism. At least in the early years, this may be [End Page 213] related to a general trend to interpret the past in terms of a relatively inflexible, modern concept of nation. These earlier studies accordingly relied largely on the same, mostly Swedish, sources. Later work has generally not expanded the framework of the discourse, however.1 A number of important early modern texts that show the much broader appeal of Gothicism are not accounted for in the standard narrative of the rise of the phenomenon in the early modern period. I will show that the Scandinavian interpretation has been allowed to overshadow a much broader Gothic tradition that encompassed a broad part of Europe, including the German lands and Spain. Gothicism has thus been reduced to just one aspect of its original scope, and the scholarly nuances and historical jockeying that shaped the narrative have largely been lost. This reduction has had consequences not only for the familiar version of the story, largely derived from Swedish texts, but also for our understanding of history writing in the sixteenth and seventeenth centuries.The Gothicism debate may be unique in its breadth and in the intensity of interest that it elicited. It is also remarkable for the variety of approaches to historical writing that scholars applied as each tried to develop a more impregnable argument than the last. Linguistic, geographical, archaeological, antiquarian, and textual methods intersected freely, inviting shifting and overlapping versions of the Gothic narrative that sometimes trace the linguistic legacy of the tribe, sometimes the ethnic legacy, and sometimes bind the two together. All of these narratives were basically concerned with the antiquity of the Goths and their legacy in early modern Europe. The implications of this were different for various regions and rulers, but most wanted to claim for themselves the strength, prestige, and antiquity of the tribe that toppled the Roman Empire, thus enhancing their own often dubious historical legitimacy. [End Page 214]This debate often took a polemical character, particularly in the political sphere, where there were fierce disputes over which region was the true homeland of the tribe, and thus which area (or people or ruling dynasty) could claim the greatest honor from their exploits and claim to be their most direct descendants. But there was also a very clear corollary to this contentious discourse. As scholars from across a wide area produced mountains of evidence pointing to the Gothic origins of their own regions, it became possible to think of Gothic origins not only in terms of difference—which kingdom, dynasty, or geographic region could claim the greatest antiquity and eminence—but also in terms of shared history and common identity across political, geographical, and linguistic boundaries. In general, the polemic was restricted to the origins of the Goths. Many writers seem to have accepted that the Gothic lands cumulatively formed a sort of historical unity that was lost through later political divisions, but which could still be traced through other means. Particularly in the Holy Roman Empire—a largely political boundary encompassing many smaller states ruled by famously fractious princes of different confessions and languages—the more conceptual notion of deep-seated Gothic origins opened the way for a different alignment of identity both within and without the boundaries of the Empire, and especially with Scandinavia, which contributed such rich arguments to the debate.There are several significant problems in a study of the early-modern view of the Goths beyond the historiographical ones outlined above. They were widely associated with other tribes... (shrink)

Ausgehend von den Konkurrenzbegriffen Niklas Luhmanns und Georg Simmels untersucht der Artikel, inwieweit sich Konkurrenzverhältnisse an frühneuzeitlichen Universitäten in den Begriffen von Konkurrenzbedingungen moderner Ökonomie interpretieren lassen. Dieses wird anhand von Gelehrtenstreiten, dem Disputationswesen sowie Konflikten über professorale Privilegien und Universitätsgründungen diskutiert. Die zentrale These lautet, dass ökonomische Bedingungen sehr wohl eine wesentliche Rolle für Konkurrenzverhältnisse spielten, diese aber eben dem feudalistischen Wirtschaftssystem entsprachen und auf Erlangung bzw. Wahrung von Pfründen und Privilegien ausgerichtet waren. Ein moderner Konkurrenzbegriff ist demnach nur (...) sehr begrenzt anwendbar. (shrink)

Early modern natural philosophers put forward the ontological program that was called "mechanical philosophy" and they gave mechanical explanations for all kinds of phenomena, such as gravity, magnetism, the colors of the rainbow, the circulation of the blood, the motion of the heart and the development of animals. For a generation of historians, the mechanical philosophy was regarded as the main alternative to Aristotelian orthodoxy during the so-called Scientific Revolution and mechanical explanations were presented as paving the way (...) for the use of experiments and mathematics in the understanding of natural phenomena. However, the historical category of mechanical philosophy was later criticized as being too broad, while early modern mechanical explanations were condemned by more epistemologically oriented minds for being incompatible with, or at least not necessarily connected to, the use of experiments and mathematics. In the last ten years, just as the new mechanistic literature emerged in philosophy of science, there has been a reevaluation of early modern mechanical explanations in a domain that had been until then considered as peripheral to the so-called Scientific Revolution, namely the domain of biology, anatomy, physiology and medicine. Although they were neither confirmed nor predictive, some early modern explanations in these domains appear to have a cognitive value similar to the value of contemporary mechanistic explanations. (shrink)

The generative and governing "idea" of radical modernity is spawned by the technique of mathematical construction deployed and interpreted by the major early-modern thinkers and their legatees. ;Chapter I is a survey of this legacy as it appears in Vico, Kant, Fichte, Marx and Nietzsche and in the post-Nietzschean inheritance of contemporary philosophy, hyperbolic in the case of Derrida et al., elliptical, in the case of Carnap and Goodman. ;In Chapter II I try to show how the pre- (...) class='Hi'>modern mathematical tradition, represented by Euclid, aimed at keeping the enticements of technical facility in check by means of didactic phronesis and how the post-Kantian interpretation of "existence" in Euclid as constructibility betrays his usage and self-understanding. I suggest that his focus in the postulates and elsewhere is on the undistorted iterability of graphic evocations of the items already intelligible thanks to the definitions or to the pre-understanding shared by the teacher and student. ;In Chapter III, devoted to Descartes the principal claims of modern constructivism are brought to sight. After examining Descartes' fabulous autobiography and its emphasis on self-origination, I turn to the style, contents and under-pinnings of the Geometry in an effort to extract from that text what he once referred to as "the metaphysics of geometry." The latter yields the conditions of successful problem-solving, i.e., dimensional homogeneity and kinematic continuity. These conditions, in turn, find their justification in Descartes' theses in the Rules concerning order, measure and the uniformity of "mental" activity. In the final section I apply the lessons learned from the Geometry and the Rules to one critical issue in the later Meditations, the transition from essence to existence. Descartes' "solution" generates a sequence of perplexities with Hobbes, Leibniz, Kant and other radical moderns continue to wrestle. (shrink)

This article seeks the origin, in the theories of Ibn al-Haytham (Alhazen), Descartes, and Berkeley, of two-stage theories of spatial perception, which hold that visual perception involves both an immediate representation of the proximal stimulus in a two-dimensional ‘‘sensory core’’ and also a subsequent perception of the three dimensional world. The works of Ibn al-Haytham, Descartes, and Berkeley already frame the major theoretical options that guided visual theory into the twentieth century. The field of visual perception was the first area (...) of what we now call psychology to apply mathematics, through geometrical models as used by Euclid, Ptolemy, Ibn al-Haytham, and Descartes (among others). The article shows that Kepler’s discovery of the retinal image, which revolutionized visual anatomy and entailed fundamental changes in visual physiology, did not alter the basic structure of theories of spatial vision. These changes in visual physiology are advanced especially in Descartes' Dioptrics and his L'Homme. Berkeley develops a radically empirist theory vision, according to which visual perception of depth is learned through associative processes that rely on the sense of touch. But Descartes and Berkeley share the assertion that there is a two-dimensional sensory core that is in principle available to consciousness. They also share the observation that we don't usually perceived this core, but find depth and distance to be phenomenally immediate, a point they struggle to accommodate theoretically. If our interpretation is correct, it was not a change in the theory of the psychology of vision that engendered the idea of a sensory core, but rather the introduction of the theory into a new metaphysical context. (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)

SUMMARYEarly modern artists who did not have access to Roman Antiquity or needed quick access to it could refer to prints after monuments such as those issued by Antoine Lafréry. But Du Choul's Discours sur la castrametation et discipline militaire des Romains [ … ] De la Religion des anciens Romains was also successful among artists, particularly painters. It was in vernacular language and widely available in French, Spanish and Italian; it was affordable and compact in format ; it (...) had plenty of woodcuts; and it focused on the Roman military life, besides the monuments, for which information could also be found in travel guides. The present essay aims to study how Du Choul's volume circulated, with several examples from Spain and Italy. The Discours will be used as a case study of artists as readers, and I will study aspects previously overlooked such as the book's market price in the early modern period, examples of use of copies of the book by artists and, more broadly, the troubles and challenges encountered by these readers when representing subjects from ancient history. (shrink)

Original explorers often see a puzzling conceptual landscape more vividly than jaded later travelers. This essay surveys several ways in which Descartes and Leibniz recognized descriptive problems within applied mathematics more clearly than later commentators have appreciated.

According to a grand narrative that long ago ceased to be told, there was a seventeenth century Scientific Revolution, during which a few heroes conquered nature thanks to mathematics. When this grand narrative was brought into question, our perspectives on the question of mathematization should have changed. It seems, however, that they were instead set aside, both because of a general distrust towards sweeping narratives that are always subject to the suspicion that they overlook the unyielding complexity of real (...) history, and because of a shift in our interests. The more obscure and idiosyncratic they are, an alchemist, a patron of the sciences or a lunatic collector is nowadays honored in journals of the history of sciences. As for the general issues involved in the question of mathematization, they are rejected as obsolete, or reserved for specialized journals in the history of mathematics. The article « Forms of mathematization » examines in general the notion of mathematization, distinguishes different forms of mathematization and defends a renewed study of the forms of mathematization in the history of the early sciences. (shrink)

My title, of course, is an exaggeration. The world no more became mathematical in the seventeenth century than it became ironic in the nineteenth. Either it was mathematical all along, and seventeenth-century philosophers discovered it was, or, if it wasn’t, it could not have been made so by a few books. What became mathematical was physics, and whether that has any bearing on the furniture of the universe is one topic of this paper. Garber says, and I agree, that for (...) Descartes bodies are the things of geometry made real ( Ref). That is a claim about the world: what God created, and what we know in physics, is nothing other than res extensa and its modes. Others, including Marion, hold that in modern science, here represented at its origins by Descartes, representation displaces beings: the knower no longer confronts Being or beings but rather a system of signs, a “code” as Marion calls it, to which the knower stands in the relation of subject to object. The Meditations, or perhaps even the Regulæ, are the first step toward the transcendental idealism of Kant. Most of this paper will be devoted to a more concrete question. Physics in the seventeenth century increasingly became a matter of applying mathematical knowledge to the solution of physical problems. The “mixed sciences” of astronomy, optics, and music, sciences then distinct from physics, became models of understanding for all of natural philosophy. My interest here is in one aspect of that development: how particular physical situations are transformed into mathematical. I will look at the work of Descartes and Isaac Beeckman, contrasting their visions of “physico-mathematics”. (shrink)

The emergence of mathematical probability has something to do with dice games: all the early discussions (Cardano, Galileo, Pascal) suggest as much. Although this has long been recognized, the problem is that gambling at dice has been a popular pastime since antiquity. Why, then, did gamblers wait until the sixteenth century ce to calculate the math of dicing? Many theories have been offerred, but there may be a simple solution: early-modern gamblers played different sorts of dice games (...) than in antiquity. While ancients diced at communal risk, early-moderns diced at individualized risk. These individual wager-games, for example, the game “hazard”, incentivized solving questions like “how likely is it to roll a triple-six?”. Before the early modern period there was no gambling game to which working out mathematical probability would have brought an advantage. (shrink)

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

This paper offers a caution that emphasis upon the importance of mathematics in recent historiography is in danger of obscuring the historical fact that, for the most part, mathematics was not seen as important in the pre-modern period. The paper proceeds by following a single case study, and in so doing offers the first account of the mathematical writings of Jean Fernel (1497–1558), better known as a leading medical innovator of the 16th century. After establishing Fernel's (...) class='Hi'>early commitment to mathematics, and attempt to forge a career as a cosmographer, it goes on to explain his abandonment of mathematics for a career in medicine. The ‘mathematization of the world picture’ is usually explained in terms of the perceived usefulness of mathematics, but Fernel's case shows that for many pre-modern thinkers mathematics was not regarded as a useful pursuit. The paper should serve as a reminder, therefore, that the take-up of mathematics by natural philosophers was by no means inevitable, but had to be carefully managed by early modern mathematical practitioners. The case of Fernel indicates that perhaps he was not the only would-be mathematical practitioner to abandon mathematics in favour of a calling that was more appreciated by contemporaries. (shrink)

Georg Agricola’s De Re Metallica (1556) is an impressive work about mining arts and sciences, still used as a reference work to understand the mining world of his time and to analyze the relationships between scholars and practitioners. This essay begins by studying how Agricola presents the underground surveying, also known as geometria subterranea. How does the author’s mathematics relate to contemporary geometry or to actual surveying practices? Second, the essay contrasts his scholarly approach with sixteenth-century administrative and technical (...) documents, as well as mining sermons. Comparing the practical geometry contained in the De Re Metallica with different kinds of sources helps to question the relationships between the arts and natural and exact sciences in the practical settings of early modern Europe. Finally, it argues for a distinction between the inquiry on the historical progress of mathematics in civil life and philosophical questions about the rise of new sciences. (shrink)

This chapter reviews references to Pythagoras by the authors of the pre-Platonic period. It shows that in the course of the fourth century, studies in mathematics, particularly geometry and arithmetic, became a constant element of the tradition of Pythagoras; astronomy and harmonics are less frequently mentioned. Mathematics did not displace metempsychosis and wonders, nor did the tradition of Pythagoras the politician which emerged concurrently with it, yet they did edge them aside, completing the ambivalent, contradictory image of Pythagoras, (...) which was retained by his Neoplatonic biographers and passed from them into modern scholarship. It concludes that Pythagoras the mathematician is as little a product of the Academy as Pythagoras the philosopher. (shrink)

Semantic analysis in early analytic philosophy belongs to a long tradition of adopting geometrical methodologies to the solution of philosophical problems. In particular, it adapts Descartes’ development of formalization as a mechanism of analytic representation, for its application in natural language semantics. This article aims to trace the mathematical roots of Frege, Russel and Carnap’s analytic method. Special attention is paid to the formal character of modern analysis introduced by Descartes. The goal is to identify the particular conception (...) of “form” developed by the analytic tradition, from Descartes to early analytic philosophy, and to determine its relation to similar notions, like ‘function’ and ‘syntax’. Finally, I focus on how Frege, Russell and Carnap’s methods of semantic analysis fit the general characterization of formal analysis previously developed. (shrink)

This paper discerns two types of mathematization, a foundational and an explorative one. The foundational perspective is well-established, but we argue that the explorative type is essential when approaching the problem of applicability and how it influences our conception of mathematics. The first part of the paper argues that a philosophical transformation made explorative mathematization possible. This transformation took place in early modernity when sense acquired partial independence from reference. The second part of the paper discusses a series (...) of examples from the history of mathematics that highlight the complementary nature of the foundational and exploratory types of mathematization. (shrink)

Most scholars think of David Hilbert's program as the most demanding and ideologically motivated attempt to provide a foundation for mathematics, and because they see technical obstacles in the way of realizing the program's goals, they regard it as a failure. Against this view, Curtis Franks argues that Hilbert's deepest and most central insight was that mathematical techniques and practices do not need grounding in any philosophical principles. He weaves together an original historical account, philosophical analysis, and his own (...) development of the meta-mathematics of weak systems of arithmetic to show that the true philosophical significance of Hilbert's program is that it makes the autonomy of mathematics evident. The result is a vision of the early history of modern logic that highlights the rich interaction between its conceptual problems and technical development. (shrink)

Contents: Acknowledgements. Conventions. Preface. Biographical sketch. 1 Introduction. 2 The Contributions. 3 Early work in analysis. 4 The Theory of Science . 5. Later mathematical studies. A On Kantian Intuitions. B The Bolzano-Cauchy Theorem.

Pierre Duhem and Jacques Maritain, influenced by positivist philosophies of science that prevailed in the late nineteenth and early twentieth centuries, adopted markedly non-realist views about the mathematical theories of the modern physical sciences. The philosophies of science they developed were a hybrid of Thomism and positivism. This paper argues that the ideas of Duhem and Maritain about the relation of the mathematical theories of modern physics to physical reality are inadequate in light of the insights (...) class='Hi'>modern physics has yielded about the physical world. (shrink)

This article deals with physico-mathematical approaches to anatomy in post-Harveyan physiology. But rather than looking at questions of iatromechanics and animal locomotion, which often attracted this approach, I look at the problem of how blood returned to the heart – a part of the circulation today known as venous return but poorly researched in the early modern period. I follow the venous return mechanisms proposed by lesser-known authors in the mechanization of anatomy, such as Jean Pecquet (1622–1674) and (...) Nicolaus Steno (1638–1686), alongside the more famous Giovanni Alfonso Borelli (1608–1679). Their mechanisms differed only in small details. Yet, these minor differences highlight significant aspects of the mechanization of the life sciences in the seventeenth century. First, they relied more on observations than hitherto acknowledged, even if only indirectly. Second, their mechanisms drew more from the physico-mathematical disciplines than from the trending corpuscularian philosophies of their time. Finally, these mechanisms led to a more accurate understanding of the circulation that remains valid today, thus revealing their cognitive benefits. In short, through the single problem of how blood returned to the heart, this article portrays the increasing complexity of anatomy in the early modern period. (shrink)

The study is focused on the relation between theology and mathematics in the situation of increasing secularization. My main concern in the second part of this paper is the early-twentieth-century foundational crisis of mathematics. The hypothesis that pure mathematics partially fulfilled the functions of theology at that time is tested on the views of the leading figures of the three main foundationalist programs: Russell, Hilbert and Brouwer.

Although the use of mathematical models is ubiquitous in modern science, the involvement of mathematical modeling in the sciences is rarely seen as cases of interdisciplinary research. Often, mathematics is “applied” in the sciences, but mathematics also features in open-ended, truly interdisciplinary collaborations. The present paper addresses the role of mathematical models in the open-ended process of conceptualizing new phenomena. It does so by suggesting a notion of cultures of mathematization, stressing the potential role of the mathematical (...) model as a boundary object around which negotiations of different desiderata can take place. This framework is then illustrated by a case study of the early efforts to produce a mathematical model for quasi-crystals in the first two decades after Dan Shechtman’s discovery of this new phenomenon in 1984. (shrink)

This paper argues for two related theses. The first is that mathematical abstraction can play an important role in shaping the way we think about and hence understand certain phenomena, an enterprise that extends well beyond simply representing those phenomena for the purpose of calculating/predicting their behaviour. The second is that much of our contemporary understanding and interpretation of natural selection has resulted from the way it has been described in the context of statistics and mathematics. I argue for (...) these claims by tracing attempts to understand the basis of natural selection from its early formulation as a statistical theory to its later development by R.A. Fisher, one of the founders of modern population genetics. Not only did these developments put natural selection of a firm theoretical foundation but its mathematization changed the way it was understood as a biological process. Instead of simply clarifying its status, mathematical techniques were responsible for redefining or reconceptualising selection. As a corollary I show how a highly idealised mathematical law that seemingly fails to describe any concrete system can nevertheless contain a great deal of accurate information that can enhance our understanding far beyond simply predictive capabilities. (shrink)

Medieval and Renaissance humanist political discourse and Machiavelli -- Machiavelli and Spanish imperialist discourse in the sixteenth century -- Machiavelli and the foundations of the Spanish reason-of-state tradition : Giovanni Botero and Pedro de Ribadeneyra -- Machiavellian discourse in the Hispanic Baroque reason-of-state tradition -- Juan Pablo Mártir Rizo's rereading of the Prince -- Conclusion.

The main topic of this essay is symbolic mathematics or the method of symbolic construction, which I trace to the end of the sixteenth century when Franciscus Vieta invented the algebraic symbolism and started to use the word ‘symbolic’ in the relevant, non-ontological sense. This approach has played an important role for many of the great inventions in modern mathematics such as the introduction of the decimal place-value system of numeration, Descartes’ analytic geometry, and Leibniz’s infinitesimal calculus. (...) It was also central for the rigorization movement in mathematics in the late nineteenth century, as well as for the mathematics of modern physics in the 20th century.However, the nature of symbolic mathematics has been concealed and confused due to the strong influence of the heritage from the Euclidean and Aristotelian traditions. This essay sheds some light on what has been concealed by approaching some of the crucial issues from a historical perspective. Furthermore, I argue that the conception of modern mathematics as symbolic mathematics was essential to Wittgenstein’s approach to the foundations and nature of mathematics. This connection between Wittgenstein’s thought and symbolic mathematics provides the resources for countering the still prevalent view that he defended an uttrely idiosyncratic conception, disconnected from the progress of serious science. Instead, his project can be seen as clarifying ideas that have been crucial to the development of mathematics since early modernity. (shrink)

ArgumentThis paper reconstructs Wronski's philosophical foundations of mathematics. It uses his critique of Lagrange's algebraic analysis as a vignette to introduce the problems that he raised, and argues that these problems have not been properly appreciated by his contemporaries and subsequent commentators. The paper goes on to reconstruct Wronski's mathematical law of creation and his notions of theory and techne, in order to put his objections to Lagrange in their philosophical context. Finally, Wronski's proof of his universal law is (...) reviewed in terms of the above reconstruction. I argue that Wronski's philosophical approach poses an alternative to the views of his contemporary mainstream mathematicians, which brings up the contingency of their choices, and bridges the foundational concerns of early modernity with those of the twentieth-century foundations crisis. I also argue that Wronski's views may be useful to contemporary philosophy of mathematical practice, if they are read against their metaphysical grain. (shrink)

In both the early modern period and in contemporary debates, philosophers have argued that there are analogies between mathematics and morality that imply that the ontology and epistemology of morality are crucially similar to the ontology and epistemology of mathematics. I describe arguments for the math‐moral analogy in four early modern philosophers (Locke, Cudworth, Clarke, and Balguy) and in three contemporary philosophers (Clarke‐Doane, Peacocke, and Roberts). I argue that these arguments fail to establish important (...) ontological and epistemological similarities between morality and mathematics. There are analogies between the two areas, but the disanalogies are more significant, undermining the attempt to confer on morality the same ontological and epistemological status that mathematics possesses. (shrink)

Wittgenstein's _Tractatus Logico-Philosophicus_, first published in 1921, has had a profound influence on modern philosophic thought. _Prototractatus_ is a facsimile reproduction of an early version of _Tractatus_, only discovered in 1965. The original text has a parallel English translation and the text is edited to indicate all relevant deviations from the final version.

The Enlightenment critique of the esprit de système and its tendency towards eclecticism have often been interpreted as symptoms of speculative shallowness. The article analyses the origins of this prejudice, with special reference to the early French Enlightenment (1700-1750). It then attempts to counter such a preconception by providing a relevant counterexample. The case study presented is that of Pierre-Louis Moreau de Maupertuis, usually considered a typical instance of the anti-systematic proto-positivist philosophe. In discussing his philosophy, epistemological and cosmological (...) issues in particular are emphasized, showing how Maupertuis’ thoroughly empiricist theory of knowledge led him to apply mathematics to all domains of human knowledge in order to reduce their uncertainty. It is concluded that Maupertuis is the author of a rich and interesting work, which brings some novelty to the philosophical scene of his own times, if not to the whole history of modern philosophy. (shrink)

This paper examines the ethical and religious dimensions of mathematical practice in the early modern era by offering an interpretation of Antoine Arnauld and Pierre Nicole’s Nouveaux éléments de géométrie. According to these important figures of seventeenth-century French philosophy and theology, mathematics could achieve extra-mathematical or non-mathematical goals; that is, mathematics could foster practices of moral self-improvement, deepen the mathematician’s piety and cultivate epistemic virtues. The Nouveaux éléments de géométrie, which I contend offers the most robust (...) account of the virtues cultivated by mathematics in the period, was envisaged by its authors to cultivate moral, Christian and epistemic virtues that could serve in the fulfilment of moral and Christian obligations. In this paper, I set out the goals of mathematical inquiry for the Port-Royalists and describe the specific virtues they believed a revised edition of the Elements of Euclid could foster. I show that Arnauld and Nicole believed that an acquaintance with mathematics could render a student of Euclid more just, truth-loving, attentive and humble, and better able to discern truth from falsity. (shrink)

The object of this paper is to reply to Morrison's ([2000]) claim that while ‘structural unity’ was achieved at the level of the mathematical models of population genetics in the early synthesis, there was explanatory disunity. I argue to the contrary, that the early synthesis effected by the founders of theoretical population genetics was unifying and explanatory both. Defending this requires a reconsideration of Morrison's notion of explanation. In Morrison's view, all and only answers to ‘why’ questions which (...) include the ‘cause or mechanism’ for some phenomenon count as explanatory. In my view, mathematical demonstrations that answer ‘how possibly’ and ‘why necessarily’ questions may also count as explanatory. The authors of the synthesis explained how evolution was possible on a Mendelian system of inheritance, answered skepticism about the sufficiency of selection, and thus explained why and how a Darwinian research program was warranted. While today we take many of these claims as obvious, they required argument, and part of the explanatory work of the formal sciences is providing such arguments. Surely, Fisher and Wright had competing views as to the optimal means of generating adaptation. Nevertheless, they had common opponents and a common unifying and explanatory goal that their mathematical demonstrations served. Introduction: Morrison's challenge Fisher v. Wright revisited The early synthesis Conclusion: unification and explanation reconciled. (shrink)

This chapter can be considered as made up of two parts, a general discussion of the notion of mathematical practice and the limits of its use, comprised by the first three sections, and a particular case study that is presented in order to exemplify the idea of the web of practices, which occupies the remaining three. The presentation of my approach to the notion of mathematical practice is brief and synthetic but more articulated theoretically than in a previous book (Ferreirós (...) 2016). Considerations in Sect. 3 about mathematical cognition and ethnomathematics have to do with the “limits” of mathematical practice, i.e., with practices that do not fall under that rubric.The case study concerns the evolution of the function concept, which offers a complex example of the web-of-practices approach. We start with early Modern developments that led to the eighteenth-century idea of function (as “analytical expression”) and the nineteenth-century “logical” idea of function, showing how complex is the spiderweb of interrelated mathematical and scientific practices knotted around the function concept. In the last section, I consider the difficult question of the most basic practices and cognitive experiences that form the basis of the idea of function. I propose that two conceptions – a dynamic and an algebraic or operational one – coalesced to give rise to the early Modern idea of function. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Handbook of Cognitive Mathematics ed. by Marcel DanesiNathan HaydonMarcel Danesi (Ed) Handbook of Cognitive Mathematics Cham, Switzerland: Springer International, 2022, vii + 1383, including indexFor one acquainted with C.S. Peirce, it is hard to see Springer's recent Handbook of Cognitive Mathematics (editor: Marcel Danesi) through none other than a Peircean lens. Short for the cognitive science of mathematics, such a modern, scientific pursuit (...) into the nature and study of mathematical practice would no doubt be found agreeable to Peirce. The fact that references to Peirce appear often throughout the Handbook is a welcome find, with Peirce's ideas being a key subject of half a dozen chapters, and where a reader of any other chapter may well find further connections to Peirce's ideas. After spending time with the Handbook, it is clear that cognitive mathematics has not only embraced some of Pierce's ideas but may be at an important forefront of Peirce studies. In the end, the field may well be an area a Peircean should pay attention to.The book itself is pitched as a reference volume to the field (p. v) with the necessary background to familiarize oneself with the aims and results. The connection to cognitive science may call to mind detailed cognitive models [Ch. 10–14], theories on the origins of numeracy and other theories behind the biological and evolutionary requirements of mathematical thought [Ch.15–18], and the like. These are all present. But a key theme of the Handbook is to situate mathematics not just within more traditional 'cognitive' faculties and the more formal, i.e. algebraic, presentations of mathematics, but also to place mathematics within other human faculties and practices, from the arts to language [Ch. 19–22], within education and learning more broadly [Ch. 23–26], and in relation to the significant, though at first perhaps less quantitative parts of mathematics, like the use of metaphor, gesture, analogy, [End Page 243] abstraction, as well as further cultural and ethnographic considerations [Ch. 5–8]. The Handbook has explicitly taken a broader, more interdisciplinary approach (p. vi–vii) towards the scientific aspects of mathematical practice—choosing to regulate the study not by antecedently drawn opinions about what mathematics is (or has traditionally been taken to be) but by what future quantifiable and diverse study may come to bear on the practice and have to say about those engaging in it.This broad interdisciplinarity has a pragmatist ring, where theory cannot so easily be separated from the normative, social, and, more generally, the more thoroughly human aspects that we encounter and employ when we engage in it. Two further commitments of cognitive mathematics steer us even closer towards Peirce's views. The first—going back, for example, to Lakoff and Núñez's Where Mathematics Comes From (2000), which is taken to be a key early work in shaping the field—is that mathematics is taken to be a language like any other and as such must be learned and situated within our other cognitive faculties (p. vi, Ch. 4). The second—and largely, one imagines, following from a concern for seeing how mathematics is actually practiced—is that mathematics is essentially diagrammatic, involving as it does experimenting in diagrams and employing other signs (one may think here even of the mathematician sketching equations or geometric figures on a sheet of paper). What is perhaps initially most noteworthy about Peirce's philosophy of mathematics is his early concern for both these aspects of the practice.The book (more like a tome of over 1300 pages) is too large to be covered here in all its aspects, hence the review will focus on those aspects a Peircean or Pragmatist may find of interest. In particular, this review focuses on several of the themes above—situating mathematics more broadly within our everyday (cognitive) lives, the necessarily diagrammatic, fallible, and creative aspects of the practice, and, finally, what these aspects may have to say about Peirce's philosophy more generally, particularly with respect to (scientific) inquiry.A Peircean reader may well begin with Pietarinen's "Pragmaticism as Philosophy of Cognitive Mathematics" [Ch. 39] and "Peirce on Mathematical... (shrink)

The metaphysics of early embryos is a hotly debated topic in contemporary bioethics and metaphysics. Many contemporary Aristotelians believe that a human being is present from the moment of conception. At the same time, certain findings in modern embryology about the formation of identical twins challenge this belief. It becomes much harder when these theories are taken into account to understand the continued identity over time of the embryo(s) given the twinning process. In this article, I will consider (...) the philosophical implications of two models of monozygotic twinning within an Aristotelian metaphysical schema one of which is the standard, or traditional, model. The other of which is a new model recently put forward by Herranz. For the sake of completeness, I will also consider the philosophical implications of chimeras for the Aristotelian position. I will explain how Aristotelians can understand the process of twinning while holding on to their belief that a human being is present from the moment of conception. Summary: I will argue that a human being is present from the moment of conception. I will argue for this on Aristotelian grounds, and I will then defend this claim from criticisms based on a number of findings in modern embryology. (shrink)