Aquinas's Concept of Infinity

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. ¹ 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. ² 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. 7, ar. 1, where Aquinas argues for the infinity of God.

1. 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. ³ 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. ⁴

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...