St. Thomas on mathematical infinity as a sign of the divine existence

A student asked me today whether the infinity of numbers constitutes a certain proof for the existence of God. My initial answer was no, since mathematical infinity is characterized by imperfection and lack. Later I remembered that St. Thomas disagreed:

When our intellect understands something, it extends infinitely. A sign of this is that when any finite quantity is given, our intellect can think a greater one. But there would be no reason for this this order of our intellect to an infinite unless there were some infinitely intelligible thing. And so it is necessary that some infinite intelligible thing exists, which must be the greatest of things: and this we call God. (Summa Contra Gentiles, 1:43)

(n.b. I translated “frustra esset” as “there would be no reason for”. This is a slightly novel traslation, but I think it’s the right one. The Latin 101 suggestion would probably be to translate “frustra” as “vain”, but Modern English simply doesn’t say “vain” as in “It would be in vain”, and so we are alienated from what St. Thomas was saying. I think this translation should be kept in mind when we encounter the common Ancient and Medieval understanding of what it means to be “in vain”, since for them to call something “in vain” should immediately lead to the conclusion that it couldn’t happen.)

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1 Comment

  1. June 4, 2007 at 8:54 am

    You might be interested in this discussion:

    http://realphysics.blogspot.com/2007/02/comprehending-infinite.html

    I think you are correct that the infinity of natural numbers does show the existence of an actual (formal) infinite, which must be God, and the only nuance I would add is that one must be careful not to identify this passive and potential infinity with God Himself. IOW, God’s infinity is not the infinity of natural numbers, but rather, it is the active infinity that makes the passive (potential) infinity of counting possible.


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