This book is at once an analytical study of one of the most important mathematical texts of antiquity, the Mathematical Collection of the fourth-century AD mathematician Pappus of Alexandria, and also an examination of the work's wider cultural setting. An important first chapter looks at the mathematicians of the period and how mathematics was perceived by people at large. The central chapters of the book analyse sections of the Collection, identifying features typical of Pappus's mathematical practice. The final chapter draws (...) together the various threads and presents a fuller description of Pappus's mathematical 'agenda'. This is one of few books to deal extensively with the mathematics of Late Antiquity. It sees Pappus's text as part of a wider context and relates it to other contemporary cultural practices and opens avenues to research into the public understanding of mathematics and mathematical disciplines in antiquity. (shrink)

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

This paper introduces a logical analysis of convex combinations within the framework of Łukasiewicz real-valued logic. This provides a natural link between the fields of many-valued logics and decision theory under uncertainty, where the notion of convexity plays a central role. We set out to explore such a link by defining convex operators on MV-algebras, which are the equivalent algebraic semantics of Łukasiewicz logic. This gives us a formal language to reason about the expected value of bounded random variables. As (...) an illustration of the applicability of our framework we present a logical version of the Anscombe–Aumann representation result. (shrink)

Greek mathematics is usually seen as having reached its height in a “golden age” around 300 b.c., after which it declined, reaching a rather sad stage in late antiquity. In this latter period Pappus of Alexandria stands out as one of the last competent mathematicians, although even his Mathematical Collection has been valued by historians mainly for its wealth of information on earlier mathematical achievements. In her readable book, Serafina Cuomo sets out to correct the conventional view of mathematics (...) in late antiquity: her general goal is “to show that the mathematics of late antiquity deserves a place both in the history of science and in the history of antiquity” . To that end she focuses on Pappus's Collection, for which she attempts “to produce a historical analysis … and at the same time to explore its wider cultural contexts” . Thus her text is not intended primarily as a discussion of the mathematics contained in the Collection; instead Cuomo looks into the historical circumstances in which Pappus produced this work and the image he wanted to convey both of mathematics and of himself.The first chapter presents some general background on the public's perception of mathematics and its practitioners in late antiquity. Emphasizing evidence “apart from what we find in the books of the famous mathematicians” , Cuomo looks at astrological treatises such as the Mathesis by Julius Firmicus Maternus and considers the social and fiscal status of various “technical” professions, such as those of land surveyor, architect, and public administrator. She uses Diocletian's edict on wages and salaries and examines the role of mathematics in education to show that, “far from being invisible or confined to ivory towers” , mathematics was perceived as an integral part of life and regarded positively. All this material is very interesting, but how relevant is it? Cuomo's broadening of the perspective to include more than just the works of famous mathematicians is certainly commendable. But should one equate the mere skill of calculating with numbers, or even that of applying some “higher” mathematics, with theoretical, philosophical, mathematical thought? Of course, Cuomo is aware of this problem. Taking it to be irrelevant for her purpose, she makes a good point: claiming affiliation with mathematics to enhance one's status, as those professional practitioners would do, presupposes a positive image of mathematics among the public . Nevertheless, such evidence would probably not have swayed previous historians' opinion of the decline of mathematics in late antiquity.Chapters 2–4 are devoted to Book 5 of the Collection, which covers isoperimetric problems and Platonic solids; Book 8, on mechanics; Book 3, on cube duplication; and Book 4, on special curves and their classification. In Chapter 2 Cuomo argues that Pappus used the widely known problems discussed in Book 5 to address a general audience so as to promote mathematics and show that “mathematicians are better qualified than philosophers … to talk about isoperimetry and the five Platonic bodies” . At the same time, pointing out the complementarity of mathematics and mechanics , Pappus stresses the usefulness of mathematics in educational and material matters. In Chapter 4 Cuomo examines how Pappus, addressing experts, “marks out his territory within the mathematical field itself” .In the final chapter Cuomo tries “to get a clearer grasp of Pappus' mathematical agenda” . She concludes that Pappus is not just a compiler admiring the past; rather, he selected his subjects and manipulated the mathematical tradition to serve his own immediate goals in the present.This book contains an abundance of interesting historical material, although, regrettably, the mathematical examples are only roughly sketched. Cuomo shows that we can learn much from looking at Pappus's Collection in its own right. Her book deserves careful study. (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)