Is God a Mathematician_ - Mario Livio [18]
Plato’s views formed the basis for what has become known in philosophy in general, and in discussions of the nature of mathematics in particular, as Platonism. Platonism in its broadest sense espouses a belief in some abstract eternal and immutable realities that are entirely independent of the transient world perceived by our senses. According to Platonism, the real existence of mathematical objects is as much an objective fact as is the existence of the universe itself. Not only do the natural numbers, circles, and squares exist, but so do imaginary numbers, functions, fractals, non-Euclidean geometries, and infinite sets, as well as a variety of theorems about these entities. In short, every mathematical concept or “objectively true” statement (to be defined later) ever formulated or imagined, and an infinity of concepts and statements not yet discovered, are absolute entities, or universals, that can neither be created nor destroyed. They exist independently of our knowledge of them. Needless to say, these objects are not physical—they live in an autonomous world of timeless essences. Platonism views mathematicians as explorers of foreign lands; they can only discover mathematical truths, not invent them. In the same way that America was already there long before Columbus (or Leif Ericson) discovered it, mathematical theorems existed in the Platonic world before the Babylonians ever initiated mathematical studies. To Plato, the only things that truly and wholly exist are those abstract forms and ideas of mathematics, since only in mathematics, he maintained, could we gain absolutely certain and objective knowledge. Consequently, in Plato’s mind, mathematics becomes closely associated with the divine. In the dialogue Timaeus, the creator god uses mathematics to fashion the world, and in The Republic, knowledge of mathematics is taken to be a crucial step on the pathway to knowing the divine forms. Plato does not use mathematics for the formulation of some laws of nature that are testable by experiments. Rather, for him, the mathematical character of the world is simply a consequence of the fact that “God always geometrizes.”
Plato extended his ideas on “true forms” to other disciplines as well, in particular to astronomy. He argued that in true astronomy “we must leave the heavens alone” and not attempt to account for the arrangements and the apparent motions of the visible stars. Instead, Plato regarded true astronomy as a science dealing with the laws of motion in some ideal, mathematical world, for which the observable heaven is a mere illustration (in the same way that geometrical figures drawn on papyrus only illustrate the true figures).
Plato’s suggestions for astronomical research are considered controversial even by some of the most devout Platonists. Defenders of his ideas argue that what Plato really means is not that true astronomy should concern itself with some ideal heaven that has nothing to do with the observable one, but that it should deal with the real motions of celestial bodies as opposed to the apparent motions as seen from Earth. Others point out, however, that too literal an adoption of Plato’s dictum would have seriously impeded the development of observational astronomy as a science. Be the interpretation of Plato’s attitude toward astronomy as it may, Platonism has become one of the leading dogmas when it comes to the foundations of mathematics.
But does the Platonic world of mathematics really exist?