It takes one to know two

Counting up, counting down

One is the simplest number. But it wasn't the first invented. At least it wasn't invented by itself.

Numbers are required only when you have multiples of something. The earth has one sun and one moon. I have one nose and one mouth. But the earth has several neighbor planets and I have ten fingers and ten toes.

The number two presupposes the number one. But the reverse is also true. One wasn't labeled as such until there was two. Before that it was just “the."

Numbers were originally for counting and keeping track. I head off to the meadow in the morning with seven sheep. I hope to return in the evening with seven sheep. I don't require a special word for seven. I don't even need the concept. I can put a pebble in my pocket for each sheep when I go out. When I return, I take out a pebble for each sheep that enters the fold. If I run out of pebbles when I run out of sheep, the sheep are all accounted for.

Early systems of enumeration were like this. People cut notches on sticks, made wedge marks on clay, tied knots in string. The common system of tallies — four strokes with a fifth crossing them — is very old. These groups of five were probably motivated by the five fingers on the hand.

Giving names to numbers makes them easier to talk about. Societies that never had large quantities of items didn't require a lot of named numbers. “One, two, three, many” might suffice.

The Egyptians named numbers up to twelve. After that the numbers repeated with slight variants. English retains this usage. “Thirteen" is "three and ten.” And so on through nineteen. The Mayans named numbers up through twenty.

Once repetition is allowed, there's no end to the counting. When this first occurred to someone is hard to tell. Archimedes tried to estimate the number of grains of sand it would take to fill the universe. In doing so he extended the Greek system of enumeration past where it had gone before. And he showed how it might be extended indefinitely.

For strict counting purposes, zero was unnecessary. If you had nothing of something, you had nothing to count. “None" sufficed. Nor were negative numbers necessary or even useful.

But at some point somebody extrapolated from counting to other actions, such as taking steps. Steps forward might be followed by steps back. Three steps forward followed by one step back would land you on two. Three steps forward followed by two steps back would land you on one. Three steps forward followed by three steps back would land you on — there ought to be a number for that. Let's call it zero.

What about three steps forward followed by four steps back? That also should have a name.

Counting had already lent itself to addition. Three sheep plus four sheep equaled seven sheep. And to subtraction. Seven sheep minus four sheep equaled three sheep.

You can't have negative sheep. But you can have negative steps, as in steps backward. Seven steps forward followed by eight steps back is the equivalent of one step backward. Call that a negative one step.

On second thought, you could have negative sheep. A sheep that you own could be a positive sheep, and a sheep that you owe — to your neighbor, perhaps — could be a negative sheep. If you owe more than you own, your sheep balance could be a negative number.

The adoption of zero and negative numbers lent a pleasing tidiness to the system of counting. You could count up or count down, step forward or step back, and you never broke the system. You always landed on another number. Mathematicians would come to say that the system was closed under addition and subtraction.

Closed but not unbounded. There was no limit to how many steps forward you could take, nor to how many steps backward. Some unrecorded person was the first to imagine a number line, with zero in the middle, positive numbers extending indefinitely to the right and negative numbers indefinitely to the left.

Not everyone was comfortable with the indefiniteness. The physical world is limited. A human life is limited. Nothing goes on forever.

Nor did the counting go on forever, actually. It simply was never compelled to stop by running out of numbers. A life might end. The universe might end. But there would always be one more number. And another after that.

For the adventurous of mind, this could be oddly comforting. You'd probably never need such astronomical numbers. But if you did, they'd be there.