Five Golden Rules: Great Theories of 20th-Century Mathematics--and Why They MatterIn Five Golden Rules, John L. Casti serves as curator to a brilliant collection of 20th-century mathematical theories, leading us on a fascinating journey of discovery and insight. Probing the frontiers of modern mathematics, Casti examines the origins of some of the most important findings of this century. This is a tale of mystery and logic, elegance and reason; it is the story of five monumental mathematical breakthroughs and how they shape our lives. All those intrigued by the mathematical process, nonacademics and professionals alike, will find this an enlightening, eye-opening, and entertaining work. High school algebra or geometry - and enthusiasm - are the only prerequisites. From the theorem that provided the impetus for modern computers to the calculations that sent the first men to the Moon, these breakthroughs have transformed our lives. Casti illustrates each theorem with a dazzling array of real-world problems it has helped solve - how to calculate the shape of space, optimize investment returns, even chart the course of the development of organisms. Along the way, we meet the leading thinkers of the day: John von Neumann, L. E. J. Brouwer, Marston Morse, and Alan Turing, among others. And we come to understand the combination of circumstances that led each to such revolutionary discoveries as the Minimax Theorem, which spawned the exciting field of game theory, and the Simplex Method, which underpins the powerful tools of optimization theory. |
Contents
Deadly Games Games of Strategy TwoPerson ZeroSum | 40 |
Morses Theorem Singularity Theory | 85 |
Thats the Way the Paper Crumples A Taylors Tale Tugging | 128 |
Copyright | |
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Common terms and phrases
action algorithm arithmetic B-II B-III behavior Brouwer Fixed-Point Theorem Busy Beaver called catastrophe theory chapter choice Chow-Down closed surfaces codimension computation constraints continuous transformation cooperative criterion function cross-cap curve degenerate critical point equations equilibrium example exist fact finite number fixed point fixed-point property formal system function ƒ game theory geometry given Gödel Halting Problem idea input integer involving labeled logical look mathematical mathematicians means minimal Minimax Theorem mixed strategy Morse function Morse's Theorem node non-Morse function nondegenerate critical point notion number of steps optimal mixed strategy original parameters payoff matrix plane play player possible Prisoner's Dilemma provable pure strategies rational reader real numbers result rules saddle point shown in Figure shows simple singularity theory situation smooth functions solution solve sphere square statement string Suppose symbols Thom Classification topological space topologically equivalent turns types values variables what's zero-sum