Is God a Mathematician? (21 page)

Figure 41

Figure 42

Figure 43

Let me parenthetically note that apparently Gauss feared that the radically new geometry would be regarded by the Kantian philosophers, to whom he referred as “the Boetians” (synonymous with “stupid” for the ancient Greeks), as philosophical heresy. Gauss then continued:

Figure 44

While Farkas was quite pleased with Gauss’s praise, which he took to be “very fine,” János was absolutely devastated. For almost a decade he refused to believe that Gauss’s claim to priority was not false, and his relationship with his father (whom he suspected of prematurely communicating the results to Gauss) was seriously strained. When he finally realized that Gauss had actually started working on the problem as early as 1799, János became deeply embittered, and his subsequent mathematical work (he left some twenty thousand pages of manuscript when he died) became rather lackluster by comparison.

There is very little doubt, however, that Gauss had indeed given considerable thought to non-Euclidean geometry. In a diary entry from September 1799 he wrote: “
In principiis geometriae egregios progressus fecimus
” (“About the principles of geometry we obtained wonderful achievements”). Then, in 1813, he noted: “In the theory
of parallel lines we are now no further than Euclid was. This is the
partie honteuse
[shameful part] of mathematics, which sooner or later must get a very different form.” A few years later, in a letter written on April 28, 1817, he stated: “I am coming more and more to the conviction that the necessity of our [Euclidean] geometry cannot be proved.” Finally, and contrary to Kant’s views, Gauss concluded that Euclidean geometry could not be viewed as a universal truth, and that rather “one would have to rank [Euclidean] geometry not with arithmetic, which stands
a priori,
but approximately with mechanics.” Similar conclusions were reached independently by Ferdinand Schweikart (1780–1859), a professor of jurisprudence, and the latter informed Gauss of his work sometime in 1818 or 1819. Since neither Gauss nor Schweikart actually published his results, however, the priority of first publication is traditionally credited to Lobachevsky and Bolyai, even though the two can hardly be regarded as the sole “creators” of non-Euclidean geometry.

Hyperbolic geometry broke on the world of mathematics like a thunderbolt, dealing a tremendous blow to the perception of Euclidean geometry as the only, infallible description of space. Prior to the Gauss-Lobachevsky-Bolyai work, Euclidean geometry
was,
in effect, the natural world. The fact that one could select a different set of axioms and construct a different type of geometry raised for the first time the suspicion that mathematics is, after all, a human invention, rather than a discovery of truths that exist independently of the human mind. At the same time, the collapse of the immediate connection between Euclidean geometry and true physical space exposed what appeared to be fatal deficiencies in the idea of mathematics as the language of the universe.

Euclidean geometry’s privileged status went from bad to worse when one of Gauss’s students, Bernhard Riemann, showed that hyperbolic geometry was not the only non-Euclidean geometry possible. In a brilliant lecture delivered in Göttingen on June 10, 1854 (figure 45 shows the first page of the published lecture), Riemann presented his views “On the Hypotheses That Lie at the Foundations of Geometry.” He started by saying that “geometry presupposes the concept of space, as well as assuming the basic principles
for constructions in space. It gives only nominal definitions of these things, while their essential specifications appear in the form of axioms.” However, he noted, “The relationship between these presuppositions is left in the dark; we do not see whether, or to what extent, any connection between them is necessary, or
a priori
whether any connection between them is even possible.” Among the possible geometrical theories Riemann discussed
elliptic geometry
, of the type that one would encounter on the surface of a sphere (figure 41c). Note that in such a geometry the shortest distance between two points is not a straight line, it is rather a segment of a great circle, whose center coincides with the center of the sphere. Airlines take advantage of this fact—flights from the United States to Europe do not follow what would appear as a straight line on the map, but rather a
great circle that initially bears northward. You can easily check that any two great circles meet at two diametrically opposite points. For instance, two meridians on Earth, which appear to be parallel at the Equator, meet at the two poles. Consequently, unlike in Euclidean geometry, where there is exactly one parallel line through an external point, and hyperbolic geometry, in which there are at least two parallels, there are
no
parallel lines at all in the elliptic geometry on a sphere. Riemann took the non-Euclidean concepts one step further and introduced geometries in curved spaces in three, four, and even more dimensions. One of the key concepts expanded upon by Riemann was that of the
curvature
—the rate at which a curve or a surface curves. For instance, the surface of an eggshell curves more gently around its girth than along a curve passing through one of its pointy edges. Riemann proceeded to give a precise mathematical definition of curvature in spaces of any number of dimensions. In doing so he solidified the marriage between algebra and geometry that had been initiated by Descartes. In Riemann’s work equations in any number of variables found their geometrical counterparts, and new concepts from the advanced geometries became partners of equations.

Figure 45

Euclidean geometry’s eminence was not the only victim of the new horizons that the nineteenth century opened for geometry. Kant’s ideas of space did not survive much longer. Recall that Kant asserted that information from our senses is organized exclusively along Euclidean templates before it is recorded in our consciousness. Geometers of the nineteenth century quickly developed intuition in the non-Euclidean geometries and learned to experience the world along those lines. The Euclidean perception of space turned out to be learned after all, rather than intuitive. All of these dramatic developments led the great French mathematician Henri Poincaré (1854–1912) to conclude that the axioms of geometry are “neither synthetic
a priori
intuitions nor experimental facts. They are
conventions
[emphasis added]. Our choice among all possible conventions is guided by experimental facts, but it remains free.” In other words, Poincaré regarded the axioms only as “definitions in disguise.”

Poincaré’s views were inspired not just by the non-Euclidean geometries described so far, but also by the proliferation of other new
geometries, which before the end of the nineteenth century seemed to be almost getting out of hand. In
projective geometry
(such as that obtained when an image on celluloid film is projected onto a screen), for instance, one could literally interchange the roles of points and lines, so that theorems about points and lines (in this order) became theorems about lines and points. In
differential geometry,
mathematicians used calculus to study the local geometrical properties of various mathematical spaces, such as the surface of a sphere or a torus. These and other geometries appeared, at first blush at least, to be ingenious inventions of imaginative mathematical minds, rather than accurate descriptions of physical space. How then could one still defend the concept of God as a mathematician? After all, if “God ever geometrizes” (a phrase attributed to Plato by the historian Plutarch), which of these many geometries does the divine practice?

The rapidly deepening recognition of the shortcomings of the classical Euclidean geometry forced mathematicians to take a serious look at the foundations of mathematics in general, and at the relationship between mathematics and logic in particular. We shall return to this important topic in chapter 7. Here let me only note that the very notion of the self-evidency of axioms had been shattered. Consequently, while the nineteenth century witnessed other significant developments in algebra and in analysis, the revolution in geometry probably had the most influential effects on the views of the nature of mathematics.