Is God a Mathematician_ - Mario Livio [77]
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