Wednesday, 7 April 2010

Question 7 - The Infinity of God

Why this Question Matters.

God is self-subsistent being and is the efficient cause of the being of all created things. It is time for Aquinas to start reeling off the consequences of these facts. This question and the next discuss God’s limitlessness and his omnipresence. These are two facts about God that, by their nature, go together. This question continues the process of purging our minds of false and simplistic ideas about God that see Him only as being rather like us but more powerful and not subject to our weaknesses. We might recall the words of the Fourth Lateran Council (1215 AD) that "between Creator and creature no similitude can be expressed without implying an even greater dissimilitude."


The Thread of the Argument

A1: Aquinas asks whether God is infinite; a word he uses in the sense of “without limit”. As we will see in the third and fourth articles, Aquinas did not like the idea of an actual infinity when considering material things; therefore he starts from the position of considering the infinite in God negatively.

Indeed, Aquinas spends most of the answer of this article talking about material things, which are composites of matter and form. He argues that in such a composition, each of the principles limits the other: form limits matter by determining it to be this type of thing; matter limits form to be the form of this thing. So, for example, the form of a horse limits this matter to be a horse rather than, say, a table; matter limits the form to be the form of this particular horse, rather than of any other horse.

Forms that are not informing matter are therefore, in this sense, unlimited. If we generalize from material forms to the highest form of all, that of being, then we see that God must be without limit as He is self-subsistent being. In God, the form of being is not in-forming anything and is therefore not limited.

A2: Given the way that Aquinas approaches the limitlessness of God in the first article, we might suspect that God’s limitlessness is unique in some important sense. God as self-subsistent being is not limited in any way; it is of God’s very essence to be unlimited. But before we can attribute such essential limitlessness uniquely to God, we must enquire into the possible limitlessness of other things.

If we think about an existing material thing composed of matter and form, we can see that in some sense we can attribute limitlessness to it. For although the substantial form of the thing limits it to be the sort of thing that it is, it can the subject of an unlimited number of accidental forms. So, for example, a lump of wood is in potency to receive any number of accidental forms of shape; the woodcarver’s profession depends upon this.

So, seen from the point of view of matter, things are limited in certain respects but unlimited in others. If we look from the point of view of form, however, we see that material things are completely limited to be what they are; the form is limited to be the form of this thing. What about forms that do not inform matter? Angels are purely spiritual beings and their substantial forms do not inform matter. So, in a sense, the substantial form of an angel is unlimited. However, we have to recognize that even such subsistent forms are not identical with their existence; their existence is limited by the nature that they have.

The conclusion of these arguments is that anything other than God can be unlimited in certain respects but has to be limited in others. Therefore it is only God that is limitless by essence.

A3: This article and the next consider aspects of the possible limitlessness of created things. The next article will consider the possibility of their being an unlimited number of things; here we look at the possible unlimited size of things.

Aquinas argues that a material body considered as a natural object (rather than through some sort of mathematical abstraction) has a determinate substantial form. Following on from a substantial form are a group of accidental forms that such a body has to have; one of these is the size of the body which has a determinate value that has to lie in a range of particular values between a maximum and a minimum. Therefore a natural body cannot be unlimited in size. In support of this, Aquinas points out the problems associated with the possibility of rectilinear or rotational motion for an unlimited body.

Aquinas claims that similar reasoning applies to mathematical bodies; if we imagine such a thing in actual existence, then it too will have a determinate shape that lies within finite boundaries.

A4: Think of a number, N. One can always think of a larger number N+1; therefore the numbers are unlimited. Similarly if one has a collection of N things, then one can always add another to the collection to obtain N+1 things. Therefore it seems that there is no limit to the number of things.

Aquinas makes a distinction between what is called an intrinsically unlimited number of things and a number of things that happens to be unlimited. The former refers to a situation where something needs the existence of an unlimited number of other things in order to exist itself; the latter to there just happening to be an unlimited number of things with no thing depending on this unlimited collection. Aquinas refers to those who are of the opinion that the former is impossible whereas the latter is possible. The problem with the intrinsically unlimited is that in order for the dependent thing to exist an infinite number of other things would have had to have happened prior to its coming to be. Aquinas claims that to traverse this infinite number of actions is impossible. (This is rather like the argument that time cannot have existed forever: for if so, an infinite amount of time would have to have been traversed to get to now.)

Aquinas makes the stronger claim that there cannot even be a number of things that happen to be unlimited. He argues that any collection of things must be subject to a definite enumeration. Since all actual numbers are finite, there cannot be an infinite number of things in a collection. He is willing to allow for the fact that a collection can be potentially infinite in the sense that whatever actual number is in the collection, more can be added to it. The addition of a finite number to a finite number will always remain finite.


Summary and Handy Concepts


  • Form limits matter in the sense of removing its potential to be many things by making it a specific type of thing. Similarly, matter limits form as the form becomes the form of some specific thing.
  • God is unlimited, or infinite, because His form of self-subsistent being is subject to no limitations at all.
  • Created things can be unlimited in certain respects, but cannot be unlimited by essence in the way that God is.
  • In the reply to the first objection to the second article, Aquinas states that although God’s power is unlimited, it is only unlimited in respect to things that are logically possible.
  • In the material world, bodies are finite in extent.
  • Aquinas is willing to consider conceptually the notion of infinity but he rules out the notion of a completed infinity existing in nature. He distinguishes between the unlimited or potentially infinite (think of the integers: for any integer there is a larger one) and the actually infinite (any actual integer is a finite integer).


Difficulties

  • In the first article Aquinas says that “that which is the most formal of all things is esse itself” and “the notion of form is most fully realised in existence itself”. Aquinas’s idea here is that the most fundamental thing that there is, is existence itself. Reference to Ia.q3.a4 and Ia.q4.a3 ad 1 may help clarify these phrases.
  • In the third article Aquinas argues that material bodies are finite in size, supporting his argument by reference to the problems that the motion of an infinite body would present. Modern physicists would be sympathetic to such arguments, but might approach the problems from the point of view of inertia and energy. More problematic, though, is Aquinas’s assertion about “mathematical bodies”. He doesn’t make clear here what he means by such things, but appears to demand that such a thing is, at least in theory, realizable as a physical body. If one thinks about mathematical abstractions from bodies, then the postulation and manipulation of actually infinite things is everyday fare in modern mathematics.
  • Similarly, in the fourth article, although Aquinas has a notion of the potentially unlimited, he is unwilling to countenance the possibility of an actually infinite collection of things. His argument is simply that any collection must have an enumeration and that all actual numbers are finite. In the light of modern mathematical ideas (see our document on Infinity) this argument doesn’t really work. Indeed, we might even argue that the infinite could apply, in principle, to physical things in the following way. One can advance the idea that there might be an infinite regress where bodies are made of atoms that are made of nuclear particles that are made of quarks that are made of more fundamental particles and so on. This may be false as a matter of contingent fact, but it is not clear that it is false in principle.


Revised 31/03/12

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