Tuesday, 19 May 2009

The East gave us nothing, and we ought to show more gratitude

At a crucial point in the high middle ages, the Islamic world, through its presence in Moorish Spain, provided an essential spur to the development of Western culture as we know it. Christendom had lost its access to the great works of classical antiquity; Arab scholars provided translations that would launch a major intellectual reawakening in the West. As important was the intellectual debate around leading figures in Cordoba such as Averroes, that gave rise to the idea that rational thought could be separated from faith, a vital step in the development of science.

But Islam also brought us another and vital gift: nothingness. Absence of presence. Or, more prosaically, zero.

One of the great difficulties in Roman mathematics is the symbols they used: how do you multiply LXII by XVIII? With positional notation, using a limited set of digits, where the column in which a figure appears determines whether it counts units, tens, hundreds or more, it becomes literally child’s play. Or at least elementary school work.

The problem is how to write a number like ninety or one hundred and two. You need a symbol to say ‘nothing in this column’. This is the first use of zero, in numbers such as 90 or 102. Several centuries before Christ, the Sumerians had this concept, using a double-wedge symbol to represent a null value in their positional notation for numbers.

A thousand years later, another great step was taken. The Jain and Hindu religions in India had stressed the need to empty oneself, in order to open oneself to the divine. Emptiness became not a negative but a positive quality, a potential, a readiness to receive and take the form of something provided from outside. Indian mathematicians applied the same thinking to numbers, treating zero not simply as an empty placeholder, but as a number in its own right, of the same status as the others but with its own properties.

Intuitively, that’s not an easy idea. After all, 5 is the number representing how many rotten apples make five. Say in a group of parliamentarians. 1 is the number of one rotten apple. But what on earth is zero? No rotten apples? In parliament? What sense is there in that?

When Islam came in contact with Indian mathematics, it absorbed the new notation turning it into the system that Christendom later modified to give us our existing ‘Arab’ numerals. Western Europe however resisted the devilish ‘Arab’ number for nothingness for some centuries. Treating nothing as a positive might be OK for Islam, but Christians wouldn’t fall into that trap.

The breakthrough came in Renaissance Italy, where banking was born. Florence developed double-entry book-keeping. In that system it is essential that all the values ultimately add up to nothing: when they do, your books balance (which means that they’re either accurate or you’ve cooked them really well). So zero was vital: it represented the crucial point of balance in accounting.

I appreciate that in today’s climate we might have mixed feelings about an invention that helped modern banking develop. Fortunately, zero had other applications.

In the sixteenth century, the Scotsman John Napier realised that certain equations were easier to solve if instead of showing them as something = something else, you showed them as something – something else = 0. Remembers those ghastly quadratic equations at school? They were never easy but at least there was a way of solving them written that way.

At the beginning of the eighteenth century, Newton and Leibniz working separately, even in rather demeaning rivalry, both came up with what we now call calculus. To work out how fast we’re travelling, we can measure the distance we cover in an hour and calculate an average, but that doesn’t tell us our speed right now. So let’s measure the distance we do over a minute. That’s more accurate but it’s still only an average. So let’s take the distance we cover in a second. Much more accurate – but still an average. We could go on, measuring a smaller and smaller distance and dividing it by a smaller and smaller time and getting more accurate at each stage. Newton and Leibniz’s extraordinary insight was that you could take this process to its end point, when the distance reaches zero and the time reaches zero. You can’t divide zero by zero – but the two thinkers showed you could work out its value without doing the actual division.

Now we’ve reached somewhere really vital. Because what we’re measuring when we use this approach is a rate of change (a speed is just a rate of change of distance over time). These thinkers had opened up a field of scientific study concerned with change – with what things are becoming, not with what things are. In other words, they provided the mathematical tools to underlie a view of the world that is dynamic, rather than the static outlook of earlier cultures, concerned above all with keeping things as they were. Our society, where the only constant is change, perturbing but also exciting, is one where the kind of understanding Newton and Leibniz developed is more needed than ever.

And that requires that gift of zero, of nothing, that we received from the East.

Useful gift, wasn’t it?

And a postscript

For a last word about absence as a positive, who better to call on but that great logician Charles Dodgson, also known as Lewis Carroll? In Through the Looking Glass the White King asks Alice to look out on the road to see whether either of his messengers is on it.

‘I see nobody on the road,’ said Alice.

‘I only wish I had such eyes,’ the King remarked in a fretful tone. ‘To be able to see Nobody! And at that distance, too! Why, it's as much as I can do to see real people, by this light!’

A little later, one of the messengers arrives. The King asks him who he met on the road.

‘Nobody,’ said the Messenger.

‘Quite right,’ said the King: ‘this young lady saw him too. So of course Nobody walks slower than you.’

‘I do my best,’ the Messenger said in a sulky tone. ‘I'm sure nobody walks much faster than I do!’

‘He can't do that,’ said the King, ‘or else he'd have been here first.’


Mark Reynolds said...

Fittingly, I've nothing to say to this post.

(It was very interesting though).

David Beeson said...

No comment.

I thought of just not making a comment but decided that your remark deserved a positive statement of no comment.