The History of Mathematical Proof in Ancient Traditions

ed. Karine Chemla, 2012

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.

Nearly twenty-five hundred years ago the Greek thinker Heraclitus supposedly uttered the cryptic words Phusis kruptesthai philei. How the aphorism, usually translated as "Nature loves to hide," has haunted Western culture ever since is the subject of this engaging study by Pierre Hadot. Taking the allegorical figure of the veiled goddess Isis as a guide, and drawing on the work of both the ancients and later thinkers such as Goethe, Rilke, Wittgenstein, and Heidegger, Hadot traces successive interpretations of Heraclitus' words. Over time, Hadot finds, "Nature loves to hide" has meant that all that lives tends to die; that Nature wraps herself in myths; and (for Heidegger) that Being unveils as it veils itself. Meanwhile the pronouncement has been used to explain everything from the opacity of the natural world to our modern angst. From these kaleidoscopic exegeses and usages emerge two contradictory approaches to nature: the Promethean, or experimental-questing, approach, which embraces technology as a means of tearing the veil from Nature and revealing her secrets; and the Orphic, or contemplative-poetic, approach, according to which such a denuding of Nature is a grave trespass. In place of these two attitudes Hadot proposes one suggested by the Romantic vision of Rousseau, Goethe, and Schelling, who saw in the veiled Isis an allegorical expression of the sublime. "Nature is art and art is nature," Hadot writes, inviting us to embrace Isis and all she represents: art makes us intensely aware of how completely we ourselves are not merely surrounded by nature but also part of nature.

Ludic Proof: Greek Mathematics and the Alexandrian Aesthetic

Reviel Netz, 2008

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.

Seeing the Face, Seeing the Soul: Polemon's Physiognomy from Classical Antiquity to Medieval Islam

ed. Simon Swain, 2007

Polemon of Laodicea (near modern Denizli, south-west Turkey) was a wealthy Greek aristocrat and a key member of the intellectual movement known as the Second Sophistic. Among his works was the Physiognomy, a manual on how to tell character from appearance, thus enabling its readers to choose friends and avoid enemies on sight. Its formula of detailed instruction and personal reminiscence proved so successful that the book was re-edited in the fourth century by Adamantius in Greek, translated and adapted by an unknown Latin author of the same era, and translated in the early Middle Ages into Syriac and Arabic. The main surviving versions in Greek and Latin are translated into English for the first time. The texts and translations are introduced by a series of masterly studies that tell the story of the origins, function, and legacy of Polemon's work, a legacy especially rich in Islam. The story of the Physiognomy is the story of how one man's obsession with identifying enemies came to be taken up in the fascinating transmission of Greek thought into Arabic.