The problem of thinking in straight lines

As we grow up we are trained to think about things in a linear way. But it can leave us ill-equipped in our complex, fast moving modern world, harm our finances and even cause problems with artificial intelligence.

This is how the problem often starts: "If Jane pays £5 for 10 grapefruits, how many grapefruits does she get for £50?"

Answering the question, it transpires that the idealised world of mathematics is the only place you can buy 100 grapefruits and no-one bats an eyelid.

To find the answer to the question, many of us have been conditioned to use linear reasoning to assume that for 10 times as much money, Jane gets herself 10 times as many grapefruits.

The word "linear" describes a special relationship between two variables – an input and an output. If a relationship is linear, a change in one quantity by a fixed amount will always produce a fixed change in the other quantity. This is a good model for all sorts of real-world relationships. With a fixed exchange-rate, a pound Sterling might be worth two New Zealand dollars, £10 would be worth NZ$20 and £100 would be worth NZ$200. This is a special type of linear relationship. As you increase the pounds you want to exchange, the number of dollars you get back increases in direct proportion – if I double the input I also double the output.

If I can buy three chocolate bars for £2, then surely, I can buy six chocolate bars for £4. The number of bars I can purchase scales linearly with the money I'm prepared to spend. Linearity assumes there are no three-for-two offers on the table. (And of course, in reality exchange rates vary wildly with the changing fortunes of the financial market.)

Not all linear relationships are in direct proportion though. To convert from Celsius to Fahrenheit you need to multiply the Celsius temperature by 1.8 and add 32. Doubling the input doesn't double the output in this relationship, but because it is linear a fixed change in the input always corresponds to a fixed change in the output. A rise of 5C is always a rise of 9F no matter what temperature you start from. These relationships can be represented as straight lines, which is why we call them linear.

Perhaps I have laboured the point a little about these linear relationships, especially since linearity is such a familiar idea. But herein lies the problem: we are so familiar with the concept of linearity that we impose our linear frame of reference on data we observe in the real world.

This is linearity bias in its simplest form. As I explore in my new book How to Expect the Unexpected, many systems do not obey these simple linear relationships. For example, if I leave money in my bank account or forget to pay off a debt then that sum of money will grow non-linearly (specifically it will grow exponentially) – interest accruing on the interest. The more money I have (or owe) the faster it will grow. Because many of us are subject to linearity bias we underestimate how quickly these sums of money will grow, which makes saving for the future seem less attractive, but also makes taking on debt seem more attractive. Individuals with higher levels of linearity bias have been found to have higher debt-to-income ratios (the amount of debt they take on relative to their income).

It seems that the most important explanation for our over-reliance on linearity comes from the mathematics classroom itself. Investigations into the origins of this bias have shown that our propensity to assume linearity is present long before we leave school. These studies pose students questions in which linearity is not the right tool to use in order to see how they respond. These so-called pseudo-linearity problems might take the form:

"Laura is a sprinter. Her best time to run 100m (328ft) is 13 seconds, how long will it take her to run 1km (3,280ft)?"

It is not possible to ascertain the correct answer from the information in the problem. However, most students still reach for the linear solution, without any concern for the unrealistic nature of their underlying assumptions. They scale up the time to run 100m by a factor of 10, to account for the distance being 10 times longer, giving a time of 130 seconds to run 1km. Clearly this can only ever be a lower bound on the true answer since it neglects to take into account the fact that no athlete can sustain their best 100m pace over the course of 1km. Indeed, the linear answer would see Laura utterly destroying the world record for running 1km – two minutes and 11 seconds.

A compounding factor is the lack of acknowledgement in maths classes that the real world is usually not as simple as a maths problem. Even artificial intelligence is picking up these mistakes: ChatGPT, a chatbot designed to mimic human interactions, has learned these same biases. When I asked it "It takes three towels three hours to dry on the line, how long does it take nine towels to dry?" it responded with the answer "nine hours" reasoning that if you triple the number of towels, you triple the amount of time it takes for them to dry. Really, if your drying line is long enough, it shouldn't take any longer for nine towels to dry in parallel than three.

I'm baking with the kids and we want to make twice as many cupcakes as the recipe suggests then we need to use twice as much of each of the ingredients. The ingredients combine linearly to make twice as much mixture. This seems only right. But to propose this should hold true of every phenomenon in our world would be to deny the existence and the magic of emergent phenomena – for example, that no single molecule of H20 is wet or the unique fractals that snowflakes form, not by adding individual crystals together, but as one complex superstructure. Even our own lives are so much more than the simple sum of atoms and molecules which comprise our physical embodiments.

Although most of the time we are unaware of them, many of the most important relationships that we experience every day are nonlinear. But we have the idea of linearity drilled into us so early on and so often that sometimes we forget that other relationships can even exist. Our overfamiliarity with linear relationships means that, when something occurs that is nonlinear, it can catch us off-guard and confound our expectations.

By making the implicit assumption that inputs scale linearly with outputs, we are liable to find that our predictions can be way off the mark and that our plans can blow up in our faces. We live in a nonlinear world, but we are so used to thinking in straight lines that we often don't even notice it.

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