Objection 1. It would seem that our intellect can know the infinite. For God excels all infinite things. But our intellect can know God, as we have said above (Question 12, Article 1). Much more, therefore, can our intellect know all other infinite things.
Objection 2. Further, our intellect can naturally know "genera" and "species." But there is an infinity of species in some genera, as in number, proportion, and figure. Therefore our intellect can know the infinite.
Objection 3. Further, if one body can coexist with another in the same place, there is nothing to prevent an infinite number of bodies being in one place. But one intelligible species can exist with another in the same intellect, for many things can be habitually known at the same time. Therefore our intellect can have an habitual knowledge of an infinite number of things.
Objection 4. Further, as the intellect is not a corporeal faculty, as we have said (76, 1), it appears to be an infinite power. But an infinite power has a capacity for an infinite object. Therefore our intellect can know the infinite.
On the contrary, It is said (Phys. i, 4) that "the infinite, considered as such, is unknown."
I answer that, Since a faculty and its object are proportional to each other, the intellect must be related to the infinite, as is its object, which is the quiddity of a material thing. Now in material things the infinite does not exist actually, but only potentially, in the sense of one succeeding another, as is said Phys. iii, 6. Therefore infinity is potentially in our mind through its considering successively one thing after another: because never does our intellect understand so many things, that it cannot understand more. On the other hand, our intellect cannot understand the infinite either actually or habitually. Not actually, for our intellect cannot know actually at the same time, except what it knows through one species. But the infinite is not represented by one species, for if it were it would be something whole and complete. Consequently it cannot be understood except by a successive consideration of one part after another, as is clear from its definition (Phys. iii, 6): for the infinite is that "from which, however much we may take, there always remains something to be taken." Thus the infinite could not be known actually, unless all its parts were counted: which is impossible.
For the same reason we cannot have habitual knowledge of the infinite: because in us habitual knowledge results from actual consideration: since by understanding we acquire knowledge, as is said Ethic. ii, 1. Wherefore it would not be possible for us to have a habit of an infinity of things distinctly known, unless we had already considered the entire infinity thereof, counting them according to the succession of our knowledge: which is impossible. And therefore neither actually nor habitually can our intellect know the infinite, but only potentially as explained above.
Reply to Objection 1. As we have said above (Question 7, Article 1), God is called infinite, because He is a form unlimited by matter; whereas in material things, the term 'infinite' is applied to that which is deprived of any formal term. And form being known in itself, whereas matter cannot be known without form, it follows that the material infinite is in itself unknowable. But the formal infinite, God, is of Himself known; but He is unknown to us by reason of our feeble intellect, which in its present state has a natural aptitude for material objects only. Therefore we cannot know God in our present life except through material effects. In the future life this defect of intellect will be removed by the state of glory, when we shall be able to see the Essence of God Himself, but without being able to comprehend Him.
Reply to Objection 2. The nature of our mind is to know species abstracted from phantasms; therefore it cannot know actually or habitually species of numbers or figures that are not in the imagination, except in a general way and in their universal principles; and this is to know them potentially and confusedly.
Reply to Objection 3. If two or more bodies were in the same place, there would be no need for them to occupy the place successively, in order for the things placed to be counted according to this succession of occupation. On the other hand, the intelligible species enter into our intellect successively; since many things cannot be actually understood at the same time: and therefore there must be a definite and not an infinite number of species in our intellect.
Reply to Objection 4. As our intellect is infinite in power, so does it know the infinite. For its power is indeed infinite inasmuch as it is not terminated by corporeal matter. Moreover it can know the universal, which is abstracted from individual matter, and which consequently is not limited to one individual, but, considered in itself, extends to an infinite number of individuals.
Source: Summa Theologica (New Advent)