Is God a Mathematician_ - Mario Livio [76]
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.
On Space, Numbers, and Humans
Before mathematicians could turn to the overarching topic of the foundations of mathematics, however, a few “smaller” issues required immediate attention. First, the fact that non-Euclidean geometries had been formulated and published did not necessarily mean that these were legitimate offspring of mathematics. There was the ever-present fear of inconsistency—the possibility that carrying these geometries to their ultimate logical consequences would produce unresolvable contradictions. By the 1870s, the Italian Eugenio Beltrami (1835–1900) and the German Felix Klein (1849–1925) had demonstrated that as long as Euclidean geometry was consistent, so were non-Euclidean geometries. This still left open the bigger question of the solidity of the foundations of Euclidean geometry. Then there was the important matter of relevance. Most mathematicians regarded the new geometries as amusing curiosities at best. Whereas Euclidean geometry derived much of its historical power from being seen as the description of real space, the non-Euclidean geometries had been perceived initially as not having any connection whatsoever to physical reality. Consequently, the non-Euclidean geometries were treated by many mathematicians as Euclidean geometry’s poor cousins. Henri Poincaré was a bit more accommodating than most, but even he insisted that if humans were to be transported to a world in which the accepted geometry was non-Euclidean, then it was still “certain that we should not find it more convenient to make a change” from Euclidean to non-Euclidean geometry. Two questions therefore loomed large: (1) Could geometry (in particular) and other branches of mathematics (in general) be established on solid axiomatic logical foundations? and (2) What was the relationship, if any, between mathematics and the physical world?
Some mathematicians adopted a pragmatic approach with respect to the validation of the foundations of geometry. Disappointed by the realization that what they regarded as absolute truths turned out to be more experience-based than rigorous, they turned to arithmetic—the mathematics of numbers. Descartes’ analytic geometry, in which points in the plane were identified with ordered pairs of numbers, circles with pairs satisfying a certain equation (see chapter 4), and so on, provided just the necessary tools for the re-erection of the foundations of geometry on the basis of numbers. The German mathematician Jacob Jacobi (1804–51) presumably expressed those shifting tides when he replaced Plato’s “God ever geometrizes” by his own motto: “God ever arithmetizes.” In some sense, however, these efforts only transported the problem to a different branch of mathematics. While the great German mathematician David Hilbert (1862–1943) did succeed in demonstrating that Euclidean geometry was