How conkers created Israel, cats cause wars, and painkillers are bad for religion

November 22, 2010

I’ve recently been reading a book called Chaos, by James Gleick.  It is a nice, easy-to-read overview of chaos theory in all its forms.  Chaos theory is not really a proper mathematical field, more of an ideology, which has applications in all walks of life.   The phrase seems to be bandied about less these days, perhaps because the ideas have become so accepted that it is no longer considered a theory, but just “how things are”.  It takes the form of turbulence, entropy and unpredictability; it has great influence on the weather, the traffic, the stock markets…indeed it is hard to imagine how science worked before the notion of chaos.  As one physicist in the book puts it:

“Relativity eliminated the Newtonian illusion of absolute space and time; quantum theory eliminated the Newtonian dream of a controllable measurement process; and chaos eliminates the Laplacian fantasy of deterministic predicability”.

Poor physicists! Always having their work eliminated by something or other.  Luckily this doesn’t happen in mathematics.  Chaos in mathematics is studied in the form of dynamical systems,  in which small perturbations in initial conditions can have a dramatic long-term effect.  This sensitivity is known in popular culture as the “butterfly effect”, from a paper by Edward Lorenz – a pioneer of chaos theory – titled: Predictability: Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas?

Lorenz was  a meteorologist, and first noticed chaotic effects whilst running weather simulations.  Weather is notoriously chaotic (see, for example, long-term forecasts by the Met Office for evidence of this), and one day, whilst trying to restart a simulation where he had left off, he fed in data which had been output from the middle of a previous session.  He noticed that the outcome was wildly different from his previous results, a consequence of the computer having rounded his output to what he had thought was an insignificantly fewer number of decimal points.    Gleick goes into weather patterns in some depth, as well as delving into such interesting topics as the fractal – and by implication, infinite – nature of coastlines (the closer you get the more little “bays” there are), and the chaotic behaviour a human heart displays while fibrillating (basically what a defibrillator does is to reset a chaotic system with a massive jolt of electricity).*

But I am not actually going to talk about chaos theory today.  Well, not quite.  Instead I am going to share a few odd and interesting freakonomics-style chains of events I’ve learnt about recently.  They all involve seemingly insignificant things – conkers, diclofenac and a cat parasite, to be precise  – which have (arguably) had a huge impact on world events.   In that sense you could possibly claim that this was some kind of chaos in practice.  But that would be quite a tenuous way to try and link it with what I’ve written so far, so I won’t.

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All models are wrong (or: lies, damned lies and statistics)*

November 10, 2010

The statistician is seen with a certain amount of disdain (or possibly sympathy) by their pure mathematical brethren. And it is with that firmly in mind that I (as a fledgling statistician) take the reins of this worthy blog.

We have some idea of what mathematics is from Adam’s posts; but what is statistics?  Statistics is applied maths with uncertainty. In statistics mathematical techniques are used to model and quantify our uncertainty about reality. Modelling climate change, predicting the outcome of elections, wrecking the financial system and ensuring the casino always wins: statistics is everywhere. And uncertainty is the key to statistics.

In order to get across an understanding of what uncertainty is I will try to describe some of the different kinds we face and how statistics deals with them.  The five levels in the following taxonomy lie on a continuum running from complete certainty to complete uncertainty, and provide a means of measuring the range and limitations of statistics in different situations.** The further we go along this continuum the less effective statistics is at prediction and inference, and many problems in statistics and quantitative social sciences like economics come from not recognising just how far along the continuum we are.

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How google does what it does

October 29, 2010

This post is inspired by a friend of mine, who very late one night recently made a valiant attempt (given the circumstances) to explain to me how the google website-ranking system works.  I was surprised to hear that at its heart it is a simple – although quite clever – application of linear algebra.  A couple of days later, in one of those funny occurrences of suddenly encountering the same concept/thing multiple times after having spent a lifetime never hearing about it, I actually came across a very similar piece of theory in my research, and read a bit more about it.  So I thought I’d explain it for the interested among you.  However, I am swiftly learning that attempting to write an entire introduction to a mathematical subject in one blog post is a foolish thing to do!  Usually what seems to happen is that I spend the entire post writing about some fundamental aspect of the subject, and then give up and explain the rest very quickly and inadequately.  So I’m afraid I’m going to have to assume you know basic linear algebra…if not, then you might want to stop reading now.

Now, other than indexing websites and providing a portal through which to access them, clearly the most crucial aspect of a search engine is the ordering system it uses to list the sites.  There needs to be some way of assigning an “importance score” to each webpage, such that the ones which people are most likely to want view come first.  Arguably the sole reason google are as successful as they are is a very effective method of doing this invented by Larry Page while he was at university, conveniently called PageRank.  The system uses the links to a page to determine its score, and crucially, it measures not just the number of these links but their “quality”;  that is, it assigns higher importance to links coming from pages which themselves have a high score.

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A year’s work, lessons learnt

October 11, 2010

I’m back!  And rather surprisingly, I seem to have gained a lot of readers in my absence.   Having not even logged on to WordPress for a few months, I have returned to see that my google reader subscription rate has doubled, and the number of people visiting the blog has increased by more than at any point since I started writing it.  I’m not really sure what lesson to take from this.  Probably it is just the natural result of a gradual snowballing effect: over time more people click on your site, the google rankings go up, causing more people to click on your site…

Then again it’s  possible that people just prefer it when I don’t write anything!  Well I’m sorry those people, but I intend to start again.  Although possibly even more erratically than before.

Anyway, I will explain the terrible sequence of events which led me to abandon blog shortly, but first, a shameless Rupert Murdoch-style using of one of my products to promote another! (I would do this at the end, but am rather doubtful as to how many people actually make it to the end of my posts).  Having been introduced to Vietnamese coffee by my father-in-law a while ago, I have utterly fallen in love with it, and realised that it is very difficult to find here in the UK.  So I have set up a (very) small business selling it.  The website is here.  Try it!  You won’t be disappointed.

Sorry about that.  Now, this is what happened.  Having struggled with the proof of a knotty mathematical problem for the better part of a year, I was advised by my supervisor to publish what I had.  So I put the paper on the arXiv (an online preprint archive), not really expecting to achieve anything, but generally wanting to share the knowledge out of a spirit of altruism (and self-promotion).  Within a few hours of it appearing, a certain Peter Mueller had read it, and proved the last part of the conjecture!  (So there you go doubters: people do read your preprints).  This was wonderful news; we invited him to be a co-author, and set about writing a final draft. I also wrote a whole long blog post about how great this was, and what it all meant.  But then a couple of days later Prof. Mueller sent me some rather less good news: he had found a  mistake in my work, which completely invalidated the whole thing…

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Fully homomorphic data encryption

May 28, 2010

In this day and age of free-flowing information everywhere, cryptography is a very hot topic. There has always been demand for better ways of encoding sensitive messages, but it is only really since the advent of the internet that this has taken on great importance in everyday life. Now that it looks likely computing will be increasingly outsourced to “cloud servers” in the future, a whole new form of encryption is necessary, and recent advances which have been made in the subject look set to make this kind of internet-based computing possible.

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Phi

May 15, 2010

One problem mathematicians constantly struggle with is that there are just not enough letters in the world: we long ago exhausted the entire Roman and Greek alphabets (and even some of the Hebrew one), and as a result many letters are used in a bewildering number of different contexts. Well, I will straight away put your troubled minds to rest by stating that, continuing my occasional series on important mathematical constants (am I allowed to call two posts a series?), when I say “phi”(\phi) I mean the number\phi ,otherwise known as the Golden Ratio, or the Divine Proportion. But first, here are some numbers:

1,1,2,3,5,8,13,21,34,55,89,144,...

This is the Fibonacci sequence, and it pops up all over the place.  It is generated by a very simple rule, which I won’t reveal in case you haven’t seen it before (try to work it out).  Now, before I start enthusing about the prevalence of the Fibonacci sequence and the golden ratio in nature, first a disclaimer: one problem with looking for appearances of these sort of things is that people can end up getting a bit obsessed and start seeing them everywhere.  I don’t doubt that some of the claims are little more than a mixture of conspiracy theory, coincidence and approximation; and I have tried to filter out the more wacky theories.  Some things are clearly more than coincidence though!

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A Serious Man, and A Beautiful Mind

May 6, 2010

You may have noticed that I am not exactly diligent about referencing the pictures and diagrams  I use on this blog.  In fact the majority of them come from google images, and I usually have no idea where they were originally used (one of the good things about writing a blog is the lack of academic rigour required).  However, last night I saw a Coen brothers film called “A Serious Man”, and am pleased to be able to reveal that the header above is actually a screenshot from that film.

As you might guess from the picture, the film itself is not really all that serious, and how Larry Gopnik – the physics professor depicted –  managed to write on a board that size is left to the viewer to ponder.  Larry’s brother Arthur is also a mathematician, although a rather less functional one.  He is quite a strange man , and spends most of the film lying on the couch writing what he calls his “Mentaculus”, which is supposed to be a “probability map of the universe”.  This may or may not be a serious endeavour; we are only given a brief glimpse at its contents, and it appears to be no more than a collection of very intricate doodles.

The Mentaculus

Writing a probability map of the universe is a noble but rather misguided aim; the notion itself doesn’t really make sense, and it would clearly be impossible to pull off.  But the urge to explain the workings of the world around us is a major drive for mathematicians.  Unfortunately, the real world is messy and chaotic, and there is very little that we can do to accurately describe how things work in practice outside of rough models.

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