Anaxagoras of Clazomenae (c. 500–c. 428 bc), a pluralist, believed that because nothing can really come into being, everything must be contained in everything, but in the form of infinitely small parts. In the beginning, all of these particles had existed in an even mixture, in which nothing could be distinguished, much like the indefinite apeiron of Anaximander. But then nous, or intelligence, began at one point to set these particles into a whirling motion, foreseeing that in this way they would become separated from one another and then recombine in the most various ways so as to produce gradually the world in which human beings live. In contrast to the forces assumed by Empedocles, the nous of Anaxagoras is not blind but foresees and intends the production of the cosmos, including living and intelligent beings; however, it does not interfere with the process after having started the whirling motion. This is a strange combination of a mechanical and a nonmechanical explanation of the world.
By far of greatest importance for the later development of philosophy and physical science was an attempt by the atomists Leucippus (flourished 5th century bc) and Democritus (c. 460–c. 370 bc) to solve the Parmenidean problem. Leucippus found the solution in the assumption that, contrary to Parmenides’ argument, the nothing does in a way exist—as empty space. There are, then, two fundamental principles of the physical world, empty space and filled space—the latter consisting of atoms that, in contrast to those of modern physics, are real atoms; that is, they are absolutely indivisible because nothing can penetrate to split them. On these foundations, laid by Leucippus, Democritus appears to have built a whole system, aiming at a complete explanation of the varied phenomena of the visible world by means of an analysis of its atomic structure. This system begins with elementary physical problems, such as why a hard body can be lighter than a softer one. The explanation is that the heavier body contains more atoms, which are equally distributed and of round shape; the lighter body, however, has fewer atoms, most of which have hooks by which they form rigid gratings. The system ends with educational and ethical questions. A sound and cheerful person, useful to his fellows, is literally well composed. Although destructive passions involve violent, long-distance atomic motions, education can help to contain them, creating a better composure. Democritus also developed a theory of the evolution of culture, which influenced later thinkers. Civilization, he thought, is produced by the needs of life, which compel human beings to work and to make inventions. When life becomes too easy because all needs are met, there is a danger that civilization will decay as people become unruly and negligent. Epistemology of appearance
All of the post-Parmenidean philosophers, like Parmenides himself, presupposed that the real world is different from the one that human beings perceive. Thus arose the problems of epistemology, or theory of knowledge. According to Anaxagoras, everything is contained in everything. But this is not what people perceive. He solved this problem by postulating that, if there is a much greater amount of one kind of particle in a thing than of all other kinds, the latter are not perceived at all. The observation was then made that sometimes different persons or kinds of animals have different perceptions of the same things. He explained this phenomenon by assuming that like is perceived by like. If, therefore, in the sense organ of one person there is less of one kind of stuff than of another, that person will perceive the former less keenly than the latter. This reasoning was also used to explain why some animals see better at night and others during the day. According to Democritus, atoms have no sensible qualities, such as taste, smell, or colour, at all. Thus, he tried to reduce all of them to tactile qualities (explaining a bright white colour, for instance, as sharp atoms hitting the eye like needles), and he made a most elaborate attempt to reconstruct the atomic structure of things on the basis of their apparent sensible qualities.
Also of very great importance in the history of epistemology was Zeno of Elea (c. 495–c. 430 bc), a younger friend of Parmenides. Parmenides had, of course, been severely criticized because of the strange consequences of his doctrine: that in reality there is no motion and no plurality because there is just one solid being. To support him, however, Zeno tried to show that the assumption that there is motion and plurality leads to consequences that are no less strange. This he did by means of his famous paradoxes, saying that the flying arrow rests since it can neither move in the place in which it is nor in a place in which it is not, and that Achilles cannot outrun a turtle because, when he has reached its starting point, the turtle will have moved to a further point, and so on ad infinitum—that, in fact, he cannot even start running, for, before traversing the stretch to the starting point of the turtle, he will have to traverse half of it, and again half of that, and so on ad infinitum. All of these paradoxes are derived from the problem of the continuum. Although they have often been dismissed as logical nonsense, many attempts have also been made to dispose of them by means of mathematical theorems, such as the theory of convergent series or the theory of sets. In the end, however, the logical difficulties raised in Zeno’s arguments have always come back with a vengeance, for the human mind is so constructed that it can look at a continuum in two ways that are not quite reconcilable. Metaphysics of number
All of the philosophies mentioned so far are in various ways historically akin to one another. Toward the end of the 6th century bc, however, there arose, quite independently, another kind of philosophy, which only later entered into interrelation with the developments just mentioned: the philosophy of Pythagoras of Samos (c. 580–c. 500 bc; see also Pythagoreanism). Pythagoras traveled extensively in the Middle East and in Egypt and, after his return to Samos, emigrated to southern Italy because of his dislike of the tyranny of Polycrates (c. 535–522 bc). At Croton and Metapontum he founded a philosophical society with strict rules and soon gained considerable political influence. He appears to have brought his doctrine of the transmigration of souls from the Middle East. Much more important for the history of philosophy and science, however, was his doctrine that “all things are numbers,” which means that the essence and structure of all things can be determined by finding the numerical relations they express. Originally, this, too, was a very broad generalization made on the basis of comparatively few observations: for instance, that the same harmonies can be produced with different instruments—strings, pipes, disks, etc.—by means of the same numerical ratios—1:2, 2:3, 3:4—in one-dimensional extensions; the observation that certain regularities exist in the movements of the celestial bodies; and the discovery that the form of a triangle is determined by the ratio of the lengths of its sides. But because the followers of Pythagoras tried to apply their principle everywhere with the greatest of accuracy, one of them—Hippasus of Metapontum (flourished 5th century bc)—made one of the most fundamental discoveries in the entire history of science: that the side and diagonal of simple figures such as the square and the regular pentagon are incommensurable (i.e., their quantitative relation cannot be expressed as a ratio of integers). At first sight this discovery seemed to destroy the very basis of the Pythagorean philosophy, and the school thus split into two sects, one of which engaged in rather abstruse numerical speculations, while the other succeeded in overcoming the difficulty by ingenious mathematical inventions. Pythagorean philosophy also exerted a great influence on the later development of Plato’s thought.