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Patterns and proofs

Keep it simple, stupid: maths doesn’t have to be ‘complex’

Lost in translation? Wallpoper via Wikimedia Commons

When we go to the cinema, we expect certain things of big-screen scientists. Most of us will get annoyed if a film gets its basic facts wrong, for example. Directors are aware of this, so generally try to avoid schoolboy errors. After all, nobody wants to get pulled up on accuracy by a young child.

Yet according to David Kirby, in his book Lab Coats in Hollywood, filmmakers believe that some inaccuracies are necessary. They think that without certain (incorrect) stereotypes – like bubbling test tubes in a lab – the audience won’t buy into the story.

The “inaccurate-but-necessary stereotype” crops up in other media too. When researchers recently looked at how plants regulate their starch consumption, newspapers reported that the plants were using “complex maths”. While searching for food, it seems bees [solve complex maths problems](]( too. And if you want to create codes, you’ll need “complicated maths”.

We also get stories about people who’ve strung together calculations, creating “complex algorithms” that can guide passengers to taxis, help campers put up a tent or point gamblers towards correct results.

This is not to criticise the research behind these stories. There are many examples of mathematical models and algorithms that are innovative and even ingenious. They are just rarely described as such. When maths is involved, the temptation is to open the thesaurus at “complex”.

Why does this happen? One reason is the language barrier. Mathematical notation can look a bit alien (sometimes even to mathematicians). But almost anything will appear complicated when translated into an unknown language. It doesn’t necessarily mean the symbols represent a difficult concept.

Take Rolle’s Theorem. Using mathematical notation, this can be written as follows:

Lots of symbols, but the basic idea is fairly straightforward. Say we have two points - call them a and b – on a horizontal line. Then we draw a smooth line (i.e. no jagged zig-zags) between them without taking the pen off the paper. It might look like this:

Rolle’s Theorem states that there will be at least one point between a and b where the line above (or below) that point has a slope of zero:

That’s all there is to it. A bit underwhelming? Perhaps. Complicated? Hardly.

Symbols contribute to maths’ reputation for complexity, but it’s more than just a language problem. Mathematical results can also be counter-intuitive. Who wouldn’t be surprised to learn that in a room of 23 people, there is a 50% chance two of them share a birthday? When unexpected results like these are pulled apart, however, there is often a neatly formed insight lurking inside. Despite the popular image of Jackson Pollock-esque blackboards, mathematicians generally prefer elegant solutions to messy, confusing ones.

Some might argue such scientific stereotypes aren’t a problem. Yes the makers of Jurassic Park placed bubbling test tubes next to real molecular biology equipment, but did it really ruin the story?

Errant test tubes are one thing; presenting a subject as complex and impenetrable is far more problematic. First, it makes people suspicious of mathematical ideas. If we don’t understand something, our instinct is often not to trust it. As a result, useful work can get lumped together with the bad.

Whether in finance or science, mathematical models are an increasingly important part of modern life. While we should always keep a healthy scepticism about new models, we should direct it at the specific quirks and caveats of these models. If we presume all mathematical approaches come from the same complicated, suspicious heap, then harmful models will escape incisive criticism and useful ones will be ignored.

Blanket suspicion isn’t the only problem. Equally concerning is when people put too much faith in seductive symbols and scribbles. On several occasions, I’ve heard people call a maths PhD thesis brilliant because no one could understand it. To them, complicated meant clever.

Some people like to encourage this view. If you work in a competitive industry, there are benefits to having others in awe of your work. It can give you power, and help you avoid criticism. But when your marvellous theory runs into trouble, admiration can quickly turn to anger, and we arrive back at previous problem.

Mathematics is not some kind of opaque, untrustworthy black magic. Nor is it an infallible solution to every dilemma. It’s just a set of ideas, which can help us understand our world. As with any subject, some bits are difficult and some are surprisingly easy. But in the words of mathematician Stan Gudder, “The essence of mathematics is not to make simple things complicated, but to make complicated things simple.”

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