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. János Bolyai, Non-Euclidean Geonietrv, and the Nature 0/ Space gives us a rich account of what happened and how.


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.


Jdnos Bolyai, Non-Euclidean Geonietry, and the Nature of Space is the first in a new series of books published by the Burndy Library at MIT’s Dibner Institute for the History of Science and Technology. The series is intended to make available to a wide audience the resources contained in the library. Accordingly, the volume reproduces in facsimile two items from the library’s collection: Bolyai’s original Latin publication (which was published in 1832 as an appendix to a much longer mathematical work by his father) and the 1896 English translation by George B. Halsted (an American mathematician whose writings did much to popularize the new geometry). These are preceded by a long “preface” by Jeremy Gray (a mathematician and historian of mathematics, at the Open University) which is in fact a very full account of the story of the parallel postulate, the discovery of non-Euclidian geometry, and the impact of these ideas from the mid-l9th to the early 20th centuries. Gray’s preface is a delight. He tells us of various purported “proofs” of the parallel postulate, of Kant’s controversial argument that Euclidcan geometry provided an example of synthetic a priori knowledge, and much more. Having set up the context, he turns to Bolyai’s own work and attempts to guide the reader through the contents of the papers reproduced later in the book. This shifts the discussion from overview to detailed mathematics; non-mathematicians are likely to find this section difficult. Should they persist, howeverer, they will reach Gray’s fascinating account of the reception of non-Euclidean geometry by mathematicians, physicists, philosophers, and even artists, which does not make serious mathematical demands on the reader.


Parallel WorldsAny 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.




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