I use recent work on Kant and diagrammatic reasoning to develop a reconsideration of central aspects of Kant’s philosophy of geometry and its relation to spatial intuition. In particular, I reconsider in this light the relations between geometrical concepts and their schemata, and the relationship between pure and empirical intuition. I argue that diagrammatic interpretations of Kant’s theory of geometrical intuition can, at best, capture only part of what Kant’s conception involves and that, for example, they cannot explain why Kant (...) takes geometrical constructions in the style of Euclid to provide us with an a priori framework for physical space. I attempt, along the way, to shed new light on the relationship between Kant’s theory of space and the debate between Newton and Leibniz to which he was reacting, and also on the role of geometry and spatial intuition in the transcendental deduction of the categories. (shrink)

This influential 1888 publication explained the real numbers, and their construction and properties, from first principles.

This thought-provoking classic investigates how the Renaissance spirit fundamentally questioned and undermined medieval thought. Of value to students of literature, political theory, history of religious and Reformation thought, and the history of science.

This is the definitive work on the first and greatest of Japan's twentieth-century philosophers, Nishida Kitaro. Interspersed throughout the narrative of Nishida's life and thought is a generous selection of the philosopher's own essays, letters, and short presentations, newly translated into English.

It is a remarkable fact that Hilbert's programmatic papers from the 1920s still shape, almost exclusively, the standard contemporary perspective of his views concerning (the foundations of) mathematics; even his own, quite different work on the foundations of geometry and arithmetic from the late 1890s is often understood from that vantage point. My essay pursues one main goal, namely, to contrast Hilbert's formal axiomatic method from the early 1920s with his existential axiomatic approach from the 1890s. Such a contrast illuminates (...) the circuitous beginnings of the finitist consistency program and connects the complex emergence of existential axiomatics with transformations in mathematics and philosophy during the 19th century; the sheer complexity and methodological difficulties of the latter development are partially reflected in the well known, but not well understood correspondence between Frege and Hilbert. Taking seriously the goal of formalizing mathematics in an effective logical framework leads also to contemporary tasks, not just historical and systematic insights; those are briefly described as “one direction” for fascinating work. (shrink)

This English translation of Intuition and Reflection in Self-Consciousness evokes the movement and flavor of the original, clarifies its obscurities, and eliminates the repetitions.

What conceptual blind spot kept the ancient Greeks (unlike the Indians and Maya) from developing a concept of zero? Why did St. Augustine equate nothingness with the Devil? What tortuous means did 17th-century scientists employ in their attempts to create a vacuum? And why do contemporary quantum physicists believe that the void is actually seething with subatomic activity? You’ll find the answers in this dizzyingly erudite and elegantly explained book by the English cosmologist John D. Barrow. Ranging through mathematics, theology, (...) philosophy, literature, particle physics, and cosmology, The Book of Nothing explores the enduring hold that vacuity has exercised on the human imagination. Combining high-wire speculation with a wealth of reference that takes in Freddy Mercury and Shakespeare alongside Isaac Newton, Albert Einstein, and Stephen Hawking, the result is a fascinating excursion to the vanishing point of our knowledge. (shrink)

Cette étude consiste en une introduction en neuf étapes à la pensée du philosophe japonais Nishida Kitarô . Le néant nous livre des traces diffuses, que nous aimierions bien effacer à la lumière de l’être. Cependant, une fois appréhendé logiquement et approché phénoménologiquement, le néant révèle une structuration topologique remarquable, qui met en question les problématiques de la métaphysique et de la morale, en particulier concernant la distinction entre l’espace et le lieu, le vide et le plein et les rapports (...) de l’un et du multiple.This essay is a nine steps introduction to the Japanese philosopher Nishida Kitaro’s thought . Nothingness shows us some diffuse traces we would like to clear out with the light of being. However, once logically apprehended and phenomenologically approached, nothingness reveals a remarkable topological structure, which puts into question the problems of metaphysics and moral, in particular concerning the distinction between space and place, fullness and void and the relation of the one and the many. (shrink)

Reverse mathematics is a new take on an old idea: asking which axioms are necessary to prove a given theorem. This question was first asked about the parallel axiom in Euclid’s geometry and later about the axiom of choice in set theory. Obviously, such questions can be asked in many fields of mathematics, but in recent decades, it has proved fruitful to focus on subsystems of second-order arithmetic, where much of mainstream mathematics resides. It has been found that many basic (...) theorems of analysis and topology, as well as certain parts of infinite algebra and combinatorics, can be proved in such systems. And, remarkably, almost all the basic theorems fall into one of five particular systems: a base system RCA0 of “constructive mathematics” and four others defined by certain axioms about real numbers. Moreover, many of the theorems not provable in RCA0 turn out to be equivalent to one of these defining axioms, so we know precisely which axiom is needed to prove them. Thus, after some motivational remarks about the parallel axiom and the axiom of choice, we will concentrate on the study of RCA0 and its extensions, which is what “reverse mathematics” is generally taken to mean today. (shrink)

This clear and elegant translation reveals how a modern Japanese thinker dared to show the basic flaw of Western epistemology. In unmasking this limitation, Matsuo presents an Eastern view of a unified experience, and provides an epistemological basis for comparative philosophy. Matsuo notes that while early Greek thought began by focusing on the right counsel ("Know thyself"), since then Western thought has been influenced by empiricistic analysis fired by the rise of scientific philosophy. The author thus turns to Eastern epistemology, (...) in particular Buddhist thought, for clues to the unified experience. The seminal idea of emptiness (śūnyatā) plays a distinct role in this discovery. The concept of emptiness encompasses the whole dimension of perception where there is no room for separation into mind and body and/or any other form of dichotomy. Once it is known that the total dimension of perception—the logic of unity—functions in each and every person, then and only then can the field of comparative thought and philosophy be cleared of al preconceptions and move into a more fruitful exchange of ideas. Until such a time, Matsuo claims, we are hopelessly engaged in merely refining the epistemological process without ever being able to understand the very basis of intelligence. (shrink)