In Mathematical Mountaintops, John Casti brilliantly recreates the solutions to the five greatest mathematical problems of all time: The Four-Color Map Problem, Fermat's Last Theorem, The Continuum Hypothesis, Kepler's Conjecture, and Hilbert's Tenth Problem.
Cash recounts these mathematical quests with great verve. In retelling the story of Hilbert's Tenth Problem, for instance, he sweeps from Britain to New York to Leningrad and introduces us to such luminaries as Alan Turing, before turning to the young Soviet researcher who credited hie breakthrough to a 700 year-old Italian problem about rabbits. He describes how Fermat's Last Theorem tantalized generations of scientists, who tried for three centuries to answer it, and relates how the final solution was greeted with unprecedented front-page headlines, prize money, and international celebration - before a flaw (soon resolved) turned up. Casti's account of the struggle to solve Kepler's Conjecture wittily reveals how the "proof of the obvious" sometimes eludes us for centuries. And his discussion of The Continuum Hypothesis movingly portrays the tragic figure of Georg Cantor, the troubled genius who created the first truly original mathematics since the Greeks, yet died insane in an institution. Casti closes with a preview of the "Magnificent Seven" - the greatest unsolved mathematical mysteries, each of which carries a million-dollar bounty from the Clay Mathematics Institute - including the Poincare Conjecture, the Yang-Mille Existence and Mass Gap (why physicists can't isolate quarks), and the Riemann Hypothesis ("the granddaddy of all mathematical mysteries").
Mathematical Mountain tops is a brilliant account of mathematicians in action-seeking hidden patterns and structures, forging elegant chains of reasoning - as they struggle with problems that challenged the greatest minds for decades, if not centuries.