Commentaria in Summam contra Gentiles, Liber I

Fifth conclusion, responding to the fifth objection¹, is: God knows infinite things.

Concerning which, Saint Thomas makes four points: first, he proves it; second, he shows the difference between the divine and the human intellect with respect to infinite things²; third, he answers the foundation of the objection³; fourth, he shows to which science the knowledge of infinite things is attributed⁴.

I. As to the first, he argues first as follows. God is the cause of infinite things, if beings are infinite: since He is the cause of all things that are. Therefore He has knowledge of infinite things: because, in knowing Himself to be the cause of things, He knows other things besides Himself.

This argument proves that he is cognizant of infinities, not absolutely, but under a condition, namely if they are infinite.

II. Second. The divine power extends itself to infinite things, since it is infinite. Therefore etc. — The consequence is proved. For a power is not known perfectly unless all the things to which it can extend are known, since the quantity of a power is in some way measured by them. But God knows His own power perfectly.

Note that Saint Thomas says the quantity of a power is attended to in a certain way according to its objects, but not that it is attended to according to them simply, because, as is said in Prima, q. 42, a. 1, ad 1, the quantity of a power is attended to first of all in its root, namely in the very perfection of the form or nature: and thus it is called spiritual magnitude in the way heat is called great on account of its perfection. In the effects of the form, however, it is attended to secondarily, namely according to the comparison of the power with its objects. And because, as is said in I Sent., d. 17, q. 2, a. 1, in this way a power is said to have quantity from its order to something extrinsic, not because it has any division in itself; therefore the quantity of a power is not attended to simply according to its objects, but only secundum quid; simply, however, it is attended to according to the perfection of the form.

III. Third. In natural things there is an infinite in potency, as is said in Physics III. Therefore God knows infinites. — The consequence is evident, because God's knowledge extends to all things that are in any way whatsoever, whether in act or in potency. — It is confirmed by the example of unity: if whatever is in it in potency were known, since every number is in potency within it.

Note that unity is in potency to every number because it is in potency to infinite additions of numbers, from which infinite species of numbers result. For a unity can be added to unity, and thus the number two results; two can be added, and thus three results; and so on to infinity. And therefore unity, which is posited as the foundation of all numbers according to their ordered procession, is in potency to every number. On account of this, if anyone were to know whatever is contained in it potentially, he would know the infinite species of numbers. In the same way, if God knows whatever is in the potency of unity, He knows infinite things. And if He knows whatever is in the potency of the continuum, which contains infinitely many parts and infinitely many divisions potentially, He knows infinite things.

IV. Fourth. From God as an infinite exemplar, things having infinite⁵ perfections can be known: because neither one of them nor several exemplars can equal His perfection, since He is of infinite perfection. Therefore etc. — The consequence is evident. Because through His essence, as a mediating exemplar, He knows other things.

It must be rightly understood that God can act from an infinite exemplar: namely, in such a way that, whatever perfection is produced, another can be produced; and once that is produced, yet another; and so on to infinity—not, however, because infinitely many perfections could be produced by God simultaneously in act. For this reason he added that there always remains a new mode in which something exemplated can imitate it.

V. To the fifth. God's understanding is infinite, as is His esse. Therefore, etc. — The consequence is proved. For, as the finite is to the finite, so the infinite is to the infinite. But our finite understanding can grasp finite things. Therefore God's infinite understanding grasps infinite things.

But that comparison seems not to hold. For it does not conflict with the finite that it be taken as a whole; but it does conflict with the infinite to be taken as a whole, since the infinite is that of which, for those receiving its quantity, there always remains something still to be received⁶.

It is answered that it is repugnant to the infinite, taken as a whole in itself, to be grasped by an intellect if it is taken successively. And thus it is understood that something always remains to be taken: for one proceeding successively in the knowledge of the infinite never arrives at knowledge of the whole, but something always remains to be taken. It does not, however, repugn the infinite to be comprehended as a whole by an infinite intellect, simultaneously: namely, in such a way that nothing of it remains to be understood or lies outside the infinite intellect.

VI. Sixth. God knows himself perfectly. Therefore, etc. — The consequence is proved. Because the intellect knowing the greatest intelligible does not know lesser things less, but more, as is said in III De Anima. Now infinite beings, however infinite they may be, would be of lesser infinity than God; and in some respect finite, because they would have received esse. Therefore, if God understands himself, he can also know all infinite things.

Note that one infinite simply cannot have greater esse than another infinite simply: but an infinite simply is indeed greater than an infinite in a certain respect; just as, if there were a body infinite in every dimension, it would be greater than a body infinite in length alone. Therefore, since God is infinite simply, while all created things, however they may be posited as infinite, are infinite only in a certain respect, on account of their limitation to a certain genus or species, Saint Thomas says that they are of lesser infinitude than God.

VII. Seventh. The divine intellect, in respect of efficacy or perfection, is infinite. Therefore by one thing, which is its essence, it can know infinite things. — The consequence is proved: because the more efficacious and limpid an intellect is in knowing, the more things it can know from one; just as every power, the stronger it is, the more unified it is.

VIII. Eighth. No intelligible perfection is lacking to the divine intellect: since it is simply perfect, as the essence is. But that to which our intellect is in potency is an intelligible perfection of it. Now our intellect is in potency with respect to infinite intelligible species, since the species of numbers and of figures are infinite. Therefore, etc.

It is confirmed. For otherwise our intellect would be cognizant of more things than the divine intellect, or the divine intellect would not know in act all the things of which it is cognizant in potency. The consequence is proved. Because our intellect can multiply to infinity the species of numbers. Therefore, if God cannot know those infinite species, our intellect will be able to know more than the divine intellect. But if He can know them and yet does not know them in act, then He does not know all at once the things of which He is cognizant.

Note first that, because the intelligible species to which our intellect is in potency is the intelligible perfection of the intellect, it follows that all the species of our intellect to which it is in potency must exist in the divine intellect, yet according to the mode of the divine intellect, and not according to the mode of our intellect. And because by the infinite species to which our intellect is in potency, infinite things are naturally able to be understood, therefore God can, by means of those species existing in himself, understand infinite things.

Note secondly that our intellect is said to be in potency with respect to infinite intelligible species, not because it could be informed by infinite intelligible species at once—for this is repugnant to the infinite, namely that it be in act—but because, however many species it has, it can still receive others to infinity, if it endures to infinity and knows through the reception of species. In the same way we must understand that it can multiply the species of numbers to infinity. Hence the potentiality of the intellect is in some way infinite in potency, not by a potentiality reducible wholly into perfect act, but reducible into an act mixed with potency, as is said of the infinity of numbers and of the division of the continuum.

IX. Ninth. The divine intellect knows all things simultaneously. Therefore, etc. — The consequence is proved. For the infinite is repugnant to cognition only insofar as it is repugnant to enumeration, by which it is known successively; for in that way its cognition cannot be completed, nor its enumeration. But it can well be known totally by some intellect.

X. Tenth. Plurality makes no difference in God's knowledge. Therefore neither does infinity. Therefore it makes no difference to the divine intellect whether the cognition is of finite or infinite things. Therefore, since He knows finite things, nothing prevents Him from also knowing infinite ones. — The antecedent is evident, because many things are known by God at once. — But the consequence is proved as follows: since every quantity consists in a certain multiplication of parts, and number holds the position of first among quantities, if plurality makes no difference, neither will that which follows upon quantity, such as infinity, make any difference.

Note that, as is said in the commentary on Boethius's De Trinitate, q. 4, a. 2, ad 6 ⁷, formally indeed discrete quantity is prior to continuous quantity: but materially continuous quantity is prior to discrete, since number is caused by the division of the continuum. And therefore, when Saint Thomas says that number is first among quantities, this must be understood of formal and perfectional priority: as is clear from the principles of discrete and continuous quantity; for unity is simpler and more abstract, and thereby more perfect than the point, which adds to indivisibility position in the continuum.

It is also evident that, as St. Thomas here suggests, that from which the ratio of quantity is taken, namely the multiplicity of parts, belongs more perfectly to number than to other quantities. Indeed, it does not seem to belong to the continuum except by reason of number potentially contained in the continuum.

2. A doubt arises, however, concerning this argument. For although something may not be repugnant to another by reason of some common feature, it may nevertheless be repugnant to it by reason of its own proper nature: just as being divided into equal parts is not repugnant to an odd number by reason of quantity or number, yet it is repugnant to it insofar as it is an odd number. So someone might say that, although being known is not repugnant to infinite things by reason of plurality, it is nevertheless repugnant to them by reason of infinitude. And thus, although plurality makes no difference in divine cognition, infinitude may nevertheless make a difference: namely, so that God can know finite things but not infinite ones. And there is a great difference between God's knowing finite things and His knowing infinite things.

It is answered that, where the reason for the incompatibility of the prior and the posterior is the same, if the reason for the prior's incompatibility is removed from something, the posterior will not be incompatible either. On this account the Philosopher says, Physics IV ⁸, that if two bodies could be in the same place, infinitely many could be as well. Since, therefore, the reason is the same in our intellect why it cannot understand many things at once and why it cannot understand infinite things—namely, because it would understand them through several species, yet it cannot use several species at once—if from some intellect there is removed the fact that what it understands it knows through several species, then just as it could understand many things through one species at once, so too it could understand infinite things, as far as the knowing intellect is concerned. And thus, since a plurality of intelligibles makes no difference in the divine cognition, because whatever God understands He understands at once through His essence, neither will infinitude make any difference, as far as the divine intellect is concerned; this is what St. Thomas intends to conclude by means of this argument. For just as God knows many things at once through His essence, so through the same essence He can know infinite things at once. And if to this is added what was assumed in the preceding argument, namely, that being known successively is the only thing incompatible with the infinite's being known, it follows that there is no incompatibility preventing the divine intellect from knowing infinite things: neither, I say, on the part of the intellect nor on the part of the knowable. Hence this argument is a confirmation of the preceding one.

It is confirmed by the authority of the Psalm, where it is said: "Of his wisdom there is no number."

XI. As to the second point ⁹, Saint Thomas deduces from what has been said the difference between the divine intellect and ours in the knowledge of the infinite. And he sets forth a fourfold difference, which is the cause of this diversity.

The first is that our intellect is simply finite.

The second reason is this: because our intellect, since it knows diverse things through diverse species, cannot know infinite things through one act of cognition. But the divine intellect can do so well.

The third follows from the preceding, because our intellect cannot know many things at once, and thus it cannot know infinite things except successively, by counting them. But the divine intellect beholds many things at once, as it were through a single species of things seen.

The fourth is this: because the divine intellect is of things that are and of things that are not, whereas our intellect is not.

It must be noted that the first difference, namely that our intellect is finite while the divine intellect is infinite, taken by itself is not a cause of diversity with respect to the cognition of infinite things, but only together with others: for, according to the teaching of St. Thomas ¹⁰, the soul of Christ knows infinite things in the Word, and yet its intellect is finite.

Or it may also be said that, because our intellect is finite, it cannot know the simply infinite, nor even the soul of Christ, as is said in III Sent. ¹¹ but only the infinite in a certain respect, when the soul knows through something one, as the soul of Christ does through the Word. The divine intellect, however, which is infinite, can know what is simply infinite.

XII. As to the third ¹², the objection is answered. And because the objection rests upon the authority of the Philosopher, who says that the infinite, insofar as it is infinite, is unknown, it is said that it is unknown if one wished to know it by measuring its parts, which would be to know the infinite insofar as it is infinite. But God does not know it thus; rather He knows it all at once. Hence He does not know the infinite, so to speak, as infinite: but it stands to His knowledge as if it were finite.

Notice first, from the doctrine of Saint Thomas in the First Part, q. XIV, a. 12, ad 1, [that] since it belongs to the notion of quantity that there be an order of parts, and the infinite follows quantity, to know the infinite in the mode of the infinite is to know it by numbering part after part. And therefore Saint Thomas says here that this is the proper notion of quantity, namely, that it be known through the numbering of its parts. And because the numbering of the infinite could never be completed, therefore the infinite, insofar as it is infinite, is unknown.

It must be noted, second, that Saint Thomas said that God does not know the infinite insofar as it is infinite—with the addition, so to speak—because, as is said in De Veritate, q. 2, a. 2, ad 7, of God himself, if emphasis is laid on the word, the infinite is finite to no one. But for this reason it can in some way be called finite when compared to divine cognition, because it is known by it as perfectly as the finite is known by a finite intellect; and because it is not known successively but all at once, like the finite.

Finally, it must be considered that when we say God perfectly knows an infinite multitude, this is not to be understood as though his knowledge reached some ultimate part, for it is repugnant to the infinite, insofar as it is infinite, to have an ultimate part; but rather because no part of it is hidden from his knowledge. And just as, given any part of the infinite, there is still a further part beyond it to infinity, so, whatever part he sees, God sees a further one; and although none remains unknown to him, he nevertheless sees that none is ultimate. And this is what it is for the infinite to be perfectly known by God.

XIII. As to the fourth ¹³, since we posit a twofold knowledge in God, namely of vision and of simple intelligence, Saint Thomas says that God knows infinite things by the knowledge of simple intelligence, because He knows those things that are in His power and in that of a creature, which are infinite; but not by the knowledge of vision, because those things that are at some time in act are not infinite.

From which it follows that the major premise of the argument by which it was thus argued ¹⁴ — "Singulars are infinite; but the infinite is unknown; therefore it does not know singulars" — is false. For singulars are not infinite, since generation, according to the faith, is not infinite on either side.

2. But a doubt arises: for St. Thomas seems to contradict himself. For in the Questions on Truth, q. XX, a. 4, ad 1, and in the First Part, q. XIV, a. 12, he holds that God can not only, speaking absolutely, know infinite things by the science of vision, but also does in fact know them now, because he knows the infinite thoughts of the damned, if the souls of the damned are everlasting and have changeable thoughts. The opposite of this is said here.

It is answered that in this passage he is speaking of subsistent natures, because the argument presupposed that singular subsistent things are infinite. For of these it is not true that the science of vision knows infinitely many: since God will not produce infinitely many subsistent things. — But in the Questions on Truth and in the First Part he speaks absolutely, so that even the science of vision extends to accidents and to things that do not subsist per se. For thus the infinite science of vision knows infinitely many things: as the infinite parts of a continuum, and the infinite thoughts of the damned.