Twentieth-century developments in logic and mathematics have led many people to view Euclid's proofs as inherently informal, especially due to the use of diagrams in proofs. In Euclid and His Twentieth-Century Rivals, Nathaniel Miller discusses the history of diagrams in Euclidean Geometry, develops a formal system for working with them, and concludes that they can indeed be used rigorously. Miller also introduces a diagrammatic computer proof system, based on this formal system. This volume will be of interest to mathematicians, computer scientists, and anyone interested in the use of diagrams in geometry.
Proceedings of the 4th International Conference on Theory and Application of Diagrams, Stanford, CA, USA in June 2006. 13 revised full papers, 9 revised short papers, and 12 extended abstracts are presented together with 2 keynote papers and 2 tutorial papers. The papers are organized in topical sections on diagram comprehension by humans and machines, notations: history, design and formalization, diagrams and education, reasoning with diagrams by humans and machines, and psychological issues in comprehension, production and communication.
Diagrams is an international and interdisciplinary conference series, covering all aspects of research on the theory and application of diagrams. Recent technological advances have enabled the large-scale adoption of d- grams in a diverse range of areas. Increasingly sophisticated visual represen- tions are emerging and, to enable e?ective communication, insight is required into how diagrams are used and when they are appropriate for use. The per- sive, everyday use of diagrams for communicating information and ideas serves to illustrate the importance of providing a sound understanding of the role that diagrams can, and do, play. Research in the ?eld of diagrams aims to improve our understanding of the role of diagrams, sketches and other visualizations in communication, computation, cognition, creative thought, and problem solving. These concerns have triggered a surge of interest in the study of diagrams. The study of diagrammatic communication as a whole must be pursued as an interdisciplinary endeavour.Diagrams 2008 was the ?fth event in this conf- ence series, which was launched in Edinburghduring September 2000.Diagrams attracts a large number of researchers from virtually all related ?elds, placing the conference as a major international event in the area. Diagrams is the only conference that provides a united forum for all areas that are concerned with the study of diagrams: for example, architecture, - ti?cial intelligence, cartography, cognitive science, computer science, education, graphicdesign,historyofscience,human-computerinteraction,linguistics,logic, mathematics,philosophy,psychology,andsoftwaremodelling.Weseeissuesfrom all of these ?elds discussed in the papers collected in the present volume.
A sweeping cultural history of one of the most influential mathematical books ever written Euclid's Elements of Geometry is one of the fountainheads of mathematics—and of culture. Written around 300 BCE, it has traveled widely across the centuries, generating countless new ideas and inspiring such figures as Isaac Newton, Bertrand Russell, Abraham Lincoln, and Albert Einstein. Encounters with Euclid tells the story of this incomparable mathematical masterpiece, taking readers from its origins in the ancient world to its continuing influence today. In this lively and informative book, Benjamin Wardhaugh explains how Euclid’s text journeyed from antiquity to the Renaissance, introducing some of the many readers, copyists, and editors who left their mark on the Elements before handing it on. He shows how some read the book as a work of philosophy, while others viewed it as a practical guide to life. He examines the many different contexts in which Euclid's book and his geometry were put to use, from the Neoplatonic school at Athens and the artisans' studios of medieval Baghdad to the Jesuit mission in China and the workshops of Restoration London. Wardhaugh shows how the Elements inspired ideas in theology, art, and music, and how the book has acquired new relevance to the strange geometries of dark matter and curved space. Encounters with Euclid traces the life and afterlives of one of the most remarkable works of mathematics ever written, revealing its lasting role in the timeless search for order and reason in an unruly world.
This book constitutes the refereed proceedings of the 7th International Conference on Theory and Application of Diagrams, Diagrams 2012, held in Canaterbury, UK, in July 2012. The 16 long papers, 6 short papers and 21 poster abstracts presented were carefully reviewed and selected from 83 submissions. The papers are organized in keynotes, tutorial, workshops, graduate student symposium and topical sections on psychological and cognitive issues, diagram layout, diagrams and data analysis, Venn and Euler diagrams, reasoning with diagrams, investigating aesthetics, applications of diagrams.
Like Douglas Hofstadter’s Gödel, Escher, Bach, and David Berlinski’s A Tour of the Calculus, Euclid in the Rainforest combines the literary with the mathematical to explore logic—the one indispensable tool in man’s quest to understand the world. Underpinning both math and science, it is the foundation of every major advancement in knowledge since the time of the ancient Greeks. Through adventure stories and historical narratives populated with a rich and quirky cast of characters, Mazur artfully reveals the less-than-airtight nature of logic and the muddled relationship between math and the real world. Ultimately, Mazur argues, logical reasoning is not purely robotic. At its most basic level, it is a creative process guided by our intuitions and beliefs about the world.
Induction, which involves a leap from the particular to the universal, has always been a puzzling phenomenon for those attempting to investigate the origins of knowledge. Although traditionally accepted as the engine of first principles, the authority of inductive reasoning has been undermined in the modern age by empiricist criticisms that derive notably from Hume, who insisted that induction is an invalid line of reasoning that ends in unreliable future predictions. The present volume challenges this Humean orthodoxy. It begins with a thorough consideration of Hume’s original position and continues with a series of state-of-the-art essays that critique the received view while offering positive alternatives. The experts assembled here draw on a perennial historical tradition that stretches as far back as Socrates and extends through such luminaries as Aristotle, Aquinas, Whewell, Goethe, Lonergan, and Rescher. They inquire into the creative moment of intellectual insight that makes induction possible, consider relevant episodes from the history of science, advance scholarly exegeses of historical interpretations of inductive reasoning, and reflect critically on the scientific and logical ramifications of epistemological and metaphysical realism.
Mathematicians at every level use diagrams to prove theorems. Mathematical Reasoning with Diagrams investigates the possibilities of mechanizing this sort of diagrammatic reasoning in a formal computer proof system, even offering a semi-automatic formal proof system—called Diamond—which allows users to prove arithmetical theorems using diagrams.
