by Mario Livio
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.
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 consistent as long as arithmetic was consistent, the consistency of the latter was far from unambiguously established at that point.
On the relationship between mathematics and the physical world, a new sentiment was in the air. For many centuries, the interpretation of mathematics as a reading of the cosmos had been dramatically and continuously enhanced. The mathematization of the sciences by Galileo, Descartes, Newton, the Bernoullis, Pascal, Lagrange, Quetelet, and others was taken as strong evidence for an underlying mathematical design in nature. One could clearly argue that if mathematics wasn’t the language of the cosmos, why did it work as well as it did in explaining things ranging from the basic laws of nature to human characteristics?
To be sure, mathematicians did realize that mathematics dealt only with rather abstract Platonic forms, but those were regarded as reasonable idealizations of the actual physical elements. In fact, the feeling that the book of nature was written in the language of mathematics was so deeply rooted that many mathematicians absolutely refused even to consider mathematical concepts and structures that were not directly related to the physical world. This was the case, for instance, with the colorful Gerolamo Cardano (1501–76). Cardano was an accomplished mathematician, renowned physician, and compulsive gambler. In 1545 he published one of the most influential books in the history of algebra—the Ars Magna (The Great Art). In this comprehensive treatise Cardano explored in great detail solutions to algebraic equations, from the simple quadratic equation (in which the unknown appears to the second power: x2) to pioneering solutions to the cubic (involving x3), and quartic (involving x4) equations. In classical mathematics, however, quantities were often interpreted as geometrical elements. For instance, the value of the unknown x was identified with a line segment of that length, the second power x2 was an area, and the third power x3 was a solid having the corresponding volume. Consequently, in the first chapter of the Ars Magna, Cardano explains:
We conclude our detailed consideration with the cubic, others being merely mentioned, even if generally, in passing. For as positio [the first power] refers to a line, quadratum [the square] to a surface, and cubum [the cube] to a solid body, it would be very foolish for us to go beyond this point. Nature does not permit it. Thus, it will be seen, all those matters up to and including the cubic are fully demonstrated, but the others which we will add, either by necessity or out of curiosity, we do not go beyond barely setting out.
In other words, Cardano argues that since the physical world as perceived by our senses contains only three dimensions, it would be silly for mathematicians to concern themselves with a higher number of dimensions, or with equations of a higher degree.
A similar opinion was expressed by the English mathematician John Wallis (1616–1703), from whose work Arithmetica Infinitorum Newton learned methods of analysis. In another important book, Treatise of Algebra, Wallis first proclaimed: “Nature, in propriety of Speech, doth not admit more than three (local) dimensions.” He then elaborated:
A Line drawn into a Line, shall make a Plane or Surface; this drawn into a Line, shall make a Solid. But if this Solid be drawn into a Line, or this Plane into a Plane, what shall it make? A Plano-Plane? This is a Monster in Nature, and less possible than a Chimera [a fire-breathing monster in Greek mythology, composed of a serpent, lion, and goat] or a Centaure [in Greek mythology, a being having the upper portion of a man and the body and legs of a horse]. For Length, Breadth and Thickness, take up the whole of Space. Nor can our Fansie imagine how there should be a Fourth Local Dimension beyond these Three.
Again, Wallis’s logic here was clear: There was no point in even imagining a geometry that did not describe real space.
Opinions eventually started to change. Mathematicians of the eighteenth century were the first to consider time as a potential fourth dimension. In an article entitled “Dimension,” published in 1754, the physicist Jean D’Alembert (1717–83) wrote:
I stated above that it is impossible to conceive of more than three dimensions. A man of parts, of my acquaintance, holds that one may however look upon duration as a fourth dimension, and that the product of time and solidity is in a way a product of four dimensions. This idea may be challenged but it seems to me to have some merit other than that of mere novelty.
The great mathematician Joseph Lagrange went even one step further, stating more assertively in 1797:
Since a position of a point in space depends upon three rectangular coordinates these coordinates in the problems of mechanics are conceived as being functions of t [time]. Thus we may regard mechanics as a geometry of four dimensions, and mechanical analysis as an extension of geometrical analysis.
These bold ideas opened the door for extensions of mathematics that had previously been considered inconceivable—geometries in any number of dimensions—which totally ignored the question of whether they had any relation to physical space.
Kant may have been wrong in believing that our senses of spatial perception follow exclusively Euclidean molds, but there is no question that our perception operates most naturally and intuitively in no more than three dimensions. We can relatively easily imagine how our three-dimensional world would look in Plato’s two-dimensional universe of shadows, but going beyond three to a higher number of dimensions truly requires a mathematician’s imagination.
Some of the groundbreaking work in the treatment of n-dimensional geometry—geometry in an arbitrary number of dimensions—was carried out by Hermann Günther Grassmann (1809–77). Grassmann, one of twelve children, and himself the father of eleven, was a school-teacher who never had any university mathematical training. During his lifetime, he received more recognition for his work in linguistics (in particular for his studies of Sanskrit and Gothic) than for his achievements in mathematics. One of his biographers wrote: “It seems to be Grassmann’s fate to be rediscovered from time to time, each time as if he had been virtually forgotten since his death.” Yet, Grassmann was responsible for the creation of an abstract science of “spaces,” inside which the usual geometry was only a special case. Grassmann published his pioneering ideas (originating a branch of mathematics known as linear algebra) in 1844, in a book commonly known as the Ausdehnungslehre (meaning Theory of Extension; the full title read: Linear Extension Theory: A New Branch of Mathematics).
In the foreword to the book Grassmann wrote: “Geometry can in no way be viewed…as a branch of mathematics; instead, geometry relates to something already given in nature, namely, space. I also had realized that there must be a branch of mathematics which yields in a purely abstract way laws similar to those of geometry.”
This was a radically new view of the nature of mathematics. To Grassmann, the traditional geometry—the heritage of the ancient Greeks—deals with physical space and therefore cannot be taken as a true branch of abstract mathematics. Mathematics to him was rather an abstract construct of the human brain that does not necessarily have any application to the real world.
It is fascinating to follow the seemingly trivial train of thought that set Grassmann on the road to his theory of geometric algebra. He started with the simple formula AB BC AC, which appears in any geometry book in the discussion of lengths of line segments (see figure 46a). Here, however, Grassmann noticed something interesting. He discovered that this formula remains valid irrespective of the order of the points A, B, C as long as one does not interpret AB, BC, and so on merely as lengths, but also assigns to them “direction,” such that BA AB. For instance, if C lies between A and B (as in Figure 46b), then AB AC CB, but since CB BC, we find that AB AC BC and the original formula AB BC AC is recovered simply by adding BC to both sides.
This was quite interesting in itself, but Grassmann’s extension contained even more surprises. Note that if w
e were dealing with algebra instead of geometry, then an expression such as AB usually would denote the product A B. In that case, Grassmann’s suggestion of BA AB violates one of the sacrosanct laws of arithmetic—that two quantities multiplied together produce the same result irrespective of the order in which the quantities are taken. Grassmann faced up squarely to this disturbing possibility and invented a new consistent algebra (known as exterior algebra) that allowed for several processes of multiplication and at the same time could handle geometry in any number of dimensions.
Figure 46
By the 1860s n-dimensional geometry was spreading like mushrooms after a rainstorm. Not only had Riemann’s seminal lecture established spaces of any curvature and of arbitrary numbers of dimensions as a fundamental area of research, but other mathematicians, such as Arthur Cayley and James Sylvester in England, and Ludwig Schläfli in Switzerland, were adding their own original contributions to the field. Mathematicians started to feel that they were being freed from the restrictions that for centuries had tied mathematics only to the concepts of space and number. Those ties had historically been taken so seriously that even as late as the eighteenth century, the prolific Swiss mathematician Leonhard Euler (1707–83) expressed his view that “mathematics, in general, is the science of quantity; or, the science that investigates the means of measuring quantity.” It was only in the nineteenth century that the winds of change started to blow.
First, the introduction of abstract geometric spaces and of the notion of infinity (in both geometry and the theory of sets) had blurred the meaning of “quantity” and of “measurement” beyond recognition. Second, the rapidly multiplying studies of mathematical abstractions helped to distance mathematics even further from physical reality, while breathing life and “existence” into the abstractions themselves.