On Euclid and the Genealogy of Proof

I argue for an interpretation of Euclid’s postulates as principles grounding the science of measurement. Euclid’s Elements can then be viewed as an application of these basic principles of measurement to what I call general measurements—that is, metric comparisons between objects that are only partially specified. As a consequence, rather than being viewed as a tool for the production of certainty, mathematical proof can then be interpreted as the tool with which such general measurements are performed. This gives, I argue, a more satisfying story of the origin of proof in Ancient Greece, and of the status of Euclid’s postulates.

Kant's Mathematical World

Kant's Mathematical World aims to transform our understanding of Kant's philosophy of mathematics and his account of the mathematical character of the world. Daniel Sutherland reconstructs Kant's project of explaining both mathematical cognition and our cognition of the world in terms of our most basic cognitive capacities. He situates Kant in a long mathematical tradition with roots in Euclid's Elements, and thereby recovers the very different way of thinking about mathematics which existed prior to its 'arithmetization' in the nineteenth century. He shows that Kant thought of mathematics as a science of magnitudes and their measurement, and all objects of experience as extensive magnitudes whose real properties have intensive magnitudes, thus tying mathematics directly to the world. His book will appeal to anyone interested in Kant's critical philosophy -- either his account of the world of experience, or his philosophy of mathematics, or how the two inform each other.

The Material World as an Aid for Proclus’ Imagination

In this paper, I seek to answer whether the physical world plays a role in Proclus’s geometry as described in his commentary on Euclid's Elements, which is carefully situated and contextualized within his Neoplatonic ontological system. I argue that while geometry is formally independent from the sense world, the geometer is inspired by and utilizes things ontologically below true geometricals while making geometrical arguments, which is enabled by the structure of the Proclean system. This presents a novel approach that resolves the difference between Proclus’ progressive view of the abstract nature of geometry and his use of geometry in practice in the sense world.

Euclid’s elements publication in Vilnius

200 years ago in Vilnius the famous Elements by Euclid was set in print in the Polish language. The translationwas made by professorof mathematics Jozef Czech (1762–1810) and published in J. Zawadski’s printing house in 1807. This publication consisted of 8 books, i.e., the first six, eleventh and twelfth books on planimetry and stereometry. In 1817 the Elements was published once more. The fact that Euclid’s Elements appeared in print inVilnius can be discussedfrom two viewpoints. First, the society of the former Commonwealth of the Two Nations could get access to the heritage of antiquity. Second, the treatise was translated and published later then in other European countries. As a result, the Elements did not become the only textbook on geometry.

Methods of Teaching or Discovery? Analysis and Synthesis from Zabarella to Spinoza

This contribution looks directly at the so-called novatores and their own appropriation and reworking of the traditional methodological and pedagogic approaches. It shows how academic approaches and established tradition worked not only as a polemical target but also as a crucial resource that nourished the growth of alternatives to academic and Aristotelian approaches. This point is developed by discussing in detail the problem of method in Spinoza, and by connecting it with its scholastic background. By the mid-seventeenth century proponents of controversial philosophies appropriated more familiar didactic genres to convey their radical doctrines. For instance, the first book of Thomas Hobbes’s De Corpore follows the familiar order of standard Scholastic Aristotelian logic textbooks, and Baruch Spinoza’s Ethics emulates Euclid’s Elements, by presenting astounding conclusions about nature and extension more geometrico. There is a long-standing debate regarding whether Spinoza’s geometrical method is a method of discovery or merely a method of presentation. This contribution examines Spinoza’s reflections on method in the Treatise on the Emendation of the Intellect in the context of contemporaneous conceptions of analysis and synthesis found in the works of Zabarella, Burgersdijk, Descartes, and Hobbes to identify the most plausible readings of his method in the Ethics.