Examines the ancient cosmic science of the female megalithic astronomers.

I present a systematic interpretation of the foundational purpose of constructions in ancient Greek geometry. I argue that Greek geometers were committed to an operationalist foundational program, according to which all of mathematics—including its entire ontology and epistemology—is based entirely on concrete physical constructions. On this reading, key foundational aspects of Greek geometry are analogous to core tenets of 20th-century operationalist/positivist/constructivist/intuitionist philosophy of science and mathematics. Operationalism provides coherent answers to a range of traditional philosophical problems regarding (...) classical mathematics, such as the epistemic warrant and generality of diagrammatic reasoning, superposition, and the relation between constructivism and proof by contradiction. Alleged logical flaws in Euclid can be interpreted as sound operationalist reasoning. Operationalism also provides a compelling philosophical motivation for the otherwise inexplicable Greek obsession with cube duplication, angle trisection, and circle quadrature. Operationalism makes coherent sense of numerous specific choices made in this tradition, and suggests new interpretations of several solutions to these problems. In particular, I argue that: Archytas’s cube duplication was originally a single-motion machine; Diocles’s cissoid was originally traced by a linkage device; Greek conic section theory was thoroughly constructive, based on the conic compass; in a few cases, string-based constructions of conic sections were used instead; pointwise constructions of curves were rejected in foundational contexts by Greek mathematicians, with good reason. Operationalism enables us to view the classical geometrical tradition as a more unified and philosophically aware enterprise than has hitherto been recognised. (shrink)

in this paper we return to Marshall Clagett’s view about the existence of an ancient Greek geometry of motion. It can be read in two ways. As a basic presentation of ancient Greek geometry of motion, followed by some aspects of its further development in landmark works by Galileo and Newton. Conversely, it can be read as a basic presentation of aspects of Galileo’s and Newton’s mathematics that can be considered as developments of a geometry (...) of motion that was first conceived by ancient Greek mathematicians. (shrink)

With In Focus Sacred Geometry, learn the fascinating history behind this ancient tradition as well as how to decipher the geometrical symbols, formulas, and patterns based on mathematical patterns. People have searched for the meaning behind mathematical patterns for thousands of years. At its core, sacred geometry seeks to find the universal patterns that are found and applied to the objects surrounding us, such as the designs found in temples, churches, mosques, monuments, art, architecture, and nature. Learn (...) the fundamental principles behind: Interpreting the sacred symbolism and principles behind shapes Calculating numbers used in sacred geometry Applying the golden ratio and Fibonacci’s Sequence to objects Translating symbolic and geometrical letters and alphabets This accessible and beautifully designed guide to sacred geometry includes a frameable poster of the main sacred geometrical shapes and their unique, innate arrangements. The In Focus series applies a modern approach to teaching the classic body, mind, and spirit subjects. Authored by experts in their respective fields, these beginner's guides feature smartly designed visual material that clearly illustrates key topics within each subject. As a bonus, each book includes reference cards or a poster, held in an envelope inside the back cover, that give you a quick, go-to guide containing the most important information on the subject. (shrink)

Geometry, etymologically the “science of measuring the Earth”, is a mathematical formalization of space. Just as formal concepts of number may be rooted in an evolutionary ancient system for perceiving numerical quantity, the fathers of geometry may have been inspired by their perception of space. Is the spatial content of formal Euclidean geometry universally present in the way humans perceive space, or is Euclidean geometry a mental construction, specific to those who have received appropriate instruction? (...) The spatial content of the formal theories of geometry may depart from spatial perception for two reasons: first, because in geometry, only some of the features of spatial figures are theoretically relevant; and second, because some geometric concepts go beyond any possible perceptual experience. Focusing in turn on these two aspects of geometry, we will present several lines of research on US adults and children from the age of three years, and participants from an Amazonian culture, the Mundurucu. Almost all the aspects of geometry tested proved to be shared between these two cultures. Nevertheless, some aspects involve a process of mental construction where explicit instruction seem to play a role in the US, but that can still take place in the absence of instruction in geometry. (shrink)

The purpose of this work is to address the relation existing between ancient Greek practical geometry and ancient Greek pure geometry. In the first part of the work, we will consider practical and pure geometry and how pure geometry can be seen, in some respects, as arising from an idealization of practical geometry. From an analysis of relevant extant texts, we will make explicit the idealizations at play in pure geometry in relation (...) to practical geometry, some of which are basically explicit in definitions, like that of segments in Euclid’s Elements. Then, we will address how in pure geometry we, so to speak, “refer back” to practical geometry. This occurs in two ways. One, in the propositions of pure geometry. The other, when applying pure geometry. In this case, geometrical objects can represent practical figures like, e.g., a practical segment. (shrink)

Peg Rawes examines a "minor tradition" of aesthetic geometries in ontological philosophy. Developed through Kant’s aesthetic subject she explores a trajectory of geometric thinking and geometric figurations--reflective subjects, folds, passages, plenums, envelopes and horizons--in ancient Greek, post-Cartesian and twentieth-century Continental philosophies, through which productive understandings of space and embodies subjectivities are constructed. Six chapters, explore the construction of these aesthetic geometric methods and figures in a series of "geometric" texts by Kant, Plato, Proclus, Spinoza, Leibniz, Bergson, Husserl and Deleuze. (...) In each text, geometry is expressed as a uniquely embodies aesthetic activity because each respective geometric method and figure is imbued with aesthetic sensibility and geometric sense (rather than as disembodies scientific methods). An ontology of aesthetic geometric methods and figures is therefore traced from Kant’s Critical writings, back to Plato and Proclus Greek philosophy, Spinoza and Leibniz’s post-Cartesian philosophies, and forwards to Bergson’s "duration" and Husserl’s "horizons" towards Deleuze’s philosophy of sense. (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)

We present an approach in which ancient Greek mathematical proofs by Hippocrates of Chios and Euclid are addressed as a form of (guided) intentional reasoning. Schematically, in a proof, we start with a sentence that works as a premise; this sentence is followed by another, the conclusion of what we might take to be an inferential step. That goes on until the last conclusion is reached. Guided by the text, we go through small inferential steps; in each one, we (...) go through an autonomous reasoning process linking the premise to the conclusion. The reasoning process is accompanied by a metareasoning process. Metareasoning gives rise to a feeling-knowing of correctness. In each step/cycle of the proof, we have a feeling-knowing of correctness. Overall, we reach a feeling of correctness for the whole proof. We suggest that this approach allows us to address the issues of how a proof functions, for us, as an enabler to ascertain the correctness of its argument and how we ascertain this correctness. (shrink)

Philosophers have studied geometry since ancient times. Geometrical knowledge has often played the role of a laboratory for the philosopher's conceptual experiments dedicated to the ideation of powerful theories of knowledge. Lorenzo Magnani's new book Philosophy and Geometry illustrates the rich intrigue of this fascinating story of human knowledge, providing a new analysis of the ideas of many scholars (including Plato, Proclus, Kant, and Poincaré), and discussing conventionalist and neopositivist perspectives and the problem of the origins of (...) geometry. The book also ties together the concerns of philosophers of science and cognitive scientists, showing, for example, the connections between geometrical reasoning and cognition as well as the results of recent logical and computational models of geometrical reasoning. All the topics are dealt with using a novel combination of both historical and contemporary perspectives. Philosophy and Geometry is a valuable contribution to the renaissance of research in the field. (shrink)

Some ancient Greek coins from the island state of Aegina depict peculiar geometric designs. Hitherto they have been interpreted as anticipations of some Euclidean propositions. But this paper proposes geometrical constructions which establish connections to pre-Euclidean treatments of incommensurability. The earlier Aeginetan coin design from about 500 bc onwards appears as an attempt not only to deal with incommensurability but also to conceal it. It might be related to Plato’s dialogue Timaeus. The newer design from 404 bc onwards reveals (...) incommensurability, namely in the context of ‘doubling the square’. It thereby covers the same topic but a different geometry as passages in Plato’s dialogue Meno (385 bc). This coin design incorporates important elements of ancient Greek geometrical analysis of the fifth century bc like the gnomon, Hippocrates’ squaring of the lunule (ca. 430 bc), and a geometrical version of monetary equivalence. Through this venue, the design’s conceptual lineage might be traced as far back as Heraclitus’ cosmology of about 500 bc. (shrink)

Even though the discovery of the regular polyhedra is attributed to the Pythagoreans, there is some fascinating evidence that they may have been known in prehistoric Scotland. In the Ashmolean Museum at Oxford University there are five rounded stones with regularly spaced bumps. The high points of each bump mark the vertices of each of the regular polyhedra. The stone balls also appear to demonstrate the duals of three of the regular polyhedra. For example, if the six faces of the (...) cube become points, they become the six vertices of the octahedron. If the eight faces of the octahedron become points, they become the eight vertices of a cube. Remarkably, groves in the Ashmolean stones indicate each of the duals, including the fact that the tetrahedron is its own dual. Instead of assuming that these ancient Scots were experts in solid geometry, we have hypothesized that they must have discovered all of this by sphere stacking, which will be demonstrate later in this article. (shrink)

For many centuries, philosophers and scientists have pondered the origins and nature of human intuitions about the properties of points, lines, and figures on the Euclidean plane, with most hypothesizing that a system of Euclidean concepts either is innate or is assembled by general learning processes. Recent research from cognitive and developmental psychology, cognitive anthropology, animal cognition, and cognitive neuroscience suggests a different view. Knowledge of geometry may be founded on at least two distinct, evolutionarily ancient, core cognitive (...) systems for representing the shapes of large-scale, navigable surface layouts and of small-scale, movable forms and objects. Each of these systems applies to some but not all perceptible arrays and captures some but not all of the three fundamental Euclidean relationships of distance (or length), angle, and direction (or sense). Like natural number (Carey, 2009), Euclidean geometry may be constructed through the productive combination of representations from these core systems, through the use of uniquely human symbolic systems. (shrink)

Aristotle’s philosophy of geometry is widely interpreted as a reaction against a Platonic realist conception of mathematics. Here I argue to the contrary that Aristotle is concerned primarily with the methodological question of how universal inferences are warranted by particular geometrical constructions. His answer hinges on the concept of abstraction, an operation of “taking away” certain features of material particulars that makes perspicuous universal relations among magnitudes. On my reading, abstraction is a diagrammatic procedure for Aristotle, and it is (...) through imagined variation of diagrams that one can universalize spatial inferences. In order to shift the discussion of Aristotle’s work on geometry from ontological to methodological questions, I argue in the first section that geometrical properties for Aristotle are necessary accidents of the particulars, canonically diagrams, in which they inhere. I then consider a line of research that shows how diagrams in ancient geometry are not illustrations of the textual proof but are in part constitutive of geometrical inference, an insight I claim can shed light on Aristotle’s philosophical project. Next I substantiate this assertion by giving an alternative interpretation of abstraction according to which it is a diagrammatic procedure that allows a part of a particular diagram to stand in for a universal. In the last section, I consider Aristotle’s account of mathematical cognition, arguing that there is evidence for the view that imagination is necessary both for the construction of figures, and for their presentation as abstractions subject to universal inference. (shrink)

We show how an epistemology informed by cognitive science promises to shed light on an ancient problem in the philosophy of mathematics: the problem of exactness. The problem of exactness arises because geometrical knowledge is thought to concern perfect geometrical forms, whereas the embodiment of such forms in the natural world may be imperfect. There thus arises an apparent mismatch between mathematical concepts and physical reality. We propose that the problem can be solved by emphasizing the ways in which (...) the brain can transform and organize its perceptual intake. It is not necessary for a geometrical form to be perfectly instantiated in order for perception of such a form to be the basis of a geometrical concept. (shrink)

The work of David R. Lachterman, The Ethics of Geometry, subtitled A Genealogy of Modernity, concerns essentially the status of geometry in Euclid’s Elements and in Descartes’s Geometry. It is a remarkable work, at once by the declared breadth of its ambitions and by the very great precision of its analyses, which are always supported by a prodigious philosophical culture. David Lachterman’s concern is to grasp, by way of an in-depth commentary of certain, particularly crucial passages of (...) these two foundational works, the change in geometry’s status from the one to the other, and this, as a sort of symptom of what the Moderns, since Descartes, will always experience as a necessary, inaugural rupture with regard to tradition. This change is underlaid, according to the author, by a profound change in the philosophical ethos, and in particular, in the geometer’s ethos. One must understand ethos, Lachterman explains, in the manner of Aristotle: those characteristic means that human beings have by which to act in the world or to behave in relation to one another, or to themselves. There is thus an “ethics” of geometry, namely, in the manner and the style of doing geometry, of behaving as a mathematician both toward apprentices and toward the veritable nature of the “entities” which are to be taught or learnt, and which give their discipline its name. That there is a difference of ethos between Euclid and Descartes implies, according to Lachterman, that there is also a difference in the source of intelligibility of the “mathematical,” understood in the most general sense. This in turn implies a deeper difference in the mode of being in general: it suffices to note that, in the ancient case, the source of intelligibility of the geometric figure lies in the nature of the figure itself, and that in the modern case, the same source is found in the “strategies” and “tactics” suited to bringing the figure to its visibility or to its embodiment, 315 in order to notice that this change in the mode of access to intelligibility must have had profound repercussions upon philosophical thought. It is in this sense that the author’s design is also, at the same time, not genetic, but genealogical. The break between ancient and modern mathematics pleas in favor of a consideration of the rupture of modern thought relative to ancient thought; thus, implicitly—and it is one of the great merits of this book that it shows this—against all homogenizing levelling, in Heidegger’s fashion, of the history of thought down to the history of “the” discipline of metaphysics. If there is something Heideggerian, in the best sense of the term, in Lachterman’s way of considering ethos as coextensive with modes of being, there is, on the other hand, and we ought to be very glad of it, something anti-Heideggerian in his original manner of bringing out the effective novelty of modern thought. For, according to him, modern thought is no longer to be defined, explicitly or exclusively, by the so-called “metaphysics of subjectivity.” Instead, it is defined by a self-regulated, operative, and constructive productivity of thought—and this as much with regard to itself as to its objects. To put it briefly, whereas ancient thought is rather polarized by the logico-eidetic, modern thought is polarized by a sort of methodical constructivism, the status of which is, in reality, very complex. If Descartes speaks of “the construction of a problem,” while Leibniz speaks of the “construction of an equation,” and Kant of the “construction of a concept,” everything depends—as we might guess—upon the multiple meanings that the term ‘construction’ may take on. Is it to be taken in the sense of an absolute creation, a quasi ex nihilo creation, which would render mathematics divine? Is it the creation of artefacts, or again, is it the methodical exploration of problems posed to thought by the existence of objects or corresponding ideas, themselves supposed to be somewhere in the divine understanding? We already discern the breadth and the depth of the debate, and we sense that the essential will be played out in Lachterman’s commentaries on the famous Cartesian solution of the problem of Pappus—the birth certificate, as we know, of analytic geometry. (shrink)

Some philosophers hold that mathematics depends on idealising assumptions. While these thinkers typically emphasise the role of idealisation in set theory, Edmund Husserl argues that idealisation is constitutive of the early Greek geometry that is codified by Euclid. This paper takes up Husserl's idea by investigating three major developments of Greek geometry: Thalean analogical idealisation, Hippocratean dynamic idealisation, and Archimedean mechanical idealisation. I argue that these idealisations are not, as Husserl held, primarily a matter of ‘smoothing out’ sensory (...) reality to produce ideal, ‘perfect’ figures. Rather, Greek geometry depends on assuming some falsehoods – equidistance from a source of illumination, a perfectly unwavering hand, or a machine that weights abstract objects – in order to make complex problems tractable. Although these idealisations were rarely discussed explicitly in antiquity, they can be systematically reconstructed from our extant sources. (shrink)

Philosophy in the period immediately after Aristotle is sometimes thought to be marked by the decline of natural philosophy and philosophical disinterest in contemporary achievements in the sciences. But in one area at least, the early third century B.C.E. was a time of productive interaction between such disparate fields as epistemology, physics and geometry. Debates between the sceptics and the dogmatic philosophical schools focus on epistemological problems about the possibility of self-evident appearances, but there is evidence from Euclid's day (...) of a quite different response. The sceptical challenge provoked the development of theories explaining error formation, showing how illusions can be studied systematically and are subject to prediction. Such theories do not legitimate claims about the nature of the underlying entities perceived, but provide justification for forming expectations about future perceptions. While it overtly focuses on purely geometrical considerations, the Euclidean model of optics nonetheless provides support for certain views about the nature of vision and the physics of light. Moreover, by offering a model in which the image received is not thought to be a perspicuous mirroring of the object seen, Euclid may have helped promote a view of perception as something reconstructed from information received, not as a mere form transferred into the eye. The ancient sceptic may indeed have fulfilled his promise to promote inquiry by focusing attention on problems that escape the attention of a hasty theorist. (shrink)

This chapter contains section titled: Discussion.

When ancient mathematical treatises lack expositions of numerical techniques, what purposes could ancient mathematical theories be expected to serve? Ancient writers only rarely address questions of this sort directly. Possible answers are suggested by surveying geometry, mechanics, optics, and spherics to discover how the mathematical treatments imply positions on this issue. This survey shows the ways in which these ancient theoretical inquiries reflect practical activity in their fields. This account, in turn, suggests that the authors (...) may have intended their theorems not to predict, but to explain phenomena. We may then consider what kind of explanations they were seeking. (shrink)

This article is primarily concerned with Aristotle’s theory of the syllogistic, and the investigation of the hypothesis that logical symbolism and methodology were in these early stages of a geometrical nature; with the gradual algebraization that occurred historically being one of the main reasons that some of the earlier passages on logic may often appear enigmatic. The article begins with a brief introduction that underlines the importance of geometric thought in ancient Greek science, and continues with a short exposition (...) of Aristotle’s views and methods in regard to logic. We then offer an interpretation of syllogisms, as well as of the main proof methods that are utilized by Aristotle, in terms of diagrams that can be seen as analogous to those of Euclidean geometry; where the various proofs proceed by appropriately manipulating an initially drawn diagram, in a way parallel to constructions that occur in Euclid’s Elements. In this way, logic is presented as following, to a large degree, methodological tenets established by the practice of geometry. Finally, we present a diagrammatic decision procedure that allows one to directly read off the validity of syllogisms from a corresponding diagram, in a way such that no further manipulation of it is necessitated. (shrink)

Greek ancient and early modern geometry necessarily uses diagrams. Among other things, these enter geometrical analysis. The paper distinguishes two sorts of geometrical analysis and shows that in one of them, dubbed “intra-confgurational” analysis, some diagrams necessarily enter as outcomes of a purely material gesture, namely not as result of a codifed constructive procedure, but as result of a free-hand drawing.

The foci of this penetrating study are Euclid's geometry and Descartes' mathematics. It is a contribution to the history of mathematics, but it is much more, for the differing approaches to mathematics in the ancient and the modern worlds is shown to have deep consequences for both doing and knowing. The investigation is centered on the nature of geometrical construction in ancient and modern mathematics, and, by extension, the crucial importance of construction to the reality and self-understanding (...) of modernity. (shrink)

In lieu of an abstract, here is a brief excerpt of the content: Unmasking the Maxim: An Ancient Genre And Why It Matters Now W. ROBERT CONNOR We live surrounded by maxims, often without even noticing them. They are easily dismissed as platitudes, banalities or harmless clichés, but even in an age of big data and number crunching we put them to work almost every day. A Silicon Valley whiz kid says, Move Fast and Break Things. Investors try to (...) Buy Low and Sell High. Investigative reporters Follow the Money. Others Follow their Bliss. The lavish host thinks, The More the Merrier, while the Modernist architect is sure that Less is More. Captains of Industry whisper to themselves, Me First; captains of sinking ships shout, Women and Children First. Be Prepared was the Boy Scouts’ marching song, while the Proud Boys Stand Down and Stand By, awaiting their marching orders. Maxims are perhaps the smallest members in a large family of speech acts, which among the Greeks included proverbs, oracles, riddles, blessings, curses and lamentations. Not all of these continue in use, but maxims have found companions—mottos, mantras, advertising tag lines and jingles, slogans, political rallying cries, hashtags, three- or four-letter acronyms and 280 character tweets from the goddess Twitter. Familiarity, however, can breed contempt, or at least inattentiveness to the full range of their influence. Sometimes what sounds at first like empty verbiage turns into action. Bruce Lee adapted an old Taoist saying when he said, Be Water, and thereby provided what was for a while a surprisingly effective strategy for protesters in Hong Kong. arion 28.3 winter 2021 6 unmasking the maxim Maxims empower, for good or ill. In the hands of bigots or ideologues, they can turn deadly. Sic Semper Tyrannis shouted John Wilkes Booth after shooting Abraham Lincoln. A manufacturer of assault rifles urged potential customers to Earn Your Man Card. How better to earn it than to obey 8chan, a web site favored by white nationalists, with its own maxim, Embrace Infamy. A devotee of the site did just that, perpetrating a mass shooting in El Paso, Texas. Maxims, then, should not be lightly dismissed. Since the ancient Greeks so relished maxims, they should surely be interrogated to help us better understand the power of these often underestimated speech acts. Before turning to the Greeks, however, the English word maxim needs a closer look. english “maxim” english borrowed maxim from a Latin phrase and boiled it down for everyday use. In Latin writings about logic, the expression maxima propositio, biggest proposition, denoted a statement that did not need to be proved, but could provide the basis for proof of other, lesser propositions. This phraseology goes back at least to Boethius in the sixth century of our era. It’s the equivalent of the generalizations that serve as the major premises in syllogistic logic. If this makes maxims sound like the axioms of geometry, that is historically right. The starting points of plane geometry are propositions that deserve to be accepted; even without proof that they are worthy of belief. That’s the meaning of the term axiōma. No one needs to prove that things equal to the same thing are equal to each other. Think about it; after a while it seems self-evident. It is worthy of assent. We might equally well call such an axiom a maxim; indeed, the earliest (1426) attested use of maxim in English is described in the Oxford English Dictionary as “an axiom; a self-evident proposition assumed as a premise in mathematical, or dialectical reasoning.” W. Robert Connor 7 This usage is now obsolete, but Thomas Jefferson understood the idea behind it, since he believed that there were such truths, waiting to be put to work in building a society where all people were equal and possess certain inalienable rights. He did not feel he had to prove these propositions; they were in his view self-evident. In using these ideas in his draft of the Declaration of Independence, he was, in effect, transferring to politics what Euclid had done in geometry. It was a swift, strategic stroke on his part, for, since self-evident truths require no argument or explanation, they focus discussion not on... (shrink)

This book treats so‐called Greek mathematics, developed in the Greek‐speaking world between about 600 b.c. and 600 a.d. It consists of four parts: early Greek mathematics, Hellenistic mathematics, Graeco‐Roman mathematics, and late ancient mathematics. Each part is divided into two chapters, “The Evidence” and “The Questions.”This separation of evidence and questions is significant. Serafina Cuomo has refused to follow the familiar method of weaving an apparently seamless history of Greek mathematics out of fragmentary and heterogeneous documents and conjectures about (...) them. The chapters of questions, where she points to issues that remain open, are very suggestive. For example, most important documents about the early development of Greek mathematics derive from a single lost work, the History of Geometry by Eudemus. Cuomo dares to cast doubt on its authenticity. Though her reservations seem extreme, we should remember that no document is neutral: Eudemus's history is a compilation, which involved choices, and the fragments we have now are the result of selection, partly intentional, partly by chance.Another merit of this book is that it considers a wide range of activity as mathematics. Practical mathematics—such as land surveying and accounting, with their sociopolitical importance—is emphasized.Cuomo's chief claim is that the standard historiography that associates the development of Greek mathematics with Plato's philosophy is only the version promulgated by Proclus; other descriptions are also possible. In fact, Pappus, Iamblichus, and others had their own versions. This claim is reasonable and contributes to a better understanding of Greek mathematics and the authors of late antiquity.In her citations, the author tries to let the text speak for itself, allowing as much as a full page to a passage or a proposition, and she refrains from using modern symbolism to explain mathematical content. Though this attitude is admirable, its cost is not negligible. Readers expecting to acquire a basic knowledge of Greek mathematics may find themselves at a loss when faced with highly technical propositions presented without elucidation.What this compact book does not include should also be mentioned. In contrast to her enthusiasm for the social and political dimensions of ancient mathematics, the author seems somewhat indifferent to its technical and theoretical aspects. Archimedes and Apollonius command only 16 pages—less than 7 percent of the text—whereas T. L. Heath dedicated 150 pages of his 1,000‐page history to them . Though documents showing the importance of land surveying are frequently quoted, little is said about the practice and the technical development of this art. The technical details of Ptolemy's works are practically dismissed—but should he not have an especially important role in alternative versions of the history of ancient mathematics because of his ingenious reconciliation of rigorous theory and the limitations imposed by reality in fields like astronomy and geography?The scantiness of the technical ingredients makes Ancient Mathematics more a history of discourses about mathematics than a history of mathematics—though this is to some extent inevitable given the character of late ancient mathematics, as Cuomo correctly emphasizes. The plan of the series to which this book belongs may be too modest to accommodate the author's ambition. Another problem attributable to the publisher is that the notes appear at the end of each chapter, so checking the references is annoying. Short references could be put in parentheses, and footnotes would be more convenient for longer notes. (shrink)

This paper considers the nature of time and temporality in Homer. It argues that any exploration of narrative and time must, as its central tenet, take into account the irreducible plurality and interconnectedness of memory, the event, and experienced time. Drawing on notions of complexity, emergence, and stochastic behavior in science as well as phenomenological traditions in the discussion and analysis of time, temporality, and change, and offering extensive readings of Homer, of Homeric epithets and formulae, and of key passages (...) in the Iliad and Odyssey, the paper argues against chronological notions of linear time and progression and in favor of a complex, dynamic temporal “geometries” of Homeric temporality. The paper concludes by briefly extending the argument to the wider domain of ancient time in general. Homer is a fundamental point of reference in the ancient world. Thus, Homeric temporality—irreducibly complex—affects the cognition and perception of time throughout the whole of antiquity. (shrink)

This radical, profoundly scholarly book explores the purposes and nature of proof in a range of historical settings. It overturns the view that the first mathematical proofs were in Greek geometry and rested on the logical insights of Aristotle by showing how much of that view is an artefact of nineteenth-century historical scholarship. It documents the existence of proofs in ancient mathematical writings about numbers and shows that practitioners of mathematics in Mesopotamian, Chinese and Indian cultures knew how (...) to prove the correctness of algorithms, which are much more prominent outside the limited range of surviving classical Greek texts that historians have taken as the paradigm of ancient mathematics. It opens the way to providing the first comprehensive, textually based history of proof. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Ancient and Modern HermeneuticsDavid HalliburtonAncient and Modern Hermeneutics, by Gerald L. Bruns; xii & 318 pp. New Haven: Yale University Press, 1992, $37.50.Modern hermeneutics, Bruns explains, has mainly gone in two directions. One is toward the transcendental ground-swells of Husserl, who remains committed to idealities, as exemplified in geometry. The second direction, taken by Heidegger, Gadamer, and Bruns (not to mention Sartre, Merleau-Ponty, Levinas, and the (...) Pragmatists) hews to the worldly rootedness of understanding; here each and all enjoy the capacity to comprehend—if not perfectly, at least perfectly well.Bruns divides his admirable opus into two large sections. Part One, on the Ancients, consists of essays on Socrates, Thucydides, the Hebrew Bible, allegory, the Jewish Midrash, and what may be broadly viewed as a correlative Islamaic text. He argues that Socrates’ ironic inclinations are not forensic ploys but manifestations of an authentic ontological disposition. (This ontological motif sounds throughout Bruns’s text.) Bruns’s Socrates has but to carry out the phusis that he himself is—in the undertow of which he cannot but be ironic even as, in being ironic, he cannot but be, like phusis, elusive and even impenetrable: “The point about Socrates is that we never know how to take him” (p. 32).Bruns handles ingeniously the problem of speeches that Thucydides “reported” but could not have heard. He argues that Thucydides provides a sort of simulacrum of the speeches. On this view, the historian does not attempt to report what the speakers actually said; instead he portrays implicitly the concrete historical situations (the modifiers are mine) in which orators orate: “the true author... would not be its original speaker nor its subsequent scribes but the situation that called for the speech to be made” (p. 58).In his challenging chapter on allegory, especially in Philo, Bruns approaches the former’s mystical commentary on the cherubim by problematizing familiar, restrictive distinctions. “In this case there can be no easy or gross distinction between the exoteric and the esoteric, the literal-minded and the wise....” To Philo “understanding is a form of recognition, of being turned toward and received into what before had been hidden or strange” (p. 102).Part Two, on the Moderns, treats of hermeneutic issues raised by Wordsworth [End Page 158] but even more paradigmatically by Luther; the tragic concept of experience; the question of tradition; radical hermeneutics; and the quarrel between philosophy and poetry. The journey ends in Bruns’s attempt to hypostatize an originary concept of freedom.To Bruns the liminal figure for modernity is Luther. It is a high hermeneutic moment that each of Luther’s students receives his own Bible with wide margins in which to copy the master’s—or conceivably his own—commentaries. Here Scripture is not merely grasped cognitively but is immediately, and, with full Deweyan and Diltheyan overtones, experienced. Luther for his part postulates that the text interprets itself. Concurring, and returning to the ontological motif noted above, Bruns proposes the one’s relation to a text “is ontological rather than simply exegetical. Luther’s distinction between law and gospel is a way of extending this ontological relation from the notion of a legal indictment to that of a spiritual one in which one’s self-understanding or self-identity is reconstituted by the text” (p. 146).Later Bruns deftly contrasts Hegel’s Umkehrung, the “movement toward enlightenment and the self-certainty of absolute consciousness” (p. 157), with Heidegger’s resolute refusal to appropriate or manipulate the world into which he, with everyone else, has been thrown. This attitude is somewhat akin to negative capability or wise passiveness—no English word or phrase gets it right—and requires a kind of letting-be, or letting-go. Bruns puts the matter about as well as one can, in short compass: “Thinking... is not reasoning or questioning but listening; its goal is not conceptual representation by Gelassenheit, or a letting go, releasement, and ‘openness to the mystery’” (p. 158).In his chapter on romantic hermeneutics, Bruns returns to Luther who, though stressing the empowerment of the individual reader, nonetheless regards the text as a vital force entering into and transforming the... (shrink)

In his collection of anecdotes, Lives, Opinions, and Remarkable Sayings of the Most Famous Ancient Philosophers, Diogenes Laertius devotes a chapter to the life of Zeno of Elea. Zeno's reputation is based on his celebrated paradoxes, amply discussed by Aristotle: a moving body will never reach its (pre-defined) telos, since it first has to cover half (or more than half) the remaining distance; the faster will never catch up with the slower, since it first has to get to the (...) point from which the slower has just left. Zeno's style is laconic, like that of an Aesop fable. Maxima e minimis: there is no superfluous word. Everything needed to arrive at the conclusion is explicitly stated. (shrink)