Monday, 9 January 2017

Bellos II

One of the advantages of being without the interweb for a few days is that you get the chance to read some books instead. I think I read more books during the 12 days of Christmas than I managed in the rest of 2016. A few years ago someone, quite possibly me, gave Dau.II a copy of Alex’s Adventures in Numberland by Alex Bellos. I guess that happened because people has decided that Dau.II is a bit of a quant despite, or perhaps because of, never having been taught The Calculus or matrix multiplication in school. Maths is often taught by anxious teachers who are at the edge of the competence and their fear and loathing is transmitted like a virus to their pupils. Bellos is an enthusiast but one who is capable of bringing the reader along with him. Mathematicians, as any expert, have to beware of the Curse of Knowledge which leaves followers bruised in the wake of a turbo-turmoil of ‘explanation’. Adventures in Numberland is to be recommended for channelling the enthusiasm and giving it sufficient context and backstory that the reader is effectively edutained. I’ve hammered Gödel Escher Bach – the eternal golden braid because the author, Douglas Hofstadter, rapidly disappeared up his own chuff and beat a few of his hobby-horses to death before the reader’s aghast eyes. Nobody asked him for a sequel.

If you have a contract to write a book – of so many pages – about mathematics, you’ve going to leave a lot on the cutting room floor. Nobody, least of all your publisher or agent (who will have been educated in the Arts Block), is going to expect a comprehensive overview of the history and effects of mathematical understanding. For that you may go back 80 years to Mathematics for the Million by Lancelot Hogben [prev]. The first chapter of that book is called Mathematics, the mirror of civilisation, which sets out the pretension of the book.  It’s actually brilliant and should be given to any teenager you know who is vaguely numerate. It might serve as an antidote to the dreadful, dull dull dull experience the poor youngster is currently getting in school. Bellos didn’t need to write a history of math, so he could cherry-pick conundrums which were neat, quirky, peculiar and to which he knew the answer.

In 2014, on the wave of his first book’s success, Bellos was persuaded to write a Son of Alex: Alex Through the Looking Glass – how life reflects numbers and numbers reflect life. The title gave me pause because it was too clever by ‘alf, in the style of Douglas Hofstadter. It was one of the boxful of books that Dau.II has no space for in her one tiny storage closet. We agree that Bellos II is his Second Eleven. The best players in his mind have already been out on the field in his first book. If there is to be a sequel, then there is no need to be comprehensive in the first volume and if neither book is comprehensive there is no need to read the whole thing to find out what happened in the end. As it happens there aren’t even eleven chapters, only 10, in the second book, although some chunky appendices make up effectively an eleventh chapter. I did read all the chapters but not all of them with equal care and attention.

The best was an investigation of the quirks of numerology that is expressed by Benford’s Law of First Digits [prev]. This is the peculiarity that if you look at any string of numerical data that spans several orders of magnitude – population of cities or countries; all numbers that have appeared in The Blob or in the last newspaper you read; the tallies in the Domesday Book – 30% of them will begin with 1 and 18% will start with 2 but only 5% will start with either a 7, 8 or 9. A random number is 5x more likely to start with 1 than with 9. Now that is counter-intuitive but the phenomenon occurs so widely in such disparate groups of numbers that it is effectively a natural law. You can monetise this by betting in the pub on, say, the cash total for a random receipt in your neighbour’s wallet. If you give even odds that this number begins with 1,2, or 3 vs 4, 5, 6, 7, 8, or 9 you’ll make money steadily.
One interesting aspect of the law is that it is scale invariant. If you multiply all the numbers by 2.54 converting whatever is being measured from inches to centimetres the numbers all change but they are still in the Benford proportion.  Let’s do it, for illustrative purposes, doubling each number
1st Digit Ns
1st Digit 2Ns
In other words if your number begins with a 1X its double is either in the 20s or 30s. All the long tail become 1s under the doubled regime.

This consistency is not apparent if you *add* a fixed amount to the numbers in your dataset. And you can use this information for doing forensic accounting: if the numbers in the accounts of Sue, Gabbitt and Runne, solicitors, are not Benford-like you can suspect fraud and dig deeper. It’s like Dau.II and her restaurant’s relationship with the Food Safety People. If her records are all in order and up-to-date, the inspector is likely to put the mess in the kitchen down to a busy day. If the written records look patchy or have the toilets checked at the precisely same time each day with the same signature, then the inspector might look more closely behind the fridge.

Discrepancies also leapt from the pages of the returns for the Iranian presidential election in 2009, when Mahmoud Ahmadinejad saw off the reformist challenger Mir Hossein Mousavi. When you look at the ballot-box totals, Mousavi’s are Benford compliant but Ahmadinejad’s are not. Presumably Ahmadinejad’s machine had stuffed extra votes for their man into some or all of the boxes. They would have been okay if they’d doubled the count for their man in each constituency.

Note: The title is as a witty but irrelevant play on Dau.I’s favourite band Bell-X1

No comments:

Post a Comment