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)

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)

Important study focuses on the revival and assimilation of ancient Greek mathematics in the 13th–16th centuries, via Arabic science, and the 16th-century development of symbolic algebra. This brought about the crucial change in the concept of number that made possible modern science — in which the symbolic "form" of a mathematical statement is completely inseparable from its "content" of physical meaning. Includes a translation of Vieta's Introduction to the Analytical Art. 1968 edition. Bibliography.

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)

James Gow's A Short History of Greek Mathematics provided the first full account of the subject available in English, and it today remains a clear and thorough guide to early arithmetic and geometry. Beginning with the origins of the numerical system and proceeding through the theorems of Pythagoras, Euclid, Archimedes and many others, the Short History offers in-depth analysis and useful translations of individual texts as well as a broad historical overview of the development of mathematics. Parts (...) I and II concern Greek arithmetic, including the origin of alphabetic numerals and the nomenclature for operations; Part III constitutes a complete history of Greek geometry, from its earliest precursors in Egypt and Babylon through to the innovations of the Ionic, Sophistic, and Academic schools and their followers. Particular attention is given to Pythagorus, Euclid, Archimedes, and Ptolemy, but a host of lesser-known thinkers receive deserved attention as well. (shrink)

This book provides a comprehensive overview of the key themes in Greek and Roman science, medicine, mathematics and technology. A distinguished team of specialists engage with topics including the role of observation and experiment, Presocratic natural philosophy, ancient creationism, and the special style of ancient Greek mathematical texts, while several chapters confront key questions in the philosophy of science such as the relationship between evidence and explanation. The volume will spark renewed discussion about the character (...) of 'ancient' versus 'modern' science, and will broaden readers' understanding of the rich traditions of ancient Greco-Roman natural philosophy, science, medicine and mathematics. (shrink)

The ancient Greeks played a fundamental role in the history of mathematics and their ideas were reused and developed in subsequent periods all the way down to the scientific revolution and beyond. In this, the first complete history for a century. Reviel Netz offers a panoramic view of the rise and influence of Greek mathematics and its significance in world history. He explores the Near Eastern antecedents and the social and intellectual developments underlying the subject's beginnings (...) in Greece in the fifth century BCE. He leads the reader through the proofs and arguments of key figures like Archytas, Euclid and Archimedes, and considers the totality of the Greek mathematical achievement which also includes, in addition to pure mathematics, such applied fields as optics, music, mechanics and, above all, astronomy. This is the story not only of a major historical development, but of some of the finest mathematics ever created. (shrink)

The prevailing narrative in the history of science maintains that the ancient Greeks did not have a concept of a ‘law of nature’. This paper overturns that narrative and shows that some ancient Greek philosophers did have an idea of laws of nature and, moreover, they referred to them as ‘laws of nature’. This paper analyzes specific examples of laws of nature in texts by Plato, Aristotle, Philo of Alexandria, Nicomachus of Gerasa, and Galen. These examples emerged (...) out of the closely intertwined Platonic and Pythagorean traditions, and these philosophers’ texts make reference to laws of nature when describing arithmetical methods, arithmological doctrines, or medical theories. Nicomachus’ laws of nature are especially noteworthy, because they have features that historians look for in the search for the origin of the modern concept of laws of nature. Nicomachus’ laws of nature are mathematical, universal, and necessary. This paper raises the possibility that the ancient Platonic and Pythagorean traditions influenced the subsequent development of the idea of laws of nature in medieval and early modern Europe, including the conception of laws of nature deployed by Johannes Kepler and Isaac Newton. (shrink)

This book endeavours to pinpoint the relations between musical, and especially instrumental, practice and the evolving conceptions of pitch systems. It traces the development of ancient melodic notation from reconstructed origins, through various adaptations necessitated by changing musical styles and newly invented instruments, to its final canonical form. It thus emerges how closely ancient harmonic theory depended on the culturally dominant instruments, the lyre and the aulos. These threads are followed down to late antiquity, when details recorded by (...) Ptolemy permit an exceptionally clear view. Dr Hagel discusses the textual and pictorial evidence, introducing mathematical approaches wherever feasible, but also contributes to the interpretation of instruments in the archaeological record and occasionally is able to outline the general features of instruments not directly attested. The book will be indispensable to all those interested in Greek music, technology and performance culture and the general history of musicology. (shrink)

A large part of the difficulty of writing "conceptual history"—to borrow a term from Reviel Netz —is that once an illuminating new conceptual framework is articulated, it begins to seem self-evident and commonsensical to later thinkers. The historian's task of problematizing the obvious, and showing us the moves by which commonsense came to be created historically, is an arduous and challenging one, requiring resources of imagination, patience, and attention to detail. Sattler displays all those qualities in this dense and demanding (...) study, and a review needs to begin by acknowledging the sheer... (shrink)

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)

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 essay examines the quantitative aspects of Greco-Roman science, represented by a group of established disci¬plines, which since the fourth century BC were called mathēmata or mathē¬ma¬tikai epistē¬mai. In the group of mathēmata that in Antiquity normally comprised mathematics, mathematical astronomy, harmonics, mechanics and optics, we have also included geography. Using a dataset based on The Encyclopaedia of Ancient Natural Scientists, our essay considers a community of mathēmatikoi (as they called themselves), or ancient scientists (as they are (...) defined for the purposes of the present paper) from a sociological point of view, focusing on the size of the scientific population known to us and its disciplinary, temporal and geographical distribution. A diachronic comparison of neighboring and partly overlapping communities, ancient scientists and philosophers, allows the pattern of their interrelationship to be traced. An examination of centers of science throughout ancient history reveals that there were five major centers – Athens, Alexandria, Rhodes, Rome and Byzantium/Constantinople – that appear and replace one another in succession as leaders. These conclusions serve to reopen the issue of the place of mathēmata and mathēmatikoi in ancient society. (shrink)

We may take tables for granted. However, due to a variety of factors, tables were a rarity in the history of ancient Greek culture, used only limitedly in very special contexts and generally in a non-systematic way, except in astronomy. In this paper I present the main types of tables that can be found in ancient Greek texts: non-ruled columnar lists (accounts and other types of informal tables), ruled columnar lists (mostly astronomical tables), and symmetric tables (...) (mainly Pythagorean displays of numbers). (shrink)

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)

ArgumentThis article presents ancient documents on the subject of homeomeric lines. On the basis of such documents, the article reconstructs a definition of the notion as well as a proof of the result, which is left unproved in extant sources, that there are only three homeomeric lines: the straight line, the circumference, and the cylindrical helix. A point of particular historiographic interest is that homeomeric lines were the only class of lines defined directly as the extension of a mathematical (...) property, a move that is unparalleled in Greek mathematics. The far-reaching connections between mathematical homeomery and key issues in the ancient cosmological debate are extensively discussed here. An analysis of its relevance as a foundational theme will be presented in a companion paper in a future issue of Science in Context. (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)

This is a translation of Jacob Klein's study "Die Griechische Logistik und die Entstehung der Algebra" which appeared in 1934-1936. His principal thesis is that the Renaissance mathematicians of the sixteenth century did not simply continue the work of the Greek and Arab mathematicians but in the process of developing ancient mathematics introduced a radically new conception of number which has since guided modern mathematical thought. The central figure in this revolution is Vieta. Klein traces the influence (...) of Vieta's ideas upon Stevin, Descartes, Wallis, and other figures of the scientific revolution, after discussing the conception of number and arithmetic in Plato, Aristotle, and other Greek sources. Persons reading this book with a primary interest in the philosophical ideas involved will be frustrated by the mass of historical detail which often obscures rather than illuminates the philosophical issues. But the book deserves its reputation as an important historical study.--R. H. K. (shrink)

Among the many instruments devised by students of mathematical sciences in ancient Greece, the monochord provides one of the best opportunities to examine the methodologies of those who employed it in their investigations. Consisting of a single string which could be divided at measured points by means of movable bridges, it was used to demonstrate theorems about the arithmetical relationships between pitched sounds in music. This book traces the history of the monochord and its multiple uses down to Ptolemy, (...) bringing together all the relevant evidence in one comprehensive study. By comparing the monochord with a number of other ancient scientific instruments and their uses, David Creese shows how the investigation of music in ancient Greece not only shares in the patterns of demonstrative and argumentative instrument use common to other sciences, but also goes beyond them in offering the possibility of a rigorous empiricism unparalleled in Greek science. (shrink)

With mathematical reconstructions and philosophical arguments I show that Toomer's 1973 paper never contained any conclusive evidence for his claims that Hipparchus had a 3438'-based chord table, and that the Indians used that table to compute their sine tables. Recalculating Toomer's reconstructions with a 3600' radius -- i.e. the radius of the chord table in Ptolemy's Almagest, expressed in 'minutes' instead of 'degrees' -- generates Hipparchan-like ratios similar to those produced by a 3438' radius. It is therefore possible that the (...) radius of Hipparchus's chord table was 3600', and that the Indians independently constructed their 3438'-based sine table. (shrink)

The paper shows that, contrary to what has been held since the sixteenth-century mathematician Christoph Clavius, there is no application of consequentia mirabilis (CM) in Greek mathematical works. This is shown by means of a detailed discussion of the logical structure of the proofs where CM is allegedly employed. The point is further enlarged to a critical assessment of the unsound methodology applied by many interpreters in seeking for specific logical rules at work in ancient mathematical texts.

Analogies between hearing and seeing already existed in ancient Greek theories of perception. The present paper follows the evolution of such analogies until the rise of 17th century optics, with due regard to the diversity of their origins and nature but with particular emphasis on their bearing on the physical concepts of light and sound. Whereas the old Greek analogies were only side effects of the unifying concepts of perception, the analogies of the 17th century played an (...) important role in constructing optical theories by imitation of acoustic theories, or vice versa. This transition depended on several factors including the changing relations between optics, music, mathematics, and physics, the diversity of early modern concepts of sound, and the rise of a new physics based on experimentation and mechanical explanation. (shrink)

ArgumentThis article is the sequel to an article published in the previous issue ofScience in Contextthat dealt with homeomeric lines (Acerbi 2010). The present article deals with foundational issues in Greek mathematics. It considers two key characters in the study of mathematical homeomery, namely, Apollonius and Geminus, and analyzes in detail their approaches to foundational themes as they are attested in ancient sources. The main historiographical result of this paper is to show thatthere wasa well-establishedmathematicalfield of discourse (...) in “foundations of mathematics,” a fact that is by no means obvious. The paper argues that the authors involved in this field of discourse set up a variety of philosophical, scholarly, and mathematical tools that they used in developing their investigations. (shrink)

Scientific observation has led to the discovery of recurring patterns in nature. Symmetry is the property of an object showing regularity in parts on a plane or around an axis. There are several types of symmetries observed in the natural world and the most common are mirror symmetry, radial symmetry, and translational symmetry. Symmetries can be continuous or discrete. A discrete symmetry is a symmetry that describes non-continuous changes in an object. A continuous symmetry is a repetition of an object (...) an infinite number of times. A consequence of continuous symmetry is the existence of conservation laws. A natural system with discrete and continuous symmetries displays several physical properties, such as the existence of long-range order. Geometric shapes exhibit symmetry as mirror reflections of each other. Symmetry in nature has been a model of beauty since the beginning of civilization. Since the earliest times, nature itself has manifestly been a model, evincing regularity in sundry forms and occurrences–from minerals and plants to the anatomy of living beings, to the regularly recurring stellar constellations. For Ancient Greek philosophers, proportion, symmetry, and harmony, were the basics to determine whether something is beautiful or not. Ancient Greek philosophers were known for bringing logic and rational thinking to phenomena that were previously explained by mythology and Gods. They observed the natural world around them and used their knowledge to answer questions about the proportion in observable objects and the origin of Earth. The Greek words _summetria_ and _summetros_ appear frequently in the Timaeus, defined as parts with each other and with the whole. Plato has a very rough concept of symmetry, and when he uses “beauty” to characterize the so-called Platonic Solids in the Timaeus, he seems to be emphasizing their regularity and indirectly their symmetry. Plato believes the four elements (earth, air, fire, water) have been constructed by the Demiurge, or a divine craftsman who appoints order in an otherwise chaotic universe. Minerals are representative examples of beauty, order, and symmetry in inorganic materials. Well-formed minerals (crystals) are a collection of equivalent faces related by symmetry. The goal of this paper is to relate the perspective of symmetry in ancient Greek philosophy with the modern scientific evidence of the geometric description of crystal structures. The five Platonic solids are ideal models of geometrical patterns that occur throughout the world of minerals. In this paper, minerals serve as a model of connecting the symmetry theory in Plato’s philosophy and modern advances in mineralogy. (shrink)

Analogies between hearing and seeing already existed in ancient Greek theories of perception. The present paper follows the evolution of such analogies until the rise of 17th century optics, with due regard to the diversity of their origins and nature but with particular emphasis on their bearing on the physical concepts of light and sound. Whereas the old Greek analogies were only side effects of the unifying concepts of perception, the analogies of the 17th century played an (...) important role in constructing optical theories by imitation of acoustic theories, or vice versa. This transition depended on several factors including the changing relations between optics, music, mathematics, and physics, the diversity of early modern concepts of sound, and the rise of a new physics based on experimentation and mechanical explanation. (shrink)

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)

Did science and philosophy develop differently in ancient Greece and ancient China? If so, can we say why? This book consists of a series of detailed studies of cosmology, natural philosophy, mathematics and medicine that suggest the answer to the first question is yes. To answer the second, the author relates the science produced in each ancient civilization first to the values of the society in question and then to the institutions within which the scientists and (...) philosophers worked. (shrink)

Over the past few decades, the question regarding the proper understanding of Diophantus’s method has attracted much scholarly attention. “Modern algebra”, “algebraic geometry”, “arithmetic”, “analysis and synthesis”, have been suggested by historians as suitable contexts for describing Diophantus’s resolutory procedures, while the category of “premodern algebra” has recently been proposed by other historians to this end. The aim of this paper is to provide arguments against the idea of contextualizing Diophantus’s modus operandi within the conceptual framework of the ancient (...) analysis and to examine the few instances, in the preserved books of the Arithmetica, which might be regarded as linked to practices belonging to the field of analysis. (shrink)

This paper aims to show that in ancient Greek architecture, it is possible to find a genesis of the geometric modeling of visual perception present in propositions of Euclid's Optics, considering mathematical knowledge as a human wisdom expression. Let us start by emphasizing that mathematical thinking is not exclusively rooted in mathematical disciplines, but also includes the broad spectrum of human activities, including activities that come from everyday life. Based on this, we present a socio-cultural characterization of human (...) experience as the source and sustainer of mathematical knowledge. Thereafter, on the basis of a content and contextual analysis of _Euclid's Optics_, we explain the use and development of geometry in the study of various optical effects of visual perception in the architectural art embodied in the Acropolis of Athens; and we focus our analysis on one of the most important of these effects, the Acropolis of Athens. We focus our analysis on one of the most emblematic architectural works of ancient Greek culture: the Parthenon of Athens. (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.

The aim of this text is to present the evolution of the relation between the concept of number and magnitude in ancient Greek mathematics. We will briefly revise the Pythagorean program and its crisis with the discovery of incommensurable magnitudes. Next, we move to the work of Eudoxus and present its advances. He improved the Pythagorean theory of proportions, so that it could also treat incommensurable magnitudes. We will see that, as the time passed by, the existence (...) of incommensurable magnitudes was no longer something strange. Already in the period of Plato and Aristotle, their existence was common place: up to the point of being considered absurd that all magnitudes were commensurable. Aristotle criticized the Pythagorean program and defended that, though belonging to the same category, number and magnitude are of distinct species: number is discrete and magnitude is continuous. We finish by presenting briefly how the concept of number was amplified throughout the centuries until it came also to include the notion of continuity. (shrink)

Whatever one may think of Schmidt’s intuition, it is still nothing but intuition, and the variety of syntactic structures which εἶναι admits of is neither articulated nor unified. Kahn, on the other hand, by the use of Transformational Grammar, is able to a large extent to generate in a regular way from a posited notion of "kernel sentence" all the Greek sentences in which εἶναι occurs. Kahn’s original plan was "to correlate every intuitive difference of meaning in the use (...) of εἰμί with a formal description of the corresponding sentence-type", but he admits that he cannot always do so. Whether this is a failure inherent in Transformational Grammar itself, or in the version Kahn uses, can for the moment be left aside. First, an example of the success and another of the failure of the technique, as Kahn practices it, are in order. Kahn formulates the rule for the recognition of periphrasis somewhat as follows: εἶναι is used periphrastically with the participle if and only if it is impossible to obtain two kernel sentences, one of which has a finite form of εἶναι and the other a finite form of the participle, but if two kernel sentences can be so obtained, the usage is not periphrastic. This rule is both simple and elegant; it will no doubt become in time a standard part of Greek grammar. The enchantment, however, which mathematical clarity can cast is best illustrated in the case of another construction. ἔστιν ὅστις... is not uncommon. It looks like the existential operator of modern logic, and the fact that it occurs far more often with οὐκ than without seems to be, linguistically, irrelevant. Now, Kahn asserts that, though he looked hard for examples, he could only find one in which the second clause has the copula, and he offers a proof as to why this should be the case. He is mistaken. Euripides has οὐκ ἔστι θνητῶν ὅστις ἕστ’ ἐλεύθερος, and Sophocles καὶ οὐδὲν τούτων ὅ τι μὴ Ζεύς, and there are several examples in just one passage of Plato’s Charmides. The Sophoclean example is important since it illustrates a double "zeroing" of εἶναι, and whereas for traditional grammar such nominal sentences are treated as primary, with the insertion of ἐστί as a secondary development, in Kahn’s use of Transformational Grammar no distinction between the presence or the absence of the verb can be allowed. For deep structure, the verb is always present, and it might be no more than an apparent paradox that a verb, whose primitive meaning is said to designate presence, can in its absence make its presence equally felt. Can a verb which is almost always eliminable be the word for reality and truth? Or is it because "being" is the only word that cannot be just a word that it can so easily be suppressed in speech? (shrink)

Two Reasons to Study Ancient Greek Philosophy Ancient Greek philosophy began with Thales, who correctly predicted an eclipse that occurred in 585 BCE, and culminated in the monumental works of Aristotle, who died in 322.1 (Unless otherwise noted, all dates in this book are BCE.) The simple fact that these thinkers lived over 2,000 years ago should provoke a question: in the age of the microchip and the engineered gene, why bother with them? One good answer (...) immediately springs to mind: to become educated. The Greeks were the intellectual ancestors of western culture. They laid the foundations for all future developments in the natural sciences, medicine, mathematics, history, architecture, sculpture, tragic and comic drama, lyric and epic poetry, as well as philosophy. To the extent that one must know one's heritage in order to know oneself, it is imperative to study the ancient Greeks. (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.

The paper begins with an exposition of Aristotle’s own philosophy of mathematics. It is claimed that this is based on two postulates. The first is the embodiment postulate, which states that mathematical objects exist not in a separate world, but embodied in the material world. The second is that infinity is always potential and never actual. It is argued that Aristotle’s philosophy gave an adequate account of ancient Greek mathematics; but that his second postulate does not (...) apply to modern mathematics, which assumes the existence of the actual infinite. However, it is claimed that the embodiment postulate does still hold in contemporary mathematics, and this is argued in detail by considering the natural numbers and the sets of ZFC. (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)

Ancient Greek mathematics developed the original feature of being deductive mathematics. This article attempts to give a explanation f or this achievement. The focus is on the use of a fixed system of linguistic formulae in Greek mathematical texts. It is shown that the structure of this system was especially adapted for the easy computation of operations of substitution on such formulae, that is, of replacing one element in a fixed formula by another, and it (...) is further argued that such operations of substitution were the main logical tool required by Greek mathematical deduction. The conclusion explains why, assuming the validity of the description above, this historical level is the best explanatory level for the phenomenon of Greek mathematical deduction. (shrink)