What is a Simple System?

by William Silvert

Abstract

It is not systems that are simple or complex, it is the models that we construct.

Introduction

When we call a system simple, are we really talking about the system itself, or about our models of the system? A system that seems simple to one scientist may seem very complex to another.

For example, in the famous (and possibly apocryphal) story of Newton and the falling apple, to Newton the apple was a simple object, but to a botanist an apple is a very complex system. So is the apple a simple or a complex system? The answer must be that simplicity, like beauty, is in the eye of the beholder (or modeller).

Simple and Complex Models

Some interesting examples of how a system can seem either simple or complex depending on how it is modelled are provided by physics and mathematics. There is a classic test of whether someone is a physicist or mathematician based on the following problem: two trains travelling at 100 km/h are approaching each other on the same track. When they are 200 km apart a bee which has been sitting on the headlamp of one train flies to the other at a speed of 200 km/h and continues to fly back and forth between the two trains until it is crushed in the ensuing collision. How far does the bee fly?

This is a mathematically complex problem involving calculation of where the bee encounters the second train, then where it again encounters the first train and so on. Since the bee travels twice as fast as the trains, at each step it covers 2/3 of the distance between them at the start of that leg of its journey. Thus its first flight is 2/3 of 200 km, and when it reaches the second train they have each travelled 1/3 of 200 km and thus on the second leg it flies (2/3)´(1/3)´200 km and so on. Working out the entire series is a difficult task and is left as an exercise for the reader.

Physicists however are lazy people who do not go through such lengthy calculations if they can possible avoid it. A physicist will simply note that the two trains will collide in one hour, so the bee flies 200 km.

When John von Neumann was posed this problem he immediately answered “200 km”. When he was then laughingly told that even though he claimed to be a mathematician his quick response proved that he was a physicist he angrily retorted, “Nonsense, I summed the infinite series in my head!” Whether the story is true or not, it shows that a problem – or system – can be either complex or simple, depending on how you approach it.

Another example, painfully familiar to physics students, is the calculation of the gravitational field of an object with spherical symmetry. The earth is approximately such an object, with a dense molten iron core surrounded by lighter rocks and minerals. So is a solid bowling ball, or a hollow tennis ball, or an apple. The calculation of the gravitational field can be formulated as the three-dimensional vector integral of the density over the entire volume of the sphere. However it can also be shown that the gravitational field of a spherically symmetric object is simply GM/r², where G is the gravitational constant, M is the total mass of the object, and r is the distance to the centre. In other word, no matter how complex the mass distribution inside the object (assuming that it is spherically symmetric), the gravitational field is what you would get if the mass were all concentrated at its centre. What could be simpler?

Another example is an old test question – if you burn a candle inside a sealed jar on a scale, does the system within the jar gain or lose weight, or remain the same? This is certainly a complex matter – the carbohydrates in the wick and candle wax combine with oxygen in the air to form water and carbon dioxide, but as the oxygen gets depleted some carbon monoxide is generated, and some uncombusted carbon will appear as black smudges on the sides of the jar. The candle may also contain impurities or be covered with metallic sprinkles, which adds to the confusion. And yet, this is such a simple problem that the answer is obvious. So can we meaningfully ask whether the system is simple or complex, or is it just the models that we are talking about?

Conclusion

This journal is clearly misnamed – it should be the Journal of Simple Models, or at least the Journal of Simple Systems and Simple Models. However, just as all systems are complex, all models are simple, and for basically the same reason. Systems are complex because no agency would fund research on a simple system. Models are simple because any criticism of the model can be rebutted with words, “This is just a simple model, we haven’t taken that into account.” Who says you can’t win?

Address all correspondence to the editor, william@silvert.org. As for copyright, who would steal a simple idea? Just be sure to acknowledge the source.