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

to why he thought that this was the necessary training for the would-be politicians:
    What we require is that those who take office should not be lovers of rule. Otherwise there will be a contest with rival lovers. What others, then, will you compel to undertake the guardianship of the city than those who have most intelligence of the principles that are the means of good government and who possess distinctions of another kind and a life that is preferable to political life?
    Refreshing, isn’t it? In fact, while such a demanding program was probably impractical even in Plato’s time, George Washington agreed that an education in mathematics and philosophy was not a bad idea for the politicians-to-be:
    The science of figures, to a certain degree, is not only indispensably requisite in every walk of civilized life; but investigation of mathematical truths accustoms the mind to method and correctness in reasoning, and is an employment peculiarly worthy of rational being. In a clouded state of existence, where so many things appear precarious to the bewildered research, it is here that the rational faculties find foundation to rest upon. From the high ground of mathematical and philosophical demonstration, we are insensibly led to far nobler speculations and sublimer meditations.
    For the question of the nature of mathematics, even more important than Plato the mathematician or the math stimulator was Plato the philosopher of mathematics. There his trail-blazing ideas put him not only above all the mathematicians and philosophers of his generation, but identified him as an influential figure for the following millennia.
    Plato’s vision of what mathematics truly is makes strong reference to his famous Allegory of the Cave. There he emphasizes the doubtful validity of the information provided through the human senses. What we perceive as the real world, Plato says, is no more real than shadows projected onto the walls of a cavern. Here is the remarkable passage from The Republic:
    See human beings as though they were in an underground cave-like dwelling with an entrance, a long one, open to the light across the whole width of the cave. They are in it from childhood with their legs and necks in bonds so that they are fixed, seeing only in front of them, unable because of the bond to turn their heads all the way around. Their light is from a fire burning far above and behind them. Between the fire and the prisoners there is a road above, along which we see a wall, built like the partitions puppet-handlers set in front of the human beings and over which they show the puppets…Then also see along this wall human beings carrying all sorts of artifacts, which project above the wall, and statues of men and other animals wrought from stone, wood, and every kind of material…do you suppose such men would have seen anything of themselves and one another, other than the shadows cast by the fire on the side of the cave facing them?
    According to Plato, we, humans in general, are no different from those prisoners in the cave who mistake the shadows for reality. (Figure 9 shows an engraving by Jan Saenredam from 1604 illustrating the allegory.) In particular, Plato stresses, mathematical truths refer not to circles, triangles, and squares that can be drawn on a piece of papyrus, or marked with a stick in the sand, but to abstract objects that dwell in an ideal world that is the home of true forms and perfections. This Platonic world of mathematical forms is distinct from the physical world, and it is in this first world that mathematical propositions, such as the Pythagorean theorem, hold true. The right triangle we might draw on paper is but an imperfect copy—an approximation—of the true, abstract triangle.
    Another fundamental issue that Plato examined in some detail concerned the nature of mathematical proof as a process that is based on postulates and axioms . Axioms are basic assertions whose validity is assumed to be