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The intellectual and human story of a mathematical proof that transformed our ideas about mathematics. In 1824 a young Norwegian named Niels Henrik Abel proved conclusively that algebraic equations of the fifth order are not solvable in radicals. In this book Peter Pesic shows what an important event this was in the history of thought. He also presents it as a remarkable human story. Abel was twenty-one when he self-published his proof, and he died five years later, poor and depressed, just before the proof started to receive wide acclaim. Abel's attempts to reach out to the mathematical elite of the day had been spurned, and he was unable to find a position that would allow him to work in peace and marry his fiancé. But Pesic's story begins long before Abel and continues to the present day, for Abel's proof changed how we think about mathematics and its relation to the "real" world. Starting with the Greeks, who invented the idea of mathematical proof, Pesic shows how mathematics found its sources in the real world (the shapes of things, the accounting needs of merchants) and then reached beyond those sources toward something more universal. The Pythagoreans' attempts to deal with irrational numbers foreshadowed the slow emergence of abstract mathematics. Pesic focuses on the contested development of algebra-which even Newton resisted-and the gradual acceptance of the usefulness and perhaps even beauty of abstractions that seem to invoke realities with dimensions outside human experience. Pesic tells this story as a history of ideas, with mathematical details incorporated in boxes. The book also includes a new annotated translation of Abel's original proof. [Publisher]

Equations, Roots of --- Abel, Niels Henrik, --- Equations, Roots of. --- Algebra --- Équations, Racines des --- Algèbre --- History --- Histoire --- Abel, Niels Henrik --- Équations, Racines des. --- Histoire. --- Abel, Niels Henrik, - 1802-1829 --- Équations, Racines des. --- Algèbre

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"The separateness and connection of individuals is perhaps the central question of human life: What, exactly, is my individuality? To what degree is it unique? To what degree can it be shared, and how? To the many philosophical and literary speculations about these topics over time, modern science has added the curious twist of quantum theory, which requires that the elementary particles of which everything consists have no individuality at all. All aspects of chemistry depend on this lack of individuality, as do many branches of physics. From where, then, does our individuality come?" "In Seeing Double, Peter Pesic invites readers to explore this intriguing set of questions. He draws on literary and historical examples that open the mind (from Homer to Martin Guerre to Kafka), philosophical analyses that have helped to make our thinking and speech more precise, and scientific work that has enabled us to characterize the phenomena of nature. Though he does not try to be all-inclusive, Pesic presents a broad range of ideas, building toward a specific point of view: that the crux of modern quantum theory is its clash with our ordinary concept of individuality. This represents a departure from the usual understanding of quantum theory. Pesic argues that what is bizarre about quantum theory becomes more intelligible as we reconsider what we mean by individuality and identity in ordinary experience. In turn, quantum identity opens a new perspective on us." --Jacket.

Identity (Psychology) --- Individuality. --- Psychology and literature. --- Psychology and philosophy. --- Philosophy and science. --- Individuality --- Psychology and literature --- Psychology and philosophy --- Philosophy and science --- Psychology --- Social Sciences --- Literature and psychology --- Personal identity --- Science and philosophy --- Philosophy and psychology --- Literature --- Conformity --- Likes and dislikes --- Personality --- Self --- Ego (Psychology) --- Science --- Philosophy --- PHILOSOPHY/General --- PHYSICAL SCIENCES/General --- Identity (Psychology).

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An exploration of polyphony and the perspective it offers on our own polyphonic brains. Polyphony - the interweaving of simultaneous sounds - is a crucial aspect of music that has deep implications for how we understand the mind. Peter Pesic examines the history and significance of "polyphonicity" - of "many-voicedness" - in human experience. He presents the emergence of Western polyphony, its flowering, its horizons, and the perspective it offers on our own polyphonic brains. When we listen to polyphonic music, how is it that we can hear several different things at once? How does a single mind experience those things as a unity (a motet, a fugue) rather than an incoherent jumble? Pesic argues that polyphony raises fundamental issues for philosophy, theology, literature, psychology, and neuroscience - all searching for the apparent unity of consciousness in the midst of multiple simultaneous experiences. After tracing the development of polyphony in Western music from ninth-century church music through the experimental compositions of Glenn Gould and John Cage, Pesic considers the analogous activity within the brain, the polyphonic "music of the hemispheres" that shapes brain states from sleep to awakening. He discusses how neuroscientists draw on concepts from polyphony to describe the "neural orchestra" of the brain. Pesic's story begins with ancient conceptions of God's mind and ends with the polyphonic personhood of the human brain and body. An enhanced e-book edition allows the sound examples to be played by a touch.

Counterpoint --- Music --- Music psychology --- Psychological aspects&delete& --- History --- Philosophy and aesthetics&delete& --- Psychological aspects --- Psychology --- Art music --- Art music, Western --- Classical music --- Musical compositions --- Musical works --- Serious music --- Western art music --- Western music (Western countries) --- Polyphony --- Music theory --- Aesthetics --- History. --- Philosophy and aesthetics --- Psychological aspects. --- SCIENCE, TECHNOLOGY & SOCIETY/History of Science --- ARTS/Music & Sound Studies --- NEUROSCIENCE/General

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"Sounding Bodies presents the ancient Greek connections between music and medicine, their reception leading to the "sonic turn" in the eighteenth century, new kinds of sonic intervention in psychic disorders and new biological applications of sound"--

Wave mechanics (vibration & acoustics) --- History of medicine --- 20th century & contemporary classical music --- Wave mechanics (vibration and acoustics) --- History of music --- Science --- Music and science --- History. --- SCIENCE / Acoustics & Sound --- MEDICAL / History --- MUSIC / History & Criticism --- Science and music

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Aus den Rezensionen zur englischen Auflage: "Die Leser von Pesics faszinierendem kleinen Buch werden zu dem unausweichlichen Urteil kommen: Niels [Henrik] Abel hat sich der Genialität im fünften Grade schuldig gemacht." William Dunham, Muhlenberg College und Autor von "Journey through Genius: The Great Theorems of Mathematics "Peter Pesic schreibt über Abels Werk mit Begeisterung und Einfühlungsvermögen, und ruft Erinnerungen an die großartigen Momente in der Entwicklung der Algebra wach." Barry Mazur, Gerhard Gade University Professor, Harvard University "Ein einzigartiges Buch. Peter Pesics Chronik des langen Weges der Mathematiker zum Verständnis, wann eine Gleichung gelöst werden kann - und wann nicht - ist amüsant, einleuchtend und leserfreundlich. Der Autor bemüht sich sehr, auch weniger bekannte Namen wie Viète und Ruffini gebührend zu würdigen und verlangt von seinen Lesern nicht mehr als Basiswissen in der Algebra - wovon ein Großteil angenehmerweise getrennt vom Haupttext plaziert wurde." Tony Rothman, Department of Physics, Bryn Mawr College "Peter Pesics Geschichte über die Entstehung der Mathematik ist genauso spannend wie ein Roman." Economist.

Mathematics. --- History. --- Algebra. --- History of Mathematical Sciences. --- Calculus.