Three Scientific Revolutions: How They Transformed Our Conceptions of Reality Read Free Page A

“arithmogeometric units”—e.g., an extended line drawn from two points, plane figures such as triangles and rectangles from several lines, a circle from a joined curved line, and three-dimensional spatial objects such as pyramids cubes, spheres, and complex polyhedra from plane figures. As Aristotle states, based on these inquires “the Pythagoreans . . . construct the whole universe out of numbers—only not numbers consisting of abstract units: they suppose the units to have spatial magnitude.” 4
    Thus the Pythagoreans were able to represent the four elements of the physical world—earth, air, fire, and water—by four polyhedra: the earth by the 4-sided pyramid or tetrahedron, air by the 6-sided cube, fire by the 8-sided octahedron, water by the 20-sided icosahedron, and the universe itself by the 12-sided dodecahedron. Because Plato apparently assigned different polyhedra to the four elements, explaining their disintegration and reconfiguration as due to the separation and recombination of their constituent plane figures, they came to be known as “the five Platonic solids.” Kepler in the early seventeenth century began his astronomical theorizing in his Mysterium Cosmographicum (The Cosmographic Mystery) with the five polyhedra of Pythagoras perhaps as revised by Plato. Other of their astronomical contributions also were extremely important, such as Eudoxus of Cnidus who made the determination of the solar year to be 365 days and five hours, along with originating the long-prevailing view that the celestial bodies revolve on a series of concentric spheres with the earth in the center.
    His pupil Callippus of Cyzicus increased his number of spheres to thirty-four to account for certain astronomical irregularities that were adopted by Aristotle. But Philolaus of Croton, in 259 BCE, astutely assigned “an oblique circular motion” to the earth around a central fire while Heraclides of Pontus and Ecphantos of Syracuse attributed to it an axial rotation from west to east to explain the apparent rising and setting of the sun, along with determining that Mercury and Venus revolve around the sun. This culminated in Aristarchus of Samos’s prescient sun-centered astronomical theory in the third century BCE, though eclipsed by Ptolemy’s geocentrism until Copernicus’s adoption of heliocentrism.
    These celestial innovations were complemented by such empirical theories as Empedocles’ conception of the four elements, earth, air, fire, and water, as basic; Anaxagoras’ rejection of Empedocles’ four elements as too limited, declaring that the original mixture consisted of an infinite number of infinitely divisible particles that were representative of all the diversity of things, but too minute to be discernable except for air and aither; Leucippus’ and Democritus’ astute atomic theory that the underlying matter of the universe consisted of solid, indivisible, insensible particles that varied in their size, shapes, solidity, and motions, excluding sensory qualities. 5
    However, deriding such empirical explanations Plato, in his famous “allegory of the cave,” described sensory knowledge as mere reflections of the imperfect material objects in the physical world or “Receptacle,” declaring that mathematics could free one from these perceptual illusions to ascend to the intelligible world of perfect archetypes, the “Realm of Forms,” culminating in the “Form of the Good” and the “Demiurge.” Apparently the latter was the creator of the real world by imposing the ideal archetypes on the imperfect Receptacle. 6 It was Plato’s philosophy that was the most influential during the medieval period because of its easy conformity with Christianity, interpreting his Demiurge as God.
    Yet it was not Plato’s philosophy but that of his pupil Aristotle that would prove the most dominant from the thirteenth to