(1.7.3)

1. The two remaining articles in this question concern infinity of matter: on the part of magnitude (i.e. size or extent) and multitude (i.e. countable number). These articles are both of interest to those concerned with the philosophy of mathematics and cosmology.

2. In support of the supposition that there can be an infinite magnitude (i.e. a material thing of infinite size), Thomas offers four objections. First, because the concept of an infinite magnitude is used in mathematics, e.g. in geometry when we postulate infinite lines. Second, because there is no contradiction between the nature of magnitude and the notion of limitlessness, and whatever is not strictly contradictory in nature is possible. Third, because a continuum is infinitely divisible, and if something can be divided into infinite parts, it can be made of infinite parts, and thus can be added up to an infinite magnitude. Fourth, because time, being the quantity of motion, receives its quantity from the magnitude of motion, and therefore since it is possible to have an infinite time, it must be possible for there to be an infinite magnitude.

3. For his Sed Contra, he makes a curious argument. He says that every body must be finite, because to be a body is to have a surface, and to be a surface is to limit the extent of a body. I am not sure that this is a valid argument, though perhaps one could salvage it.

4. In the Corpus he begins by distinguishing between infinity of magnitude and infinity of essence. He established in the previous article that nothing but God can be infinite through its essence, and now wants to show that nothing can be infinite in magnitude. He divides his consideration of this question into two parts: first, regarding natural material things; second, regarding the objects of mathematics.

5. In the case of natural material things, he observes that every actually existing material thing has a substantial form, and is thus determined by that form to a particular essence, with particular accidents, etc. Thus material things have a particular place, and a particular quantity. To be infinite is to lack determination as to quantity (since infinity signifies a lack of limitation, and quantities are only determinate by being limited), and therefore no material being can be actually infinite in magnitude, though as we have discussed they can have a kind of potential infinity.

6. He argues for the same conclusion again using the example of motion. Natural material bodies are capable of motion, and he says that an infinite body could not move, since it would occupy every place. This does not seem to be true, since there could be an infinite plane of some finite thickness, still capable of motion. However, he provides another argument from motion, on the basis of the idea of circular motion. Every body is capable of being rotated. But if we have an infinite body, the rotation of the body would require some parts of it to traverse infinite distances, which is impossible. Moreover, if two points in the body are infinitely distant, then they could never be connected to each other, which seems to contradict the idea that they are part of the same body.

7. As for mathematical objects, he observes that mathematical objects are abstracted from particulars, and are therefore potentially real. If one actually existed, it would have to exist under a particular form, with particular accidents, as said above. And thus the mathematical body would have to have a finite quantity for the same reasons just given.

8. In response to the first objection, Thomas points out that geometers don’t deal with actually infinite lines: rather, they take lines that are actually finite and mentally abstract their quantity from them as necessary for the construction or demonstration at hand.

9. To the second he responds that infinity is not opposed to the notion of magnitude generally speaking, but it is opposed to the notion of any particular species of magnitude (i.e. any sort of thing that has magnitude), and thus because infinity is impossible in every species, it must be impossible in the genus, even if the notion of the genus doesn’t exclude it by itself.

10. His response to the third involves a clever distinction. He observes that the division of a whole into parts is the resolution of a thing with one form into the quantity of matter which makes it up. In this way there is a potential infinity, because one can continue to divide the matter however one likes. But when we add one thing to another, each addition increases the form of the whole, and this happens to a determinate extent, because the forms of existing things are determinate. So while on the side of matter there is a potential infinity because of its divisibility, on the formal side there is no actual infinity, because every addition remains an addition of determinate quantity, and therefore finite.

11. Finally, he resolves the fourth objection by pointing out that movement and time are successive, and not simultaneous wholes. Thus though one can speak of the “whole time” or the “entire journey”, only one given moment of that time or that journey is actual at each instant, so that even though a length of time or a path of movement could be potentially infinite, this would not produce any actual infinity.

OUTLINE OF ARTICLE

OBJECTIONS

- Mathematicians use infinite objects.

- There is nothing contradictory about the notion of an infinite magnitude.

- If a magnitude can be infinitely divisible, it ought to be capable of infinity by addition.

- An infinite stretch of time is possible, so an infinite magnitude must be possible as well.

CORPUS

- Actual material things exist in determinate substantial forms with determinate accidents, and therefore have finite quantities.

- This is true also of mathematical objects, if they existed actually and not just in the mind.

REPLIES

- Mathematicians abstract the actual quantity from real objects and consider them only under the aspects convenient to the task at hand.

- Infinity is opposed to every particular kind of magnitude, and therefore must be opposed to magnitude in general.

- The infinite divisibility of a thing is on the part of the potency of matter, but an infinity by addition would require an infinity on the part of form, which is impossible as explained in the corpus.

- Time and movement are successive and not simultaneous wholes: though an infinite time could pass, only part of that time would be actual at any given moment.

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