Is God a Mathematician_ - Mario Livio [78]
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