The use of perspective in Renaissance painting caused a revolution in the history of seeing, allowing artists to depict the world from a spectator’s point of view. But the theory of perspective that changed the course of Western art originated elsewhere—it was formulated in Baghdad by the eleventh-century mathematician Ibn al Haithan, known in the West as Alhazen. Using the metaphor of the mutual gaze, or exchanged glances, Hans Belting—preeminent historian and theorist of medieval, Renaissance, and contemporary art—narrates the historical encounter between science and art, between Arab Baghdad and Renaissance Florence, that has had a lasting effect on the culture of the West.

In this lavishly illustrated study, Belting deals with the double history of perspective, as a visual theory based on geometrical abstraction (in the Middle East) and as pictorial theory (in Europe). How could geometrical abstraction be reconceived as a theory for making pictures? During the Middle Ages, Arab mathematics, free from religious discourse, gave rise to a theory of perspective that, later in the West, was transformed into art when European painters adopted the human gaze as their focal point. In the Islamic world, where theology and the visual arts remained closely intertwined, the science of perspective did not become the cornerstone of Islamic art. Florence and Baghdad addresses a provocative question that reaches beyond the realm of aesthetics and mathematics: What happens when Muslims and Christians look upon each other and find their way of viewing the world transformed as a result?

No one has figured more prominently in the study of the German philosopher Gottlob Frege than Michael Dummett. His magisterial Frege: Philosophy of Language is a sustained, systematic analysis of Frege's thought, omitting only the issues in philosophy of mathematics. In this work Dummett discusses, section by section, Frege's masterpiece The Foundations of Arithmetic and Frege's treatment of real numbers in the second volume of Basic Laws of Arithmetic, establishing what parts of the philosopher's views can be salvaged and employed in new theorizing, and what must be abandoned, either as incorrectly argued or as untenable in the light of technical developments.

Gottlob Frege (1848-1925) was a logician, mathematician, and philosopher whose work had enormous impact on Bertrand Russell and later on the young Ludwig Wittgenstein, making Frege one of the central influences on twentieth-century Anglo-American philosophy; he is considered the founder of analytic philosophy. His philosophy of mathematics contains deep insights and remains a useful and necessary point of departure for anyone seriously studying or working in the field.

Widespread interest in Frege’s general philosophical writings is, relatively speaking, a fairly recent phenomenon. But it is only very recently that his philosophy of mathematics has begun to attract the attention it now enjoys. This interest has been elicited by the discovery of the remarkable mathematical properties of Frege’s contextual definition of number and of the unique character of his proposals for a theory of the real numbers.

This collection of essays addresses three main developments in recent work on Frege’s philosophy of mathematics: the emerging interest in the intellectual background to his logicism; the rediscovery of Frege’s theorem; and the reevaluation of the mathematical content of The Basic Laws of Arithmetic. Each essay attempts a sympathetic, if not uncritical, reconstruction, evaluation, or extension of a facet of Frege’s theory of arithmetic. Together they form an accessible and authoritative introduction to aspects of Frege’s thought that have, until now, been largely missed by the philosophical community.