This is the first complete English translation of Pavel Florensky's original and ambitious attempt to arrive at a geometric representation of imaginary numbers, in a context that had already captured the attention of other mathematicians, including Gauss, Argan, Cauchy and Bellavitis. Florensky did not limit his attempt solely to complex projective geometry, but extended it to encompass Ptolemaic-Dantean cosmology and Einstein's Principle of Relativity, as well as a new epistemological theory. The resulting treatise combines various disciplines and explores the relationship between an immanent realm of knowledge and a transcendent one.
The author, a Nobel Laureate and one of the 20th century's most important logicians, asks and answers basic questions about the intersection of philosophy and higher mathematics. 1897 edition.
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Today complex numbers have such widespread practical use--from electrical engineering to aeronautics--that few people would expect the story behind their derivation to be filled with adventure and enigma. In An Imaginary Tale, Paul Nahin tells the 2000-year-old history of one of mathematics' most elusive numbers, the square root of minus one, also known as i. He recreates the baffling mathematical problems that conjured it up, and the colorful characters who tried to solve them. In 1878, when two brothers stole a mathematical papyrus from the ancient Egyptian burial site in the Valley of Kings, they led scholars to the earliest known occurrence of the square root of a negative number. The papyrus offered a specific numerical example of how to calculate the volume of a truncated square pyramid, which implied the need for i. In the first century, the mathematician-engineer Heron of Alexandria encountered I in a separate project, but fudged the arithmetic; medieval mathematicians stumbled upon the concept while grappling with the meaning of negative numbers, but dismissed their square roots as nonsense. By the time of Descartes, a theoretical use for these elusive square roots--now called "imaginary numbers"--was suspected, but efforts to solve them led to intense, bitter debates. The notorious i finally won acceptance and was put to use in complex analysis and theoretical physics in Napoleonic times. Addressing readers with both a general and scholarly interest in mathematics, Nahin weaves into this narrative entertaining historical facts and mathematical discussions, including the application of complex numbers and functions to important problems, such as Kepler's laws of planetary motion and ac electrical circuits. This book can be read as an engaging history, almost a biography, of one of the most evasive and pervasive "numbers" in all of mathematics. Some images inside the book are unavailable due to digital copyright restrictions.
With wit and clarity, the authors progress from simple arithmetic to calculus and non-Euclidean geometry. Their subjects: geometry, plane and fancy; puzzles that made mathematical history; tantalizing paradoxes; more. Includes 169 figures.
This book offers an outline of the foundations of a theory of constitutional semiotics. It provides a systematic account of the concept of constitutional semiotics and its role in the representation and signification of meaning in constitution, constitutional law, and constitutionalism. The book explores the constitutional signification of meaning that is stretched between rational entrenchment and constitutional imagination. It provides a critical assessment of the rationalist entrapment of constitutional modernity and justifies the need to turn to 'shadow constitutionalisms': textual, symbolic-imaginary and visual constitutionalism. The book puts forward innovative incentives for constitutional analysis based on constitutional semiotics as a paradigm for representation of meaning in rational, textual, symbolic-imaginary and visual constitutionalism. The book focuses on the textual, imaginative, and visual discourse of constitutionalism, which is built upon collective constitutional imaginaries and on the peculiar normativity of constitutional geometry and constitutional mythology as borderline phenomena entrenched in rational, textual, symbolic-imaginary and visual constitutionalism. The book analyses concepts such as: constitutional text and texture, authoritative constitutional narratives and authoritative constitutional narrators, constitutional semiotic community, constitutional utopia, constitutional taboo, normative ideology and normative ideas, constitutional myth and mythology, constitutional symbolism, constitutional code and constitutional geometric form. It explores the textual entrenchment of constitutionalism and its repercussions for representation and signification of meaning.
