Thursday, May 07, 2009

The Monster Zero

I have a lot of projects just now, which is usual - and you might be interested in one of the newest, which may end up being a commentary on pedagogy: the "science" of teaching, especially as it deals with mathematics and the sciences. No, I am not going to take a hint from Fr. Jaki and write Chesterton a Seer of Mathematics - though that would make a very interesting collection. It would send all our lit'ry friends screaming down the hall. But it is certain that he mentions mathematics with at least some reasonable skills, even if he was no specialist - but then that is Jaki's argument in Chesterton a Seer of Science. How else could someone write something as wonderful as this:
You cannot evade the issue of God; whether you talk about pigs or the binomial theory, you are still talking about Him.
[GKC Daily News Dec. 12 1903 quoted in Maycock The Man Who Was Orthodox]
No - but if we are serious about God and Chesterton we cannot evade mathematics, just as we cannot evade technology...

Nor can you evade clicking here if you wish to read more...

Quite some time ago, I wrote a column on my own blogg linking numbers and Chesterton and Aquinas and that old whine about the Barbie doll that said "Math is hard". As usual my rambling discourse was larded with several Chesterton quotes - and part of this argument gets to the root of the matter: the mystical and difficult topic of epistemology - the science of knowledge itself, as well as pedagogy: What do we know, how do we know it, how do we impart that knowledge to others?

Since I have only a limited time today, and I do not expect to begin the writing of this massive text just now, I must try to take another and Chestertonian approach. Rather than proceeding formally, I would try to do what Chesterton di, and propose sample ideas which may help get to the larger abstractions. For example:

One of the issues which comes up in any discussion of mathematics is the spectacular thing called "zero". It is easy enough to begin talking about the history of the idea, or about the convenience it provides - the grand simplification we have in our "base ten" scheme of writing numbers is a fantastic advance over what the Romans had, and even the Greeks for all their mathematical cleverness (think Euclid!) If you doubt this, just get out a nice big sheet of paper and try multiplying LXXVII by CIX. Or, if that is too easy, try multiplying oz by rq. Take your time... then try multiplying 77 by 109, and see how much easier a time you have!

But there is another aspect of zero which is lots more interesting than its history or its convenience in numeric notation. It is a place where something deeper lurks - at least according to some writers, it touches on metaphysics. They begin with the idea that it is a kind of word - a "something" - which stands for what they call "nothing"... but I cannot go there. For one, I think there is a difference between the number zero and the idea of nothing, since zero is just as "real" a thing as every other number. But I was not just being a rebel - at least I didn't think I was. (Besides, there's a time when it may be good to be a rebel: we saw that not so long ago when we read Orthodoxy, remember?)

So I sat and thought about this, trying to figure out what was going on. I remembered the technical definition of zero as a number - I mean a value - and not as a digit (a symbol) which is a convenience for writing numbers.

You mean there's a real definition, Doctor?

Oh, yes. Zero is the identity in what we call the monoid of the integers under the operation of addition. (Yes, zero is also the identity for the reals, and so forth, but we'll leave that for another day.) What does that mean? It means you can add zero to any number and not change that number. Obvious, you say - but there didn't have to be such a thing. There are other things in mathematics called "semigroups" which do NOT have zero! Sure they are strange, but contrary to Barbie, math is easy - St. Thomas Aquinas himself said so.

OK... but.. what does all this blather about zero have to do with Chesterton, Doctor?

Well, I wondered whether GKC had anything at all to say about "zero" and it turns out that "zero" comes up very rarely in the AMBER collection. But when it does, it sheds a marvellous light on the topic. Just consider this,
The great psychological discovery of Paganism, which turned it into Christianity, can be expressed with some accuracy in one phrase. The pagan set out, with admirable sense, to enjoy himself. By the end of his civilization he had discovered that a man cannot enjoy himself and continue to enjoy anything else. Mr. Lowes Dickinson has pointed out in words too excellent to need any further elucidation, the absurd shallowness of those who imagine that the pagan enjoyed himself only in a materialistic sense. Of course, he enjoyed himself, not only intellectually even, he enjoyed himself morally, he enjoyed himself spiritually. But it was himself that he was enjoying; on the face of it, a very natural thing to do. Now, the psychological discovery is merely this, that whereas it had been supposed that the fullest possible enjoyment is to be found by extending our ego to infinity, the truth is that the fullest possible enjoyment is to be found by reducing our ego to zero.
[GKC Heretics CW1:127]

...something of which Christmas is the best traditional symbol. It was then no more than a notion about the point at which extremes meet, and the most common thing becomes a cosmic and mystical thing. I did not want so much to alter the place and use of things as to weight them with a new dimension; to deepen them by going down to the potential nothing; to lift them to infinity by measuring from zero. The most logical form of this is in thanks to a Creator, but at every stage I felt that such praises could never rise too high; because they could not even reach the height of our own thanks for unthinkable existence, or horror of more unthinkable non-existence. And the commonest things, as much as the most complex, could thus leap up like fountains of praise....
[GKC G. K.'s Weekly Dec 13, 1934 quoted in M. Ward, Gilbert Keith Chesterton]

It is commonly in a somewhat cynical sense that men have said, "Blessed is he that expecteth nothing, for he shall not be disappointed." It was in a wholly happy and enthusiastic sense that St. Francis said, "Blessed is he who expecteth nothing, for he shall enjoy everything." It was by this deliberate idea of starting from zero, from the dark nothingness of his own deserts, that he did come to enjoy even earthly things as few people have enjoyed them; and they are in themselves the best working example of the idea. For there is no way in which a man can earn a star or deserve a sunset. But there is more than this involved, and more indeed than is easily to be expressed in words. It is not only true that the less a man thinks of himself, the more he thinks of his good luck and of all the gifts of God. It is also true that he sees more of the things themselves when he sees more of their origin; for their origin is a part of them and indeed the most important part of them. Thus they become more extraordinary by being explained. He has more wonder at them but less fear of them; for a thing is really wonderful when it is significant and not when it is insignificant; and a monster, shapeless or dumb or merely destructive, may be larger than the mountains, but is still in a literal sense insignificant. For a mystic like St. Francis the monsters had a meaning, that is, they had delivered their message. They spoke no longer in an unknown tongue. That is the meaning of all those stories, whether legendary or historical in which he appears as a magician speaking the language of beasts and birds. The mystic will have nothing to do with mere mystery; mere mystery is generally a mystery of iniquity.
[GKC St. Francis of Assisi CW2:73-4]
That last might well set the tone for my text, if it is ever written - just read this bit again:
...he sees more of the things themselves when he sees more of their origin; for their origin is a part of them and indeed the most important part of them. Thus they become more extraordinary by being explained. He has more wonder at them but less fear of them; for a thing is really wonderful when it is significant and not when it is insignificant...
Perhaps by now zero has become just a tiny bit extraordinary to you - which, if it is nothing at all, is quite a trick.

P.S. Having reread this before I posted it, I find it hilarious that he uses the word "significant" there - it is a clue to understanding the use of zero as a digit, for the zero in "10" or "100" or "1000" is quite significant for a mere nothing... Hee hee.


  1. That last part always strikes me too. It reminds me of what Cardinal Ratzinger writes about Genesis. That one of the most important messages of Genesis isn't to tell us the precise means by which something was created, but that great news that the world is created by God, and that it is good. We need not fear the evil spirit of the Moon, but rejoice in the wonder of it's creator. It is also in a sense comparable to Job. He doesn't understand things and is confused. God tells him that he made everything. And, somehow, that fear is erased, but not his wonder.

  2. The last bit reminds me of the "origin" issues of comic books. The origins or the genesis of any hero or villain is always the best part of the story. The backstory always enriches the enjoyment of the present narrative.


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