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Visions of Infinity

the Great Mathematical Problems
nftaussig
Oct 03, 2013nftaussig rated this title 3 out of 5 stars
Ian Stewart, an emeritus professor of mathematics at the University of Warwick, makes a valiant attempt to explain fourteen of the most important mathematical problems to a lay audience. For the most part, he succeeds. However, some of the problems are sufficiently abstract (particularly, the Hodge Conjecture) that even a gifted expositor such as Stewart cannot state them in terms a lay audience can understand. However, to his credit, he does try. Some of the problems such as the Goldbach conjecture that every even number larger than 2 can be expressed as the sum of two primes are readily understood, if difficult to solve. Others, such as the Hodge Conjecture, can only be understood by a specialist. In explaining these problems, Stewart gives the reader a sense of what it is that mathematicians do. Specifically, he demonstrates how mathematicians accumulate evidence that a conjecture is true, and illustrates that crucial insights sometimes come from drawing upon seemingly unrelated branches of mathematics (such as the use of number theory in constructing regular polygons). He also conveys why the problems discussed are difficult, while not always managing to explain their importance. Readers with mathematical training will find the descriptions in the book maddeningly vague, and will, no doubt, notice errors in the text (k^2 + k + 41 is not prime if k = 40) and imprecise definitions in the glossary (he fails to specify that the integers in a Pythagorean triple must be nonzero). There is also a particularly unfortunate analogy between Ernest Rutherford's determining the shape of an atom by bombarding it with alpha particles (nuclei of helium atoms) and shooting bullets into a dark field to see what is there.