Is God a Mathematician_ - Mario Livio [73]
Let me pause here for a moment to allow for the meaning of the word “choosing” to sink in. For millennia, Euclidean geometry had been regarded as unique and inevitable—the sole true description of space. The fact that one could choose the axioms and obtain an equally valid description turned the entire concept on its ear. The certain, carefully constructed deductive scheme suddenly became more similar to a game, in which the axioms simply played the role of the rules. You could change the axioms and play a different game. The impact of this realization on the understanding of the nature of mathematics cannot be overemphasized.
Figure 40
Quite a few creative mathematicians prepared the ground for the final assault on Euclidean geometry. Particularly notable among them were the Jesuit priest Girolamo Saccheri (1667–1733), who investigated the consequences of replacing the fifth postulate by a different statement, and the German mathematicians Georg Klügel (1739–1812) and Johann Heinrich Lambert (1728–1777), who were the first to realize that alternative geometries to the Euclidean could exist. Still, somebody had to put the last nail in the coffin of the idea of Euclidean geometry being the one and only representation of space. That honor was shared by three mathematicians, one from Russia, one from Hungary, and one from Germany.
Strange New Worlds
The first to publish an entire treatise on a new type of geometry—one that could be constructed on a surface shaped like a curved saddle (figure 41a)—was the Russian Nikolai Ivanovich Lobachevsky (1792–1856; figure 42). In this kind of geometry (now known as hyperbolic geometry), Euclid’s fifth postulate is replaced by the statement that given a line in a plane and a point not on this line, there are at least two lines through the point parallel to the given line. Another important difference between Lobachevskian geometry and Euclidean geometry is that while in the latter the angles in a triangle always add up to 180 degrees (figure 41b), in the former the sum is always less than 180 degrees. Because Lobachevsky’s work appeared in the rather obscure Kazan Bulletin, it went almost entirely unnoticed until French and German translations started to appear in the late 1830s. Unaware of Lobachevsky’s work, a young Hungarian mathematician, János Bolyai (1802–60), formulated a similar geometry during the 1820s. Bursting with youthful enthusiasm, he wrote in 1823 to his father (the mathematician Farkas Bolyai; figure 43): “I have found things so magnificent that I was astounded…I have created a different new world out of nothing.” By 1825, János was already able to present to the elder Bolyai the first draft of his new geometry. The manuscript was entitled The Science Absolute of Space. In spite of the young man’s exuberance, the father was not entirely convinced of the soundness of János’s ideas. Nevertheless, he decided to publish the new geometry as an appendix to his own two-volume treatise on the foundations of geometry, algebra, and analysis (the supposedly inviting title of which read Essay on the Elements of Mathematics for Studious Youths). A copy of the book was sent in June 1831 to Farkas’s friend Carl Friedrich Gauss (1777–1855; figure 44), who was not only the most prominent mathematician of the time, but who is also considered by many, along with Archimedes and Newton, to be one of the three greatest of all time. That book was somehow lost in the chaos created by a cholera epidemic, and Farkas had to send a second copy. Gauss sent out a reply on March 6, 1832,