Parallel WORLDS FERNANDO Q. GOUVEA The reviewer is in Dept. Of Mathematics, Colby College, Mayflower Hill 5830 Waterville, ME 04901 . Try E-mail at: fqgouvea@colby.edu The discovery of non-Euclidean geometry surely counts as one of the crucial
turnings in the history of human thought. It was a slow turn, begun in the early
l830s and only really completed at the dawn of the 20th century. In the process, it
caused a transformation in how scientists and philosophers thought about mathematics, about the space surrounding us, and about the relation between the
two. The story begins, as many mathematical stories do, with Euclid’s magisterial compendium of Greek mathematics, The Elements. Euclid based his account of geometry on five assumptions that he seems to have regarded as self-evident properties of space. Four of these assumptions are fairly simple, but the fifth, known as the “parallel postulate,” is quite complicated. Many of Euclid’s readers over the centuries have wondered whether it really needed to be that complicated. Even Euclid seems to have felt that there was something unusual about this postulate, as he carefully avoided using it until it became absolutely necessary. Dissatisfaction with the parallel postulate led many mathematicians to attempt to prove it on the basis of the other four postulates or to find some simpler fact that could be reasonably described as a “self-evident” truth about the space in which we live and from which the parallel postulate could be deduced. Everyone shared the assumptions that Euclidean geometry provided a true description of actual space and that what was missing was simply a full understanding of how the parallel postulate fit into the picture. In the early 19th century, János Bolyai and Nikolai Ivanovich Lobachevskii both started to investigate the problem. Independently, each looked carefully at what sort of geometry would result if one did not assume the parallel postulate. (Bolyai called this a “geometry of absolute space) They concluded that a geometry in which the parallel postulate did not hold true—a non-Euclidean geometry was in fact possible and free from contradictions. Both published their work in obscure places, making it difficult for their achievements to be absorbed by the mathematics community. But slowly it dawned on mathematicians that a true intellectual revolution had occurred. If geometry does not have to be Euclidean, then in what sense is it a description of the actual space in which we live? Was it that our space is truly Euclidean, and these new geometries were purely imaginary worlds? Or could it be that one of these other geometries actually described our space better? The latter was possible because being non-Euclidean turned out to be a large-scale characteristic: space could “look Euclidean” on a small scale even if it was non-Euclidean overall. Such questions led mathematicians to back away from any claim that their geometries (now in the plural) described actual space. Instead, they came to view their work as simply describing possible geometries and so to the idea that mathematics at best provides models of what reality might be like rather than somehow accessing reality directly. Deciding which of the possible models came closest to describing our actual physical space became a question for physicists.
Any author who attempts to reach an audience that includes specialists and nonspecialists alike—-as the preface specifics that books in the Burndy Library series should is faced with difficult decisions. Though Gray often speaks successfully to both kinds of reader, the results are not uniform; at times, the reader will need a little bit of patience when dealing with material meant for others. Those who make that effort will find that they have acquired a much deeper understanding of the “non-Euclidean revolution” and its far-reaching consequences. Return to the words of wisdom, space index..
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