...a blog by Richard Flowers

Friday, June 09, 2006

Day 1981: Mysteries of Doctor Who #4: So, is my Butterfly Net REALLY a threat to the Entire Universe?


Science fiction has a habit of picking a fancy and sticking with it: the nature of the relationship between creator and created as applied to the shiny robot; alien invaders among us who are really (SHOCK!) Communists; or uploading your mind into a whole new environment called cyberspace are all examples.

Doctor Who is no different, whether it is the "let's remake every horror story ever" era of the mid seventies; the "footlights fuelled fantasy whimsy pastiche" era of the late seventies; or the "GUNS with a capital GUNS" era or the mid eighties.

When it came to the nineteen-nineties (oh yes there WAS Doctor Who in the nineteen-nineties) the particular piece of zeitgeist that the NEW ADVENTURES surfed most often was THE BUTTERFLY EFFECT.

The Butterfly Effect is the most widely known (and most widely misquoted) example from the mathematic theory called the study of systems governed by non-linear equations by people who know what they are talking about but not how the real world works, or CHAOS THEORY by people who want to sell books.

Most people probably remember from school physics classes some stuff about how the world is kind of run by some LAWS that were made up by mathematicians, especially Sir ISAAC NEWTON-BOMB.

Sir Isaac basically said that moving things carry on moving pretty much as they are unless you PUSH them. And then how much they ACCELERATE depends on how HARD you push.

At the time this was revolutionary, because people thought then (and often think now) that it is how FAST you go that depends on how hard you push, but this is not right: how fast depends on how hard you push AND how long you push for.

(People tended to be distracted by the fact that FRICTION causes things to slow down again when you stop pushing sideways, and GRAVITY tends to pull things down to Earth with a bump a lot when you stop pushing upwards.)

Anyway, that is a CLASSIC example of a LINEAR SYSTEM. The outcome is determined by one input (push) and how much output you get out is PROPORTIONAL to how much input you put in.

Also, it is nice and easy to solve using ALGEBRA. Or even counting on your feet! If you know where you are at the start and how fast you are going, then all you really need to do is some MULTIPLICATION to know where you will get to at the end and how quickly.

Unfortunately the maths gets very, very hard if you try and solve more complicated problems, like when there is more than one input.

If you take a system like the WEATHER, then there are a lot of "inputs". There is which way the wind is blowing determines how the clouds move, how many clouds there are determines how much the sun warms up the sea beneath; how warm the sea is determines how much rain becomes clouds; the shape of the surface of the land is determines how the wind and clouds move; how cold it is where the clouds end up determines whether it is rain or snow; how much snow falls changes how much sunlight is reflected back and in turn how warm or cold it is; and so on and so on.

This is called a NON-LINEAR system because there are MANY inputs and they do not respond in the nice easy PROPORTIONAL way of giving back in relation to how much you put in.

Nevertheless, you could still work out a BIG EQUATION with lots of letters and numbers and try to make a MODEL of how the weather works.

In fact, this is EXACTLY what the weather men do with their computers. When they aren't playing TOMB RAIDER, anyway.

But here is the problem: you want to use your model to predict what the weather is going to be like tomorrow, and the day after, and the day after that.

So you tell the computer lots of things about what the weather is like today and it puts all of these into the big equation and it says "bleep bleep this is tomorrow's weather".

Yes, JUST like that.

Some scientists were doing some experiments just like this to test their weather computer. It was in the nineteen-seventies which was a while ago and so they had to run printouts on paper to see the results. They would run one every half an hour. At the end of the day, they had to switch off the computer and because it did not have a HARD DISC DRIVE or indeed any kind of storage memory, that would be the end of the experiment for the day and the next day they had to start all over again by feeding in the last set of numbers that they had printed out the night before.

However, in order to check that the machine was working alright, what they did was in fact feed in the figures not of the LAST print out, but the LAST-BUT-ONE print out and then they expected to get a check after half an hour that the first printout of that day was the same as the last from the day before.

Except it wasn’t.

It was quite a lot different.

This puzzled them a lot for a very long time and they took their computer to pieces and couldn't find anything wrong so they put it back together again and still it didn't work out the same.

And then they had a brainwave: the figures on the printout that they were typing in the next morning were NOT the same as the figures that the computer had been thinking of at the time of the printout. In order to save paper and indeed themselves from going bonkers, the figures were ROUNDED to a few decimal places.

BUT the computer had been thinking of figures that were VERY, VERY SLIGHTLY different.

When they put in the last-but-one-time's figures, they were actually putting in an APPROXIMATION of the REAL numbers that the computer had been thinking of.

And here's the thing: when you are doing LINEAR equations, if you get the starting figure a LITTLE bit wrong; then you get the finishing figure a LITTLE bit wrong. Because, being LINEAR, the wrongness in the output is PROPORTIONAL to the wrongness in the input.

But when you are doing NON-LINEAR equations the mathematics gets more COMPLICATED – not just in the sense that you can tie all four of your legs together if you try solving them by counting on you feet, but also because you can no longer be sure that a SMALL error at the start will end up with a SMALL error at the end.

Worse than that, it is often impossible even to predict whether a small error at the start will give you a big error or a small error at the end!

People remember that Chaos Theory is the branch of mathematics that gave us all those exciting computer pictures of FRACTALS and MANDELBROT SETS.

What the Mandelbrot Set is – in one sense – is a map of the kinds of errors you get in one simple non-linear system: if your starting error is inside the set then your ending error is small; if your starting error is outside the set, the your ending error is HUGE. But, as everyone knows, the edge of the set is INCREDIBLY fiddly twiddley and the detail seems to go on forever. In fact it's not just seeming, it REALLY DOES go on forever: you can expand it ANY NUMBER OF TIMES and still find more twiddles.

Okay, you are saying, so where does the BUTTERFLY come in.

Well, if you are going to build a weather computer, you need to give it all that information about today's weather – so where do you get that information from?

What you do is this, you set up lots of weather stations all over the world and take readings on all the input things like cloud cover and temperature and rainfall and wind speed and how many cows are lying down.

But that's REALLY an approximation, because you are going to have to guess what the input values were for places that you don't have weather stations.

And we've already seen the problem with approximations: the small errors quickly blow up out of all proportion and in random and unpredictable ways. Your weather computer's prediction for even a FEW WEEKS into the future is predicting heat waves when it turns out you get hurricanes that week instead.

Even if you make you approximation really, really, really good it can only ever be an approximation because you cannot cover the entire surface of the Earth in weather stations. There'd be no room for elephants!

But suppose, just as an experiment, you COULD cover the entire surface of the globe in weather stations, one every metre say. You would have a very, very good approximation indeed. But you still would not be able to take into account every air current in the Earth's atmosphere.

A BUTTERFLY, flying between two of your weather stations flaps its wings in a way that your equation doesn't predict because it doesn't even know the butterfly is there. So there is STILL, even in this insane experiment, STILL a difference, an error between the numbers that you tell your weather computer and the ACTUAL starting position of the Earth's atmosphere.

And the chaos maths means that that error, tiny as it is, has an entirely unpredictable effect on the difference between your prediction and the weather that really happens.

THAT is what is meant by the famous quote:

"The flap of a butterfly's wings in Brazil causes a tornado in Tokyo"

The butterfly ON ITS OWN does not cause the tornado: it is only one tiny contributing factor. The point is that you did not PREDICT the tornado because you did not include the wing flap in what you told the weather computer.

The Butterfly Effect is that PREDICTIONS in NON-LINEAR systems GO WRONG.

However, that is NOT the end of the stick that was grasped by the writers of many of the New Adventures.


They got the idea that changing something tiny makes a big difference in the outcome.

In particular, if you go back in time and change something – typically murdering an unfortunate butterfly JUST TO MAKE THE POINT – then when you return to the future, shock gasp, the Federation has ceased to exist and/or everyone you ever loved is now mauve in hue and/or Mr Balloon is an honest and entirely believable figure of wisdom benevolently guiding mankind with no thought of benefit for himself.

[R: steady on, Millennium, the world couldn't be THAT different!]

This, of course, misses half the point.

The Butterfly Effect is about PREDICTIONS.

If you travel back in time from the future, then the future is NOT A PREDICTION.

It is a bit of your STARTING information.

Because of the uncertain nature of memory never mind history, you could argue that HOW the future got from the point that you travelled to in the past to where you started from IS a prediction and so that COULD change. Not that you'd notice, because your start point (in the future) is fixed by BEING the START point, not the END POINT!

Anyway, after all those Butterflies affecting the Seventh Doctor, they only went and gave the eighth Doctor a whole ROOM FULL of the things!

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