Aquinas’ Concept of Infinity

John Tomarchio. Journal of the History of Philosophy. Volume 40, Issue 2. April 2002.

Much has been written of late about Aquinas’s concept of divine infinity, but the attention given to his other metaphysical uses of the term ‘infinite’ has been peripheral-sometimes to ill effect in the interpretation of his concept of divine infinity. The intent of this article is to offer an explication of Aquinas’s analogical concept of infinity. What Aquinas means when he calls beings or their principles infinite is sufficiently complex that it needs to be analyzed in its own right before his arguments for things’ actually being infinite can be evaluated.

There is no one place in his writings where Aquinas gives a full treatment of the various metaphysical senses of infinity, and so his view has to be reconstructed from the analysis of sundry texts in juxtaposition. As my intent is to reconstruct his overall theory, I will take up the relevant texts in a logical rather than historical order. I will first address the kind of relative infinity Aquinas predicates of certain principles of things, namely, the matter, form, and existence of finite beings, and then the kind of actual infinity he predicates of certain things, namely angelic substances and God. In the course of doing so, I will identify four global distinctions that run through all these predications and constitute the framework of his overall analogy. I will conclude the article with a recapitulation of this overall analogy.

Aquinas predicates infinity both of beings and of principles of beings, and both with respect to their quidditative perfections or what they are (signified by their genera, species, differences, and attributes), and with respect to their actuality or how they exist (signified by such categories as substance, accident, and relation). To begin to parse these distinctions, let us turn to Summa theologiae Ia, qu., ar. I, where Aquinas argues for the infinity of God.

The Relative Infinity of Matter and Form Considered Absolutely

Somewhat paradoxically, Aquinas begins his argument for God’s infinity with a consideration of the infinity of finite things. He predicates a certain infinity both of the matter and of the forms of finite things. To a finite thing’s matter considered in itself he assigns an infinity of formal indetermination, and to its form considered in itself an infinity of formal intension. He states that while a material thing’s form perfects its matter in delimiting it, the thing’s matter restricts the perfection of its form in delimiting it:

[S]omething is said to be infinite from the fact that it is not limited. Now matter is in a certain way limited through form, and form in a certain way through matter. Matter is in fact limited through form inasmuch as before it receives a form, matter is in potency to many forms, but when it receives one, it is limited by it. Form however is limited through matter inasmuch as a form considered in itself is common to many things, but by being received in matter it becomes the form determinately of this thing. Matter moreover is perfected by the form through which it is limited, and so the infinite as it is attributed to matter has the character of the imperfect; it is, as it were, matter possessing no form. Form however is not perfected by matter, but rather its amplitude is contracted by it, so that the infinite as it obtains on the part of a form not determined by matter has the character of the perfect.

At first glance, Aquinas seems to use the term ‘infinite’ equivocally: to signify lack of perfection in the case of matter and perfection in the case of form. On closer consideration, however, what is common to his two predications is the notion of relative indeterminacy.

This relative indeterminacy is evident only by our considering these principles of finite beings in themselves. What there is in actuality are finite things composed of determinate matter and determinate forms; viewed as complete beings, finite things are finite in all respects. Such principles of finite beings have no actuality independently either of the finite things of which they are the principles or of one another; it is for this reason that Aquinas says “as it were” in positing the notion of “matter not possessing form.”

As their actuality is dependent on one another, so is their infinity relative to one another. Matter is in one respect indeterminate relative to the forms to which it is by its very nature related, and material form is in another respect indeterminate relative to the kind of matter to which it is by its very nature related. Matter is not in itself determined to any one form, being by its very nature ordered to having some form only in actuality. H2O, for example, can be liquid, solid, or gaseous only because it is not in itself determined to any one of these accidental forms. Nevertheless, by its very nature and in actuality, H2O never exists without one or other of them. Only as thus formally undetermined in itself can it be a potency for these determinate forms.

For its part, a material form has a determinate nature in itself undetermined to any given subject-matter. For a form to be the form of this thing is for it to be, not form simply, but form restricted to this substrate and to the particular way the subject-matter has of receiving it. Thus the form of ‘fluidity’ is in itself undetermined to any given material, and for this reason can be ‘common’ to water or oil or mercury, however possessed by each in its own particular way, according to its own particular capacity for fluidity.

What is easily overlooked in this passage is that the correlative indeterminacy Aquinas ascribes to matter and material form is itself twofold. He at first states that matter is “in potency to many forms” and that the form is restricted through matter to being the form “determinately of this thing,” invoking an individuation of the form. He then says, further, that although the matter is perfected by the form it receives, the form’s “amplitude is contracted” by the matter, invoking an attenuation of the form’s perfection. With more strictness of terminology than Aquinas employs, I would distinguish between the singularity and the particularity that accrues to the form from its reception in matter. It is one thing for Aquinas to say that by means of matter a form is determined to an individual thing, to explain for example how multiple individual human beings each has its own human soul, and another to say that the perfection of a form is restricted by matter, to explain for example that a particular human being does not exhaust all the perfection possible to a human being. Put simply, using matter to explain that Bob is not Maria is different than using it to explain that Bob is no Einstein.

This distinction between the singularity and particularity of a form due to its reception in and composition with matter corresponds to a distinction Aquinas makes more generally between extensive and intensive infinity. Extensive infinity pertains to number and quantity, and intensive infinity to quality and perfection.

In qu. 2, ar. 9 of the De veritate, Aquinas distinguishes between the ordinary use of the term ‘quantity’ to refer to extension, and a metaphysical use of it to refer to degree of perfection. Arguing that “[n]othing prevents something from being infinite in one way and finite in another way,” he posits the heuristic example of an infinite white body. He says that the whiteness of the white body might be called ‘infinite’ because of the infinite extension of the body, “but its quantity per se, that is, its intensive quantity, would nonetheless be finite, and this would likewise be the case for any form of an infinite body, for every form received in any matter is limited according to the mode of the receiver, and so does not have infinite intension.”‘ In short, the infinite white body would be infinitely extended, but not infinitely white.

Aquinas thus distinguishes between an infinitely extended quantity and an infinitely intensive formality. As in Summa theologiae la 7.1 above, he distinguishes between the extensive infinity of matter as quantifiably or numerically divisible, and its privative infinity as formally or qualitatively determinable. On the one hand, a lump of clay is divisible into two parts, and as thus divided can be the principle of numerical individuation for two figures; on the other hand, the lump of clay can also be formed into a variety of shapes. Likewise, a tomato plant can convert the same quantity of nutrients into two tomatoes or into a tomato and leaves. All this is possible only because the matter in each case has neither a determinate quantity nor a determinate form in itself, although in actuality it always has some determinate quantity and some determinate form (whether natural or artificial).

This distinction between the individuation of a form and a limitation of its formal amplitude points to an important duality in Aquinas’s use of the term ‘form.’ On the one hand, he uses the term to refer to an entitative principle of things, their principle of quidditative identity: a human being’s rational soul is its ‘form.’ On the other hand, he also uses ‘form’ to refer to its nature, essence, or kind as such: a human being’s human nature is its ‘form.’ ‘Human nature,’ moreover, allows of two different kinds of conception. For example, on the one hand we can talk about what is true of ‘a human being’ or ‘the human being’: “A human being has an organic body and a rational soul,” or “The human being is a rational animal.” On the other hand, we can also talk about the ‘humanity’ or ‘humanness’ of human beings: “The crime was an attack on their very humanity, indeed on the humanity of us all.”

There are different kinds of abstraction at work here, and it would require too long an excursus to give Aquinas’s account of them. Suffice it to say that, according to Aquinas, to conceive of ‘a human being’ or ‘the human being’ is to consider what is common to human beings in a way that leaves their individual mode of existing implicit even if indeterminate or generalized, whereas to conceive of formalities such as ‘humanity’ or ‘rationality’ is to positively prescind from the individual mode of existing of human beings in actuality. For Aquinas, what there are in the world are many human beings, each of which is human in virtue of their own human form (i.e., rational soul). The formality of ‘being human’ (ratio hominis) that is multiple in these individuals is manifested as unitary in the consideration of a mind that attends to what all these human beings have in common while abstracting from what individuates and distinguishes them.” Accordingly, in the view of Aquinas, human nature is in one way multiple in the human beings that have it in common, and in another way unitary in the minds that grasp it in itself.

I will call the kind of non-precisive abstraction that gives us the concept of “the human being” generalization, and the kind of precisive abstraction that gives us the concept of “humanness” universalization, as the distinction in terminology will prove useful later in considering Aquinas’s claim for the infinity of existence. The implications of this twofold abstraction are also important for his argument for the infinity of God’s power in Super libros Sententiarum, lb. i, ds. 43, qu. i, ar. i. In this passage, Aquinas refers to Aristotle’s distinction in the fifth book of the Metaphysics between the senses of ‘limit’ as it applies to quantity and as it applies to essence. Addressing himself to the latter, he gives the following account of the relative infinity of form: “For every form if abstractly considered has infinity in its own proper formality [ratio]; as in whiteness abstractly considered, the formality of whiteness is not limited to anything, but the formality of color and the formality of existing is determined in it, and it is contracted to a determinate species” (italics added).”

We have seen that the ‘form’ of ‘whiteness’ in the sense of a principle of quidditative determination in white things can be called infinite in the sense that it is undetermined in its nature to this or that particular thing. But the formality of any such ‘form’ of ‘whiteness’ can also be considered in itself and relative to more universal formalities, for example ‘color.’ Aquinas thus moves between considering real relationships between principles of beings, namely the form and matter of things, to logical relationships among formalities, namely of ‘whiteness’ and ‘color.’ That is to say, he moves from considering the indeterminacy of the color white relative to white things, to considering the indeterminacy of ‘color’ relative to ‘whiteness.’ Though moving from real to logical relations, he remains in the order of essence, nature, quiddity, or ‘whatness.’

Four Global Distinctions

At this point, I should explicitly formulate four globally important distinctions implicit in Aquinas’s various treatments of infinity. The first is between two modes of consideration, between an entitative consideration of beings and their principles, and an absolute consideration of being and principles of being. My terminology here is based on Aquinas’s own usage, but it is not his own terminology. For example, in the De substantiis separatis, he distinguishes between a consideration of matter in itself (secundum se) and as it exists in actuality (in actu). In his commentary on the Sentences, he describes the ‘absolute’ consideration of a perfection as a consideration of it without regard for any ‘mode.’ By ‘mode’ I take him to mean ‘mode of existing.’

To consider the color white absolutely is to consider it in itself, denoting it without regard for its various modes of existing in actuality (for example, in a white statue, in white light, in the retina of an eye beholding it, in the mind of a person conceiving of it, or in the power of what can produce it). To consider the color white entitatively, as it exists in entities, is to connote its modes of existing in actuality. It is the difference between thinking of ‘whiteness’ and of ‘being white,’ or between ‘humanity’ and ‘human being.’

This distinction between two modes of consideration, namely absolute and entitative, dovetails with a distinction between two orders of being to be considered, the order of essence and the order of existence. When Aquinas in Summa theologiae Ia 7.1 predicates a privative infinity of the matter of material things and an intensive infinity of their forms, at issue are quidditative determinations. When he alludes to the reciprocal limitation of forms and matter in things, he throws a sideward glance at a second item of analysis in a second order of consideration, namely the actual existence of those form-matter composites. To say that Socrates is constituted of a rational soul and an organic body is to speak of his essence or what he is, to argue about whether his rational soul is capable of subsisting apart from his body is to speak of its existence or how it exists, whether in dependence on his body, as his color, weight, and size exist, or not.

A third global distinction that needs to be drawn is between two kinds of real properties of things. It is important to understand that the relative infinity which Aquinas ascribes to principles of finite things considered in themselves is not merely a logical infinity. Even though he holds that both material things and their principles are in actuality always finite, and that the correlative infinity of these principles is evident only in the mind, it is nonetheless real. By ‘real’ I mean that it signifies something about the thing being considered, and not simply about our way of considering it.” It is true that Aquinas also ascribes a merely logical infinity to concepts, as when he says that the specific formality ‘whiteness’ limits the generic formality of ‘color.’ This entails a merely logical distinction because there is not one reality in things, ‘whiteness,’ that we relate to a second reality in things, ‘color’; rather we logically consider the one reality in things in two different ways when we call it ‘a color’ rather than ‘white. The distinction is between our modes of consideration only and not the objects of consideration. By contrast, when we distinguish between the color whiteness and the body that is white, we distinguish between distinct realities that exist in composition with one another as well as in separation from one another in actuality.

Similarly, although the infinity of matter and of material forms can be known only by considering them as they are in themselves and not as they exist together in actuality, what we are considering is what is true of the really distinct principles of things. In fact, what is really true of them in themselves must be understood if what is true of them as they exist in actuality is to be understood. For example, we call fluid those material substances that, having no shape in themselves, have their shapes determined by something else. In actuality, there are no fluids without some determinate shape. However, by considering what is common to all fluids apart from what distinguishes them, we grasp conceptually what is true about all fluids in themselves: namely, that in themselves they are without shape, and it is precisely this feature of theirs that explains how it is that in actuality they always have a shape determined by something else. The intrinsic shapelessness of fluids is not reducible to the logical indeterminacy of our concept of fluids; rather, our concept expresses a feature of the fluids, although one never manifest in actuality, but only in thought.

Similarly, the relative infinity Aquinas predicates of matter considered in itself and of material forms considered in themselves is not reducible to the indeterminacy of our way of considering matter and form; rather this relative indeterminacy belongs to them in themselves and explains how they exist in actuality with one another. It is only because matter is really infinite in itself that the same quantity of matter can at different times take on opposite forms, as the matter of a piece of wood can become that of ash, or as a white body can turn gray; and it is only because the formality of a form is infinite in itself that it can be common to multiple forms or realized to different degrees in different individuals, as humanity is common to Socrates, Plato, and Aristotle, or a tan deeper in the one than in the others.

The fourth and final distinction to be drawn is between two kinds of determinations of things: perfective determinations in which some actuality is conferred on a receiver, and restrictive determinations in which some actuality is limited by a receiver. This distinction was at the basis of Aquinas’s contrast between form’s perfective limitation of the privative infinity of matter and matter’s restrictive limitation of the intensive infinity of form. We shall see that he will also make use of the distinction in drawing contrasts between the intensive infinity of form, the intensive infinity of existence considered in itself, and the intensive infinity of God.

The Infinity of Existence Considered Absolutely

Let us return to Aquinas’s argument for the infinity of God in Summa theologiae Ia, qu. 7, at. I. Having argued for the intensive infinity of a material form’s quidditative perfection considered absolutely, Aquinas analogically extends this notion of intensive infinity to the perfection of existence considered absolutely, calling existence “the most formal” of all perfections. He then goes further and analogically extends this notion of the infinity of existence considered absolutely to the actual existence of God:

That which is of all things the most formal is existence itself, as is apparent from the above. Therefore, since the divine existence is not existence received in anything, but is itself its own subsisting existence, as shown above, it is evident that God himself is infinite and perfect. This argument, with its two rapid analogical transpositions, presupposes the results of a previous argument that God has neither form nor matter as a principle of his being, nor an existence distinct from and received in an essence, but rather is pure existence.

The first line of the above quotation is the crux of this argument for the infinity of God’s subsisting existence and requires close analysis: “That which is of all things most formal is existence itself” (Illud autem quod est maximeformale omnium est ipsum esse).” Having earlier in the passage identified form as the perfection of matter, Aquinas now identifies existence as the perfection of form. In so doing, he introduces a new sense of perfection and so a new order of being to consider.

He has jumped from the order of essence to the order of existence. He draws an analogy between these two orders: just as a form communicates a quidditative determination to matter in the order of essence, so does existence communicate a determinate actuality to form in the order of existence. This passage thus points us to another relative sense of infinity in Aquinas’s metaphysics, the infinity of existence as such, considered absolutely.

“Existence as such considered absolutely is infinite: for it can be participated in by an infinity of things and in an infinity of ways.” “This formulation of the intensive infinity of existence as such (ipsum esse) from Summa contra gentiles 1.43 can be juxtaposed with another formulation in De veritate 29.3, where Aquinas says that the perfection of existing “can be varied in diverse things in infinite ways.”‘ This concept of the twofold infinity of existence considered as such, namely its infinite possibility of participation by an infinity of things and in an infinity of ways, requires close analysis. All the distinctions we have seen so far will serve us here: the distinction between extensive and intensive infinity; between absolute and entitative modes of consideration; between the orders of essence and of existence; between real and logical properties; and between perfective and restrictive determinations.

Let me begin with things and their principles by developing Aquinas’s bipolar analogy between their forms and their existence on the one hand, and their forms and their matter on the other hand. On the one hand, just as the form of a thing perfects its matter by conferring a quidditative perfection on the matter that the matter does not contain in itself, so too the existence of a thing perfects its form by conferring on it the actuality it does not have in itself. On the other hand, as the form in turn is restricted to the singular mode of the matter that receives it, so the existence is restricted to the quidditative mode of the form that receives it.

Aquinas sketches this bipolar relation of form with matter and with existence in the following passage of the De substantiis separatis:

[T] hose things which participate existence from the first being participate existence, not according to the universal mode of existing as it is in the principle, but rather particularly, according to a certain determinate mode of existing which belongs to this genus or this species. And each thing is adapted to one determinate mode of existing according to the mode of its substance … [T] herefore a thing composed from matter and form through its form becomes participative of existence itself from God according a certain mode of its own.”

Aquinas first compares God and creatures with respect to their existence: he contrasts the universal mode of God’s existence with the particular mode of creatures’ existence. He then represents the genera and species themselves of created things as having determinate modes of existing. But he moves quickly back again to individual things: each thing is adapted to a determinate mode of existing according to the mode of its substance. He then applies the claim to the individual principles of individual things: it is through its own form that the material composite comes to participate in existence, to possess an existence of its own, according to its own proper mode.

Aquinas thus moves among three kinds of analysis in this passage: he moves from comparing the modes of existing proper to created and uncreated being, to comparing the modes of existing proper to the species and genera of created beings, and finally to considering the mode of existing proper to an individual created being. All these various kinds of ‘modes’ are modes of existence. The backdrop to all these comparisons is existence as such.

But how does one move from considering beings and their principles to conceiving of existence as such? Just as the relative infinity of a material form’s quiddity is grasped only when it is considered in itself, so the relative infinity of existence is arrived at only when it is considered in itself. Just as one grasps a common formality by considering what beings of a like form all have in common without considering what individuates them, so one grasps the formality of existence by attending to what is true in common of the existence of all finite beings without considering what differentiates them. In the case of material form, what is left out of consideration is the individuating effects of matter; in the case of received existence, what is left out of consideration is the differentiating distinctions of essence. In a word, all the different kinds of things that exist have in common that they have existence, in however different ways.

As in the case of form and matter, the infinity of existence considered in itself is relative, relative to the essences of the beings that have existence. Furthermore, as in the case of form and matter, this intrinsic infinity of the received existence of finite things is real even if never actual in things themselves. The perfection of existence is infinite in itself and not simply as conceived of indeterminately by us. Its intrinsic infinity must in fact be understood in order to understand how there can be both many and different finite beings. Moreover, it is necessary in Aquinas’s metaphysics for proving that any finite being as a determinate kind of being is not identical with its existence and so has its existence caused by an infinite being, namely God.

There are two crucial disanalogies, however, attendant on the analogy I have been drawing between the relative infinity predicable of the matter and forms of material things and the relative infinity predicable of their existence. First, in the case of form and matter, the two principles are really distinct in such a way as to combine to form a third thing, the material composite. Socrates, for example, is identical neither with his soul nor with his body, but is the composite of both. In the case of a thing’s essence and its existence, however, what results from their union is not a third thing, but only the existing thing. This is the case because a thing’s existence is not merely the primary perfection in the set of its perfections; it is rather the perfection as such of its whole essence, the actualization of the thing as a whole, of all its perfections and operations. It is in this sense that it is infinite relative to them, being at once the actualization of all of them. It accordingly transcends and can thereby extensively embrace their whole order of quidditative determination.

The second crucial disanalogy follows upon the first. It is precisely because every thing’s actuality embraces and permeates all that it is that the order of existence embraces and permeates the whole order of essence. As a consequence, the perfection of existing does not allow of the kind of precisive abstraction that essences allow of Somewhat paradoxically, any absolute conception of existence cannot be like the concept ‘humanity’ or ‘humanness,’ but only like the concept of ‘a human being’ or ‘the human being.’ Speaking strictly, one cannot universalize individual acts of existing in the way that one can universalize individual forms. Rather, one can only generalize the individual act of existing. That is to say, one can leave indeterminate in one’s consideration the multiple things that exist or the multiple ways of existing, but one cannot positively prescind from them altogether without falsifying the conception.

The conception of all actuality must include all that is related to actuality, so that an adequate conception of existence cannot positively exclude anything at all. If existing allows of ‘a concept’ at all, it will be the fullest of all concepts, not the thinnest. To suggest an analogy for a conception of being, if I say, “Seeing is believing,” I have not precisely abstracted from who sees and believes, nor from what is seen and believed. Considering these acts ‘generally’ or ‘universally’ leaves indeterminate but still implicit and cointended the subjects, objects, and conditions of seeing and believing. Similarly, conceiving of being ‘generally’ leaves indeterminate but still implicit and cointended the subjects of existing: “Being a being means having existence in some way.” Conceiving of existing ‘universally’ leaves indeterminate but still implicit and cointended the ways of existing: “Existing means being actual in some way or in some way related to what is actual.”

Thus far we have considered the infinity of existence considered relative to things that participate in existence. But Aquinas also calls existence infinite considered relative to the ways in which existence is participated in by things: “Existence as such considered absolutely … can be participated in by an infinity of things and in an infinity of ways.” These ‘ways’ of participating in existence refer to the perfections of things considered absolutely, for example, ‘life,’ ‘wisdom,’ and ‘strength.’ In Summa theologiae, Ia 4.2, Aquinas reduces all such perfections of things to ways of having existence. “The perfections of all things pertain to the perfection of existence, for in whatever respect they are perfect it is insofar as they have existence in some way.” Aquinas does not have in mind here existence conceived of entitatively as the individual act of existing, but rather existence conceived of in itself or absolutely, according to the formality common to every act of existing. Thus conceived, existence intensively embraces within itself all other perfections. Thus conceived, it is the preeminent perfection not merely as the first of all perfections but as the plenitude of all perfections: “[E]xistence understood simply, as including in itself every perfection of existing, is preeminent over [the perfection] of life and all subsequent [perfections], for in this sense existence itself already possesses in itself all the subsequent [perfections].”

One might try to illustrate this intensive infinity of existence with an analogy with the visible spectrum of light. White light is not simply the primary color of light among the colors of light, nor some universal color potency determinable to other colors by the addition of positive determinations it does not contain in itself. Rather, white light contains intensively and so virtually within itself all the other colors of light, which can for this reason be educed from it by way of refraction, that is, by way of restrictive determinations rather than perfective ones. In a similar way, according to Aquinas, the perfection of existence considered absolutely contains within itself all other perfections of being, which considered absolutely are participations, refractions, restrictive determinations of existence. In sum, all subsequent perfections are but determinate ways of having existence.

The infinite plenitude of existence Aquinas has in mind is not God, nor some Neoplatonic hypostatization, nor some universal predicate. Then what exactly is it? It signifies what it means for anything to be. For Aquinas, for anything to be means for it to be actual. The consideration of what it is for anything to be actual leaves nothing whatever out of consideration. What is ‘common’ to all that there is, first of all, is that it in some way has existence. According to Aquinas, that is what it means to call anything a being, and conceiving of something as a being is the prerequisite of conceiving anything else about it. The fact that everything that exists is one thing, distinct from all other things, or that each is something, distinct in kind, is also ‘universally’ true. Aquinas’s expressions ens commune and ens universals refer neither to some common reality in all things nor merely to some common intelligible content of all concepts, but rather to what is commonly and universally true of all that there is.

These judgments about what is commonly and universally true of all beings form the basis of a conception of their ‘existing’ that leaves indeterminate whether it be this man’s existing or that one’s, or human existence rather than vegetative, or a substance’s existing rather than an accident’s, or of something’s existing in potency, or only as a possibility, or as impossible in itself but nonetheless conceived of by us. To grasp that it is universally true that every being has some essence actuated by some existence of which it itself cannot be the cause is to grasp at once what is most common about all beings and most proper to each being. To assert, “Being a being means having existence,” is to posit every and any being that has existence, and each one in its complete actuality. It is to speak of being in general, or ens commune. To go on to conceive of the existence itself they possess, leaving the possessors and their ways of possessing it indeterminate, is to conceive of existence simply, or else commune.

For Aquinas, the conception of existence in itself or esse commune has a universal comprehension and extension. In the following quotation of the passage in De veritate 29.3 quoted earlier, Aquinas speaks of the infinite ‘variation’ of this universal formality in diverse things:

Therefore, with respect to the formality of existing [ratio essendi], there can be nothing infinite except for that thing in which is embraced all the perfection of existing, which can be varied in diverse things in infinite ways. And in this way God alone is infinite according to essence, for his existence is not limited to any determinate perfection, but embraces within itself every mode of perfection to which the formality of being [ratio entitatis] can extend. However this infinity can belong to no creature, for the existence of any creature is limited to the perfection of its proper species.

Aquinas asserts that God’s infinity can belong to no creature. In doing so, he indicates an important contrast between the infinite variability of created existence and the infinite actuality of the creator’s existence. The infinity of created existence is incommensurable with the infinity of divine existence.

Aquinas does not make explicit in the above passage that God not only embraces the whole formality of created being, but infinitely more besides. The above passage tempts one to consider the formality of existence as common to creatures and to God, and to regard divine infinity as the preeminent mode of this formality. However, as we shall see, for Aquinas the analogical similitude between God and created beings is even more distant than the analogical similitude among created beings. The formality of existence as we conceive it is the formality of beings that have their existence from a creator, and the infinity of this formality consists in its infinite possibility of participation and variation in such beings. As the cause of such received existence, the creator intensively embraces all that can be participated in by such beings, but he also actualizes all that is unique to subsisting existence and cannot be participated or communicated. In his commentary on the De divinis nominibus, Aquinas explains that although created existence itself is infinite relative to creatures, it is not infinite relative to untreated existence; it falls short of God’s infinite existence, having its own distinct intelligible determination. Nevertheless, in De potentia 1.1 Aquinas explains how the existence of all creatures conceived of collectively “as something complete and simple but not subsisting” can provide us with the analogue for conceiving the divine infinity that transcends all created existence.

For Aquinas, ipsum esse commune refers to the participated existence of finite beings considered absolutely, that is, considered in itself or according to its common formality. The infinity of existence so considered is not merely the logical indeterminacy of the conception of existence relative to the individual beings from which the conception is formed, but refers to its infinite possibility of multiplication and variation, to its intrinsic possibility of being participated in by an infinite number of beings in an infinite variety of ways: “Existing as such, considered absolutely, is infinite because it is possible for it to be participated in by infinite [things] and in infinite ways.”

In this absolute way of considering ‘existing,’ it reveals itself as infinite in three respects: relative to the actual beings that are participants in it (e.g., human beings), to the perfections according to which they participate in it (e.g., rationality), and to the ways they participate in it (e.g., subsistence). For Aquinas to say that ‘participatable’ existence is infinite is not simply to say that another being or way of being can always be conceived; rather, there really can always be another being or way of being, and for this reason another can always be conceived. How this infinite possibility of having existence is grounded in the creative power of God or the procreative power of already existing creatures is matter for another analysis. Suffice it here to say that when Aquinas asserts that existence is in itself infinite, he means that no finite being and no form of finite being exhausts the possibility of finite existing.

Actual Infinity among Creatures

The notion of infinity common to created existence, forms, and matter is the relative indeterminacy of principles of beings. Aquinas also predicates an actual infinity of some beings. We have seen him predicate it of God’s subsisting existence; but he also predicates an actual infinity of the subsisting forms of immaterial or angelic substances.

In the following passage of the Quaestiones de veritate, Aquinas enumerates the kinds of infinity found ‘among things’:

[A] mong things we find something which is simply and in all ways infinite, as is God; and something which is in all ways finite, as are material things; and something which is in a certain way finite and in a certain way infinite, as any immaterial substance is indeed finite inasmuch as it has existence limited to a proper nature, in that no created substance, even a material one, is its own existence, but rather participates in existence; however it is infinite because of its freedom from that limitation according to which a form is limited by being received in matter, since everything received is in the receiver according to the mode of the receiver.

Note that in considering actual beings, Aquinas here states that material things are in all ways finite, even though considering the two principles of material things absolutely in Summa theologiae Ia 7.1, he states that they are relatively infinite. In any case, the explanatory crux of the passage is the principle enunciated in the last sentence, that everything received is received according to the mode of the receiver. As a material form is limited to the mode of the matter that receives it, so existence received by an immaterial form is limited to the mode of the form that receives it. As we have seen, the matter limits the form’s quiddity in number and degree, and the existence is limited according to the form’s formality and mode.

We have already looked closely at Aquinas’s argument for the infinity of God in Summa theologiae Ia 7.1. In the following article, 7.2, he contrasts the absolute infinity of God with the qualified infinity of a subsisting form’s quiddity. Invoking the opinion that biblical angels are immaterial substances, Aquinas concurs that in contrast to forms received in matter, such forms are “infinite in a certain respect, insofar as the forms are not terminated nor contracted by any matter.” He states that such a form is however itself a limitation of existence, the perfection as such of every actual form: “But because a form thus created as subsisting has existence and is not its existence, it is necessary that its very existence be received and contracted to a determinate nature. It cannot therefore be infinite simply.”

Insofar as the angel’s form is not received in any matter but rather is subsistent, its specific quiddity is not limited to any particular subject nor to any particular degree of its natural perfection. In this respect, the angelic form is both unique in its kind and infinite in the degree or completeness of its own quidditative perfection. This ‘reflexive’ infinity is however a qualified infinity, because although the angel’s quidditative perfection is not limited to or by anything else, it is itself a limited way of having existence. Accordingly, on the one hand, the qualified infinity of the angel’s form is due to its being a subsisting form; on the other hand, the finitude of the angel’s existence is due to its being a received existence. The angel’s existence is limited to the angelic form as to a particular subject of existence, and to its quiddity as to a particular mode of existence.

In all the cases of infinity that we have seen-that of matter, of material form, of participated existence, and of subsisting forms-what in every case entails limitation is the otherness of receiver and received. What is received is in itself infinite relative to how it exists in the receiver. The individuality of the receiver is constituted by a twofold otherness: it is other than what it receives and other than other receivers. The wisdom of Socrates is identical neither with all human wisdom nor with the wisdom of Solomon. No actually received perfection can be infinite because to be received is precisely to be appropriated, to be bound to the receiver. Aquinas maintains that “the perfection of one thing cannot be in another according to the determinate existence which it had in that thing.”

What is received is determined to the receiver’s existence and mode of existing. Aquinas thus highlights the metaphysical ambivalence of ‘determination.’ On the one hand, principles of determination insofar as they articulate are principles of perfection: a perfection is a certain way of having existence. On the other hand, insofar as principles of determination receive, divide, appropriate, or are other than what they diversify, they are principles of limitation and so of ‘imperfection.’ To predicate infinity of God absolutely, Aquinas needs to deny of God all the various imperfections of determination: the quidditative privation of matter; the quidditative limitations of forms; and the existential restrictions of acts of existing that inhere in forms or natures. At the same time, he wants to analogically extend determination to God insofar as it signifies perfection, actuality, and distinction.

The Infinity of God

In the fifth reply of qu. 2, ar. 2, of the De vegetate, Aquinas draws a distinction between the privative infinity of matter and the intensive infinity of God: God is not called infinite privatively, in the way that infinity pertains to quantity, but rather negatively, “because his essence is not limited by anything, for every form received in anything is bounded according to the mode of the receiver; and so since the divine existence is not received in anything, he is his own existence; accordingly his existence is not finite and for this reason his essence is called infinite.” Aquinas thus makes sense of the notion of an infinite essence through a mode of existing, namely subsistence. He constructs this concept of subsistence in the order of existence on an analogy with subsistence in the order of essence.

In the order of essence, any formal perfection considered in itself is infinite in its kind. Moreover, any form that actually subsists is actually infinite in its kind because its formal perfection is unreceived in and therefore unlimited to anything else, as in the case of angelic substances. From these qualified and reflexive senses of quidditative infinity among finite beings, Aquinas argues analogically to an unqualified sense of quidditative infinity in God. Because God’s essence is simply his existence, the infinity of plenitude predicable of a subsisting form’s essence is predicable of the very existence of God. Moreover, because his essence is simply that perfection which is the intensive plenitude of all essential perfections, the infinity of his essence is not merely reflexive, but unqualified and absolute. Furthermore, the real distinction between what exists and its mode of existing collapses in the case of God, that is, it becomes a merely logical distinction. As God’s existing is not really distinct from his essence, neither is his mode of existing. In a way, it is too imprecise to say that God is existence itself; a precise formulation of Aquinas’s concept of divine being necessarily includes the divine mode of existing: God is subsisting existence, absolute existence, infinite existence, pure actuality.

Beginning with the Summa contra gentiles, Aquinas employs formulations of God’s unique existential infinity in terms of infinite actuality as well as infinite quiddity. The following formulation of the Summa contra gentiles draws the same analogy with a subsisting form’s infinity as the above passage of the De veritate, except in terms of act:

Every act inhering in another receives termination from that in which it is: for what is in another is in it according to the mode of the receiver. Therefore an act existing in nothing is terminated by nothing: for example, if there were a whiteness existing per se, the perfection of whiteness in it would not be terminated, inasmuch as it would have whatever of the perfection of whiteness can be had. But God is an act in no way existing in another: for neither is he a form in matter, as has been proven; nor does his existence inhere in any form or nature, since he is his own existence, as was shown above. What remains is that his existence itself is infinite.

It is noteworthy that instead of appealing to what he thinks are actual cases of subsisting forms, namely angelic substances, Aquinas here employs a Platonic figure he thinks an actual impossibility to formulate exigencies of pure form that he can in turn use to construct an analogue of pure act. He first isolates the formality of whiteness from any mode of its existing or inherence in something else—“whatever of the perfection of whiteness can be had”—and then considers the theoretical possibility of its subsistence. Such a subsisting whiteness (were it possible) would actualize infinitely what the formality of whiteness constitutes as a distinct perfection, a determinate possibility of existence or determinate way of having existence. God’s subsisting existence is unqualifiedly infinite because it is the perfection whose quidditative intensity is unqualifiedly infinite.

In the Summa contra gentiles, Aquinas argues that God is infinite in goodness because of how he exists: “There can be no mode, nor can one even be conceived, in which any perfection may be more fully possessed than [the mode] of that which is perfection through its essence and whose existence is goodness. This however is God. Therefore in no way can anything better or more perfect than God be conceived. Therefore he is infinite in goodness.” In the De veritate, he establishes that God is infinite in essence because what he is comprehends all that can be: “[O] nly God is infinite according to essence, for … he contains in himself every mode of perfection to which the formality of being can extend itself.” In sum, the orders of essence and of existence converge in the unqualified infinity of God. Conceiving of God’s existence as unqualifiedly infinite thus requires a twofold negation: with respect to his mode of existing, the negation of the imperfection of inherence in a limiting subject; and with respect to what he is, the negation of the imperfection of any formal restriction. According to Aquinas, God is infinite both in what he is and in how he exists, both in quid sit and quomodo sit, both because he is the infinitely intensive perfection and because he is that perfection infinitely. In a word, “the essence of God is infinite in all ways.”

Aquinas thus analogically extends the unicity of a pure created form in the order of essence to existence, the actuality of all things and of the forms themselves: “Whatever per se subsisting created form might be posited, it must participate existence … Participated existence is limited to the capacity of the participant. Therefore only God, who is his own existence itself, is pure and infinite act.” These formulations of divine infinity in terms of act and actuality are crucial for Aquinas’s analogical concept of infinity because they allow him to explain the distinction of the divine existence from the created existence of finite beings without any appeal to an individuating principle: “God is existence itself subsisting by itself [ipsum esse per se subsistens]. And it has been shown that subsisting existence cannot but be one; just as if there were a subsisting whiteness it could not be one, since whitenesses are multiplied according to recipients. The result therefore is that all things other than God are not their existence, but participate existence.” Thus the very subsistence of God’s existence renders him both distinct, unique, and infinite. The unparticipated is of its very nature individual.

It must be said, however, that Aquinas’s use of this analogy is perilous for him. It can seem that a God who is ipsum esse subsistens is but a hypostasized mode of ipsum esse commune. It can seem like the infinity of esse commune is but the logical indeterminacy of a concept and the infinity of God the entitative analogue. It can seem as though infinite and finite existence constitute two modes of the one transcendental perfection of existence. These would all be misinterpretations of Aquinas’s analogical concept of infinity. To avoid them, one must distinguish between two analogies of being: an analogy of being among finite beings based on their participation in the common formality of ipsum esse commune, and an analogy of being between finite beings and their common cause, ipsum esse subsistens, based on the participation of these effects in their transcendent cause.

Participation in Esse Commune VS in Esse Subsistens

A passage from the Summa theologiae will help to make the necessary distinction between the participation of finite beings in esse commune and in esse subsistens. In Prima pars 4.3, Aquinas describes degrees of causal similitude. His premise is that effects necessarily resemble their cause, because agents act insofar as they are actual, and they are actual insofar as they have existence, and they have existence in accordance with their forms. Since they have being through their forms, they communicate being in accord with their forms. Accordingly, the degree of causal resemblance will follow upon the degree of communication.

Aquinas identifies three kinds of causes and degrees of causal similitude among finite beings. First, sometimes an agent produces an effect which resembles itself according to the same formality and mode (as when a human parent generates a human child, and both are called ‘human’). Second, sometimes an agent produces its similitude in its effect according to a same formality but in a different mode (as when fire heats a pot, and both are called ‘hot,’ the one essentially and the other by communication). But, third, sometimes the causal similitude involves neither the same formality nor the same mode, but only an analogically similar formality, because the agent and its effect belong to different genera (as when artists and their artwork are both called ‘intelligent’).

Now because substances give existence to accidents, accidents are called ‘beings’ analogically in this third way. Aquinas uses this analogy among all the ways of being a being based on their causal relationship to the primary way of being, namely being a substance, to construct a still more remote analogy between all finite beings and their ultimate, universal, and transcendent cause:

If there is some agent that is not contained in any genus, its effect will approach the similitude of the agent even more remotely, not in such a way as to participate in the similtude of the agent’s form according to a same formality of species or of genus, but according to a certain analogy, as existence itself is common to all things [sicut ipsum esse est commune omnibus]. And in this way those things that are from God are, insofar as they are beings, made like him, the first and universal principle of all existence.

Aquinas draws a meta-analogy between the analogy of being among finite beings based on their participation in esse commune and the analogy of being between all beings and God, the universal efficient cause of esse commune.

With regard to these two distinct analogies, it is important to note that Aquinas asserts in his commentary on Boethius’s De ttinitate that God is not part of the subject of metaphysics, namely being in general (ens commune), but rather God enters into the science only as the terminus of its investigation, as the ultimate principle and cause of the subject matter. Likewise, he writes in his commentary on the De divinis nominibus that God does not depend on else commune, is not contained under it, and does not participate in it; rather, it depends upon him, is contained under his power, and is a participation in him. In short, God does not participate in ens commune or in esse commune.

Aquinas constructs his meta-analogy between finite and infinite being thus: just as the absolute conception of existence (esse commune) comprehends whatever does or could participate in existence as the transcendental actuality of all things, forms, and perfections, so God (esse subsistens) as the transcendent cause of the infinite possibility of finite existence intensively comprehends in himself and surpasses the whole formality of finite existing and finite being:

[W]hatever of perfection is in any creature, all preexists and is contained in God in a surpassing way [secundum modum excellentem]. Moreover, not only that in which creatures share, namely existence itself, pertains to perfection, but also those things by which creatures are distinguished from each other … And thus, since it contains within itself all perfections, the essence of God is compared to all essences of things, not as what is common to what is proper … but as the perfect act to imperfect act to imperfect acts.

The notion of ‘act’ allows Aquinas to formulate an unqualified divine plenitude of perfection: what is quidditatively common to many stands opposed to what is proper to each of them, while what is quidditatively perfect contains supereminently what is imperfect. Differences or determinations insofar as they are articulations rather than limitations of being must also be ascribed to God. For Aquinas, divine infinity is not the relative infinity of what is common in abstraction from what determines it, but rather the simple and absolute infinity of the perfect being relative to which finite beings are partial, multiple, and imperfect participations. Because infinite subsisting existence is pure actuality, it contains its effects intensively:

[T] he first act is the universal principle of all acts, because it is infinite, virtually prepossessing in itself all things, as Dionysius says. It is therefore participated by things, not as a part, but according to a diffusion of a procession of itself . And received acts, which proceed from the first infinite act and are certain participations of it, are diverse.

Thus actually infinite existence, the cause of all finite existence and of the infinite possibility of finite existence, contains within its creative power that whole infinite possibility of finite existence and infinitely more.

But as God’s existing and essence are in actuality one, so is his power; the distinction among these three is not actual nor real, only logical. To say that he is infinite, that his existence is infinite, that his essence is infinite, that his power is infinite, or that he is infinite existence, are so many ways of speaking of one same thing. This is the absolutely unqualified kind of infinity that Aquinas analogically predicates of God on the basis of the analogically multiple kinds of infinity he discovers among things. In sum, the ultimate analogate of Aquinas’s analogical concept of infinity is the infinity of the infinite cause of infinity in finite things.

Recapitulation

Aquinas’s analogy of infinity as presented here is sufficiently complex to merit a reprise. First, he distinguishes between modes of consideration: considering beings or principles of being absolutely, or what is true of them in themselves, and considering them entitatively, or what is true of them as they exist in actuality. Second, he distinguishes between two orders of being to consider: considering a thing’s quiddity, or what it is, and its actuality, or how it exists, in short, between the orders of essence and existence. The above two distinctions give rise to a third, a distinction among two kinds of properties of things: real properties that pertain to things in themselves but not necessarily to how they exist in actuality; and real properties that pertain to things as they exist in actuality. Fourth, he distinguishes between kinds of determination: perfective determinations that confer on the receiver perfection it does not have in itself, and restrictive determinations that limit the perfection of the received. Aquinas uses these four distinctions globally to predicate infinity of matter in itself, of material form in itself, of existence in itself, of subsisting immaterial forms, and of subsisting existence or God, as follows.

Considered in the order of essence and in itself, matter is called infinite privatively because in itself it lacks any quiddity or formal determination. This infinity is real and not merely logical. Considered relative to form, matter is a principle of quidditative restriction in number and degree: it restricts any form it receives to a singular and particular mode of the form’s essence or formality. In actuality, matter always exists finitely as the finite principle of finite entities, being always quidditatively finite or determinate in virtue of its composition with a finite form. Thus, the material of this log before me is in itself infinite in the sense that it is able both to have once been the material of a living tree and is still able to be the material of ashes; granted that in actuality it will always be the material of something, always determined to being this or that kind of thing by this or that form, its very potential for formally multiple determinations is precisely its being really infinite or indeterminate relative to them.

Material form allows of a complementary analysis. Considered in the order of essence and in itself, any material form is called infinite negatively because, although it is quidditatively determinate, it is nevertheless intensively infinite within its own kind: its quidditative formality is not limited to any subject’s singular or particular mode. This infinity is real and not merely logical, even if never actual. Considered relative to matter, form is a principle of perfective determination, conferring on matter a determinate quidditative perfection it does not have in itself. In actuality, a material form always exists finitely as the finite principle of a finite entity; it is limited through the finite being’s matter to a particular and singular mode of existing. Thus the perfection of being “feline” is not in itself limited to this or that cat, nor to cats rather than leopards, tigers, or lions; although in actuality ‘being feline’ always entails being a determinate kind of feline and a determinate individual feline, no one feline or kind of feline exhausts the full possibility of ‘being feline.’

Aquinas analogically reenacts in the order of existence the above analyses in the order of essence. Considered in itself, existence is called infinite negatively because it is not limited to this or that being, nor to this or that kind of being, nor to this or that perfection of being, nor to this or that mode of existing. This infinity is real and not merely logical. Considered relative to finite essences and principles of finite essence (i.e., form and matter), existence is a perfective principle of determination: it determines a finite essence and its principles to actualization, conferring on them actuality they do not contain in themselves. Conversely, existence is always limited to the essence and essential principles of the being that receives it, to existence of a particular quidditative kind and categorical mode. Finite existence is always the actuality of this thing, this kind of thing, this way of existing. But no single being, nor kind of being, nor mode of existing, exhausts the full possibility of being a being, i.e., of having existence.

There can be more than one finite being, one kind of being, and one way of existing, only because created existence is really infinite in itself, even if never actually so in things. It is really infinite in its intrinsic possibility of participation and variation, a possibility that is infinite in the sense that it can never be fully exhausted. Thus, to be Socrates, to be a human being, to be an animal, to be rational, to be a substance, to be a principle of substance, to inhere in substance, to be conceived in the mind of a knower, to be conceived in the mind of a maker, are all ways of being actual or of having existence. Being actual in some way or other or related to what is actual is the formality common to all things that are in any way said to be. Aquinas’s notion of esse commune is his conception of this transcendentally universal formality of beings. The conception of this transcendental formality is the fullest rather than the thinnest of all conceptions because it is the conception of the existence of things. As the existence of each thing is the actuality of all else that is in it, so the transcendental conception of existence itself extends to all things and to all that there is in things.

Thus far Aquinas’s analogical concept of infinity predicates a relative indeterminacy of finite principles of being: of matter and material form relative to one another, and with respect to quiddity; and of received existence relative to beings, kinds of being, and ways of having being, and with respect to actuality. He next extends this concept of infinity to actually subsisting beings.

Aquinas argues that just as a material form considered absolutely in the order of essence is infinite in its quidditative intension, so is an actually subsisting form. However, because such a subsisting form is actually unlimited to any matter that would restrict the formality of the form to any particular subject or degree of its perfection, its infinity of quidditative intension pertains to it not merely in itself but as it exists in actuality. A subsisting form is accordingly both unique and infinite in its kind: being by nature subsistent and therefore unlimited to or by anything else, it is of itself individual, fully actualizing and therefore exhausting the perfection of its nature, essence, or formality. Although this infinity is absolute and actual, it is nonetheless qualified, for although this form’s own quiddity remains undetermined in itself, it still constitutes a quidditative determination of existence.

God alone is infinite in all ways. He is infinite subsisting existence, existence unlimited to any essence distinct from itself. There is no actual or real distinction between his existence and his essence; the distinction is merely logical, pertaining to our ways of conceiving of his infinite actuality. Likewise, there can be no distinction between an absolute consideration of his existence in itself and a consideration of it relative to something else. He is actually absolute existence. Conceived of by us in the order of essence, he is called infinite in his essence because he actually and intensively includes in himself all essential perfections. Conceived of by us in the order of existence, he is called infinite in existence because he does not have existence but rather is existence subsistently and so unrestrictedly. Both in what he is and in how he exists he is infinite. Moreover, just as he does not have existence but is existence subsistently, likewise he is not distinguished from beings that have existence by himself having some limit relative to them, but rather by being infinitely what they are finitely, by being the supereminently and transcendently intensive plenitude of all their determinations. It is an infinity that is unqualified, absolute, actual, in the first moment of conception negative, but in the final analysis intensive.

In this way, Aquinas moves analogically from the infinity of the principles of finite beings to the infinity of their ultimate cause. Having by degrees constructed this analogical concept of infinity and arrived at length by means of it to a metaanalogate, the unqualified infinity of divine subsisting existence, he grounds the complete meta-analogy on a relationship of causal similitude. If every effect in some way resembles its cause, and if the transcendent cause of all finite being is infinite, it should be no surprise that, in the image of their cause, all finite things are in a certain way infinite.