# Collapsarithmetic - Chris Malins

The exponential function is one of the basic building blocks of modern mathematics. In principle to understand it requires only the most basic arithmetic – if something is exponentially growing in time, it increases by a set factor every year. Tale a simple example – a species of bacteria that reproduces so that, unrestrained, its numbers double every minute. This is an exponentially growing population.

Imagine that one of these bacteria is in a bottle at 11pm, and that at midnight the bottle will be full. Now the question – when will the bottle be half full? The answer is simple – at one minute to midnight. For all of its existence up to 11.59, the bacterial colony has had more than half of its available resource (space) left. Only as one minute to midnight passes does it start to become clear how close to exhausting this resource the colony really is.

Consider now people. We are different to bacteria, insofar as many of us allegedly understand the exponential function. But there seems to be a disconnect between this knowledge that it exists, and achieving an intuition for what it means. I say this, because anything growing by the same percentage, year on year, will double over some period. Imagine that energy use were growing at a steady 7% per annum. In ten years, we would double our demand. And in each subsequent ten year period, we would use more energy than had been used in the entirety of human history. For 2% growth, the doubling period would be 35 years.

Exponential growth is intrinsically unsustainable. It is inconceivable that any real resource would be available in enough quantity to sustain even 2% growth indefinitely. Take the wind – plenty of that, you might think? If energy use grew 2% annually from now, in less than 100 years we would need to harness every breath of wind in the Earth’s atmosphere to keep us powered. A handful of years later, we would be using as much energy as absorbed by all the plants on Earth.

What about better technology? Plants and wind are just inefficient ways to harvest solar power. Well, if we could harvest every bit of solar energy that fell on the Earth’s surface, we could keep going a little longer. If we covered the entire planet, land and sea, with 100% efficient solar cells, we could meet our requirement for maybe 500 years. That is to say, in a time comparable to the duration of the Roman Empire, we would have gone from the industrial revolution to using 100% of all solar energy reaching the planet.

If we were using nuclear power instead of solar, we’d never reach this point – in less than 300 years we’d be releasing so much energy we’d be causing global warming without needing the greenhouse effect. In 500 years the average summer daytime temperature would be tending towards boiling point.

Clearly these scenarios cannot happen – natural limitations will kick in first. And that is the point – the social philosophy of growth, the idea that sustained 3% GDP growth (or growth in the use of any resource) could continue is absurd - economies built on this premise must fail. Nevertheless, this arithmetically ill-informed expectation is dominant – we rely on constant growth to afford services, pay pensions, stave off unemployment and so on.

We need to get growth under control – population growth, energy growth, fertiliser use growth, water use growth. And we need an economy where no growth is ok, where zero annual GDP growth doesn’t mean a collapse in the standard of living. Any other long-term economic vision is absurd – because it’s past one minute to midnight, and I think I’m getting uncomfortably close to the top of the bottle.

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