This is the first book-length analysis of the techniques and procedures of ancient mathematical commentaries. It focuses on examples in Chinese, Sanskrit, Akkadian and Sumerian, and Ancient Greek, presenting the general issues by constant detailed reference to these commentaries, of which substantial extracts are included in the original languages and in translation, sometimes for the first time. This makes the issues accessible to readers without specialized training in mathematics or in the languages involved. The result is a much (...) richer understanding than was hitherto possible of the crucial role of commentaries in the history of mathematics in four different linguistic areas, of the nature of mathematical commentaries in general, of the contribution that the study of mathematical commentaries can make to the history of science and to the study of commentaries in general, and of the ways in which mathematical commentaries are like and unlike other kinds of commentaries. (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)

Late ancient Platonists discuss two theories in which geometric entities xplain natural phenomena : the regular polyhedra of geometric atomism and the ccentrics and epicycles of astronomy. Simplicius explicitly compares the status of the first to the hypotheses of the astronomers. The point of omparison is the fallibility of both theories, not the reality of the entities postulated. Simplicius has strong realist commitments as far as astronomy is concerned. Syrianus and Proclus, too, do not consider the polyhedra as devoid (...) of physical reality. Proclus rejects epicycles and eccentrics, but accepts the reality of material homocentric spheres, moved by their own souls. The spheres move the astral objects contained in them, which, however, add motions caused by their own souls. The epicyclical and eccentric hypotheses are useful, as they help us to understand the complex motions resulting from the interplay of spherical motions and volitional motions of the planets. Yet astral souls do not think in accordance with human theoretical constructs, but rather grasp the complex patterns of their motions directly. Our understanding of astronomy depends upon our own cognition of intelligible patterns and their mathematical images. (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)

The theorem of Pythagoras, Euclid's "Elements", Archimedes' method to find the volume of a sphere: all parts of the invaluable legacy of ancient mathematics. But ancient mathematics was also about counting and measuring, surveying land and attributing mystical significance to the number six. This volume offers the first accessible survey of the discipline in all its variety and diversity of practices. The period covered ranges from the fifth century BC to the sixth century AD, with the focus on (...) the Mediterranean region. Topics include: * mathematics and politics in classical Greece * the formation of mathematical traditions * the self-image of mathematicians in the Graeco-Roman period * mathematics and Christianity * and the use of the mathematical past in late antiquity. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:American Journal of Philology 125.1 (2004) 140-144 [Access article in PDF] C. J. Tuplin and T. E. Rihll, eds. Science and Mathematics in Ancient Greek Culture. Foreword by Lewis Wolpert. Oxford: Oxford University Press, 2002. xvi + 379 pp. 21 black-and white ills. 3 tables. Cloth, $80. It has become something of a truism to say that, whatever their ambitions for abstraction, scientists remain profoundly caught up in (...) the life of their culture. Yet, that is a truism only at a certain level. No one is surprised to hear that Euclid had to eat and drink just like everyone else and relied on royal patronage for the leisure to pursue his studies, but it may occasion surprise to learn that the shape of his number theory should owe something to the particular way in which his eye fell upon a multiplication table. That Galen was influenced by his teachers is not news; but that his commitment to a particular school of philosophy could cause him to see and not see certain things when he was conducting dissections—here is something of interest. The papers in this volume, which were first delivered at a conference in Liverpool in 1996, have it as their common task to show how ancient forms of life influenced the pursuit and acquisition of knowledge on the part of scientists like Euclid and Galen.Heading the collection is a study by Andrew Barker of some idiosyncratic features of Greek harmonic theory. Archytas, Aristotle, and Ptolemy each attempted to characterize the physical basis of the two opposite sound qualities, oxusand barus. As Aristotle himself observed, the terms are metaphors drawn from the tactile realm, and trouble arose because in that realm they are heterogeneous opposites, i.e., the proper opposites of "sharp" and "heavy" are "dull" and "light." Barker uncovers a corresponding uncertainty in the texts about whether these sound qualities should be analyzed as something more like weight or something more like sharpness. Thus, an excessive trust in the descriptive adequacy of ordinary language terms led these thinkers to frustration.Harry Hine's paper on earthquakes and vulcanology deals with ancient knowledge of two phenomena that elicited the attention of scientist and non-scientist alike. Like Barker, Hine draws our attention to a terminological deficiency: although there was a fairly elaborate vocabulary for seismoi, neither Greek nor Latin possessed a generic noun that could properly be translated as [End Page 140] "volcano." This deficiency seems to be correlated with the fact that earthquakes form a regular part of the meteorological curriculum, while volcanoes usually received independent treatment in works such as the Aetna. Particularly intriguing is Hine's suggestion that the opinion of locals, especially sailors, may have been the source of certain scientific theses about volcanoes, e.g., that Aetna's behavior was closely correlated with the weather.The difference between ancient astronomy as practiced and as it was popularly believed to operate is the subject of Alan Bowen's paper. His central question is, when did astronomers first set out to make predictions of eclipses? The surprising answer is that, despite popular beliefs to the contrary, such prediction was never practiced by astronomers in a serious way before Ptolemy. Along with scrutiny of the misunderstanding in Herodotus, Diodorus, and Pliny, the paper offers thorough analysis of the most well-documented "prediction," that of a lunar eclipse that took place the night before the battle of Pydna in 167 B.C.E., which was supposedly foreseen by the Roman (!) general C. Sulpicius Gallus. Bowen casts doubts upon the entire tradition, showing how adherence to an ideology of the well-rounded commander led the sources to inflate his achievement: while some reports have Gallus simply assuaging his troops' fears at the event, others have him explaining its causes or even foretelling it. Bowen's skepticism is well founded, and one must regard these supposed feats of eclipse-prediction as a kind of science fiction, prescient yet false all the same.R. Hanna's subject is the format and function of the parapegma calendar first devised by... (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)

On knowledge and practices: a manifesto -- The web of practices -- Agents and frameworks -- Complementarity in mathematics -- Ancient Greek mathematics: a role for diagrams -- Advanced math: the hypothetical conception -- Arithmetic certainty -- Mathematics developed: the case of the reals -- Objectivity in mathematical knowledge -- The problem of conceptual understanding.

On knowledge and practices: a manifesto -- The web of practices -- Agents and frameworks -- Complementarity in mathematics -- Ancient Greek mathematics: a role for diagrams -- Advanced math: the hypothetical conception -- Arithmetic certainty -- Mathematics developed: the case of the reals -- Objectivity in mathematical knowledge -- The problem of conceptual understanding.

This book examines the birth of the scientific understanding of motion. It investigates which logical tools and methodological principles had to be in place to give a consistent account of motion, and which mathematical notions were introduced to gain control over conceptual problems of motion. It shows how the idea of motion raised two fundamental problems in the 5th and 4th century BCE: bringing together being and non-being, and bringing together time and space. The first problem leads to the exclusion (...) of motion from the realm of rational investigation in Parmenides, the second to Zeno's paradoxes of motion. Methodological and logical developments reacting to these puzzles are shown to be present implicitly in the atomists, and explicitly in Plato who also employs mathematical structures to make motion intelligible. With Aristotle we finally see the first outline of the fundamental framework with which we conceptualise motion today. (shrink)

Originally published in 1949. This meticulously researched book presents a comprehensive outline and discussion of Aristotle’s mathematics with the author's translations of the greek. To Aristotle, mathematics was one of the three theoretical sciences, the others being theology and the philosophy of nature. Arranged thematically, this book considers his thinking in relation to the other sciences and looks into such specifics as squaring of the circle, syllogism, parallels, incommensurability of the diagonal, angles, universal proof, gnomons, infinity, agelessness of the universe, (...) surface of water, meteorology, metaphysics and mechanics such as levers, rudders, wedges, wheels and inertia. The last few short chapters address ‘problems’ that Aristotle posed but couldn’t answer, related ethics issues and a summary of some short treatises that only briefly touch on mathematics. (shrink)

It has been argued in relation to Old Babylonian mathematical procedure texts that their validity or correctness is self-evident. One “sees” that the procedure is correct without it having, or being accompanied by, any explicit arguments for the correctness of the procedure. Even when agreeing with this view, one might still ask about how is the correctness of a procedure articulated? In this work, we present an articulation of the correctness of ancient Egyptian and Old Babylonian mathematical procedure texts (...) – mathematical texts presenting the solution of problems. We endeavor to make explicit and explain how and why the procedures are reliable over and above the fact that their correctness is intuitive. (shrink)

This paper is an updated translation of an article published in French in the Journal Lato Sensu (I, 2014, p. 70-80). We study here the so-called 'Mathematical part' of Plato's Theaetetus. Its subject concerns the incommensurability of certain magnitudes, in modern terms the question of the rationality or irrationality of the square roots of integers. As the most ancient text on the subject, and on Greek mathematics and mathematicians as well, its historical importance is enormous. The difficulty to understand (...) it lies in the close intertwining of different fields we found in it: philosophy, history and mathematics. But conversely, correctly understood, it gives some evidences both about the question of the origins of the irrationals in Greek mathematics and some points concerning Plato's thought. Taking into account the historical context and the philosophical background generally forgotten in mathematical analyses, we get a new interpretation of this text, which far from being a tribute to some mathematicians, is a radical criticism of their ways of thinking. And the mathematical lesson, far from being a tribute to some future mathematical achievements, is ending on an aporia, in accordance with the whole dialogue. Résumé. Cet article est une traduction anglaise révisée d'un article paru en français dans la revue Lato Sensu (I, 2014, p. 70-80). (shrink)

This book treats ancient logic: the logic that originated in Greece by Aristotle and the Stoics, mainly in the hundred year period beginning about 350 BCE. Ancient logic was never completely ignored by modern logic from its Boolean origin in the middle 1800s: it was prominent in Boole’s writings and it was mentioned by Frege and by Hilbert. Nevertheless, the first century of mathematical logic did not take it seriously enough to study the ancient logic texts. A (...) renaissance in ancient logic studies occurred in the early 1950s with the publication of the landmark Aristotle’s Syllogistic by Jan Łukasiewicz, Oxford UP 1951, 2nd ed. 1957. Despite its title, it treats the logic of the Stoics as well as that of Aristotle. Łukasiewicz was a distinguished mathematical logician. He had created many-valued logic and the parenthesis-free prefix notation known as Polish notation. He co-authored with Alfred Tarski’s an important paper on metatheory of propositional logic and he was one of Tarski’s the three main teachers at the University of Warsaw. Łukasiewicz’s stature was just short of that of the giants: Aristotle, Boole, Frege, Tarski and Gödel. No mathematical logician of his caliber had ever before quoted the actual teachings of ancient logicians. -/- Not only did Łukasiewicz inject fresh hypotheses, new concepts, and imaginative modern perspectives into the field, his enormous prestige and that of the Warsaw School of Logic reflected on the whole field of ancient logic studies. Suddenly, this previously somewhat dormant and obscure field became active and gained in respectability and importance in the eyes of logicians, mathematicians, linguists, analytic philosophers, and historians. Next to Aristotle himself and perhaps the Stoic logician Chrysippus, Łukasiewicz is the most prominent figure in ancient logic studies. A huge literature traces its origins to Łukasiewicz. -/- This Ancient Logic and Its Modern Interpretations, is based on the 1973 Buffalo Symposium on Modernist Interpretations of Ancient Logic, the first conference devoted entirely to critical assessment of the state of ancient logic studies. (shrink)

Mathematics often seems incomprehensible, a melee of strange symbols thrown down on a page. But while formulae, theorems, and proofs can involve highly complex concepts, the math becomes transparent when viewed as part of a bigger picture. What Is a Number? provides that picture. Robert Tubbs examines how mathematical concepts like number, geometric truth, infinity, and proof have been employed by artists, theologians, philosophers, writers, and cosmologists from ancient times to the modern era. Looking at a broad range of (...) topics -- from Pythagoras's exploration of the connection between harmonious sounds and mathematical ratios to the understanding of time in both Western and pre-Columbian thought -- Tubbs ties together seemingly disparate ideas to demonstrate the relationship between the sometimes elusive thought of artists and philosophers and the concrete logic of mathematicians. He complements his textual arguments with diagrams and illustrations. This historic and thematic study refutes the received wisdom that mathematical concepts are esoteric and divorced from other intellectual pursuits -- revealing them instead as dynamic and intrinsic to almost every human endeavor. (shrink)

Very Short Introductions: Brilliant, Sharp, Inspiring Ancient Greece is often considered to be the birthplace of science and medicine, and the explanation of natural phenomena without recourse to supernatural causes. These early natural philosophers - lovers of wisdom concerning nature - sought to explain the order and composition of the world, and how we come to know it. They were particularly interested in what exists and how it is ordered: ontology and cosmology. They were also concerned with how we (...) come to know (epistemology) and how best to live (ethics). At the same time, the scientific thinkers of early Greece and Rome were also influenced by ideas from other parts of the world, and incorporated aspects of Egyptian, Babylonian, and Indian science and mathematics in their studies. In this Very Short Introduction Liba Taub gives an overview of the major developments in early science between the 8th century BC and 6th century AD. Focussing on Greece and Rome, Taub challenges a number of modern misconceptions about science in the classical world, which has often been viewed with a modern lens and by modern scientists, such as the misconception that little empirical work was conducted, or that the Romans did not 'do' science, unlike the Greeks. Beginning with the scientific notions of Thales, Pythagoras, Parmenides and other Presocratics, she moves on to Plato and Aristotle, before considering Hellenistic science, the influence of the Stoics and Epicurean ideas, and the works of Pliny the Elder, Eratosthenes, and Ptolemy. In her sweeping discussion, Taub explores the richness and creativity of ideas concerning the natural world and its function in the ancient world, and the influence these ideas have had on later centuries. ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable. (shrink)

We can now recognize Aristotle's many accomplishments in logical theory, not the least of which is treating the deduction process itself as a subject matter and thus establishing the science of logic. Aristotle took logic to be that part of epistemolo gy used to establish knowledge of logical consequence. Prior Analytics is a metalogical treatise on his syllogistic system in which Aristotle modelled his deduction system to demonstrate certain logical relationships among its rules. Aristotle's n otion of substitution distinguishes logical (...) syntax from semantics and enabled him to distinguish validity from deducibility sufficiently to note the completeness of his logic. (shrink)

Plato is the first scientist whose work we still possess. He is our first writer to interpret the natural world mathematically, and also the first theorist of mathematics in the natural sciences. As no one else before or after, he set out why we should suppose a link between nature and mathematics, a link that has never been stronger than it is today. Mathematical Plato examines how Plato organized and justified the principles, terms, and methods of our mathematical, natural science. (...) "Roger Sworder deserves our gratitude for drawing attention to the significance of mathematics in Plato's thought and writings. He lays the principal discussions out before us with clarity. He also presents Plato as a theorist of nature: of physics and not just metaphysics, to use Aristotle's distinction. Not all readers, we should admit, will be equally convinced of the usefulness of Plato's science for today, but they will all be led more deeply into Plato's vision of reality."--ANDREW DAVISON, Westcott House, Cambridge "Here is Plato for an anti-Platonic age. The author gives careful attention to some of the most important passages in the Platonic dialogues and offers new solutions to some of Plato's most famous mathematical puzzles. He then considers the implications of these penetrating studies for the philosophy of science, and the natural sciences especially. This is a book that revivifies the core themes of Platonism and restores science to worship. It shows Roger Sworder to be one of the foremost students of Plato writing today, and places him in the noble tradition of Thomas Taylor."--RODNEY BLACKHIRST, author of Primordial Alchemy and Modern Religion: Essays on Traditional Cosmology. (shrink)

Of Euclid’s lost manuscripts, few have elicited as much scholarly attention as the Porisms, of which a couple of brief summaries by late-Antiquity commentators are extant. Despite the lack of textual sources, attempts at restoring the content of this absent volume became numerous in early-modern Europe, following the diffusion of ancient mathematical manuscripts preserved in the Arabic world. Later, one similar attempt was that of French geometer Michel Chasles (1793–1880). This paper investigates the historiographical tenets and practices involved in (...) Chasles’ restoration of the porisms, as well as the philosophical and mathematical claims tentatively buttressed therewith. Echoes of the Quarrel of the Ancients and the Moderns, and of a long-standing debate on the authority and usefulness of the past, are shown to have decisively shaped Chasles’ enterprise—and, with it, his integration of mathematical and historical research. (shrink)

This book is not a conventional history of mathematics as such, a museum of documents and scientific curiosities. Instead, it identifies this vital science with the thought of those who constructed it and in its relation to the changing cultural context in which it evolved. Particular emphasis is placed on the philosophic and logical systems, from Aristotle onward, that provide the basis for the fusion of mathematics and logic in contemporary thought. Ettore Carruccio covers the evolution of mathematics from the (...) most ancient times to our own day. In simple and non-technical language, he observes the changes that have taken place in the conception of rational theory, until we reach the lively, delicate and often disconcerting problems of modern logical analysis. The book contains an unusual wealth of detail (including specimen demonstrations) on such subjects as the critique of Euclid's fifth postulate, the rise of non-Euclidean geometry, the introduction of theories of infinite sets, the construction of abstract geometry, and-in a notably intelligible discussion-the development of modern symbolic logic and meta-mathematics. Scientific problems in general and mathematical problems in particular show their full meaning only when they are considered in the light of their own history. This book accordingly takes the reader to the heart of mathematical questions, in a way that teacher, student and layman alike will find absorbing and illuminating. The history of mathematics is a field that continues to fascinate people interested in the course of creativity, and logical inference quite part and in addition to those with direct mathematical interests. Ettore Carruccio, who until his retirement was professor of philosophy at the University of Turin. He has made many contributions to mathematical and logical theory as well as to the history of the science. Isabel Quigly was the literary editor of The Tablet for many years. (shrink)

In Plato’s eponymous dialogue, Timaeus, the main character presents the universe as an (almost) perfect sphere filled by tiny, invisible particles having the form of four regular polyhedrons. At first glance, such a construction may seem close to an atomistic theory. However, one does not find any text in Antiquity that links Timaeus’ cosmology to the atomists, while Aristotle opposes clearly Plato to the latter. Nevertheless, Plato is commonly presented in contemporary literature as some sort of atomist, sometimes as supporting (...) a form of so-called ‘mathematical atomism’. However, the term ‘atomism’ is rarely defined when applied to Plato. Since it covers many different theories, it seems that this term has almost as different meanings as different authors. The purpose of this article is to consider whether it is correct to connect Timaeus’ cosmology to some kind of ‘atomism’, however this term may be understood. Its purpose is double: to obtain a better understanding of the cosmology of the Timaeus, and to consider the different modern ‘atomistic’ interpretations of this cosmology. In short, we would like to show that such a claim, in any form whatsoever, is misleading, an impediment to the understanding of the dialogue, and more generally of Plato’s philosophy. (shrink)