Under Construction

Number

God made the integers; all else is the work of man - Leopold Kronecker - address to the Berliner Naturforscher-Versammlung (the Berlin Society of Natural Scientists), 1886

The last page left us holding one - and, tucked inside it, two. That is enough to start a world, because from one and two the whole number line builds itself, each new number forced into being by the ones already there. Nobody hands the numbers down. They make themselves, and as they go they lay down the first structure that everything later will be built upon.

Index

Counting Is Not Adding

The First Corner

The Numbers That Stay Home

Galileo's Worry

Counting Is Not Adding

We are taught counting as adding: one, and one more, and one more again. Virtualism comes at it the other way about. Two is not one with something stuck on the side; two is what you find when you notice that one was already double - the nothing, and its wholeness, held apart. Two is got by dividing one, not by adding to it.

One divided is two.

And it does not stop. One and two, set side by side, are themselves a small paradox that can only be settled by a third thing; one, two and three force a fourth; and so the line of whole numbers marches off, each step forced by the steps before it in just the way the first something was forced out of nothing. None of it is arbitrary, and none of it is invented. It is the same engine, turning over and over.

One oddity worth keeping in your pocket: each new number quietly changes all the others. To count a thing is to measure the whole of existence against a finer grain, and the grain shifts every time a fresh number arrives. The numbers are not a dead ladder leaning against the world; they are alive, and they tick.

Each new integer changes everything in the whole of Existence by altering the granularity of its nature.

The First Corner

So far the world is a line. Every number is just so many steps along the one road, and a line is all you can make by stepping. Then you reach four, and something new is quietly on offer. Four is the first number that can be read in two genuinely different ways. It is four-in-a-row, 1 × 4, the old story along the line. But it is also 2 × 2 - a square - and a square is not a longer line. It is a line that has turned a corner into a second direction.

Four is paradoxically both 1 × 4 and 2 × 2.

This is the hinge of the whole production. Up to here every number was forced; there was no freedom, only the next compelled step. At four, for the first time, the world is offered something rather than driven into it - the possibility of turning a corner into a new dimension. It need not take the offer in any given case, which is why this gentler engine is a matter of possibility rather than compulsion. But once the possibility is real, examples begin to appear, and area joins length as a thing the world can do.

The possibility of the thing leads to the probability of the thing.

Those two engines - the one that forces a new direction into being, and the one that merely permits new arrangements once the room exists - run the whole of Virtualism. They get their proper names and their full workings a little further on, under Paradox and Emergence. Number is simply the cleanest place to catch them both at work, with nothing but arithmetic on the table.

The Numbers That Stay Home

If four can turn a corner because it is 2 × 2, you might ask which numbers cannot. The answer is the primes. A prime is a number that no smaller numbers multiply up to - it cannot be laid out as a rectangle of others, only as itself in a single row. So while the composite numbers go off and build squares and blocks and higher shapes, the primes stay on the original line, unable to join in. They are the numbers that never manage to become anything but themselves.

The primes are the numbers that fail to emerge as something else.

Galileo's Worry

There is an old puzzle, sharpened by Galileo, that this view quietly dissolves. Every number has a square - 1, 4, 9, 16, and on - so you can pair each number with its square and never run out, which seems to say there are exactly as many squares as numbers. Yet the squares plainly thin out as you climb; almost every number is not a square. How can a part be as plentiful as the whole it sits inside?

The sleight of hand is the phrase all the numbers, as though they were a finished heap you could weigh against another finished heap. On this account there is no such heap. The numbers are still being produced; the line has no last number and no completed total. You cannot set two endless, unfinished lists side by side and declare them equal or unequal, because neither list is ever all there to be counted. The paradox needs a bag holding all the numbers to do its work, and there is no such bag.

Infinite is an adjective and can never be a noun.

Which leaves number looking very different from the two usual options. It is not a gift dropped in from outside, and it is not a convenient fiction we lay over the world for our own purposes. It is the world's own first structure - discovered, not decreed - and every later floor, all the supposed work of man, is built of it and is every bit as real. From here the next question is what these numbers and relationships actually are, and what it means to call any of them real.