Showing papers by "Reviel Netz published in 2004"

Barbed Wire: An Ecology of Modernity

Abstract: In this original and controversial book, historian and philosopher Reviel Netz explores the development of a controlling and pain-inducing technology - barbed wire. Surveying its development from 1874 to 1954, Netz describes its use to control cattle during the colonization of the American West and to control people in Nazi concentration camps and the Russian Gulag. Physical control over space was no longer symbolic after 1874. This is a history told from the perspective of its victims. With vivid examples of the interconnectedness of humans, animals, and the environment, this dramatic account of barbed wire presents modern history through the lens of motion being prevented. Drawing together the history of humans and animals, Netz delivers a compelling new perspective on the issues of colonialism, capitalism, warfare, globalization, violence, and suffering. Theoretically sophisticated but written with a broad readership in mind, Barbed Wire calls for nothing less than a reconsideration of modernity.

The Transformation of Mathematics in the Early Mediterranean World: From Problems to Equations

Abstract: Acknowledgements Introduction 1. The problem in the world of Archimedes 2. From Archimedes to Eutocius 3. From Archimedes to Khayyam Conclusion References Index.

Towards a Reconstruction of Archimedes' Stomachion

Abstract: and Plan of Paper The Stomachion is the least understood of Archimedes' works. This paper provides a reconstruction of its goal and structure. The nature of the evidence, including new readings from the Archimedes Palimpsest, is discussed in detail. Based on this evidence, it is argued that the Stomachion was a treatise of geometrical combina­ torics. This new interpretation is made possible thanks to recent studies showing the existence of sophisticated combinatorial research in antiquity. The key to the new interpretation, in this case, is the observation that Archimedes might have focussed not on the possibility of creating many different figures by different arrangements of the pieces but on the way in which the same overall figure is obtained by many different arrangements of the pieces. The plan of the paper is as follows. Section 1 introduces the Stomachion. Sec­ tion 2 discusses the ancient testimonies and the Arabic fragment, while Section 3 translates and discusses the Gr~ek fragment. Section 4 sums up the mathematical reconstruction offered in this paper, while Section 5 points at the possible intellec­ tual background to the work. Appendix A contains a transcription of the Greek fragment, appendix B an English translation with redrawn diagrams, appendix C a reproduction of the digitally enhanced images of the pages of the palimpsest con­ taining remains of the Stomachion.

The Limits of Text in Greek Mathematics

Abstract: This article argues for a limited role of the text in Greek mathematics, in two senses of “text”: the verbal as opposed to the visual; and the literate as opposed to the “oral” (understood in a wide sense). The Greek mathematical argument proceeds not within the confines of the verbal alone, but essentially relies upon diagrams. On the other hand, it does not use other specific techniques, such as those of the modern cross-reference, relying instead upon verbal echoes. The two, taken together, suggest a model of scientific writing radically different from what we associate with our own mathematics. In methodological terms, the article surveys its evidence in detail, and makes comments concerning the methodology of studying ancient texts through the evidence of those texts alone.

Euclid’s data

Abstract: The Data, traditionally ascribed to Euclid, the author of the Elements, deals with a limited type of results: showing that, with one geometrical object being ‘given’, another is given as well. The background to this is as follows. In theorems, the goal is to show the truth of a claim; in problems it is to perform a task after being given certain objects (e.g. being given a straight line, to construct on it an equilateral triangle). Now, the more you are ‘given’, the easier is the fulμlment of the task. The Data provides one with a mechanism for extending what one has been given by the terms of the problem. This is a tool for the solution of problems. One can note a historical development: from an early Greek mathematics focused on the solution of special problems, to late-antique and medieval mathematics focused on the systematic arrangement of theorems. This largely accounts for the rise of the Elements into its canonical rôle in our own image of Greek mathematics, and the relative obscurity of the Data. T.’s book is an outstanding contribution to the study of ancient mathematics, and may serve in correcting this historical imbalance. Formally a bilingual text with commentary, this book is really a research monograph whose argument is conducted through attention to the unfolding of the detail through the text of the Data. The translation is precise and thoughtful, and the few anachronistic symbols employed are in the spirit of the original. As for the text, T. simply follows H. Menge, Euclidis Data (Leipzig, 1896); a reasonable choice, but I regret T.’s decision not to consult the manuscript tradition for the diagrams. This is the μrst research monograph dedicated to the Data, and may be seen as a complementary volume to the treatment of the Elements in I. Mueller, Philosophy of Mathematics and Deductive Structure in Euclid’s Elements (Cambridge, MA, 1981). Like Mueller before him, T. looks for the foundational mathematical principles underlying the work. One can identify in this book two poles, a negative and a positive. The negative pole is in the argument that the Data is not concerned with doing algebraical manipulations via geometrical representations (the core of this argument is in Chapter 13, arguing for the non-algebraical nature of proposition 86, but many comments throughout the book serve the same purpose). The positive pole is in the analysis of Greek geometrical practice as a ritualized game of make-believe (the terms are mine), providing in this way a context for the expression ‘being given’. The core of this argument is in the μrst chapter, dedicated to the deμnitions, but T.’s discussion throughout is motivated by this understanding of Greek geometry. The negative pole of the book is by now familiar and usually non-controversial. The argument still needs to be made, and it is made persuasively by T. The positive pole    337

From Archimedes to Khayyam

Abstract: In this chapter we concentrate on the fate of Archimedes' problem in one eminent work of Arabic science: Omar Khayyam's Algebra (eleventh to twelfth centuries). (This is, of course, the same Omar Khayyam famous for his Persian poetry; here we concentrate on his science.) As we shall see below, this decision to focus on Khayyam is to a certain extent arbitrary: the problem had a significant history in the Arabic world before and after Khayyam. He does occupy a special position in the history of the problem. Our knowledge of Arabic treatments prior to him is in some cases derived from him alone (much as we know of early Greek treatments of the problem through the work of Eutocius). And while the later history of the problem adds much that is mathematically valuable, we can usefully end our survey with Khayyam. With him, as we shall see, the route from problems to equations is largely completed. It is also helpful to compare like with like: and it is therefore appropriate to have our survey – begun with the genius of Archimedes – end with the genius of Khayyam. Our goal in this chapter, then, is to show that Khayyam's mathematics already differs essentially from Archimedes'. This should be a deep conceptual divide, along the lines suggested by Klein and Unguru. We also need to show the historical basis for this divide, in terms of changes in the practice of mathematics from the world of Archimedes to the world of Khayyam.

The problem in the world of Archimedes

Abstract: In this chapter I discuss the Archimedean problem in its first, “Classical” stage. In section 1.1, I show how it was first obtained by Archimedes and then, in 1.2, I offer a translation of the synthetic part of Archimedes' solution. Following that, section 1.3 makes some preliminary observations on the geometrical nature of the problem as studied by Archimedes. Sections 1.4 and 1.5 follow the parallel treatments of the same problem by two later Hellenistic mathematicians, Dionysodorus and Diocles. Putting together the various treatments, I try to offer in section 1.6 an account of the nature of Ancient geometrical problems. Why were the ancient discussions geometrical rather than algebraic – why were these problems , and not equations ? The problem obtained In his Second Book on the Sphere and Cylinder , Archimedes offers a series of problems concerning spheres. The goal is to produce spheres, or segments of spheres, defined by given geometrical equalities or ratios. In Proposition 4 the problem is to cut a sphere so that its segments stand to each other in a given ratio. For instance, we know that to divide a sphere into two equal parts, the solution is to divide it along the center, or, in other words, at the center of the diameter. But what if want to have, say, one segment twice the other?

The two books On the sphere and the cylinder

Abstract: Introduction Translation and Commentary: On Sphere and Cylinder Book I On Sphere and Cylinder Book II Eutocius' Commentary to On Sphere and Cylinder Book I Eutocius' Commentary to On Sphere and Cylinder Book II Bibliography Index.