Sumit unus, sumunt mille

An observation on the mathematics of the Most Holy Eucharist

Chesterton was struck, as anyone with intelligence would be, by the remarkable poetry composed by St. Thomas Aquinas. For so much of Aquinas can seem dry if not mechanical, as he carefully goes through point by subpoint, and piles detail upon detail and argument upon counter-argument. Then, like a waterfall in a dry, rocky desert,one finds his compositions which he wrote to adorn the feast of Corpus Christi, about which he had done some of his most complex and careful work:

The Corpus Christi Office is like some old musical instrument, quaintly and carefully inlaid with many coloured stones and metals; the author has gathered remote texts about pasture and fruition like rare herbs; there is a notable lack of the loud and obvious in the harmony; and the whole is strung with two strong Latin lyrics. Father John O'Connor has translated them with an almost miraculous aptitude; but a good translator will be the first to agree that no translation is good; or, at any rate, good enough. How are we to find eight short English words which actually stand for "Sumit unus, sumunt mille; quantum isti, tantum ille"?
[GKC, St. Thomas Aquinas CW2:509]

No, I have not found those eight English words. Let me first show you the complete stanza:

A sumente non concisus
Non confractus, non divisus,
Integer accipitur.
Sumit unus, sumunt mille:
Quantum isti, tantum ille:
Nec sumptus consumitur.
[St. Thomas Aquinas, Lauda Sion, Sequence for the feast of Corpus Christi]

Here is one translation:

Whoso eateth It can never
Break the Body, rend or sever
Christ entire our hearts doth fill:
Thousands eat the Bread of Heaven,
Yes as much to one is given:
Christ, though eaten, bideth still.
[tr. Msgr. Henry, quoted in Britt, Hymns of the Breviary and Missal 179]

From the same book, a more literal translation:

By the recipient the whole (Christ) is received; He is neither cut, broken, nor divided. One receives Him, a thousand receive Him; as much as the thousand receive, so much does the one receive; though eaten He is not diminished.
[Ibid. 182]

Rather, I would like to explore the idea which Aquinas sets forth within these words, suggested by the actual Latin itself.

For, it is well-known to mathematicians that there is something which can be divided into an infinite number of pieces, each piece infinitely large, and yet still not exhausting the original.

That something is the whole numbers or counting numbers, also called the integers: 1, 2, 3, and so forth. (We don't need zero or the negatives for this particular discussion.) We call this set I.

Note: if you missed the pun, look at the first word of the third line in the Latin: Integer = "the whole".

Here are the assumptions we shall need in order to show this mathematical truth:

1. Two sets A and B are considered the same "size" when each member of A has one and only one corresponding member in B, and vice versa. (We use this way of defining "size" so that we can talk about "infinitely large" sets.)

2. A "prime number" is an integer (whole number) has no factors other than itself and one.

3. It can be proven that for any given number n, there is ALWAYS a bigger number which is prime. This means the set of prime numbers P which is 2, 3, 5, 7, 11, 13, etc has an infinite number of members.

4. I use the symbol ^ to mean "raised to the power of" - so 2^3 means two times two times two (which is eight).

OK. Now, consider the set of the powers of two: 2, 2^2, 2^3, 2^4, 2^5, etc. which we'll call D.

Here is the first amazing thing: the set D is the same size as I.

Now, you (my opponent) will say: "but but but there are fewer things in D than there are in I! For I has 3 and 5 and 7, but T doesn't have them."

And I say, no, according to our first assumption they are the SAME size, for each member 2^i in D corresponds to ONE AND ONLY ONE member i in I. Hence they are the same size.

You will note, therefore, that I have just broken the integers I into one set, infinitely large, called D, and "everything else" left over. But I am not done.

I will NOW make another set, just like D, but I will call it T - it will have the powers of three: 3, 3^2, 3^3, 3^4, 3^5, etc. You will note that none of these members are in D. Again, this set is the same size as I and the same size as D.

So now I have D and T, and I still have "everything else" in I left over.

OK. NOw! Remember we assumed that we can always find prime numbers which are bigger than a given number. So for any member p in P the set of primes, I can construct a "power set" from that number: p, p^2, p^3, p^4, p^5, etc. which we shall call p*. Notice that D is 2* and T is 3*.

Now this is where it gets amazing: Each power set has an infinite number of members - it is the SAME SIZE as I, for there is one and only one member p^i in p* for each member i in I. However, there are an INFINITE number of these power sets, for there are an infinite number of primes in P.

AND YET THERE ARE STILL numbers "left over", numbers which are pairs of primes like 6, 10, 14, 15, and so forth - and each of those numbers forms power sets which themselves are infinite.... And there are STILL more, which are triples of primes, like 30 and 42, etc, each of which have... etc...

What does any of this have to do with the Eucharist? Not a lot. But there is a tiny hint of something here. And we have not even starting talking about things like stars or rocks or trees ... or people.

To put it into non-mathematics: any mother will tell you that she multiplies her love as her children increase in number.

Oh, well, I didn't explain it at all well, I guess. But it means something like "Love one another as I have loved you." St. John has more details in his gospel (see chapter six).