The notion of infinity had long since been introduced into mathematics, but this infinity was what philosophers call a becoming. Mathematical infinity was only a quantity susceptible of growing beyond all limit; it was a variable quantity of which it could not be said that it had passed, but only that it would pass, all limits.

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.

The notion of infinity had long since been introduced into mathematics, but this infinity was what philosophers call a becoming; Mathematical infinity was only a quantity susceptible of growing beyond all limit; it was a variable quantity of which it could not be said that it had passed, but only that it would pass, all limits.

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.

After these came to be relished, an infinite scale of infinites and s (ascending and descending always by infinite steps) was imagined and proposed to be received into geometry, as of the greatest use for penetrating into its abstruse parts. Some have argued for quantities more than infinite; and others for a kind of quantities that are said to be neither finite nor infinite, but of an intermediate and indeterminate nature.

The actual infinite arises in three contexts: first when it is realized in the most complete form, in a fully independent otherworldly being, in Deo, where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto as a mathematical magnitude, number or order type.

[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.

I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking

The Infinite has to be a relative concept. Go any distance: an infinite space means that there is more to be explored.

[E]ven in the most precise part of science, in mathematics, we cannot avoid using concepts that involve contradictions. ...[I]t is well known that the concept of infinity leads to contradictions that have been analyzed, but it would be practically impossible to construct the main parts of mathematics without this concept.

We know that there is an infinite, and we know not its nature. As we know it to be false that numbers are finite, it is therefore true that there is a numerical infinity. But we know not of what kind; it is untrue that it is even, untrue that it is odd; for the addition of a unit does not change its nature; yet it is a number, and every number is odd or even (this certainly holds of every finite number). Thus we may quite well know that there is a God without knowing what He is.

[E]ven in the most precise part of science, in mathematics, we cannot avoid using concepts that involve contradictions. ...[I]t is well known that the concept of infinity leads to contradictions... but it would be practically impossible to construct... mathematics without this concept.

Aristotle says the infinite is imperfect, unfinished, and therefore unthinkable; it is formless and confused. Only as objects are delimited and distinct do they have a nature.

As one aspect of perspective and printing, mathematical or numerical infinity serves as an instance of how our various physical extensions or media act upon one another through the agency of our own senses. It is in this mode that man appears as the reproductive organ of the technological world, a fact that Samuel Butler bizarrely announced in Erewhon.