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Reasoning with the Infinite From the Closed World to the Mathematical Universe by Michel Blay University of Chicago Press, Chicago, 1998. 226 pp. $30, 25.95. ISBN 0-226-05834-4.
The Greeks were the first to confront the problem of the infinite mathematically, in the guise of Zeno's famous paradoxes of motion and the Pythagoreans' discovery of incommensurable magnitudes. If they failed to come to successful terms with the concept, they also knew that it was best avoided until mathematics could handle the concept without the uncertainties the infinite seemed destined to entail. Several millennia later, European mathematicians were no better off, as Galileo realized when he regarded the infinite as paradoxical. Similarly, George Berkeley leveled his criticism against the Newtonian calculus and its infinitesimals as nothing more than "ghosts of departed quantities." Nevertheless, successful analysis of motion or continuity cannot help but involve such problems as the infinite divisibility of space and time, and hence the infinite could not be ignored. The best-known part of the early history of the infinite and its counterpart, infinitesimals, is the development of the calculus by Newton, Leibniz, and their contemporaries. They created a useful tool for mathematics and for the newly emerging subject of mathematical physics. Progress in the 18th and 19th centuries deepened the understanding of the technical details of the calculus, as mathematical physics developed to an extraordinary degree in the hands of such adepts as Euler, Lagrange, and Laplace.
It is against this background of the struggle mathematics has waged with the infinite that Michel Blay's book must be considered. Its title adapts that of a wellknown work by another French author, Alexandre Koyre's...