Mathematics and Logic

Mark Kac and Stanislaw M. Ulam

Given the small size of the book (170 pages), itís unbelievable how much the authors managed to put in. Itís even more astonishing that virtually none of the topics is discussed in a superficial manner. Besides profound examples included in the first chapter, the later chapters touch on such topics as Jordan Curve Theorem, continuity and differentiability, axiomatization, Laplace Transform and convolution, Artinís Theory of Braids, Banachís result on nondifferentiable functions, Godelís Incompleteness Theorem, Turing machines and algorithmic decidability, non-Euclidean geometries and models, Thermodynamics, Dynamic Systems, Game Theory, entropy, and more. The book assumes a degree of mathematical culture that may be expected from a curious undergraduate student. Itís good for perusal too. The style is clear and fluid and makes itís easy to grasp a topic at a glance. Topics take a few page and are mostly independent of one another. However, the reader would make the most of the book by reading through and depending on the authorsí selection and sequencing of the topics. Authorsí profound understanding of the nature of Mathematics and its current developments allowed them to weave a rare variety of topics into a coherent whole whose evolution an interested reader should be wary of missing.

G.-C. Rota wrote in his Indiscrete Thoughts:

It took me several years to realize what Stan Ulam's real profession was. Many of us at the Los Alamos Laboratory who were associated with him knew how much he disliked being alone, how he would summon us at odd times to be rescued from the loneliness of some hotel room, or from the four walls of his office after he had exhausted his daily round of long distance calls.

One day I mustered the courage to ask him why he constantly wanted company, and his answer gave him away. "When I am alone," he admitted, "I am forced to think things out, and I see so much that I would rather not think about!" I then saw the man in his true light. The man who had the highest record of accurate guesses in mathematics, the man who could beat engineers at their game, who could size up characters and events in a flash, was a member of an all-but-extinct profession, the profession of prophet.

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