Running Counter to Inert Crystallized Opinion*December 26, 2004
Philip J. Davis
János Bolyai, Non-Euclidean Geometry, and the Nature of Space. By Jeremy J. Gray, Burndy Library (distributed by MIT Press), Cambridge, Massachusetts, 2004, 256 pages, $20.00.
Given a straight line L in the plane and a point P not lying on L, then through P there is exactly one straight line that does not intersect L. Right? Well, if you believe that---the crystallized opinion---then you are a Euclidean, for the statement is a consequence of Euclid's Fifth Postulate.
If you believe that there is an infinity of such parallel lines, you are a Bolyaian or a Lobachevskian, depending on your national predilections.
If you believe that there is no such parallel, you are a Riemannian. And if you believe that there are precisely six parallels, I'm not sure what to call you.
In any case, recognition that one might hold other beliefs (assumptions, really) about the line L and the point P blew to bits the accepted truth that Euclidean geometry had a priori validity as a description of physical space. Ultimately, it led to Poincaré's conventionalism and the idea of mathematical models, rather than rock-ribbed truths, and it threw mathematics wide open to philosophies other than those holding that its stuff was made in heaven even before the Big Bang blasted away. As Gray writes,
"When Bolyai created a whole new world out of nothing, he thereby liberated mathematics from being solely the servant of science, and this, in turn, enabled mathematics to make fresh contributions to the study of the natural world."
János Bolyai (1802-1860) was born in Koloszvar, now Cluj, a part of Transylvanian Romania, but in Bolyai's day a part of the Austro-Hungarian Hapsburg Empire. Transylvania has been a bone of contention between Hungary and Romania for ages, with governmental authority oscillating like a Ping-Pong ball over its boundaries. Apparently, Romania wants also to claim Bolyai for its own: There is a one-leu Romanian postage stamp bearing a picture of Bolyai, and there is the Babes-Bolyai University in Cluj-Napoca. (Incidentally, a Budapest friend tells me that Hungarians pronounce Bolyai's name "Bolya-ee," and not, as I have always pronounced it, with the second syllable rhyming with "sky.")
The story of the simultaneous discovery (or the creation) of non-Euclidean geometry by Bolyai and Lobachevsky, and of its consequences, has been told many times over and at different levels of mathematical sophistication and purpose. The Brown Science Library has three shelvesful of books on this specific topic. The available material ranges from short embedded chapters to the magisterial History of Non-Euclidean Geometry by B.A. Rosenfeld, in which the spotlight leaves Bolyai and Lobachevsky to spread widely over the whole unfolding drama of geometry from the Babylonians to Lie groups and beyond.
Jeremy J. Gray, a professor of mathematics at the Open University in the UK, is a historian who has given us many fine volumes on the history of mathematics. Gray's treatment here, probably a spin-off from his longer Ideas of Space (second edition, Oxford, 1989), provides a mini-history of the subject. Not too long and not too short, it is just the right length for a general educated readership. The text can be read through in an hour or so (if you don't try to follow Bolyai's path closely by working through the geometrical details provided). Gray takes us from Euclid through the various feelings of angst about parallels, through attempts to deduce mono-parallelism via the other postulates. He mentions the undeveloped feelings of Gauss about the matter, proceeding thence to Bolyai's tradition-shattering ideas, with a nod to Lobachevsky, and ultimately to the work of Riemann, Beltrami, Poincaré, and numerous other post-Bolyai/Lobachevsky worthies.
Of the many pre-Bolyai attempts at proving Euclid's parallel axiom from the others, Gray has singled out Legendre's erroneous attempt to show, without taking the Fifth, that the angle sum in a triangle cannot be less than 180°. And Legendre was certainly a big man in our business. Gray presents this problem as a bit of a puzzle for problem solvers, revealing Legendre's error only on a later page.
The Dibner Institute for the History of Science and Technology, located in Cambridge, on the campus of the Massachusetts Institute of Technology, is a tribute and an eloquent memorial to Bern Dibner, founder of the Burndy Engineering Company. The Burndy Library, a part of the Dibner, was originally begun in 1936 to house the growing collection of Mr. Dibner and now contains more than 50,000 volumes dedicated to the history of science and technology.
The "Burndy" recently launched a new series of publications. According to Benjamin Weiss, the library's curator of rare books, the books published in the series are to reflect the strengths or the interesting particularities of the Burndy collection. The unifying theme of the series is "Burndyness," rather than any particular subject or approach. Gray is the lead-off author, and as a consequence, there is more to his book than the mini-history of non-Euclidean geometry. Taking advantage of the holdings of the Burndy Library, the book is loaded with portraits of mathematicians whose images are both known and less well known. Among the latter: Clairaut, Legendre, Beltrami, Farkas (Wolfgang) Bolyai, and a new (probably conjectured) image of his son, the eponymous János. More significantly, to these and to Gray's text has been added a facsimile-30 pages long-of the original Latin publication of Bolyai, together with an English translation made years ago by George Bruce Halstead. Halstead (1853-1922) is introduced to readers via a five-page notice.
Devotees of the history of mathematics will be glad to have this archival material at hand, while historical amateurs such as myself will find it instructive to compare Bolyai's original formulation with the manner in which non-Euclidean geometry is currently presented in texts and courses. A brief scan should make us thankful for the existence of all the translators, commentators, simplifiers, generalizers, historians, popularizers who have made this and other revolutionary material, often difficult of comprehension, available to us. (One thinks, of course, of Newton, but many lesser names of yore will do as well.)
A few things I didn't know and learned from Gray's book:
(1) Bolyai squared the circle in non-Euclidean geometry. The constructions are included here.
(2) Bolyai's Nachlass, more than 10,000 manuscript pages, apparently contains little of mathematical interest. He peaked early. Disappointment that Gauss and Lobachevsky had invaded his turf? Maybe. In any case, how many bombshells can an individual expect to set off in a lifetime?
(3) The scientists, philosophers (even theologians) of the 19th century did not rush to sign on to the non-Euclidean bandwagon. I wish that Gray had told us more about this resistance movement,† but I did learn from him that Gottlob Frege (1848-1925), the famous logician and philosopher, expressed contempt for non-Euclidean geometry:
"No one can serve two masters. . . . If Euclidean geometry is true, non-Euclidean geometry is false, and if non-Euclidean
geometry is true, then Euclidean geometry is false."
Or, as Pistol asks in Henry IV, Part II, "Under which king, Bezonian? Speak or die!" Frege spoke up and declared himself as a Euclidean, whereas today we all carry multiple mathematical allegiances.
I have Gray's book to thank for the following flight of my imagination. Defining a mathematical bombshell as a development that seriously alters the philosophy of mathematics, I can think---as a rather stingy assessment---of only four such in the past four centuries: Newton's Principia (1687), Bolyai-Lobachevsky's non-Euclidean geometry (1832-1840), Cantor's set theory (1874), and Gödel's Incompleteness Theorem (1930). Figuring a bombshell interval that averages, say, 100 years, the next one should be due in about 2030. What will it consist of? Who knows? I don't think that the solution of any of the current hard unsolved mathematical problems will lead to an overturning of philosophical notions now strongly held. I have read intimations that mathematics will shortly be totally mechanized. This should rumble the philosophers. Maybe.
Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island, and can be reached at firstname.lastname@example.org.
*"Perhaps I can express more fully in verse ideas and emotions which run counter to the inert crystallized opinion---hard as a rock---which the vast body of men have vested interest in supporting."---Thomas Hardy, Notebooks, 1896.
† Take a look, for example, at J.L. Richards, Mathematical Visions: The Pursuit of Geometry in Victorian England, Academic Press, 1988.