There are two kinds of reality in the Pythagorean teaching: (1) numbers, and (2) unlimited space. The essential nature of things, the Being that abides, consists in the shaping of this unlimited space into mathematical forms. The numbers or the forms are the limited aspect of Being; space is the unlimited aspect of Being. Actual Being consists in the union of the two aspects. Being therefore has two roots, each being necessary to the other. The later Pythagoreans, indeed, called attention to the fact that their numbers were not the same as the different kinds of matter out of which the other Cosmologists conceived the world to be fashioned. Numbers are not the stuff out of which the world of nature-objects have arisen, but rather are forms of nature-objects. Numbers are the patterns or models of things; things are the copies or imitations of numbers. Unlimited space furnishes the material; numbers or mathematical forms furnish the mould; the result is a material thing. Here we find the early basis of Plato’s doctrine of Ideas, and the correlation in Aristotle of Form and matter. If we were to draw an analogy between the Pythagorean conception of numbers and any part of the preceding cosmological teaching, we should find the similarity between the numbers and the earlier efficient causes and not between the numbers and the elements. For example, Pythagorean numbers have a function more nearly like Love and Hate than like the four elements in Empedocles’ teaching. On the other hand, Pythagorean unlimited space is analogous to the Empedoclean elements. =2. The Pythagorean Dualistic World.=[13] The Pythagoreans carried out their conception of this twofold reality both in their mathematical studies and in their conceptions of natural objects. It was from such investigations that they were impressed by the dualism in everything and so reached their principle. They observed in mathematics that the number-series consists of alternate odd and even numbers. The odd numbers are limited and the even unlimited (because they could be divided). They explained the elements as determined by mathematical forms: fire has the form of a tetrahedron; earth, of the cube; air, of the octohedron; water, of the icosahedron; and an additional fifth element, the æther, of the dodecahedron. They carried this dualism further by identifying the limited form with the odd, with the perfect, and with the good; while the unlimited was identified with the even, the imperfect, and the bad. Some of the Pythagoreans even sought to trace out this dualism in the many realms of experience, and they originated a table of ten pairs of opposites: limited and unlimited; odd and even; one and many; right and left; male and female; rest and motion; straight and crooked; light and dark; good and bad; square and oblong.