Commentaria in Primam Partem Summae Theologiae

In the title, God is taken properly, as it is a name of nature: so that the question concerns the essence of God, not His power or knowledge. The word infinite, since it is a compound of in and finite, has an in that is negative, not privative. Now finite signifies the termini of quantitative perfection; since the notion of finite and infinite accords with quantity, according to the Philosopher, I Physics¹; and just as there is a twofold quantity, namely of bulk and of perfection, so there are twofold termini, and a twofold finitude and infinitude. Here, however, since quantity of bulk has already been excluded from God², we must understand infinitude of perfection.

And diligently note that, because, as is said in III Physics ³, the infinite is not a substance but an accident of things, it follows that infinity or finitude is required according to the nature of the thing. For the mode of infinity of a thing is one thing, that of power another, that of knowledge another. Quidditative infinity is indeed considered with respect to the exclusion of essential termini, namely differences and the like. And since the question at hand concerns the infinity of the divine nature, therefore the discussion is about the infinity of substantial perfection. So that the sense is: Whether God according to his essence is of such perfection that he excludes all essential termini or limits.

II. In the body he does three things: first, he reports the opinion of the ancients; second, he distinguishes concerning finitude and infinitude⁴; third, he answers the question⁵.

III. As to the first, he does two things. First is this. Among the ancients, the first principle is infinite, because from it infinite things come to be. - Second. Certain of the ancients, erring, attributed infinity of bulk to the first principle, because they posited the first principle as material. Hence it followed as a consequence that they posited quantitative infinity, since quantity follows upon matter: and thus an error concerning the genus of the principle caused an error concerning the mode of infinity.

IV. With respect to the second⁶, it is distinguished both as finite and as infinite in respect of perfection: because some perfection is found on the side of matter, and some on the side of form. And they differ in this, that finitude on the side of matter signifies perfection, while infinitude on that side signifies imperfection; but on the side of form it is the reverse: for infinitude there signifies perfection, and finitude imperfection, as is stated quite clearly in the text.

V. Concerning this part, before proceeding further, there is a doubt both against the finitude and against the infinitude of the form. Against its finitude indeed, because the form of an ox—indeed even the intellectual soul according to St. Thomas⁷—is perfected by its conjunction with matter. Therefore the finitude of a form does not signify imperfection, but perfection. — Against its infinitude truly, because negation posits no perfection; nor does any thing acquire perfection from mere separation from another. Therefore infinity on the part of the form does not signify perfection.

VI. To the first of these it is said that "form" can be taken in two ways: namely, absolutely, insofar as it is form; and in a qualified sense, that is, insofar as it is such a form, for instance, one informing matter. A form of this kind is perfected by its union with matter; but not form insofar as it is form. And since the discussion here concerns form absolutely, according to its full breadth—whose inferior part is admittedly that which is limited by matter, while the other part remains free and cannot be confined by any limits of matter—the proposed inference must be denied.

To the second, it is said that, although negation or separation does not signify formally a perfection, it nevertheless signifies it fundamentally: for the foundation of the negation of unibility with matter signifies a great perfection of the form. And this suffices.

VII. As to the third point⁸, there is one conclusion, answering the question affirmatively: God is infinite. — Proof. God is his own subsistent act of being (esse): therefore his act of being is not received in anything: therefore he is infinite and perfect, that is, of infinite perfection, namely an infinity holding itself on the side of form. — The antecedent is clear from the third question⁹. Now the first consequence is self-evident. The second, however, is proved from the fact that being (esse) is the most formal of all things.

VIII. Concerning this procedure, a doubt arises from Scotus, in Book I of the Sentences, distinction II, in answering the first question: for he censures both the roots and the procedure. He says, first, that our procedure is as follows: Form is limited by matter; therefore a form which is not by nature meant to exist in matter is infinite. — He argues against the antecedent. Every form, prior by nature to its reception in matter, is in itself at such a grade of being: therefore finite or infinite. It is not, therefore, first limited by a relation to the negation of something extrinsic, say matter, or anything else. — But against the procedure itself. First, because it would follow that the essence of an angel would be infinite, since it is a form not receptive of existence in matter. Nor does it avail, he says, to claim that the nature of an angel is limited by its esse: for in our case, esse is posterior to essence, and thus the nature, in the first sign, abstracting from esse, since it is infinite, is therefore in the second sign not limitable by esse. — Second, because it commits the fallacy of the consequent, as in Book III of the Physics¹⁰: a body is limited by another body: therefore a body not limited by anything else is infinite.

IX. Concerning the same argument a doubt arises: for, once the whole is granted, it will not seem to novices that the question has been answered. For one ought to conclude that God is of such greatness of perfection that He would exceed every limit of perfection; but what has been proved is that God is not restricted by any receptive principle, which seems very far removed from the first.

X. To make these difficulties evident, it must be noted that the text covertly subdistinguishes infinity formal into the infinity of form and the infinity of being (esse) or act, which is more common than form. For just as act is twofold—namely being (esse) and form—so receptive potency is twofold, namely essence and matter. And thus reception and non-reception are twofold; and likewise finitude and infinitude are twofold—always speaking on the side of act. And just as being (esse) is an act of another kind than form, and essence a potency of another order than matter (as the difference between composition from esse and essence, and from matter and form, shows), so too the reception of esse in essence is of another kind than that of form in matter; and likewise the finitude of esse by essence differs from that of form by matter; and conversely, the finitude of essence by esse, and of matter by form.

And to speak of it point by point: esse in itself signifies a certain perfection; but one cannot imagine how great a perfection it signifies unless it is understood as applied to some nature, say to wisdom, or to Gabriel, and so on. And therefore esse, by the very fact that it is received into some essence, acquires the limits of perfection according to the mode of the receiving nature: and hence, if it is posited as joined to no quiddity at all, but as subsisting naturally in itself, it will have no essential term. Form, however, although it signifies a perfection in itself, can nevertheless be understood as limited to such a perfection even if it is not unifiable with any matter, as is evident in the case of the separate intelligences.

Hence it is clearly gathered how different is the determination of esse by essence, or conversely; and the termination of form by matter, or conversely. For in respect of the former is considered what is finite or infinite simply; but in respect of the latter, finite and infinite only in a qualified sense. Yet because esse and form agree in the character of act and of being receptive in another, therefore from the finitude and infinitude of form one can proceed to the infinitude and finitude of esse. And for this reason the text, in order to intimate the proportional identity between the finitude and infinitude of each, ascends from form to esse: but in order to intimate their diverse character, it subjoined, not that esse is the first form, but that esse is the most formal of all things.

XI. And from these the response to Scotus's objections is easily clear¹¹. For I say first that that antecedent, form is limited by matter, does not speak of limitation simply, but of such, namely in respect of material terms: and therefore the objection poses no obstacle. - And likewise the instances collapse against the argument: for it is true that the nature of an angel is infinite secundum quid. And if one presses: how then from that foundation does the text prove the infinity simply of God himself? the response is clear from what has been said. For the text leads us by hand from the infinity of form, which is secundum quid, to the infinity of the most formal of all things, namely esse, which is simply: and from that it infers God as infinite simply.

XII. And since, in the same way, one could argue about esse, following Scotus, as he argues about form; therefore I say secondly¹² that infinite, like immaterial, is a negative predicate, yet fundamentally positive. And just as the positive contained in the term immaterial is best proved from the natural negation of matter, both in act and in potency; so the positive contained in the signification of infinite simply is best demonstrated to befit God, from the negation of essential terms, as is done in the text. Hence I concede that esse, prior by nature in itself, is of such perfection—say finite or infinite—as it is receptible or irreceptible in this or that: though I say with this that these two mutually follow upon one another; such that if esse is altogether irreceptible, it is infinite simply, and conversely; and likewise, if it is finite, it is receptible, and conversely. And on this account one can argue from the destruction of the antecedent to the destruction of the consequent, without sophism, as happens in convertible terms. – And thus the response to Scotus is clear.

XIII. To the objection raised by the novices¹³, the answer can now be evident from what has been said. For it has already been shown¹⁴ that the total negation of the reception of esse itself is grounded upon an infinite perfection simply speaking.

And note how formal and drawn from its own principles the doctrine of St. Thomas is. For, treating of the infinity of essence, he considered only the essential terms; nor did he digress to the concomitants by which the infinity of power or intellect or will is attended upon, as others less perspicacious do.

In the title, note: the term infinite by essence does not signify the same thing as a thing of infinite perfection, because the term infinite is not restricted here to the infinite that holds on the side of form, which denotes perfection; but it is taken commonly, so as to include also the infinite on the side of matter, as is clear from the course of the body of the article and from the third argument. Hence "infinite by essence" signifies the same thing as a thing of an altogether indeterminate essence, that is, one determined neither to any genus nor to any species—and this whether per se or reductively. And this is a question about singular things, which can only be found to exist in the nature of things. I say this on account of things taken in abstraction, which may happen to be abstracted to such a degree that according to some account they are not restricted to any genus. Nevertheless, according to the thing itself, even that account, existing in this thing, is determined to some genus and species: for being (ens) in man exists in the human species, etc. So that the sense of the question is: Whether, besides God, some thing existing actually or potentially in the nature of things is of an essence indeterminate simply.

II. In the body there is a single conclusion, answering the question in the negative: No thing other than God is infinite simply; although it may be infinite in a certain respect.

This conclusion has two parts, as is evident: and in the text each part is proved. But because the infinite is twofold—namely on the side of act and on the side of potency—he first proves each part of the conclusion separately in the material infinite; then in the formal. Concerning that one, since matter is always bounded by some form, it is not infinite simply speaking; and yet, because it remains in potency to infinitely many forms, there is given an infinite in a certain respect. Concerning this one, however, because a created form is either in matter, and consequently bounded by it, or subsisting without matter and receiving esse. And this latter, because it is without matter, is therefore infinite in a certain respect—that is, with respect to material limits; yet because it has received esse, it is finite simply speaking. For esse so fitted to this one determinate nature, and to no other, must be limited simply speaking; otherwise it would contain within itself every perfection of every nature. – And from the parts of the universe sufficiently enumerated, the universal conclusion proposed has been proved: and quite clearly, if what precedes¹ be recalled.

Take note, novice, that not by doubting, but in the philosophical manner of deferring to his own treatise on the Angels to be treated below², he says that certain persons hold the opinion that angels are subsistent substances.

In the title, it can be taken simply, according to logical possibility. – In the body, three things: first, he assigns the cause of the question, namely that it is not vain but necessary; second, he divides the body¹; third, he answers the question raised².

II. As to the first point, the necessity of the question at issue is shown from the extraneousness of infinity in respect of magnitude to infinity of essence: namely, that it is one thing to be infinite in essence and another to be infinite in quantity. And this is proved in the case of natural things. – Here note that between these infinities there is a twofold difference. One is a difference of separation, whereby one is not the other. And this is evident, since in God there is found infinity of essence without quantity, while among these things there can perhaps be found infinity of quantity without infinity of essence, as is clear from the conditional laid down in the text. – The other is a difference of consequence, because one does not entail the other: namely, neither does what is infinite in essence entail infinity of quantity, nor—and this was more open to doubt—does infinite magnitude entail infinite essence, as is shown in the text, if fire were infinite. And with regard to this difference the text is to be interpreted: quoniam est admodum ad propositum.

III. Concerning this part, a doubt immediately arises that troubles the minds of beginners: because, as is said in VIII Physics³, in an infinite magnitude there must be infinite power. Therefore an infinite essence: for power does not exceed essence. Hence, if there were an infinite fire, it would be of infinite power; and consequently of infinite essence. It is therefore false what is said in the text, that if there were an infinite magnitude, it would not have an infinite essence.

To this it is briefly said that the objection equivocates. For infinite essence can be understood in two ways. First, of the infinity proper to the essence as essence: which, as is clear from what was said⁴, is nothing other than indetermination, that is, a lack of essential terms, namely genus and difference. Second, of the infinity taken commonly, or quantitative, etc. In the case at hand, the objection proceeds from infinity in the second way: but the text speaks of infinity in the first way. And therefore significantly the text says infinite according to essence, and does not say infinite essence: for essential infinity is better explicated by the phrase "according to essence", as is clear.

IV. As to the second⁵, the distinction is this. "Body" can be taken in two ways: physically, for the subject of three-dimensional extension; and mathematically, for that very three-dimensional extension considered absolutely. The text is clear enough.

Note here that that famous distinction of body, posited by Albert the Great in Book I of the Physics⁶, concerning threefold body—namely, mathematical, physical, and metaphysical—although it is not found explicitly in St. Thomas, is nevertheless not only not opposed to his doctrine but accords with it. For in him as well, a being composed of matter and substantial form, insofar as it precedes the aptitude of itself for quantity and motion, is a body belonging to metaphysical consideration. But since with respect to infinity according to magnitude, the physical and metaphysical body are not placed in different classes; therefore here the substratum of three dimensions—namely, that composed of matter and form—constitutes only one single member; and it is called physical body because by its natural form it receives its species, as is evident inductively.

V. As to the third point⁷, the question is answered, in accordance with the two members of the distinction laid down, by two conclusions⁸, in the negative. The first is: No natural body can be infinite in magnitude.

It is proved in two ways⁹. First a priori. Accidents follow upon the substantial form; therefore determinate accidents follow upon a determinate form; therefore a determinate quantity, both in greater and in lesser. Therefore every natural body has a determinate quantity both in greater and in lesser. Therefore it cannot be actually infinite. — The second inference is proved: because quantity is one of the accidents that follow upon the substantial form. The third, however, is proved: because every natural body has a determinate substantial form.

VI. Concerning the first consequence, a doubt arises as to what sense it has. For either ly determinate accidents imports a generic or specific determination: and then the sense is that such accidents follow a form of such a species, say fire, according to species, say heat, light, such a figure, levity, etc. In this sense the consequence is excellent, but it does not infer the proposed conclusion: for from such a sense the second consequence does not follow, therefore a determinate quantity, both in greater and in lesser; for a greater and a lesser quantity are not of different species, as is evident. — Or it also signifies a gradual determination, with respect to qualities, and a determination of extension, with respect to quantity. And then the sense is that a determinate form is followed not only by these accidents according to species, such that not those; but by accidents in such a degree or extension, such that not in another. And according to this sense it is without doubt here intended.

But it can still be understood in two ways. First, that determinate accidents follow a certain form, not only in the whole, but in all and each of its parts: for instance, that a certain quantity follows the form of water, not only in some whole body of water, but in whatever integral parts. And this sense is foreign to truth and to our purpose: since there is no smallest part of water or of whiteness, as is said in the book On Sense and What Is Sensed¹⁰; but, given any part whatsoever, there is given a smaller one existing in it, in homogeneous things. — Secondly, that determinate accidents follow a certain form in the whole and in parts capable of existing separately. And this sense is directly intended: since in merely indwelling parts there is not the proper form itself, but a part of the form, both in act and in potency; whereas the text says that determinate accidents and a determinate quantity follow a determinate form. But whence this consequence holds in this sense, or what is its foundation, is hidden. And therefore it turns into doubt: first, with respect to the determination of quantity in the lesser; second, with respect to the determination of the same in the greater¹¹.

VII. Scotus, in II Sent., dist. II, qu. IX, endeavors to prove the opposite, and to maintain that there is no minimum in homogeneous substances capable of existing per se: but, given any quantity of water whatsoever, a smaller one can exist per se. And thus its form does not determine for itself a definite minimum quantity, as is said in the text.

He therefore argues from the reason of quantity (Metaphysics V), thus: It is no more essential to quantity that it be able to be divided into parts, than that each of the parts into which it is divided can be this something. Therefore each part of any quantity whatsoever of water can exist per se. — The antecedent is clear from Metaphysics V¹² The consequence, moreover, is manifest in itself. And nevertheless the consequent is confirmed in four ways. First, because the parts are of the same nature as the whole, as regards matter and form. Therefore to exist per se, which does not conflict with the whole, does not conflict with a part. — Second, because every part has a nature: hence it does not conflict with it to be an individual of the same species as the whole. Therefore it can be an individual; and consequently can exist per se. — Third, because the parts are naturally prior to the whole. Therefore it does not conflict with them to be prior in time, etc. — Fourth, because, given that least water, all corruption being set aside, even that of the container, if it is divided, then either after the division—whereas before there was one water in act and many in potency—there will be several waters in act, and the point is established; or not. And if not, then the water has been annihilated without a corruptive cause. This is proved: because division alone does not corrupt. This is proved: because it divides into homogeneous parts, out of which the whole is constituted; therefore it divides into waters; or else water is composed of non-waters, as integral parts. Nor, he says, is there any repugnance on the side of small quantity: because the form of water was already present with so small a quantity.

VIII. Against the determination of quantity upward¹³, an argument is drawn from reason and from authority. From reason, indeed: given any fire whatsoever, if combustible material properly disposed is placed near it, it is certain that it will burn. That fire, therefore, grows, insofar as it lies within itself, to infinity. — And this is confirmed by the authority of Aristotle, II On the Soul, text 41¹⁴: fire would be increased to infinity if combustible material were added to infinity. Therefore there is no greatest fire.

IX. To make this difficulty evident, note first that we do not intend here to treat of the maximum and minimum in its entirety, since that would be beyond our purpose; but only whether there is given a maximum and minimum from within. For this is what the reason assumed in the text requires, wishing that from within, namely from the substantial form, each natural substance claims for itself a definite quantity, both in excess and in deficiency. And therefore we have brought forward only these few reasons in this disputation. There will be, God willing, a special question on the maximum and minimum universally. — Note second that we have two reasons from Aristotle for this. One is in I Physics, text 36¹⁵, namely: the quantity of the whole arises from the quantities of the parts, etc. But this reason, as is clear, proceeds from a supposition: for it supposes that a maximum is given, and concludes that a minimum is given, otherwise the whole would not be of a definite quantity. The other is in II On the Soul, text 41¹⁶, namely: of all things constituted by nature the measure is determinate, etc. And this reason, as Scotus says¹⁷, speaks of ensouled things, in which alone there is a place for growth properly so called, of which the discussion there is concerned.

True, whatever may be said of that text, St. Thomas universalized it: and what is said there of the soul he took to hold of every substantial form, namely, that it claims for itself a certain quantity. And he proved it according to the doctrine handed down in V Metaphysics¹⁸ and II Physics¹⁹, namely, that causes are compared with their effects, and conversely, and are measured by them. For from this it follows that, if quantity follows naturally upon substantial form in general, then a determinate quantity follows a determinate form. But quantity is determined in no other way than according to its terms: for the corporeity of man and of water does not differ in species, so that one could not say that a determinate quantity follows a determinate form according to species. Therefore there follows a quantity determined according to its terms. — And even if quantities differed in species, the ground of the inference would still stand, from the fact that a determinate form is the power of a determinate act, and hence nothing consequent upon it can be infinite: for if it had an infinite heat, rarity, quantity, or any accident whatsoever, then from a finite form there would be an infinite power; which is unintelligible. Since therefore accidents follow physical form in general, it follows that certain accidents follow a certain form, and hence a thing of a certain power: not only such that these follow but not those; but also such that they follow in such a degree or term, highest or middle, and not in a greater or lesser one. — And thus we have the ground on which the inference of the text holds²⁰, not only concerning quantity, but concerning all accidents.

X. Against Scotus's argument,²¹, the antecedent is granted, but the consequence is denied. And the reason is that both things—namely, being divisible into parts, and those parts being able to be a this-something—if understood really, do not belong to quantity through itself positively, but non-repugnantly, as is clear in the case of the quantity of the heavens: for that quantity can neither be divided by actual division, nor can any part of it exist separately, and yet it is truly a quantity. Both conditions, therefore, posited in the definition of quantity, can be impeded from being exercised by the natural form to which the quantity is joined. Hence that definition belongs to quantity as such: and yet it is repugnant to some quantity. — And if against this Scotus's argument be brought forward: Whenever two things follow from something through itself, they are impossible, and those two things are impossible; but the conditions of quantity and of a minimal form are incompossible; therefore quantity and a minimal form are incompossible: — the answer is, first, by applying the whole argument to the nature of the heavens, which does not admit division. Second (since to multiply is not to solve), that that major premise is true of things following through themselves positively: but not non-repugnantly. Hence the capacity for infinite speed belongs to motion through itself non-repugnantly, and is even demonstrated of it; yet it is repugnant to any natural motion, as is clear in II On the Heavens, text. 39²²: and therefore motion and naturalness are not incompossible.

It could nevertheless be said otherwise, that those conditions are not understood according to actual division, but according to designation: and that thus every quantum is divisible, and any part can be this something, according to designation. And then nothing follows against our thesis.

XI. To the first confirmation, the inference is denied. For the form of water requires some condition in that which is perfectible per se in the first way, which it does not require as part of it: and one of these conditions is such a quantity. And the same judgment holds for so much heat, etc.

To the second, likewise, it is said: that communicating in a nature is not sufficient for being able to be an individual of the same species; but one must add the conditions required for existing of oneself. For whatever has the nature of water together with the conditions necessary for existing of itself can be an individual of water; and not otherwise.

To the third, the consequence is denied, insofar as it concerns only the non-existent parts.

To the last argument it is said that, if the smallest fire alone were posited and divided, it would not only be corrupted but annihilated; although both the antecedent and the consequent are impossible. Nor need one seek any other destructive cause than division: since even if division as such is not corruptive, yet division of such a thing, namely of a natural minimum, is universally corruptive; and in such a case annihilative. Nor is the minimum fire one in act and many in potency, except in the way in which parts are said to be in potency in the whole: namely because they are by an esse that is not their own but the whole's; just as the parts of the heavens are in potency in the whole.

XII. To the things objected against the maximum²³, it is answered that a maximum fire can be understood to be given in two ways: in one way, speaking of an individual of fire that is one per se; in another way, of one that is one as it were by aggregation. And I say that a maximum and minimum fire is given intrinsically in an individual that is one per se; although a maximum that is one as it were by aggregation is not given, except from without (namely, because the nature of the universe does not allow fire to surpass the other elements, etc.). Now that is called an individual of fire one per se which is constituted of one thing necessarily in act both in form and in matter, such that by mere division it could not become two fires. Nor does this consist in the indivisible: but the whole breadth from the minimum quantity of fire to the quantity of two minima, exclusively, constitutes a fire that is one per se and one in number. But the quantity of two minima, and greater, makes one as it were by aggregation. And the same judgment holds concerning water and the other elements, and homogeneous things of this kind. – Nor, moved by these considerations, should you understand that a plant is not one per se, which is one in act and many in potency: since the parts of a plant are organic, and for the natural completion of its individual there is required that it have all its parts, etc. It is not so in homogeneous things of this kind, as is evident. – Therefore, when combustible material is added to any fire, etc., it must be granted that it would burn, and that it would grow infinitely: but it would not be one fire per se and one in number. – And thus the answer to the authority of Aristotle is clear.

XIII. Concerning the proof of the second consequence²⁴, a doubt arises: because something false is assumed, namely that quantity is one of the accidents following the substantial form; since it is evident that quantity is an accident consequent upon matter. – Nor does it avail to say that quantity is considered in two ways, namely absolutely and as terminated; and that although absolutely it is consequent upon matter, its termination nevertheless follows the form. For these things, although true, are nevertheless not to the point. Because the text, from the fact that quantity follows the form, infers that therefore it is terminated: whence, if it were assumed that quantity with respect to its termination follows the form, there would be a begging of the question.

XIV. To this it seems one should reply that quantity can be taken in two ways. In one way, according to what is in it of act: and thus it follows form. In another way, according to what is in it of potency: and thus it follows matter. And since most of what pertains to it belongs to the potency of matter, as is clear from the definition of the quantitative given in Book V Metaphysics²⁵, therefore absolutely speaking it is posited as an accident consequent upon the composite by reason of matter. And since the discussion at hand concerns quantity in act, because it concerns the infinite in act; therefore in the passage at hand, using "quantity" with respect to act, he attributes it to form; and from the fact that it, insofar as it is in act, follows form, he infers: therefore it is terminated, like the other things consequent upon form.

XV. Secondly, the conclusion is proved²⁶ from the notion of mobility. Every natural body has some natural motion, as is shown in I On the Heavens, text 5²⁷; no infinite body is naturally movable: therefore. – The minor is proved by using the distinction of natural motion made in I On the Heavens²⁸. For it cannot be moved by rectilinear motion, because every such body can be outside its own place, at least as to its parts. Nor by circular motion, because lines drawn from the center of a circle, the farther they are drawn out, the more they diverge from one another; therefore in an infinite body they would diverge from one another infinitely; therefore one would never reach the place of another; therefore one part of an infinite circular body could never come to the place where another is; therefore it implies that it be moved. And this argument is taken from I On the Heavens, text 35²⁹.

XVI. The second conclusion³⁰ is: No mathematical body can be infinite in act. – It is proved. If it is a body in act, therefore it has a form; therefore a figure; therefore a limit. Therefore it is not infinite. – The first consequence is proved: because nothing is in act except through its form. The second, however: because the form of a body, of the genus of quantity as such, is figure. The third, moreover, follows from the definition of figure. And this argument is taken from Book III Physics, text 40³¹.

XVII. In response to the second objection, note that, if it is examined carefully, he denies that the infinite is a passion of quantity, although he concedes that it is not repugnant to it. And if against this the fifteenth text of Book I Physic.³² is adduced, namely, that the notion of finite and infinite befits quantity, the answer is ready at hand: that this is not said because quantity lays claim to infinity for itself; but conversely, because infinity lays claim to quantity for itself, since the infinite is not intelligible except as a quantum: just as it is also said that the notion of void befits place. For by this the philosophers intend that if these are found to exist, they must be found in quantity and place, etc.

XVIII. In response to the third objection, note that, although the response is taken from Book III of the Physics, from which the other responses are also taken (the first indeed from text 71³³, the fourth from text 74³⁴, and this one from text 66³⁵): although, I say, this is so, nevertheless the words of St. Thomas in the Third Part, question VII, article 12³⁶; and in Book III of the Sentences, distinction XIII, question I, article 2, question III, ad 1, stand against this response; indeed even here, in the following article, in the reply to the second objection, by conceding the multiplication of figures to infinity. Likewise in Book III of the Physics³⁷, Aristotle expressly concedes, in texts 59 and 60, that magnitude is increased to infinity, just as it is divided.

XIX. To this it is briefly answered, with Aristotle in the same place, that magnitude's being increased to infinity can be understood in two ways. First, on the ground that an addition can always be made to it according to proportional parts. And thus it is said there that it can be increased to infinity: but by such increase a certain determinate magnitude will never be exceeded. Second, because addition can always be made simply, so as to exceed every determinate quantity. And thus a distinction must be drawn. For we speak either of magnitude considered within the limits of the continuous, or of it as taking on the character of the discrete. In the first way, the continuum is not augmentable to infinity, whether by natural or by intellectual augmentation: and the cause of this is assigned in the text. In the second way, however, it is augmentable to infinity, just as number is: and to this belongs the increase of figures to infinity. – That the continuous as such is said not to be augmentable to infinity even by intelligible augmentation is not because the intellect cannot, given any magnitude whatever, imagine a greater one, and so on to infinity; but because in the nature of the continuum as such there is no potency for an increase of this kind. Yet in the nature of material number there is; and likewise in the nature of the continuum there is a potency for diminution to infinity, as is clear from its definition. And thus all things are consonant, when understood according to their diverse modes.

XX. Concerning the reason assigned in the text³⁸, a doubt arises from Scotus, in Book III Sentences, distinction XIII, question I. For he attacks it as follows. Every number is one in itself formally, as is said in Book VIII of the Metaphysics³⁹: therefore, when number is increased, something is added to the form; therefore, an approach to the form does not prevent increase to infinity. Therefore it is wrongly said in the text that magnitude is not increased to infinity because one approaches the whole and the form.

XXI. To this it is said that, since the number augmentable to infinity is not any number whatsoever, but the number of continua, as is clear from III Physics, text. 68⁴⁰, the augmentation of number does not accrue to number simply, but to such a number, namely that of the parts of a continuum: and hence what is clearly augmented is not the whole but the part; just as also the division of the continuum, from which it is caused. But the augmentation of the continuum accrues to the whole simply. – Therefore Scotus's first consequence must be denied: for he argues from number to such a number, that is, a material one. Which is not permissible in the matter at hand: because the term material diminishes the ratio of number with respect to its formality, since it adds materiality to the form, and mixes act with potency, as is evident.

In the title, according to any multitude whatsoever, whether of the genus of quantity or outside the genus. And from this it is clear that this question is higher and more universal than the inquiry made in III Physics¹: for there the discussion concerns only material multitude, following upon the division of the continuum; here, however, the discussion concerns multitude universally, whether material or immaterial.

II. In the body there are two parts: first, he reports the opinion of Avicenna and Algazel; second, by refuting it, he answers the question².

III. As to the first point, he sets out two propositions and explains them. The one is: it is impossible for an actually infinite multitude to exist per se. The other: it is possible for an actually infinite multitude to exist per accidens. Both are explained once their terms are made clear, namely an infinite multitude per se and an infinite multitude per accidens. They are proved as follows: because the work would never be completed if it depended on infinites per se; but not so per accidens, as is clear from the example of the letter.

IV. As to the second point, there are two conclusions answering the question, according to the diversity of act in relation to potency. The first is: It is impossible for there to be an actually infinite multitude, whether per se or per accidens. The second is: It is possible for there to be an infinite multitude in potency³.

The first conclusion is proved in two ways⁴. Every multitude actually existing is in some species of multitude: therefore it exists according to some species of number: therefore it is finite. – The antecedent is evident. The first consequence is proved: because the species of multitude are according to the species of numbers. The second: because number is a multitude measured by one.

V. Concerning the first consequence and its proof, a doubt is raised. For multitude is something more common than number, and consequently it ought to have more species than number: otherwise it would not retain the character of the more common. Therefore the species of multitude are not according to the species of number. – And this is confirmed as follows: for it can be said that the species of finite multitude are according to the species of number; but not the species of multitude absolutely. – And Aristotle attests to this in Book V Metaphysics, in the chapter on Quantity⁵, saying that finite plurality is number: as though he were excepting infinity.

VI. To this it is briefly answered that, since we naturally come to know what is unknown through what is better known, and since we experience that all the species of multitude that we know are proportional to the species of number, it is sufficiently in accord with the art that a universal proposition can be formed, namely, that every species of multitude is according to some species of number.

To the objection, however, it is said that it is one thing that all the species of multitude are species of number, which is false; and another, that all the species of multitude are according to the species of number, that is, proportional to them, which is what was assumed, and is true. Hence multitude, because it is something more universal, has far more species than number: for it has all the species of number, and all the species of immaterial multitude, which is not properly number, etc. – And by this the answer to the confirmation is clear, namely that this would be said willingly, not rationally. – The text of Aristotle, moreover, is no obstacle. Since nothing from it can be drawn for the purpose at hand except that an infinite plurality would not be a number: but this does not stand in the way of this argument, but rather confirms the second consequence, as is evident.

VII. Secondly, the same conclusion is proved⁶. Every multitude actually existing in the nature of things is created; therefore it is comprehended under some definite intention of the Creator; therefore under a definite number. Therefore it is impossible that it be infinite. – The antecedent is evident. The first consequence, however, is proved: because an agent acting not in vain accomplishes something. The remaining consequences are left as evident.

VIII. Concerning this argument a doubt is raised: for it either assumes something false, or it does not infer what is intended. For when it is assumed that every creature is comprehended under some definite intention of the Creator, this is understood either of a definite intention with respect to the certainty of the end; or with respect to the certainty of the means by which the end is attained.

If with certainty of an end, then it can be expounded in two ways: either so that the created thing itself is a certain end; or so that it has a certain end. If therefore the sense is that every created thing is comprehended under some intention of the creator as a certain end, the proposition seems manifestly false: since something created is neither the end of the creator, nor of any other creature, for instance the lowest creature. – If however the sense is that every created thing is comprehended under some intention of the creator as ordered to a certain end, the proposition is most true: because God himself, and the participation in his goodness, is the certain end of every created thing intended by the creator. But then nothing follows for the purpose: for from this it does not follow, therefore under a certain number.

But if the sense is that every created thing is comprehended under some intention of the creator, as a certain means through which the end is acquired, the proposition seems false and unproven. False indeed: because it can be comprehended under the intention of the creator in another way than this. For it can be intended not as a certain means for acquiring the end, but as accompanying or proceeding from a certain means through which the end is acquired: as is clear from Averroes, XII Metaphysics, commentary 37, where he holds that the intelligences move the spheres so as to be assimilated to God by emanating beings; and from this it follows that infinite circulations are caused by them and infinite effects, and infinite souls created, according to Avicenna. – Unproven, however: because from that means, no agent acts anything in vain, nothing else can be inferred except, therefore it acts on account of an end; which granted, it would be said that an infinite multitude is on account of an extrinsic end, namely God and the perpetual preservation of the assimilation causing it to God, as the philosophers say.

IX. Concerning the same point again, a doubt arises: because in it there seems to be committed the fallacy of the divided sense taken as composite, proceeding from any created thing to many, indeed to infinitely many, things collectively. For although any created thing whatsoever is the terminus of some definite intention and operation of the creator himself, nevertheless it does not follow that many created things must be the terminus of one intention and operation, but rather of many intentions and creations. And consequently infinitely many created things will be the termini of infinitely many intentions, just as of creations.

X. Concerning the same point, a doubt still remains: because the root of it—namely, that every created thing is comprehended under some definite intention of the creator—although true of creatures per se, would nevertheless be denied by Avicenna with respect to creatures per accidens. For an accidental infinity is intended only accidentally, as the text says; and consequently it is created only accidentally. Now a thing is called created per accidens when it must be created for the preparation of another thing, as happens with the intellectual soul, which, once nature has prepared a suitably disposed body, is always created. And since infinitely many bodies have already been prepared, infinitely many souls have been created; and these souls, as such, have happened to remain in act, because they are incorruptible. Thus their infinity is not posited as intended per se, but follows upon the eternity of human generation.

XI. For the clarification of this process⁷, note first that created can be taken properly, as distinguished against caused: and it can be taken commonly, so that it makes no difference to say caused or created. That it is taken commonly, the universal proof of the text indicates, when for its proof it assumed that an agent does not act in vain. But that it is taken properly, the very formal signification of the word shows. - Now it matters little or nothing in which way it is taken. For if it is taken properly, the words of the proof must be weighed when he says something acts, that is, attains the proper term of its operation. And by this are excluded all things consequent upon the proper and intended work of the agent: and of every created being it is verified that it is the proper term of creation. But if it is taken commonly, what is caused must be restricted to what is properly caused, as distinguished against what is consequent upon causation.

Truth, in either way, is restricted to caused things or to created things actually existing, so that collapsed things are excluded. And this is hinted at in the text, where it says of the creator's. For what has been created and perseveres is, in a certain way, continuously being created, because it is preserved: for it depends both in its coming-to-be and in its being (esse) on the creator. - But what falls under the creator's intention includes not only what is actually comprehended, but also what is comprehensible, even if it were not actually comprehended.

XII. But from the fact that the proposition assumed concerns only things existing in act, which are the proper terms of action, the solution of all objections is had.⁸ For all things existing in act the Creator can order both as a means toward the acquisition of some end, and intend as an end to be attained through something else. Since, indeed, it is posited to exist in act, its production is already posited as terminated; and consequently it can be directed as a means toward acquiring something; and likewise it could have been an end intended and acquired through some means; nor does the ratio of end, or the ratio of means to an end, conflict with that which exists in act. And therefore, since intention properly belongs only to an end and to the means through which the end is to be acquired, everything existing in act is comprehensible under a certain intention of the creator, both as end and as means. As end, indeed: both because there is no creature so lowly that it is not, or cannot be, the end of something else; and because at least it is a certain end of its own production; and because this is understood insofar as it is from itself: for nothing repugnant to the ratio of end belongs, insofar as it is from itself, to any created thing existing in act, although perhaps something may be the end of nothing. As means, however: because after it there remain many productions and noble perfections of many things, to one of which it can be ordered as a means, being already ens in actu.

Nor does the first objection against this member stand in the way. For, as is already clear from what has been said⁹, speaking formally of the created, it cannot be a proper consequent of the term of creation; rather, the created must terminate the creation. Now such a created thing must be intended either as an end or as a means to an end. — Nor is it true that this has not been proven. For from that assumed proposition, an agent does nothing in vain, the term of the interior intention is most fittingly concluded from the end of the exterior operation. For since the operation of no agent is in vain, but directed to a certain proper end, the intention of the one operating must also be with respect to a certain end: for the intention of the agent establishes the end of the operation. — And thus the response to the first objections is clear.

XIII. To the second doubt, however¹⁰, it is said that, since the whole multitude of beings in act can be taken as one created thing—because it does not exceed the universe, which has the character of a single first created object of thought—it follows that not only individual created things, but all things together, insofar as they coexist in act, are comprehensible under some one definite intention of the Creator. And therefore no sophism is committed in the matter at hand, where from the opponent's proposition that an infinite multitude exists in act there is inferred: therefore it is comprehensible under a definite intention of the Creator, therefore it is not infinite; for the infinity of a multitude is repugnant both by reason of the means for attaining the end and by reason of the end itself.

XIV. To the third doubt¹¹, it is said that what is created per accidens can be understood in two ways. In one way, as per accidens is distinguished from per se. And in this way, it is denied that anything is created per accidens: since every created thing is intended by the Creator in itself, just as it is also in itself the terminus of creation; otherwise it would not come to be from nothing. In another way, as per accidens is distinguished from per se primo, and signifies the same as through another. And in this way something is said to be created per accidens, that is, through another, at least occasionally. And in this way intellectual souls are created per accidens, both according to our opponent and according to us: though more per accidens with us than with him. But this does not prevent them from being individually intended in themselves, just as they are created in themselves. For although they have an order among themselves that is per accidens, yet when related to the Creator, they are both intended and produced in themselves: indeed, more per se than any corruptible thing whatsoever; for incorruptible things are more perfect and worthier parts of the universe, and on this account are more intended. A sign of this is that in the sphere of active things, only the species are intended per se, because they alone are perpetual. And since not only individual things, but infinitely many are posited together in act, it follows, as was said¹², that all are intended and produced per se, inasmuch as they are comprehensible under some one definite intention; indeed, even producible together in act, as far as lies in themselves.

XV. Note here that from this passage it follows that St. Thomas has removed for himself every path of reconciliation with the doctrine of Aristotle, according to his exposition. For he holds that Aristotle believed the intellective souls to be immortal, and numbered according to the number of bodies: and it is established that Aristotle held the world to be eternally generated. From these things, however, it manifestly follows that human souls are actually infinite: which is here judged impossible, and in II Cont. Gent., chap. LXXXI, it is said not to conflict with the doctrine of Aristotle in III Physic.¹³ and I Caeli¹⁴, because in both places the discussion was of the material infinite.

XVI. Concerning this conclusion¹⁵, many arguments are brought forward by Capreolus¹⁶ from Gregory and Adam: but all are reducible to three. First, then, he argues as follows. There are given infinitely many proportional parts in act in a continuum: therefore they can be given as separated. Ergo. — The inference is proved: because every absolute being (ens) that exists can be conserved by the divine power without that which is not of its essence; but the continuation of one with another is not of the essence of the parts; ergo.

Second, God, at any beginning of any proportional part of one hour, can create an angel; therefore in one hour he can create infinitely many angels in act. — And this is confirmed: because whatever God can do successively he can do simultaneously, in the case of permanent things.

Third, God is of no lesser power than infinitely many souls, if they existed, and than the fire of hell. But those souls could suffer infinite torments all at once, and the fire could inflict infinite torments, if there were infinitely many souls there. Therefore.

XVII. To the first two arguments the reply is brief: there is a fallacy of moving from the divided sense to the composite. For although continuity with another part does not belong to the essence of each single part, it nevertheless belongs to the essence of all of them together: because it belongs to their essence, taken together, that they be in potency, which is repugnant to the separation of each from the others. — And likewise it involves numbering the beginnings of all the proportional parts of an hour: for by that successive enumeration they are posited as completed at the end of the hour, and yet infinite. For if they are infinite, they will never be numbered; and if they are completed, they will not be infinite. God can therefore act in each of the beginnings, but not in all of them. — And by this the confirmation is answered: for neither simultaneously nor successively can He produce infinitely many things actually existing.

To the third, however: even granting the whole, nothing is obtained against the intended point. For it is not from a defect of divine power, but from the impossibility of the thing to be made, that there arises the fact that things cannot be infinite in act. Yet with this it stands that those conditionals are true: just as, if man is a lion, he is capable of roaring; nevertheless it does not follow, therefore God can make a man capable of roaring.

XVIII. The second conclusion¹⁷ is responsive: The infinite according to multitude is found in potency. — It is proved. The infinite is found in potency according to the division of the continuum; therefore also according to multitude. — The inference is proved: because an increase in multitude follows upon the division of magnitude. The antecedent is evident, and it is proved, because one proceeds to matter.