Mark Peterson makes an extraordinary claim in this fascinating book focused around the life and thought of Galileo: it was the mathematics of Renaissance arts, not Renaissance sciences, that became modern science. Galileo's Muse argues that painters, poets, musicians, and architects brought about a scientific revolution that eluded the philosopher-scientists of the day, steeped as they were in a medieval cosmos and its underlying philosophy.
According to Peterson, the recovery of classical science owes much to the Renaissance artists who first turned to Greek sources for inspiration and instruction. Chapters devoted to their insights into mathematics, ranging from perspective in painting to tuning in music, are interspersed with chapters about Galileo's own life and work. Himself an artist turned scientist and an avid student of Hellenistic culture, Galileo pulled together the many threads of his artistic and classical education in designing unprecedented experiments to unlock the secrets of nature.
In the last chapter, Peterson draws our attention to the Oratio de Mathematicae laudibus of 1627, delivered by one of Galileo's students. This document, Peterson argues, was penned in part by Galileo himself, as an expression of his understanding of the universality of mathematics in art and nature. It is "entirely Galilean in so many details that even if it is derivative, it must represent his thought," Peterson writes. An intellectual adventure, Galileo’s Muse offers surprising ideas that will capture the imagination of anyone—scientist, mathematician, history buff, lover of literature, or artist—who cares about the humanistic roots of modern science.
Eminently suited to classroom use as well as individual study, Roger Myerson's introductory text provides a clear and thorough examination of the models, solution concepts, results, and methodological principles of noncooperative and cooperative game theory. Myerson introduces, clarifies, and synthesizes the extraordinary advances made in the subject over the past fifteen years, presents an overview of decision theory, and comprehensively reviews the development of the fundamental models: games in extensive form and strategic form, and Bayesian games with incomplete information.
Game Theory will be useful for students at the graduate level in economics, political science, operations research, and applied mathematics. Everyone who uses game theory in research will find this book essential.
The study of group actions is more than a hundred years old but remains to this day a vibrant and widely studied topic in a variety of mathematic fields. A central development in the last fifty years is the phenomenon of rigidity, whereby one can classify actions of certain groups, such as lattices in semi-simple Lie groups. This provides a way to classify all possible symmetries of important spaces and all spaces admitting given symmetries. Paradigmatic results can be found in the seminal work of George Mostow, Gergory Margulis, and Robert J. Zimmer, among others.
The papers in Geometry, Rigidity, and Group Actions explore the role of group actions and rigidity in several areas of mathematics, including ergodic theory, dynamics, geometry, topology, and the algebraic properties of representation varieties. In some cases, the dynamics of the possible group actions are the principal focus of inquiry. In other cases, the dynamics of group actions are a tool for proving theorems about algebra, geometry, or topology. This volume contains surveys of some of the main directions in the field, as well as research articles on topics of current interest.
Amid the unrest, dislocation, and uncertainty of seventeenth-century Europe, readers seeking consolation and assurance turned to philosophical and scientific books that offered ways of conquering fears and training the mind—guidance for living a good life.
The Good Life in the Scientific Revolution presents a triptych showing how three key early modern scientists, René Descartes, Blaise Pascal, and Gottfried Leibniz, envisioned their new work as useful for cultivating virtue and for pursuing a good life. Their scientific and philosophical innovations stemmed in part from their understanding of mathematics and science as cognitive and spiritual exercises that could create a truer mental and spiritual nobility. In portraying the rich contexts surrounding Descartes’ geometry, Pascal’s arithmetical triangle, and Leibniz’s calculus, Matthew L. Jones argues that this drive for moral therapeutics guided important developments of early modern philosophy and the Scientific Revolution.
This original study considers the effects of language and meaning on the brain. Jens Erik Fenstad—an expert in the fields of recursion theory, nonstandard analysis, and natural language semantics—combines current formal semantics with a geometric structure in order to trace how common nouns, properties, natural kinds, and attractors link with brain dynamics.
In their search for truth, contemporary religious believers and modern scientific investigators hold many values in common. But in their approaches, they express two fundamentally different conceptions of how to understand and represent the world. Michael E. Hobart looks for the origin of this difference in the work of Renaissance thinkers who invented a revolutionary mathematical system—relational numeracy. By creating meaning through numbers and abstract symbols rather than words, relational numeracy allowed inquisitive minds to vault beyond the constraints of language and explore the natural world with a fresh interpretive vision.
The Great Rift is the first book to examine the religion-science divide through the history of information technology. Hobart follows numeracy as it emerged from the practical counting systems of merchants, the abstract notations of musicians, the linear perspective of artists, and the calendars and clocks of astronomers. As the technology of the alphabet and of mere counting gave way to abstract symbols, the earlier “thing-mathematics” metamorphosed into the relational mathematics of modern scientific investigation. Using these new information symbols, Galileo and his contemporaries mathematized motion and matter, separating the demonstrations of science from the linguistic logic of religious narration.
Hobart locates the great rift between science and religion not in ideological disagreement but in advances in mathematics and symbolic representation that opened new windows onto nature. In so doing, he connects the cognitive breakthroughs of the past with intellectual debates ongoing in the twenty-first century.
Elemental learning.
The splendid achievement of Greek mathematics is here illustrated in two volumes of selected mathematical works. Volume I (LCL 335) contains the divisions of mathematics; mathematics in Greek education; calculation; arithmetical notation and operations, including square root and cube root; Pythagorean arithmetic, including properties of numbers; the square root of 2; proportion and means; algebraic equations; Proclus; Thales; Pythagorean geometry; Democritus; Hippocrates of Chios; duplicating the cube and squaring the circle; trisecting angles; Theaetetus; Plato; Eudoxus of Cnidus (pyramid, cone); Aristotle (the infinite, the lever); Euclid.
Volume II (LCL 362) contains Aristarchus (distances of sun and moon); Archimedes (cylinder, sphere, cubic equations; conoids; spheroids; spiral; expression of large numbers; mechanics; hydrostatics); Eratosthenes (measurement of the earth); Apollonius (conic sections and other works); later development of geometry; trigonometry (including Ptolemy’s table of sines); mensuration: Heron of Alexandria (mensuration); Diophantus (algebra, determinate and indeterminate equations); Pappus (the revival of geometry).
Elemental learning.
The splendid achievement of Greek mathematics is here illustrated in two volumes of selected mathematical works. Volume I (LCL 335) contains the divisions of mathematics; mathematics in Greek education; calculation; arithmetical notation and operations, including square root and cube root; Pythagorean arithmetic, including properties of numbers; the square root of 2; proportion and means; algebraic equations; Proclus; Thales; Pythagorean geometry; Democritus; Hippocrates of Chios; duplicating the cube and squaring the circle; trisecting angles; Theaetetus; Plato; Eudoxus of Cnidus (pyramid, cone); Aristotle (the infinite, the lever); Euclid.
Volume II (LCL 362) contains Aristarchus (distances of sun and moon); Archimedes (cylinder, sphere, cubic equations; conoids; spheroids; spiral; expression of large numbers; mechanics; hydrostatics); Eratosthenes (measurement of the earth); Apollonius (conic sections and other works); later development of geometry; trigonometry (including Ptolemy’s table of sines); mensuration: Heron of Alexandria (mensuration); Diophantus (algebra, determinate and indeterminate equations); Pappus (the revival of geometry).
In recent years scholars from a variety of branches of mathematics have made several significant developments in the theory of group actions. Groups of Circle Diffeomorphisms systematically explores group actions on the simplest closed manifold, the circle. As the group of circle diffeomorphisms is an important subject in modern mathematics, this book will be of interest to those doing research in group theory, dynamical systems, low dimensional geometry and topology, and foliation theory. The book is mostly self-contained and also includes numerous complementary exercises, making it an excellent textbook for undergraduate and graduate students.
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