The Irresistible Lure of the Linear

Straight Line Cityscape

Straight Line Cityscape – Photo by Montag

Straight lines.

Great aren’t they?

With straight lines we build barns, houses, bridges and roads.

With ruler – or straight edge – in hand, our engineers design piston rods, pipelines, train tracks. Even artists use them to help create the illusion of perspective.

The humble straight line has been responsible for countless advances in the modern world.

Take the wheel, that shining exemplar of the non-linear. Wheels often only make motion possible through the use of spokes or axles. That is, straight lines.

No wonder then that we’re obsessed with the linear. But it’s not all good.

A word of caution, if I may.

In addition to the wonderful practicality of straight things, it might also be that they are the cause of numerous socio-political fuck-ups. The simple cause of these self-inflicted disasters is that we find it difficult to go beyond the linear.

Let me explain.

Within basic statistics (of which I claim to be no expert) there is a concept of the linear regression line. It sounds complex, but it is not. Even if you haven’t been anywhere near a mathematics or statistics course you will be aware of it.

Consider a graph with time (in years) on its horizontal axis and profit on the vertical axis (see figure below). You’ve seen this countless times before, I am sure. If this was your company, or one you had invested in, you would hope to see profit increasing over time as indicated below.

Raw Data

Note how the progress of profit is not smooth. In some years it is up on the previous period, while in others it is down. Nevertheless, the general trend is upward; that is, growth.

The linear regression line is simply a mathematically smoothed trend of the data. It’s an illustration; an easy to digest summary of the more jaggedy real-world data. The graph below shows the linear regression (trend) of the data superimposed upon the first graph.

Raw Data With Linear Trend

This trend line is very useful. Chiefly, because it allows for prediction. We can continue drawing the line such that we can say “if we continue as we are, in 2020 we’ll be making an annual profit of £2 million”. Loadsamoney! (See below).

Linear Trend With Prediction

Smoothing data into a linear regression may be a human trait. We might do this instinctively in our heads. In the same way that humans seek patterns in the world, we might also seek straight-line trends in data. Give us a bunch of facts and we’ll look for the linear regression.

Which is cool, but it may also be problematic.

The advanced statistician will tell you that there many other types of trends in addition to the linear: exponential, polynomial, power. You don’t need to worry about the details, it’s just that mathematically, the real world data may not legitimately fit a linear trend, as much as you might want to simply draw a straight line over the plot. The only valid mathematical trend might be a little more, well, curvy, like the polynomial trend line below.

Definitely not straight.

Polynomial

And indeed your “linear” trend for 2008-2016 might just be a small upward slope in the overall and much larger squiggliness of the full data set for 1990-2050.

60 year chart

Why is this important?

It is because in so many walks of life – for example business, politics, life – theorists might think in terms of the linear, when it’s not actually applicable to the situation in question.

We’re done with mathematics now. Instead here are some everyday examples of what I am talking about.

Last summer I gave a talk alongside a lovely anarchist with whom I discussed the idea of freedom. Now I’ll be honest, I am very sympathetic to the anarchist sensibility (and for those who are not aware anarchism is a legitimate political philosophy, it’s not about smashing things up), but only up to a point.

Real anarchists rightly conclude that a certain quantity of freedom is good (you’d be crazy not to think this – “more slavery please!”). The mistake they may make is to assume that ever greater freedom leads to ever greater benefits, i.e. in a linear fashion. However – and this is my point – it might be that at a particular point along the line, freedom produces no more benefits. Or even that past a certain point, greater freedoms result in negative outcomes.

Refer back to the polynomial chart above. Change the horizontal axis label from Year to Freedom, and the vertical axis from Profit to Social Good.

See?

This may be but a single instance of theory refusing to see beyond the linear.

Here’s another example. Market capitalism. For sure a lot of nice things have come out of this political philosophy. Neo-liberals and others were so enamoured of these good things that they went off on a grand linear illusion. They thought that more and more market capitalism would continue to deliver the goods (so to speak) for society. However, perhaps now in the light of banking crises, corruption and inequality we might be forgiven for re-examining the exact shape of the trend line that we once assumed was straight.

I could burden you with countless other examples, but I won’t. My blog posts are long enough as it is.

Here’s some homework.

Think of other areas in which theorists may mistakenly assume the linear trend. Add your answer as a comment below.

The best examples don’t win a prize. And I’m sorry, we can’t return your paintings.

That’s it for today, but the next time you’re about to jump onto an easy straight-line conclusion, remember that your assumed relationship might be only a little slope within a much grander and more complicated overall trend.

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