The History of Zero (Part II)

'Right,' said Colon....'I heard this wizard down the University say that the Klatchians invented nothing. That was their great contribution to maffs, he said. I said, "What?" an' he said, they come up with zero.'

'Dun't sound that clever to me,' said Nobby. 'Anyone could invent nothing. I ain't invented anything.'

-Terry Pratchett JINGO p.29

(via a comment from samart, thanks!!!)

So Brahmagupta has come up with his fascinating theories about zero in India, and various tradespeople and explorers have transported these ideas (along with their spices and so forth) back to Baghdad and the rest of the Middle East. 

The most famous of the Muslim mathematicians to use the new zero in his work was Mohammed ibn-Musa al-Khowarizmi (the gentleman on the stamp). He "invented" or at least coined the term for algebra ("al-jabr" means "completion").

In any case, al-Khowarizmi investigated linear and quadratic equations equal to zero. He made enormous strides in demonstrating the great practical use of al-jabr and ultimately of zero, not simply as a placeholder but as an extremely important number in its own right (read more about al-Khwarizmi). 

As a brief aside, in algebra one of the most exciting things about zero is its remarkable multiplicative property. Anything multiplied by zero is equal to zero, and (crucially for quadratic equations) if the product of any two numbers is zero then one of them has to be zero.

*Ahem*

Interesting though it is, this doesn't get us much further in the story. The next player to enter the scene was a little boy called Leonardo. His father was a customs officer and merchant from Pisa, and young Leonardo would accompany him on his trips in North Africa and the Middle East. Leonardo grew up to write a book called Liber abaci, in which he explains the Hindu/Arabic numerals, including zero. You've probably heard of Leonardo. He's better known as...Fibonacci! (read more about Fibonacci).

Since Fibonacci was Italian, this brought zero firmly into the European sphere of thought. As is usual with revolutionary ideas, the private sector cottoned on to the wonderful properties of zero quickly, while governments and officaldom in general was more suspicious.

However, the work of several influential mathematicians (including Descartes) gradually brought zero into common usage. Eventually Newton and Leibniz were to find it indispensible to the furtherance of calculus...  

And now...well, can you imagine maths, or life for that matter without zero?

I certainly can't!

By the way, go here for a wonderful discussion of the history of zero in general...This is where I got a lot of my inspiration for this mini-series.

The History of Zero (Part I)

0

When you come to think of it, there isn't much we can do without zero. Zero is just zero...the additive identity to some, just plain nothing to others, a placeholder in the units, tens, hundreds (and hopefullly thousands) columns to still others.

But the number zero has an amazingly chequered history, ranging from the ancient Sumerians to the classical Indian and Arabic mathematicians...so here goes:

Sumerians were farmers, so they needed to keep track of numbers of livestock etc... They had a complicated record keeping system of base 60 (a decent explanation can be found here: http://www.crystalinks.com/sumermath.html).

The Sumerian system was positional. In other words, where a particular symbol is in relation to all the other symbols tells you that symbol's value. An example in modern numerical notation is this:

431: 1 has the value 1, because it is in the units place. 3 has the value 30, because it is in the tens column. 4 has the value 400 because it is in the hundreds column and so on...

Any kind of positional system ends up requiring something to mean "empty" - for example, there is nothing in the 10s column in the number 604, so we write "0". So eventually the Babylonians, who had inherited the Sumerians' system, came up with a special symbol to mean exactly that. At first they just left the space empty, but since spaces tend to get squashed and disappear, at length they actually used a symbol as placeholder:

Well, so far so good. As a human race we have "invented" zero, at least as a "punctuation mark between numbers" (http://www.scientificamerican.com/article.cfm?id=history-of-zero). There's still a long way to go.

The next distinct step (that we know about) took place in 6th century India during the Gupta dynasty. The mathematician Brahmagupta began to work with negative and positive numbers ("debts" and "fortunes") and realised that a sum like 3 - 4 was far from meaningless, despite the fact that it is difficult to imagine it concretely.  He started to see numbers as abstract entities, not just representations of quantity (more discussion at http://www.storyofmathematics.com/indian_brahmagupta.html). 

But of course if you're dealing with negative numbers then you have to face the problems of sums like

(-1) + (+1)

 So he came up with the idea of zero, as a special number rather than just a placeholder. And he developed a whole lot of rules to go with it...

Though a lot of these statements may seem obvious, at the time these ideas were revolutionary! They were so exciting that they quickly (relatively quickly anyway) made their way over to Baghdad and the rest of the Middle East.

Bertie Russell and the Incompleteness Theorem (Part II)

Right, where was I? Bertie has just discovered his crazy uncle and a new fear - of going insane.

Ok, so the next big event in our young hero's life is that his grandmother (finally) hires a Maths tutor for him. Mathematics opened a skylight in Bertie's life. For the first time, Bertie came across the idea of proof. In a world that was full of terrors and things that go bang in the night, he realised that if something is true, you should be able to demonstrate that it is true. And thereafter you need not agonise over it any more - you've shown it to be true. Equally, if something is untrue, you should be able to disprove it and move on.

Of course the other way in which Mathematics changed Bertie's life was that it soon became clear that he was very good at it...and he was only to get better!

At that point we will leave Bertie for a few years. He grew up in more or less the ordinary way, eventually went off to university at Trinity College, Cambridge where he steadily demonstrated himself to be more and more brilliant.


But as Bertie found out more about Mathematics, something began to worry him. Everything was so logical, up to a certain point. Theorems built on theorems. Propositions were proved or disproved. Yet something was still missing. The whole edifice of Mathematics was built on axioms. In one way or another, something foundational had to be assumed.

Given that in some ways Bertie's happiness and sanity depended on the certainties of Mathematics, he did not take kindly to this, and quickly set about trying to remedy this foundational problem in the field. Between a marriage (begun and ended), several affairs and multiple academic papers; conferences and international unrest; Bertie attacked the problem with vigour. Together with his great friend Arthur Whitehead he wrote a massive tome on the fundamental principles of Mathematics. He became extremely famous. But he still couldn't settle the foundations of Mathematics to his own satisfaction.

And then something really remarkable happened. A young Mathematician called Kurt Gödel proved what became known as the Incompleteness Theorems. One (and only one) of the shattering implications of these theorems is this: there will always exist some things that are true, but not provable. And there will always exist some things that are untrue, but not disprovable. Another way of saying this is that is impossible to have an entirely internally consistent set of axioms for Mathematics.

Wait...just think about that for a second...


Can you imagine how devastating this must have been? Bertie's life goal was crushed in one (or two) little theorems...

We will leave Bertie at this point. As a brief postscript, I am happy to report that Bertie survived this setback and went on to make many more contributions to Mathematics, Philosophy and even politics (particularly as a pacifist during both World Wars).

I leave you with the thought of the Incompleteness Theorems.

May they keep you awake at night.

They truly are remarkable.

As was Bertrand Russell.