“All those… who discourse concerning nature, always subject a certain other nature of… elements, to the infinite… But no one of those who make the elements to be finite introduces infinity. Such, however, as make infinite elements, as Anaxagoras and Democritus, say that the infinite is continuous by contact. …Rationally, too, do all philosophers consider the infinite as a principle; for it cannot be in vain, nor can any other power be present with it than that of a principle: for all things are either the principle, or from the principle; but of the infinite there is no principle, since otherwise it would have an end. …it is also unbegotten and uncorruptible, as being a certain principle: for… end is the corruption of everything. …It likewise appears to comprehend and govern all things, as those assert who do not introduce other causes beside the infinite… It would seem also that this is divine: for it is immortal and indestructible, as Anaximander says, and most of the physiologists.”

“Since the science of nature is conversant with magnitudes, motion, and time, each of which must necessarily be either infinite or finite…[we] should speculate the infinite, and consider whether it is or not; and if it is what it is. …[A]ll those who appear to have touched on a philosophy of this kind… consider it as a certain principle of beings. Some, indeed, as the Pythagoreans and Plato, consider it, per se, not as being an accident to any thing else, but as having an essential subsistence… the Pythagoreans… consider the infinite as subsisting in sensibles; for they do not make number to be separate; and they assert that what is beyond the heavens is infinite; but Plato says that beyond the heavens there is not any body, nor ideas, because these are no where: he affirms, however, that the infinite is both in sensibles, and in ideas. …Plato establishes two infinities, viz. the great and the small.”

“Since… nature is a principle of motion and mutation… it is necessary that we should not be ignorant of what motion is… But motion appears to belong to things continuous; and the infinite first presents itself to the view in that which is continuous. …[F]requently …those who define the continuous, employ the nature or the infinite, as if that which is divisible to infinity is continuous.”

“[B]ecause that which is finite is always bounded with reference to something… it is necessary that there should be no end… [N]umber also appears to be infinite, and mathematical magnitudes, and that which is beyond the heavens. And since that which is beyond is infinite, body also appears to be infinite, and it would seem that there are infinite worlds; for why is there rather void here than there? …If also there is a vacuum, and an infinite place, it is necessary that there should be an infinite body: for in things which have a perpetual subsistence, capacity differs nothing from being. The speculation of the infinite is, however, attended with doubt: for many impossibilities happen both to those who do not admit that it has a subsistence, and to those who do. …It is …especially the province of a natural philosopher to consider if there be a sensible infinite magnitude.”

“It is hard to be finite upon an infinite subject, and all subjects are infinite.”

“But it is better to assume principles less in number and finite, as Empedocles makes them to be. All philosophers… make principles to be contraries… (for Parmenides makes principles to be hot and cold, and these he demominates fire and earth) as those who introduce as principles the rare and the dense. But Democritus makes the principles to be the solid and the void; of which the former, he says, has the relation of being, and the latter of non-being. …it is necessary that principles should be neither produced from each other, nor from other things; and that from these all things should be generated. But these requisites are inherent in the first contraries: for, because they are first, they are not from other things; and because they are contraries, they are not from each other.”

“Infinity, in its first form (the improper-infinite) presents itself as a variable finite [veranderliches Endliches]; in the other form (which I call the proper infinite [Eigentlich-unendliche]) it appears as a thoroughly determinate [bestimmtes] infinite.”

“[I]t is impossible that each of the elements should be infinite. For that is body which has interval on all sides; and that is infinite which has extension without bound.”

“It is only by intuition that the infinite can be apprehended. But why is this? Why cannot the infinite be apprehended by concepts? To see this we must understand that the word "infinite," in the religious sense, has nothing at all to do with that sense of the word in which it is applied to space, time, and the number series. We may call this latter the mathematical infinite to distinguish it from the religious infinite. And it is the confusion between these two which misled us into the false trail of supposing that the infinity of God's mind refers to the amount of His knowledge and that the finitude of man's mind refers to his ignorance. The religious infinite, or in other words the infinity of God, means that than 'which there is no other. In this sense neither space nor time could be infinite, since space is an "other" to time, and time is an "other" to space.”