Friday, June 20, 2014

On Infinite Actual Multitudes


1.  Finally, Thomas considers the possibility of an infinite multitude of things.  This article is very difficult, as it brings in a lot of logical and metaphysical considerations that we have not heretofore encountered.  (As an indication of this, Cajetan’s commentary on this article fills two full folio pages in the Leonine edition.)  Our treatment of it will be relatively brief.  

2.  In support of the possibility of an infinite actual multitude he proposes three objections: First, because potency is potential by virtue of its ability to become actual.  Thus if there is a potentially infinite multitude, then there can be an actually infinite multitude.

2.  Second, because it is possible for any species to be actualized, and there are infinitely many different possible shapes, so that there could be infinitely many actual figures.

3.  Third, because nothing opposes the addition of new members to a multitude, and thus there is no reason to say that one could not go on adding to a multitude to infinity.

4.  For the Sed Contra he cites a familiar verse from the book of Wisdom.

5.  In the Corpus he begins by distinguishing between an absolutely (per se) infinite multitude and an accidentally (per accidens) infinite multitude.  According to Avicenna and Algazel, it is possible for an infinite multitude to exist accidentally but not per se.  The distinction is obscure, and given that he rejects both kinds of infinity I will omit it here.

6.  Thomas’s solution to the question is that every multitude must have a determinate species, and the species of multitude are numbers, which are determinate and finite and measured relative to one.  This argument seems to suffice in itself.  One could add a variety of mathematical considerations, but I find myself incompetent to do so at present.  If one were to bring St. Thomas into conversation with modern mathematics, I think it would be necessary to come to terms with what Kant calls the “quid facti” of modern mathematical notions, i.e. their actual basis in experience and their logical genesis.

7.  He answers the first objection by pointing out that every potency is reduced to act in accord with is own being.  This means that though a given multitude has a certain infinite potency to it, that potency is reduced to act by stages, through the addition of successive determinate multitudes to the original, and is always further extensible.  Infinity signifies a lack of determination, not an absolute quantity.

8.  In his reply to the second, he draws a comparison between the the infinite species of geometrical figures and the infinity of the natural numbers.  Just as we cannot produce an infinite number simultaneously, but only by a successive counting out of numbers, the infinity of possible figures cannot be produced simultaneously, but by succession, so that even though there are infinitely many potential figures, they cannot all be made to exist simultaneously.  

9.  He answers the third objection by pointing out that while no particular species of multitude is opposed to the addition of another to it, the notion of an infinite multitude is opposed to every particular species of number, and therefore it is impossible for a multitude to be actually infinite.

10.  When we examine these arguments, we find that they all reduce to the same claim: that actually existing creatures are determinate in their essences and therefore, if they are material things, determinate in their quantity, since quantity is one of the essential features of matter.  Thus since an infinite thing is indeterminate as to its quantity, nothing infinite can exist, either by way of multitude or magnitude.


- Potency is potential because it can be made actual.  If there is an infinite potency, then there can be an infinite actuality.
- Every species can be actualized.  There are infinite species of geometrical figures.  Therefore there can be an actually infinite multitude of geometrical figures.
- There is nothing contradictory in the notion of a multitude of adding to it up to infinity.

- Every actual multitude must belong to a species of multitude, but multitudes are specified by their number, and determinate numbers are finite, and measured by one.

- Infinite potencies are made actual successively, and are infinite by the fact that they can always be made greater, not because they are ever actually infinite.
- An infinite set of potential species can be generated successively but not simultaneously.

- The notion of an infinite multitude is opposed to every particular species of multitude.

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