Is God a Mathematician_ - Mario Livio [11]
And where do the heavens fit into all of this? Pythagoras and the Pythagoreans played a role in the history of astronomy that, while not critical, was not negligible either. They were among the first to maintain that the Earth was spherical in form (probably because of the perceived mathematico-aesthetic superiority of the sphere). They were also probably the first to state that the planets, the Sun, and the Moon have an independent motion of their own from west to east, in a direction opposite to the daily (apparent) rotation of the sphere of the fixed stars. These enthusiastic observers of the midnight sky could not have missed the most obvious properties of the stellar constellations—shape and number. Each constellation is recognized by the number of stars that compose it and by the geometrical figure that these stars form. But these two characteristics were precisely the essential ingredients of the Pythagorean doctrine of numbers, as exemplified by the Tetraktys. The Pythagoreans were so enraptured by the dependency of geometrical figures, stellar constellations, and musical harmonies on numbers that numbers became both the building blocks from which the universe was constructed and the principles behind its existence. No wonder then that Pythagoras’s maxim was stated emphatically as “All things accord in number.”
We can find a testament to how seriously the Pythagoreans took this maxim in two of Aristotle’s remarks. In one place in his collected treatise Metaphysics he says: “The so-called Pythagoreans applied themselves to mathematics, and were the first to develop this science; and through studying it they came to believe that its principles are the principles of everything.” In another passage, Aristotle vividly describes the veneration of numbers and the special role of the Tetraktys: “Eurytus [a pupil of the Pythagorean Philolaus] settled what is the number of what object (e.g., this is the number of a man, that of a horse) and imitated the shapes of living things by pebbles after the manner of those who bring numbers into the form of triangle or square.” The last sentence (“the form of triangle or square”) alludes both to the Tetraktys and to yet another fascinating Pythagorean construction—the gnomon.
The word “gnomon” (a “marker”) originates from the name of a Babylonian astronomical time-measurement device, similar to a sundial. This apparatus was apparently introduced into Greece by Pythagoras’s teacher—the natural philosopher Anaximander (ca. 611–547 BC). There can be no doubt that the pupil was influenced by his tutor’s ideas in geometry and their application to cosmology—the study of the universe as a whole. Later, the term “gnomon” was used for an instrument for drawing right angles, similar to a carpenter’s square, or for the right-angled figure that, when added to a square, makes up a larger square (as in figure 2). Note that if you add, say, to a 3 × 3 square, seven pebbles in a shape that forms a right angle (a gnomon), you obtain a square composed of sixteen (4 × 4) pebbles. This is a figurative representation of the following property: In the sequence of odd integers 1, 3, 5, 7, 9,…, the sum of any number of successive members (starting from 1) always forms a square number. For instance, 1 12; 1 3 4 22; 1 3 5 9 32; 1 3 5 7 16 42; 1 3 5 7 9 25 52, and so on. The Pythagoreans regarded this intimate relation between the gnomon and the square that it “embraces” as a symbol of knowledge in general, where the knowing is “hugging” the known. Numbers were therefore not limited to a description of the physical world, but were supposed