The book contains a collection of essays by scholars and artists from a range of different fields including art, art history, architectural theory and philosophy. The essays are based on papers given at a symposium in Copenhagen in June 2008 and refer to the following considerations: When spectators confront and designers invent works of art and architecture, vital questions regarding their appearance arise. Based on multiple discourses on these subjects, contemporary positions in art, architecture and philosophy draw up new challenges, especially with regard to the creative practices. Within and between these positions emerge potentials for modes of thinking and doing with a new sensitivity. Contributors include Michael Asgaard Andersen and Henrik Oxvig, Renaud Barbaras, Andrew Benjamin, Olafur Eliasson, Sanford Kwinter, David Leatherbarrow, Martin Seel, David Summers, and Sven-Olov Wallenstein.

An introduction to awe-inspiring ideas at the brink of paradox: infinities of different sizes, time travel, probability and measure theory, and computability theory. This book introduces the reader to awe-inspiring issues at the intersection of philosophy and mathematics. It explores ideas at the brink of paradox: infinities of different sizes, time travel, probability and measure theory, computability theory, the Grandfather Paradox, Newcomb's Problem, the Principle of Countable Additivity. The goal is to present some exceptionally beautiful ideas in enough detail to enable readers to understand the ideas themselves (rather than watered-down approximations), but without supplying so much detail that they abandon the effort. The philosophical content requires a mind attuned to subtlety; the most demanding of the mathematical ideas require familiarity with college-level mathematics or mathematical proof. The book covers Cantor's revolutionary thinking about infinity, which leads to the result that some infinities are bigger than others; time travel and free will, decision theory, probability, and the Banach-Tarski Theorem, which states that it is possible to decompose a ball into a finite number of pieces and reassemble the pieces so as to get two balls that are each the same size as the original. Its investigation of computability theory leads to a proof of Gödel's Incompleteness Theorem, which yields the amazing result that arithmetic is so complex that no computer could be programmed to output every arithmetical truth and no falsehood. Each chapter is followed by an appendix with answers to exercises. A list of recommended reading points readers to more advanced discussions. The book is based on a popular course (and MOOC) taught by the author at MIT.

In a world filled with ambiguity, we want faith to act like an orderly set of truth-claims to solve the problems that life throws at us. While there are certainties in Christian faith, at the heart of the Christian story is also paradox, and Jen Pollock Michel helps readers imagine a Christian faith open to mystery. Jesus invites us to abandon the polarities of either and or in order to embrace the difficult, wondrous dissonance of and.

This “fun, brain-twisting book . . . will make you think” as it explores more than 75 paradoxes in mathematics, philosophy, physics, and the social sciences (Sean Carroll, New York Times–bestselling author of Something Deeply Hidden) Paradox is a sophisticated kind of magic trick. A magician’s purpose is to create the appearance of impossibility, to pull a rabbit from an empty hat. Yet paradox doesn’t require tangibles, like rabbits or hats. Paradox works in the abstract, with words and concepts and symbols, to create the illusion of contradiction. There are no contradictions in reality, but there can appear to be. In Sleight of Mind, Matt Cook and a few collaborators dive deeply into more than 75 paradoxes in mathematics, physics, philosophy, and the social sciences. As each paradox is discussed and resolved, Cook helps readers discover the meaning of knowledge and the proper formation of concepts—and how reason can dispel the illusion of contradiction. The journey begins with “a most ingenious paradox” from Gilbert and Sullivan’s Pirates of Penzance. Readers will then travel from Ancient Greece to cutting-edge laboratories, encounter infinity and its different sizes, and discover mathematical impossibilities inherent in elections. They will tackle conundrums in probability, induction, geometry, and game theory; perform “supertasks”; build apparent perpetual motion machines; meet twins living in different millennia; explore the strange quantum world—and much more.

"This highly innovative book is a multidisciplinary study of green and its significance from multiple perspectives: aesthetic, architectural, environmental, political, and social. It is centered on the Kingdom of Bahrain, the smallest and greenest of the Arab states in the Persian Gulf, where green has a long and deep history appearing cooling, productive, and prosperous--and a radical contrast to the hot, hostile desert. As is the case with cities around the world, green is often celebrated as a counter to gray urban environments, yet green has not always been good for cities. To have the color green manifested in arid environments is often in direct conflict with 'green' from an environmental point of view; this paradox is at the heart of the book. Given the resources required to maintain green in arid areas, including cities, the provision of green often bears significant environmental costs. In arid environments such as Bahrain, this contradiction becomes extreme and even unsustainable. Based on long-term ethnographic fieldwork, Gareth Doherty explores the landscapes of Bahrain where green represents a plethora of implicit human values and lives in dialectical tension with other culturally and environmentally significant colors and hues. The book's six chapters focus on: Blue, Red, Date-palm Green, Grass Green, Beige, and White. Implicit in his book is the argument that concepts of color and object are mutually defining and thus a discussion about green becomes a discussion about the creation of space and place"--

Stimulating, thought-provoking analysis of the most interesting intellectual inconsistencies in mathematics, physics, and language, including being led astray by algebra (De Morgan's paradox). 1982 edition.

This exploration of the scientific limits of knowledge challenges our deep-seated beliefs about our universe, our rationality, and ourselves. “A must-read for anyone studying information science.” —Publishers Weekly, starred review Many books explain what is known about the universe. This book investigates what cannot be known. Rather than exploring the amazing facts that science, mathematics, and reason have revealed to us, this work studies what science, mathematics, and reason tell us cannot be revealed. In The Outer Limits of Reason, Noson Yanofsky considers what cannot be predicted, described, or known, and what will never be understood. He discusses the limitations of computers, physics, logic, and our own intuitions about the world—including our ideas about space, time, and motion, and the complex relationship between the knower and the known. Yanofsky describes simple tasks that would take computers trillions of centuries to complete and other problems that computers can never solve: • perfectly formed English sentences that make no sense • different levels of infinity • the bizarre world of the quantum • the relevance of relativity theory • the causes of chaos theory • math problems that cannot be solved by normal means • statements that are true but cannot be proven Moving from the concrete to the abstract, from problems of everyday language to straightforward philosophical questions to the formalities of physics and mathematics, Yanofsky demonstrates a myriad of unsolvable problems and paradoxes. Exploring the various limitations of our knowledge, he shows that many of these limitations have a similar pattern and that by investigating these patterns, we can better understand the structure and limitations of reason itself. Yanofsky even attempts to look beyond the borders of reason to see what, if anything, is out there.