Is God a Mathematician? (22 page)

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:
x
²
) to pioneering solutions to the cubic (involving
x
³
), and quartic (involving
x
⁴
) 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
x
²
was an area, and the third power
x
³
was a solid having the corresponding volume. Consequently, in the first chapter of the
Ars Magna,
Cardano explains:

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:

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:

The great mathematician Joseph Lagrange went even one step further, stating more assertively in 1797:

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 we 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.