In the first part of chapter 2 of book II of the Physics Aristotle addresses the issue of the difference between mathematics and physics. In the course of his discussion he says some things about astronomy and the ‘ ‘ more physical branches of mathematics”. In this paper I discuss historical issues concerning the text, translation, and interpretation of the passage, focusing on two cruxes, the first reference to astronomy at 193b25–26 and the reference to the more physical branches at (...) 194a7–8. In section I, I criticize Ross’s interpretation of the passage and point out that his alteration of has no warrant in the Greek manuscripts. In the next three sections I treat three other interpretations, all of which depart from Ross's: in section II that of Simplicius, which I commend; in section III that of Thomas Aquinas, which is importantly influenced by a mistranslation of, and in section IV that of Ibn Rushd, which is based on an Arabic text corresponding to that printed by Ross. In the concluding section of the paper I describe the modern history of the Greek text of our passage and translations of it from the early twelfth century until the appearance of Ross's text in 1936. (shrink)

The philosophy of mind is an especially flourishing plot of philosophical terrain these days. In part this activity derives from the quality of the work and in part from the topic's location, at the intersection of science--computer science, mathematics, biology, and cognitive psychology--and metaphysics and epistemology, even ethics. If recent developments date from Ryle and his iconoclasm, the modern study of mind has an older provenance in the writings of Descartes, Hobbes, Spinoza, Locke, Hume, and other figures whose work (...) coincided with the rise of the new science, the fragmentation of religious hegemony, and the recovery of ancient traditions. In this period, of special importance to the treatment of human experience and mental phenomena was the retrieval of Epicurean and Stoic texts and ideas. It is, therefore, an opportune moment to have a sophisticated and comprehensive introduction to the philosophy of mind of the ancient Stoics and Epicureans. In this book, Julia Annas has provided just that. (shrink)

The transformation of mathematics from ancient Greece to the medieval Arab-speaking world is here approached by focusing on a single problem proposed by Archimedes and the many solutions offered. In this trajectory Reviel Netz follows the change in the task from solving a geometrical problem to its expression as an equation, still formulated geometrically, and then on to an algebraic problem, now handled by procedures that are more like rules of manipulation. From a practice of mathematics based on (...) the localized solution we see a transition to a practice of mathematics based on the systematic approach. With three chapters ranging chronologically from Hellenistic mathematics, through late Antiquity, to the medieval world, Reviel Netz offers an alternate interpretation of the historical journey of pre-modern mathematics. (shrink)

This book represents a new departure in science studies: an analysis of a scientific style of writing, situating it within the context of the contemporary style of literature. Its philosophical significance is that it provides a novel way of making sense of the notion of a scientific style. For the first time, the Hellenistic mathematical corpus - one of the most substantial extant for the period - is placed centre-stage in the discussion of Hellenistic culture as a whole. (...) Professor Netz argues that Hellenistic mathematical writings adopt a narrative strategy based on surprise, a compositional form based on a mosaic of apparently unrelated elements, and a carnivalesque profusion of detail. He further investigates how such stylistic preferences derive from, and throw light on, the style of Hellenistic poetry. This important book will be welcomed by all scholars of Hellenistic civilization as well as historians of ancient science and Western mathematics. (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 book highlights the emergence of a new mathematical rationality and the beginning of the mathematisation of physics in Classical Islam. Exchanges between mathematics, physics, linguistics, arts and music were a factor of creativity and progress in the mathematical, the physical and the social sciences. Goods and ideas travelled on a world-scale, mainly through the trade routes connecting East and Southern Asia with the Near East, allowing the transmission of Greek-Arabic medicine to Yuan Muslim China. The development of science, (...) first centred in the Near East, would gradually move to the Western side of the Mediterranean, as a result of Europe's appropriation of the Arab and Hellenistic heritage. Contributors are Paul Buell, Anas Ghrab, Hossein Masoumi Hamedani, Zeinab Karimian, Giovanna Lelli, Marouane ben Miled, Patricia Radelet-de Grave, and Roshdi Rashed. (shrink)

The five hundred years from 300 B.C. to A.D. 200 were a period during which Greek science made spectacular advances and Greek philosophy underwent dramatic changes. How much did the scientists take note of the philosophical issues bearing on their pursuits? What progress did the philosophers make with methodological and theoretical issues arising out of developments in science? What influence did philosophical criticism or philosophical ideas have on specific theories in medicine or mechanics, mathematics or astronomy? These are some (...) of the questions discussed in this series of papers by the distinguished scholars who took part in the Confe;rence Hellenistique in Paris in 1980. The result is a broad-ranging and pioneering volume which will be of importance to scholars in the history and philosophy of science and to those whose interests lie in classical philosophy. (shrink)

Mr. Brumbaugh gives several accounts in the course of his work of the main purpose of his study, and the emphasis falls now one way and now another. Readers may easily be misled by the opening sentence of the introduction, which suggests that Plato's mathematical illustrations are pointers to "diagrams which Plato had designed, and were intended to accompany and clarify his text." If that is what Mr. Brumbaugh intended, he has failed to make out his case. There is no (...) direct evidence that Plato designed diagrams; and there is some good inferential evidence to the contrary. It would be surprising, for example, if Plato's diagrams had survived that long, that Aristotle should have described the Divided Line in terms of an arithmetical progression and ignored Plato's express indication of proportion between the upper and lower segments. Mr. Brumbaugh himself faces this passage and admits that "no continuous tradition connects the figures of Hellenistic scholars with Plato's original design." But it really does not matter. If we design diagrams in the light of our knowledge of Plato's mathematical imagination, it is most certainly a help to the interpretation of the passages concerned. This, as a matter of fact, seems to be Mr. Brumbaugh's main reason for ascribing the diagrams to Plato, and while the conclusion does not follow, the premiss is correct; and the premiss, not the conclusion, is the main point at issue. (shrink)

The five hundred years from 300 B.C. to A.D. 200 were a period during which Greek science made spectacular advances and Greek philosophy underwent dramatic changes. How much did the scientists take note of the philosophical issues bearing on their pursuits? What progress did the philosophers make with methodological and theoretical issues arising out of developments in science? What influence did philosophical criticism or philosophical ideas have on specific theories in medicine or mechanics, mathematics or astronomy? These are some (...) of the questions discussed in this series of papers by the distinguished scholars who took part in the Conférence Hellenistique in Paris in 1980. The result is a broad-ranging and pioneering volume which will be of importance to scholars in the history and philosophy of science and to those whose interests lie in classical philosophy. (shrink)

ArgumentThe Fourth Postulate of Euclid’s Elements states that all right angles are equal. This principle has always been considered problematic in the deductive economy of the treatise, and even the ancient interpreters were confused about its mathematical role and its epistemological status. The present essay reconsiders the ancient testimonies on the Fourth Postulate, showing that there is no certain evidence for its authenticity, nor for its spuriousness. The paper also considers modern mathematical interpretations of this postulate, pointing out various anachronisms. (...) It further discusses the validity of the ancient proof by superposition of the Fourth Postulate. Finally, the article proposes an interpretation of the history of the concept of angle in Greek geometry between Euclid and Apollonius, and puts forward a conjecture on the interpolation of the Fourth Postulate in the Hellenistic age. The essay contributes to a general reassessment of the axiomatic foundations of ancient mathematics. (shrink)

The latest book of Reviel Netz presents a highly erudite analysis of the style of Hellenistic mathematics. Besides the Introduction and Conclusion the book is composed of four chapters. Before turning to more general remarks I would like first to outline the contents of the book.The Introduction starts with the presentation of Archimedes’ Spiral lines. In a condensed form, the author outlines his approach and calls attention to the stylistic peculiarities of that work. Most of the themes discussed (...) in more detail in the following chapters, such as the analysis of the narrative structure of the treatise, its working with surprise, the exposition of the rhetorical devices, the discussion of technical tools, and the breaking of genre-boundaries emerge here side by side in a single treatise.The first chapter, called The carnival of calculation, starts with another work of Archimedes, the Stomachion, for which Netz offers a new interpretation as a treatise in geometrical combinatorics. Then he turns to Archimedes’ On the measurement of the circle, where, in a detailed analysis of the calculations made by Archimedes, Netz presents a pattern that he identifies in several other works of Hellenistic mathematics, a pattern he calls ‘asymptotic calculation’. It consists in a combination of strict geometric arguments with calculations of proportions, some precise, some only approximate, which together with rather arbitrary simplifications lead to a result that is achieved after an abrupt and unexpected halt. Netz shows that, according to the surviving testimonies, a similar pattern can be found in many Hellenistic authors, such as Aristarchus, Hipparchus, and Apollonius of Perga. "[I]n many Hellenistic mathematical works we see calculations that give rise to values that are big and unwieldy, or else are only asymptotically found. Such calculations end up not with the satisfaction of having simplified … ". (shrink)

The “unknown heritage” is the name usually given to a problem type in whose archetype a father leaves to his first son 1 monetary unit and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\frac{1}{n}}$$\end{document} (n usually being 7 or 10) of what remains, to the second 2 units and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\frac{1}{n}}$$\end{document} of what remains, and so on. In the end, all sons get the same, and nothing remains. The earliest known (...) occurrence is in Fibonacci’s Liber abbaci, which also contains a number of much more sophisticated versions, together with a partial algebraic solution for one of these and rules for all which do not follow from his algebraic calculation. The next time the problem turns up is in Planudes’s late thirteenth century Calculus according to the Indians, Called the Great. After that the simple problem type turns up regularly in Provençal, Italian and Byzantine sources. It seems never to appear in Arabic or Indian writings, although two Arabic texts (one from c. 1190) contain more regular problems where the number of shares is given; they are clearly derived from the type known from European and Byzantine works, not its source. The sophisticated versions turn up again in Barthélemy de Romans’ Compendy de la praticque des nombres (c. 1467) and, apparently inspired from there, in the appendix to Nicolas Chuquet’s Triparty (1484). Apart from a single trace in Cardano’s Practica arithmetice et mensurandi singularis, the sophisticated versions never surface again, but the simple version spreads for a while to German practical arithmetic and, more persistently, to French polite recreational mathematics. Close examination of the texts shows that Barthélemy cannot have drawn his familiarity with the sophisticated rules from Fibonacci. It also suggests that the simple version is originally either a classical, strictly Greek or Hellenistic, or a medieval Byzantine invention; and that the sophisticated versions must have been developed before Fibonacci within an environment (located in Byzantium, Provence, or possibly in Sicily?) of which all direct traces has been lost, but whose mathematical level must have been quite advanced. (shrink)

For the purposes of this monograph, "by a degree" is meant a degree of recursive unsolvability. A degree [script bold]m is said to be minimal if 0 is the unique degree less than [script bold]m. Each of the six chapters of this self-contained monograph is devoted to the proof of an existence theorem for minimal degrees.

Kurt Gdel was the most outstanding logician of the 20th century and a giant in the field. This book is part of a five volume set that makes available all of Gdel's writings. The first three volumes, already published, consist of the papers and essays of Gdel. The final two volumes of the set deal with Gdel's correspondence with his contemporary mathematicians, this fourth volume consists of material from correspondents from A-G.

This volume presents the proceedings from the Eleventh Brazilian Logic Conference on Mathematical Logic held by the Brazilian Logic Society in Salvador, Bahia, Brazil. The conference and the volume are dedicated to the memory of professor Mario Tourasse Teixeira, an educator and researcher who contributed to the formation of several generations of Brazilian logicians. Contributions were made from leading Brazilian logicians and their Latin-American and European colleagues. All papers were selected by a careful refereeing processs and were revised and updated (...) by their authors for publication in this volume. There are three sections: Advances in Logic, Advances in Theoretical Computer Science, and Advances in Philosophical Logic. Well-known specialists present original research on several aspects of model theory, proof theory, algebraic logic, category theory, connections between logic and computer science, and topics of philosophical logic of current interest. Topics interweave proof-theoretical, semantical, foundational, and philosophical aspects with algorithmic and algebraic views, offering lively high-level research results. (shrink)

This book casts new light on the process that in the sixteenth and seventeenth centuries led to a profound transformation in the study of nature with the emergence of mechanistic philosophy, the new mixed mathematics, and the establishment of the experimental approach. It is argued that modern European science originated from Hellenistic mathematics not so much because of rediscovery of the latter but rather because its “applied” components, namely mechanics, optics, harmonics, and astronomy, and their methodologies continued (...) to be transmitted throughout the Middle Ages without serious interruption. Furthermore, it is proposed that these “applied” components played a role in their entirety; thus, for example, “new” mechanics derived not only from “old” mechanics but also from harmonics, optics, and astronomy. Unlike other texts on the subject, the role of mathematicians is stressed over that of philosophers of nature and the focus is particularly on epistemological aspects. In exploring Galilean and post-Galilean epistemology, attention is paid to the contributions of Galileo’s disciples and also the impact of his enemies. The book will appeal to both historians of science and scientists. (shrink)

Humanism, in the context of the classical world, contrasted the vitality of human life with the shadowy existence of the underworld endured after death. The buzz of ideas that permeated Athens in the fifth century is usually known as ‘Sophism’. The Sophists were attracted to Athens from throughout the Greek world, and they loved argument for its own sake. Much more important in the humanist tradition is Aristotle, who came to Athens from the northern Aegean to study with Plato in (...) Athens. It is a measure of a humanist society that it never allows dogma to impede progress, and after the fourth century the Greek tradition continued to be dynamic, especially in the Hellenistic city of Alexandria, where mathematics and science flourished into the second century CE. The way that Christianity developed can be seen in the contrasting interpretations of Christianity by the Greek theologian Origen and the Latin Augustine. (shrink)

The removal of water from mines was one of the key issues that former miners had to deal with. Roman colonists brought new technology to the Iberian Peninsula that addressed this problem. However, they did not invent this technology because it had already been applied to the growth of other endeavours in the Hellenistic society throughout the Eastern Mediterranean. In the mine, the Archimedes screw, waterwheels, bucket pulleys, and Ctesibius pumps were the primary drainage systems. In this essay, the (...) primary characteristics, and modes of operation of machines are examined. Without leaving out the most significant finds made in the southwest of the Iberian Peninsula, one of the regions with the longest history of mining exploitation. To serve as a foundation for future research in this field, this work compares the primary mining mechanisms in ancient Huelva on a qualitative and quantitative level by the implementation of a TOPSIS methodology, a multi-criteria decision analysis method. (shrink)

The Ikhwan al-Safa (Brethren of Purity), the anonymous adepts of a tenth-century esoteric fraternity based in Basra and Baghdad, hold an eminent position in the history of science and philosophy in Islam due to the wide reception and assimilation of their monumental encyclopaedia, the Rasa'il Ikhwan al-Safa (Epistles of the Brethren of Purity). This compendium contains fifty-two epistles offering synoptic accounts of the classical sciences and philosophies of the age; divided into four classificatory parts, it treats themes in mathematics, (...) logic, natural philosophy, psychology, metaphysics, and theology, in addition to didactic fables. Epistles 32 to 36 comprise the first five treatises in the third division of the Rasa'il, on the sciences of the soul and intellect. Combining Islamic revelation with Hellenistic philosophy, the Ikhwan delineate herein their metaphysical system. Epistles 32 and 33 present adaptations of the Pythagorean doctrines, and of Neo-Platonist leitmotifs, whereby a numerical analogy is applied to the unique and transcendent God, or the One, from whom all existence emanates. Epistle 34 takes up the pervasive theme of the correspondence between microcosm and macrocosm, situating the human being as the central link between the celestial and terrestrial realms. In Epistle 35, we find an explanation of the intellectual faculties of the individual human soul, whose ultimate aim is ascension to the timeless reality of pure intellect. Finally, Epistle 36 presents itself as the astrological epistle par excellence of the Rasa'il; from the coming-to-be of worms, to the emergence of religions and empires, nothing in the sub-lunar sphere escapes the determining influence of the celestial cycles. (shrink)

The purpose of the conference "On Pythagoreanism", held in Brasilia in 2011, was to bring together leading scholars from all over the world to define the status quaestionis for the ever-increasing interest and research on Pythagoreanism in the 21st century. The papers included in this volume exemplify the variety of topics and approaches now being used to understand the polyhedral image of one of the most fascinating and long-lasting intellectual phenomena in Western history. Cornelli's paper opens the volume by charting (...) the course of Pythagorean studies over the past two centuries. The remaining contributions range chronologically from Pythagoras and the early Pythagoreans of the archaic period (6th-5th centuries BCE) through the classical, hellenistic and late antique periods, to the eighteenth century. Thematically they treat the connections of Pythagoreanism with Orphism and religion, with mathematics, metaphysics and epistemology and with politics and the Pythagorean way of life. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Plato, Proclus, and the Limitations of Science S. SAMBURSKY I THE NEOPLATONICREVlV~of Plato's views on the physical world offers some highly interesting aspects to the historian of scientific ideas. There is first of all the interaction between a 600-year-old tradition and other philosophical systems that grew up during this long period and that exerted such a decisive influence on later antiquity. And there is further the magnificent development of (...) science--mathematics and astronomy in particular--in the period following Plato, reaching its climax in the Hellenistic era. This development introduced profound changes in the scientific world picture by adding a wealth of new facts to astronomy, mechanics, optics, and biology, and thus facing the later disciples of Plato with a difficult alternative : They could either assume an orthodox attitude by doubting the facts or at least belittling their significance for the Platonic doctrine; or else they could compromise by attempting to interpret the facts in the Platonic spirit--doing so by availing themselves of the well-known expedient of reading between the lines of the master's writings. It can, of course, not be denied that everything Plato has written on the physical world and on scientific cognition will always be, as it has been in antiquity, subject to interpretations. The task of a Platonic commentator is made even more difficult by his poetic style, his predilection for allegory, and his frequent use of allusions at the very moment when one expects systematic analysis. The increasing anti-metaphysical bias of modern science combined with the fact that Plato himself was neither a practicing scientist nor a creative mathematician led to the disparagement of his contribution to scientific thought in the last centuries. This applies in particular to the Timaeus,where the main tenets of PIato's scientific doctrine are expounded. However, on the whole, this negative attitude is not justified. The consistency and significance of Plato's epistemology of science is borne out by his statements in the Timaeusif they are taken together with his assertions concerning the role of mathematics in the cognition of nature and his appeal to the astronomers to "save the phenomena" of the planetary motions. Careful consideration of all these passages, regarded as a coherent whole, will show in the first place that Plato's philosophy of science does not fall short of his other great contributions to philosophy and human thought. It will further serve as a background against which the significant shift of emphasis that is found in Proclus's evaluation of the capacity of the human mind for the progressive understanding of nature will be seen. We all know of Plato's fundamental belief that the sensible world, i.e., the obiect of physical inquiry, is only an image of the inaccessible world of ideas. The intelligible world, the ultimate reality, is permanent and unchangeable, whereas [11 2 HISTORY OF PHILOSOPHY the material world is a world of transience and change. Every possible description of the physical world is therefore by necessity only a plausible story, not rising above the level of probability. This is the background of the celebrated passage in the Timaeus (29 C) : If then.., we prove unableto render an accountat all points entirely consistentwithitself and exact,you must not be surprised.If we can furnish accountsno less likely than any other, we must be content,rememberingthat I who speak and you my judges are onlyhuman, and consequently it is fittingthat we should, in these matters, accept the likelystory and look for nothingfurther. Here science has once and forever been assigned a place below the absolute truth, in view of the transient nature of the physical world and the role of the sensible world as an image of truth. Can some assertion be made as to the essence of this image? What is the ontological significance of all the changes and transmutations and what is the mechanism of that permanent generative process whose various components blend into the one perceptible unit which we call "nature"? Plato's answer is given in his allegory of the struggle between Reason and Necessity (Timaeus 48 A) : "The generation of this universe was a mixed result of the combination... (shrink)

The reception of the rational sciences, scientific practice, discourse and methodology into Arabic Islamic society proceeded in several stages of exchange with the transmitters of Iranian, Christian-Aramaic and Byzantine-Greek learning. Translation and the acquisition of knowledge from the Hellenistic heritage went hand in hand with a continuous refinement of the methods of linguistic transposition and the creation of a standardized technical language in Arabic: terminology, rhetoric, and the genres of instruction. Demonstration more geometrico, first introduced by the paradigmatic sciences- (...) class='Hi'>mathematics, astronomy, mechanics-and adopted by philosophers embracing the cosmology of Neoplatonism, was complemented and superseded by the methods of syllogistic demonstration. Faced with the establishment of philosophy as a demonstrative science, which claimed absolute and universal knowledge, even the hermeneuti. (shrink)

This concise, lively introduction to ancient Greek philosophy will help beginning students of both classical studies and philosophy get their bearings within an important yet complex array of names, schools, and ideas. The book illuminates the key period from the sixth to the third century BC, looking at the ideas that engaged the Greeks, in particular those of the Presocratics, the Sophists, Socrates, Plato, Aristotle, and the earliest Hellenistic philosophers. After chronologically mapping the main figures and their interconnections, _Introducing (...) Greek Philosophy_ focuses on themes especially relevant to philosophy today, including the origins of the universe and its mathematical structures; divine creation versus evolution and natural law; probability theory and the criteria for truth; political debates on democracy, citizen rights, and state obligations; and finally ethics, happiness, and the best way to live. (shrink)

The Science Museum, London, has recently acquired four fragments of a portable sundial with associated calendrical gearing. All the fragments are made of low zinc brass of substantially the same composition. The sundial is of a type known in other examples, some the products of recent archaeological excavations and all dated to the Late Antique or Early Byzantine period. Dating by the place names included in the latitude table, by the style of the heads of the planetary gods used to (...) identify the days of the week, and by the style of the script, all indicate that the instrument dates from the late fifth century of the Christian era or the first half of the sixth.The gearing, of which two arbors survive, carrying four toothed wheels and one ratchet, seems to have resembled that of a calendrical device described by al-Bīrūnī in about AD 1000. Like al-Bīrūnī's device and the geared calendar associated with a Persian astrolabe of AD 1221/2 now in the Museum of the History of Science, Oxford, the Byzantine instrument displayed the shape of the Moon and its age in days. Probably the calendar further resembled its Islamic counterparts in also displaying the positions of Sun and Moon in the Zodiac.The emergence of Byzantine mathematical gearing indicates that the Hellenistic tradition attested by the Antikythera machine continued to be active in the Byzantine period, and suggests that it may have influenced the Islamic tradition. Once again an artefact has shown up the inadequacy of the evidence derived from literary sources. (shrink)

L'oeuvre de Roshdi Rashed se deploie selon deux directions principales: la naissance, dans les Temps Modernes, des mathematiques sociales et de la theorie des probabilites d'une part, et l'histoire des sciences exactes a l'epoque de la Renaissance de l'Islam classique d'autre part. Cette oeuvre de longue haleine n'a cesse d'elargir la connaissance que nous avons de ces dernieres, par l'edition et la traduction de textes nouveaux, la reconstitution des traditions conceptuelles dans lesquelles ces textes s'inscrivent, ainsi que par la mise (...) en resonance de ces traditions. Elle agit en meme temps comme une genealogie de certaines formes de la mathesis moderne. Sans manifeste ni discours sur la methode, les travaux de Roshdi Rashed proposent un approfondissement constant de la tradition de l'epistemologie historique illustree par Gaston Bachelard et Georges Canguilhem. C'est pourquoi ces travaux ne laissent indifferent ni l'helleniste, ni bien sur l'arabisant specialiste de champs connexes, ni le medieviste, ni les specialistes de l'histoire des sciences modernes. C'est ce dont temoigne ce recueil d'etudes, dues a d'eminents collegues de Roshdi Rashed et a ses eleves. (shrink)

Summary The Almagest of Ptolemy (mid-second century ad) contains eleven dated reports of observations of the positions of planets made during the third century bc in Babylon and Hellenistic Egypt. The present paper investigates the character, purpose, and conventions of the observational programmes from which these reports derive, the channels of their transmission to Ptolemy's time, and the fidelity of Ptolemy's presentation of them. Like the Babylonian observational programme, about which we have considerable knowledge through cuneiform documents, the Greco-Egyptian (...) ones were not directed towards the deduction of mathematical models of celestial motion but appear to have investigated patterns, correlations, and periodicities of phenomena. Ptolemy's immediate sources most likely were not the original series of observational records, but treatises by various astronomers of the intervening four centuries, including Hipparchus. While Ptolemy does not appear to have tampered with the wording of the reports, he faced difficulties and uncertainties in interpreting them; critically, he lacked sufficiently detailed information about the ancient calendars to be able to convert the reported dates accurately into his own chronological framework based on the Egyptian calendar. (shrink)

The Academy was a philosophical school established by Plato that safeguarded the continuity and the evolution of Platonism over a period of about 300 years. Its contribution to the development of Hellenistic philosophical and scientific thinking was decisive, but it also had a major impact on the formation of most of the other philosophical trends emerging during this period. This volume surveys the evidence for the historical and social setting in which the Academy operated, as well as the various (...) shifts in the philosophical outlook of Platonism during its existence. Its contribution to the evolution of special sciences such as mathematics is also examined. The book further includes the first complete annotated translation in English of Philodemus' History of the Academy, preserved on a papyrus from Herculaneum. It thus offers a comprehensive picture of one of the most prominent and influential of all educational institutions in ancient Greece. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Dynamic Encounters between Buddhism and the West ReportLaura Langone and Alexandra S. IlievaThe following is a summary of the 2021 Postgraduate Conference titled "Dynamic Encounters between Buddhism and the West," which took place online on June 28 and 29. The conference was conceptualized, organized, and run by three AHRC funded PhD students at the University of Cambridge: Laura Langone (Faculty of Modern and Medieval Languages); Alexandra S. Ilieva (Faculty (...) of Divinity); and Harry Harland (Faculty of Law). The conference took place over two days and featured eighteen presenters across six panels, and two keynote speakers. There were speakers from four continents (Europe, North America, Asia, and Oceania). A wide range of topics in multiple fields was discussed: religious studies, neuroscience, art history, politics, philosophy, literature, philology, mathematics, art, and music. Each day consisted of three panels, each with three presentations, one keynote presentation, and a networking session.The keynote speakers were Prof. Albrecht Classen (The University of Arizona) and Prof. Graham Parkes (University of Vienna).Prof. Classen opened the conference with his presentation. He taught us that we have been globalized and transcultural since at least the Middle Ages. To support this claim, he provided us with ample evidence of Buddhist-Western exchanges in Art and Literature dotted throughout the literature and architecture of Europe. Prof. Parkes observed that following Zen Master Dōgen's thought as well as Nietzsche's philosophy we can develop a saner approach to the environment, which provides us with resources to help us to tackle the challenges of the climate crisis.Our first panel was titled "Hellenistic-Buddhist Encounters," which followed on nicely from Prof. Classen's thesis that encounters between Buddhism and the West have been ubiquitous and have dated as far back as the Hellenistic period. Our second panel, "Encounters with the Contemporary World," were impressive examples of intercultural philosophy. Each participant demonstrated a unique ability to bring Buddhist ideas into dialogue with contemporary Western philosophy, specifically ethics, philosophy of mathematics, and epistemology. The third panel, titled "Therapeutic Encounters," taught us of the therapeutic value of Buddhist meditation and philosophical inquiry. As our presenters emphasized, this therapeutic value is not just an ancient artifact to be studied anthropologically, but a relevant and applicable approach to the modern idea of well-being that therefore deserves continued scholarship.On the second day, we started with our fourth panel called "Encounters through Western Conceptual Lenses," which among others gave us insight into the reception [End Page 393] of Japanese Buddhism by the German liberal protestants of the nineteenth century. Our fifth panel—"Encounters with German Philosophy"—was an excellent example of contemporary comparative work between Buddhist philosophy, ethics, and logic, and the German thinkers Hegel and Nietzsche.We learned that Nietzsche labeled Buddhism as nihilism––a position that can be challenged through Ambedkar's reading of Buddhism. On the other hand, despite Nietzsche's criticism of Buddhism, striking similarities between his approach to life and that of Mahāyāna Buddhism can be found, both elaborating a life-affirming philosophy. We also became familiar with similarities and differences between the logic of Nishida and Hegel. Finally, our last panel, "Literary and Artistic Encounters" made for a fitting conclusion, reminding us that Buddhist-Western encounters are ongoing, not just politically and intellectually but also in art, music, and literature. These encounters are dynamic in nature and fundamentally lacking in an essential core. Instead, they are entities that echo through the ages allowing for reinterpretation and reintegration across geographic and temporal scopes.Both during and after the conference we received extremely positive feedback from the participants. The level of participation and enthusiasm was highly promising, showing that the future of intercultural studies of this sort is bright. Graduate conferences like these are key for early career scholars to set up networks and find likeminded peers to engage in ongoing dynamic encounters.In conclusion, this conference served to highlight the sophisticated emerging scholarship taking place that seeks to investigate Buddhist-Western interactions and dialogue. But, perhaps more importantly, the conference itself makes a metapoint: not only has the world been transcultural for many centuries, but the conference itself and in particular the global spatial locations of... (shrink)

The Epistles of the Brethren of Purity' is an encyclopedic compendium, probably composed in tenth-century Iraq by a society of adepts with Platonic, Pythagorean, and Shi'i tendencies. Its 52 sections ('epistles') are divided into four parts (Mathematics, Natural Philosophy, Sciences of the Soul and Intellect, and Theology). The current volume provides an edition, translation, and notes to Epistle 3 ('On Astronomia'), which forms one of the 14 sections on Mathematics. The content is a mixture of elementary astronomy and (...) astrology, but it is not a beginner's textbook. Rather, the purpose is to use examples from those disciplines to provide spiritual, moral, and soteriological guidance. Thus the Epistle uses the argument from design to show the necessity for a Creator who made the harmonious universe; this wondrous design is then employed by the authors as a model, providing humans with a paradigm for proper ethical, political, and even economic conduct; and the study of Astronomia helps the soul achieve ultimate happiness as it seeks to throw off the shackles of this mundane world and oppressive body in favour of the purity of the celestial realm.0Although by no means typical of Islamic astronomical literature, Epistle 3 of the Brethren of Purity gives a window into a fascinating and intriguing group operating during the early period of Islam who sought to continue and adopt one of the esoteric strands of Hellenistic philosophy within an Islamic context, meshing astronomy, astrology, Platonic-Pythagorean philosophy, Quranic and Biblical quotations, and anecdotes from the lives of the Abrahamic prophets. (shrink)

Sophists’ apologia. -/- Sophists were the first paid teachers ever. These ancient Greek enlighteners taught wisdom. Protagoras, Antiphon, Prodicus, Hippias, Lykophron are most famous ones. Sophists views and concerns made a unified encyclopedic system aimed at teaching common wisdom, virtue, management and public speaking. Of the contemporary “enlighters”, Deil Carnegy’s educational work seems to be the most similar to sophism. Sophists were the first intellectuals – their trade was to sell knowledge. They introduced a new type of teacher-student relationship – (...) the mutually beneficial communication on equal terms. They taught pupils how to think independently and how to persuade others, which was inseparably connected with the rise of democracy in the most advanced Greek polices. Sophists were the first to proclaim that all people are naturally equal. They put forward the idea of natural rights and social contract, working out the fundamentals of the present-day law. In addition, they developed the basics of philology, psychology, logic, and gave a scientific explanation to the origins of religions. They embodied definite positive ideals of their epoch. Sophistry flourished in the second half of the Vth through he first third of the IVth centuries B. C. The period coincides with the heyday of the entire Greek history – the so called Greek Miracle epoch. Sophists expressed some present-day concepts of Ancient Greek ideology and mentality, characteristic of its Golden Age times. The Ancient Athens of the times of Periclus were similar to the Florence of Renaissance as to the type of social structure. That is why (?) the art of eloquence (public speaking) was highly respected in Florence. The two cities had a lot of features of the present-day capitalist society, which was due to their role as the main centres of industry of their “worlds-economies”. During the subsequent Hellenic period, the Old Order was restored, based on the feudal rent recipients’ rule. The degradation of society resulted in the decline of sophistry. Sophistry represents the highest point in the evolution of Greek philosophy. It is also the most prominent school as to the novelty of its ideas. That was owing to the common propensity for innovation typical of Periclus’ Golden Age, which in its turn resulted from the establishment of the individualistic and rationalistic thinking similar to present-day mentality. Among other aspects, close attention was paid to ethnic issues; some humanistic ideas such as Master and Servant ethics and Enlightment as the primary condition of man’s freedom, received scientific grounding. The genre of philosophical dialogue was developed, as well as the rationalistic scientific gnoceology, aimed at consistently opposing any religion. Dialectics, the most outstanding achievement of the Antiquity, was also worked out by the sophists. Zeno of Helleas was one of the sophists. Democrites’ ideas were, too, quite close to dialectics. Socrates’ contemporaries regarded him ironically, as shown, for example, in the Aristophanes’ comedy “Clouds”. Nevertheless, this scholar can still be referred to as a sophist, although of somewhat weird personality. Socrates’ popularity, which came later, should be attributed to the fact that he was the first philosopher ever to have been sentenced and executed by law, rather than to any valuable scientific contributions. Sophistry as such, rather than Socrates, marks the line between pre-Socratic and post- Socratic schools. Plato, with the exception of his latest works, is considered to belong to the same philosophical school. Specifically, there are valid reasons to believe that he was paid fees by his disciples. In Plato’s Dialogues, all points of view will be proven and then disproved, so that an integral philosophical system is hard to observe. Plato’s “Idealism” is but one of the many possible theories. His ideal philosophy is a high-minded argument of wise men. As late as towards the end of his life, Plato betrayed the ideals of sophistry, turning into the world’s first proponent of totalitarianism. Sophistry is Greek classical literature. Later on, philosophy, as well as the entire Greek civilization, started to fall into decline. Over 500 subsequent years, Hellenistic-Roman scholars never came up with anything novel. The entire process of antique philosophical evolution can be subdivided into the following six stages: Phoenician (IX–VIIth centuries B. C.), archaic (VIth – the first half of IVth century B. C., classic (the epoch of sophistry: the second half of the Vth through the first decades of the IIIrd centuries B. C.), late antique (middle of the IIIrd – first decades of the VI th centuries B.C.) The fact that early in the VIth century B. C. sophistry was rejected in Athens, has a lot to do with the consequences of their defeat in Peloponnesian war, a social catastrophe. Ever since, the prevalent part of the criticism of sophistry is a criticism of sophists’ immoral ways of life and personalities, rather than the essentials of their philosophy. For instance, Aristotle’s criticism primarily concerns their pragmatism and tendency to make a profit out of teaching. However, Aristotle himself taught his disciples along similar lines, showing them how to “be able to prove both opposites”. It was Aristotle who summed up sophists’ century-long elaborations. The traditional modern interpretation of sophists as shown in Plato’s dialogues appears to be groundless. The truth of the matter is that Plato regards sophists as more serious philosophers than Socrates was. It becomes especially clear in the dialogue titled “Protagoras”, which shows the latter as gaining an undoubted victory over Socrates. Generally speaking, Plato’s works never regard sophists as weak or worthless opponents. A common opinion as to the essence of sophists’, in particular, Protagoras’s philosophy, has hitherto not been worked out. Many researchers renounce the very existence of any specific philosophical position in sophistry. The term “relativism” itself now has a bad name. However, it should be borne in mind that Protagoras doctrine emerged as a result of an attempt to get out of the ontological dead-end to which the entire Greek natural philosophy had come. To this purpose, he employed Anaxagoras’s idea about everything comprising everything, whereupon he built his physical-ontological theory, and Zeno of Helleas’s concept that all objects are such and different at the same time in the same respect. According to Protagoras’s philosophy, every object does not only seem, but is different from every other object, including human beings. Therefore, every person is his or her own measure of all other entities. The objective reality is the Ego’s relations with the outer world. Every object’s essence is defined through its interaction with ourselves. Nothing remains stagnant – every existence already contains non-existence, and vice-versa. However, the relativity of being is not to be regarded as absolute. That means, that relatively stable and therefore relatively cognizable entities do exist, among them the words we say. Doubt is not considered absolute by relativists either, which differentiates them from skeptics and agnostics. They dare to believe. Uncertainty is Godly – here, Protagoras becomes very close to the concept of apathetic theology. Gorgius shared Protagoras’s views and tried to prove them applying the same method as Zeno applied when upholding Parmenidis’s philosophy. He argued that if one accepts the opposite point of view and agrees that nothing can exist and not exist at the same time, one is bound to come to an the absurd conclusion that nothing is cognizable and nothing exists. To prove that there is a universal contradiction within the Existence itself, Protagoras and his followers in their turn put forward a number of special logical arguments, called rules of contraries, or sophisms. The most well-known of them is the Euathle’s sophism, which proves that an object can be such and different at the same time in the same respect. Generally speaking, all relationships of any entity are extraneous. The idea that all entities, including concepts, exist and do not exist at the same time, was also accepted by Plato. It should be stressed, that Plato’s philosophy as a whole is relativists’ greatest achievement. In his dispute against relativism, Aristotle put forward the main concepts of his ontology: the existence falls into the existence in possibility and the existence in reality, or substance and accidence – the ever changing “sublunary” world and the stable world of divine heavenly bodies. This division made his system self-contradictory to the point of absurd. Aristotle quite realized, that the direct disproof of relativism is hardly possible, which fact makes his criticism a mere tautology. It is very indicative in this respect that materialists, idealists and even skeptical agnostics applied the same arguments do disprove relativism. That demonstrates that the inner contradiction between any forms of “absolutism” and relativism is the most fundamental philosophical problem. Relativist ideas were repeatedly renewed throughout history, provided the social conditions were favourable. In variations, the concepts were advanced by Tzuan Tsi and ancient Chinese sophists, apostle Paul, Nicolas of Cusa, Galileo, Rene Descartes, G. Berkley, D. Hume, Ch. S. Pierce, E. Mach, A. Bogdanov, F. de Saussure, N. Bore, P. Feuerabend, U. Quain and others. However, no consistent relativist ontological system was developed after Protagoras. Meanwhile, a valid relativist theory could provide a clue to a fuller and more adequate general physical concept of the world, and to a deeper understanding of quantum mechanics. -/- Relativism as an ontological system of beliefs. -/- The relativist principle of relativity of all that exists is fundamental for the entire system of philosophy. Absolutely everything exists, but, at the same time, no existence is absolute. Anything is possible, but those entities only exist for us with which we interact this way or another, i. e., reality is interaction, “I interact – hence, I exist”. However, an overly close relationship between entities leads to non-existence – in other words, there is a certain existential optimum. For all of us, information, or perceptible heterogeneities, is real. Each of us is the center of a definite system of interactions with the world. If God is an endless Possibility, then the World is but the man’s dialogue with God. However, the ties that bind us with the world are of some definite kind, which predetermines our reality. The principle of relativity constrains itself, as any relativity is relative. On the one hand, that calls forth the Evil – caused by our own weakness. On the other hand, it makes our world relatively stable and cognizable. Generally speaking, the world is what we are. Everybody has an own Universe, but since people are fairly similar, our universes can be regarded as one Universe with common laws and regulations. It is possible for us to change the degree of objects’ existence in relation to ourselves and one another. It is equally important, though, to remain ourselves (keep following our “warrior’s path”). Since everything exists and no existence is absolute, the process of cognition should involve a criticism of everything but never a complete rebuttal of anything. The criterion of verity for any theory then is in its capacity to build itself upon a bigger number of diverse statements than in any other theory. Theoretically, all philosophies should ideally be complementary. The question of what in fact happened is meaningless unless we specify the “we”, the “where” and the “when”. The past can only exist as part of the present. When we change, our past undergoes corresponding qualitative objective changes, all of which are irreversible. The contemporary physics regards all objects of the Universe as existing for one another, which results in absurd conclusions. In fact, only what we interact with, exists. The so called “speed of light in vacuum” c is the maximum speed of direct interaction and direct exchange of signals. Specifically, at the oncoming speed c with a significant blue parallax the gamma-quantum is unable to interact with the matter. When this speed is reached, the gravitational mass of objects in relation to one another becomes equal to zero (imaginary), whereas their inertial mass equals infinity. Further acceleration will bring us into an entirely different Universe. That means that the Universe is an open system, which fact destroys all present-day cosmological models. Anything is possible in this world. Our reality is magic. Every bit of space contains a potentially limitless number of bits of other, however virtual, universes, as well as an infinite number of particles, which are still virtual for us (as postulated by quantum mechanics). In other words, there exists an infinity of different realities. With all the above in mind, the different levels of entities’ existence, in particular, of Life and Reason, should be taken into consideration. For example, to which extent a live system exists in relation to the inanimate forms of being is very indefinite, as what life is has to be determined by living creatures. An even greater compass of levels of existence can be brought about through interaction – that is the core idea of Godel’s theorem. As a result, the range of forms of existence tends towards infinity – the evolution of matter through super-intelligence will bring us up to God’s heights. A more generalized mathematical model of the physical world, including a universal operator of the co-existence of objects, can be developed based on the above considerations. The bonds between objects become weaker owing to 1) a high speed, 2) acceleration, 3) rotation, 4) strong gravity, 5) strong fields of other kinds, 6) the difference of levels of existence , 7) phase shifts. Yet, there always is an existential optimum. Such mathematical model is to operate certain non-transitive algebras. Relativist ontology assumes that the notion of “temperature” is relative, thus making it possible to ignore the Second Law of Thermodynamics. In its turn, that will allow for phase shifts. A perpetual motion machine of the second type will become a reality. Rotation could presumably be used to increase inertial and decrease gravitational mass, slowing down the time. An experimental research could shed light on the mentioned effects. (shrink)

Euphorion SH 417 deserves to be better known. A curiosity in itself—an apparent poetic reference to number theory—it is also, potentially, one of our earliest references to Euclidean material. On the authority of a late commentator on Aristotle, Euphorion, a mid-third-century b.c. Euboean poet who was also active in Athens and Antioch, is said to have mentioned perfect numbers—i.e. numbers which equal the total of all their factors, including 1 . It is a pity that the context in Euphorion does (...) not survive, and that the line is only preserved, and indeed interpreted, by so late a source. But the wording of the fragment—if Westerink's restoration of its various corruptions is plausible—would strongly suggest a reference to the notion of perfect number. The fragment has been known since Westerink published it in 1960, and was included both in Van Groningen's edition of Euphorion in 1977 and in Supplementum Hellenisticum . But its implications have still not been discussed, and when David Fowler came to gather the evidence for references to Euclidean material in and after the third century b.c. in The Mathematics of Plato's Academy, his attention, unsurprisingly, was not drawn to it. Euphorion has had a bad press, as a poet of rebarbatively obscure myth and intractable vocabulary; yet he holds some interest, and we may be missing more insights into the intellectual life of the Hellenistic period which the perverse intelligence and baneful wit of the fragments display. (shrink)