I'll sing you two-ho!
Green grow the rushes —ho,
What are your two—ho?
Two, two the lily-white Boys, clothèd all in green—ho.
One is one and all alone and ever more shall be so.
Later on in this section I shall try to explain how it is that mathematicians have made themselves comfortable with the idea of counting the infinite. The key idea is that of pairing, for example by making a tally by scratching one stone with another. ¹ This book is trying to learn you mathematics, not paleontology. I was all set to invent a little story, perhaps of a stone-age hunting party, scattered by an unexpected encounter with a tyrannosaurus†, trying to work out if anybody was missing: (Where's Og? I thought he was with Ug? etc.) But then I began to wonder if that really was how counting got started. We tend to think of numbers as dry, dusty things. I want to suggest to you that in fact they have a deep, magical power.
Imagine you are living in the stone age, somewhere in an inhospitable northern Europe. Nature is all around you, and you never feel like you are on top. At night there are wolves and bears. Sometimes there's a silvery light in the sky and you can still see enough to feel safe. Sometimes it isn't there. In the dark you feel small and vulnerable. Sometimes the hunting is good, or you can find berries and mushrooms to eat. Sometimes it is very cold, and all the animals are gone, and you wonder where the next meal will come from.
Now imagine you can count, that you can say to yourself:
"Tonight it is dark, but in two nights the new moon will be shining a little, and I won't be so frightened." "Today it is cold, but in seven days the sun will be shining a little longer each day, and I will be able to catch some squirrels to eat."
"Tonight it will still be light, but tomorrow there will be no moon. I must gather plenty of firewood." "Today it is still warm, but in one month it will be winter. I must try to gather plenty of hazelnuts"
Where you cannot control, to be able to predict brings power. It is, truly, magic.
If this seems a little implausible, think about:
- How many stories have numbers in their title: "Snow White and some Dwarfs", "Goldilocks and a few Bears", "Ali-Baba and rather a lot of Thieves" wouldn't have quite the same ring to them would they?
- How often stories have three-fold repetition of incidents: the hero or heroine who must perform three impossible tasks, and be helped by the three animals they had been kind to earlier, or resist three temptations (like Jesus in the new testament); the maiden who has, through an enchantment, been forgotten by her prince, and who must spend three nights outside his chamber to recall him to his senses.
- If you are familiar with the Jewish Bible, think about how often God's ability to count what seems to us innumerable or unknowable is cited as evidence of his wisdom and power (for example at the end of Job), or how important it seems to have been to record the ages of endless lists of ancestors whose deeds go unrecorded. Then there is the remarkable story recorded in 1 Chronicles 21, where David commits a terrible sin - counting the people of Israel, and 70,000 people die in the plague sent by God as punishment.
- How football crowds often incorporate numbers into their chants, and how many children's rhymes incorporate counting.
I hope that has made numbers seem exciting enough for you to want to read about some of the different types of number that mathematicians use.