Is God a Mathematician? by Mario Livio Read Free Book Online Page B

self-evident. For instance, the first axiom in Euclidean geometry is “Between any two points a straight line may be drawn.” In The Republic, Plato beautifully combines the concept of postulates with his notion of the world of mathematical forms:

    Figure 9
    I think you know that those who occupy themselves with geometries and calculations and the like, take for granted the odd and the even [numbers], figures, three kinds of angles, and other things cognate to these in each subject; assuming these things as known, they take them as hypotheses and thenceforward they do not feel called upon to give any explanation with regard to them either to themselves or anyone else, but treat them as manifest to every one; basing themselves on these hypotheses, they proceed at once to go through the rest of the argument till they arrive, with general assent, at the particular conclusion to which their inquiry was directed. Further you know that they make use of visible figures and argue about them, but in doing so they are not thinking about these figures but of the things which they represent; thus it is the absolute square and the absolute diameter which is the object of their argument, not the diameter which they draw…the object of the inquirer being to see their absolute counterparts which cannot be seen otherwise than by thought [emphasis added].
    Plato’s views formed the basis for what has become known in philosophy in general, and in discussions of the nature of mathematicsin 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 observableheaven is a mere illustration (in the same