The greatest shortcoming of the human race is our inability to understand the exponential function.

Dr Albert A Bartlett

Dr Albert A Bartlett, Professor Emeritus in Nuclear Physics at Colorado University, Boulder USA is an unlikely doomsayer, yet his words of mathematical wisdom, spelled out in measured, avuncular tones to his students, prophesy disaster for the human race. I have seen his lecture on You Tube where it is billed as “The most IMPORTANT video you’ll ever see.” http://www.youtube.com/watch?v=F-QA2rkpBSY. I recommend it, not only for it’s lucid explanation of the effects of exponentiality but also as an example of how a patient man should talk to idiots (the rest of us, I’m afraid). It has had over a million views but I doubt if another ten million views will have any effect on how we behave; there are simply too many of us now. Do the maths and you will see why. If there are six billion of us alive now and one million have seen the video, then 5.999 billion have not seen it. Besides, viewing is one thing, changing human behaviour is another. We don’t have a good track record when it comes to learning from our mistakes and there is a reason for this, as you will see.

Dr Bartlett explains the exponential function like this:

Imagine a flask which contains two bacteria which reproduce themselves every minute. After one minute there will be four bacteria, after two minutes there will be eight bacteria, after three minutes, sixteen bacteria and so on. After fifty-nine minutes there will be millions of bacteria but that’s OK because the flask is still only half full. After all, bacteria are quite tiny and it’s a big flask, half empty. Now Dr Bartlett asks us, ‘How long before the flask is full?’ We mathematical dummies ponder a bit and count our fingers, so he gives us the answer, ‘One minute!’ We feel a bit embarrassed; it’s so obvious! The Doctor makes the suggestion that these are very clever, industrious bacteria (you see where he is going, don’t you? There’s an analogy coming up). They discover that there are three more flasks that they can use, three times the space they started with. Phew, that was lucky! But then our optimism takes a slap in the face. Notice how I have identified us with the bacteria; most empathetic of me. ‘How long before the first extra flask is filled?’ Pause. ‘One minute!’ ‘And the next two flasks?’ Pause. ‘One more minute!’

With that, the human race is condemned by the inevitability and unconquerable truth of mathematics to a future of misery, distress and an unimaginable death rate.

Here is how it applies to the human race on earth. It’s not just humans, of course. The maths applies to anything that increases at a percentage rate, which means all life since the year dot. There are two closely linked restraints on everlasting exponential growth and they are running out of the means to increase (fuel, food, space) and death. To be really effective death should occur before you have reproduced, but any death reduces the numbers and slows down the rate of increase.

So what are the aspects of the human species that are subject to exponential growth? You at the back? Yes, population. Most of human activity is concerned with reproduction; we can’t seem to stop doing it. We don’t want to stop doing it. We are very good at it. About 10,000 years ago, our numbers could be counted in thousands and while our numbers were constrained by disease, limited food supplies and high child and parent mortality, then population growth was slow. It has only begun to increase rapidly in the last 2500 years and has rocketed in the last two centuries. Reasons: reduced child mortality, control of infectious disease, abundant food and clever use of space e.g. cities.

It’s worth reminding yourself of that flask of bacteria at this point. When things double they increase by the whole amount of what went before. The time this takes to happen is called the doubling time (easy, this maths isn’t it?). A simple way of working out the doubling time is to divide the rate of increase into 70. For example, a rate of increase of 7% per annum will give a doubling time of 10 years (70/7 =10). That’s a nice interest rate on your savings but if it is applied to oil and gas, say, it would mean that you had to discover and exploit, every ten years, as much oil and gas as has ever been discovered before. Of course, in the real world, it’s more complicated and I can already hear you say, “renewables, nuclear fission, energy efficiency, sustainable economies.” OK, they’ll slow things down a bit but, big, big but, if demand continues to increase at all there will be a doubling time and you will have to find as many resources in that time as ever were there before, and then again, and again, and again.

Here is a real time, real world fact. World population is now, give or take a few hundred million, 6 billion. A growth rate of 1% gives a doubling time of around 70 years (70/1 =70). That is a world population of 12 billion in 2080. In fact, the growth rate is about 1.2% which makes it less than 60 years to reach 12 billion. The usual argument at this point is that nations that are more prosperous have lower birth rates and that, as the world becomes more prosperous, birth rates in general will fall and population will stabilise. Sounds good, doesn’t it? Until you realise that there is a Catch 22. It’s this: in order to reduce human population growth to a zero rate of increase you have to increase the prosperity of everyone, or at least most of the world, up to current levels of prosperity in, say, Italy, which has a low birth rate. To do this you need the same level of resources that Italy uses spread across all nations of the earth. There are not enough resources to do this, and meanwhile the world population is growing anyway and using even more resources, without getting prosperous enough for the birth rate to stabilise. You need to have a growth in resource availability and use which is fast enough to overtake population growth. Remember, any growth rate is exponential and results in a doubling time.

It’s a fact that all economies assume growth as a birthright and generally a ‘good thing’. Think of the panic and gloom that occurs when we experience a recession and the joy that comes with a period of growth! Let’s do the maths again. UK economy is expected to grow by around 1.8% in 2010 and everybody say this isn’t very good. Growth in India and China is huge by comparison. It varies both in its effect on demand for resources and in its fluctuation, but generally, economic growth is a positive figure. So let’s look at an average 2% growth in the world economy. The doubling time is 70/2 = 35 years. Yes, at that rate the world economy will be twice as big in 2045. Will it be using twice the resources? Probably not. For one thing, not all wealth demands physical resources directly. I mean, look at banking. For another, we will probably find less resource depleting ways of doing things. What we won’t do, however, is use less resources. There will still be a doubling time, even if it is delayed by our ingenuity or luck. Remember, doubling time means using as much again as everything that has gone before. Let’s be generous and say that the doubling time is increased to 70 years. This means that by 2080, when the world population is 12 billion (double what it is now) the world economy will be four times as big as it is now and using twice the resources that it does now.

In 2008, total worldwide energy consumption was 474 exajoules (5×1020 J) with 80 to 90 percent derived from the combustion of fossil fuels. An exajoule is a million terawatts and a terawatt is a trillion joules. To put it in perspective, it’s the equivalent energy of 8 million of the atomic bombs that were dropped on Hiroshima. So by 2080 the world economies will need the annual energy equivalent of 16 million atomic bombs and, ha ha, most of that produced from renewable resources or else we all melt from global warming.

Now, what do you suppose the world energy demands were in, say 1710? You don’t know? Well, neither do I, but if you slide down the exponential curve you will see that it was not many atom bombs’ worth compared with today. Nevertheless, economic growth had started its inexorable path towards world resource depletion, driving population growth, and drawing on the fossil fuels for energy. But if you had been a member of the Royal Society, say Sir Isaac Newton, who was its President then, you would have laughed at the idea that in 300 years time we would be in this precarious situation. It would be another 100 years until Thomas Malthus published his Essay on the Principle of Population. He was so right in his deductions that it’s worth quoting him here.

‘The power of population is indefinitely greater than the power in the earth to produce subsistence for man. Population, when unchecked, increases in a geometrical ratio. Subsistence increases only in an arithmetical ratio. A slight acquaintance with numbers will show the immensity of the first power in comparison with the second.’[i]

Back to the flask of bacteria. If only one bacteria (nicknamed Malthus) was aware of the problem, then it might suggest that the flask was filling up, or rather, that the rate at which it was filling up was increasing and that there was a problem. No other bacteria would believe this because at 50 minutes into the hour ( the flask is full on the hour) the place would look pretty well empty. We are probably just entering the last, 59th, minute now, and we don’t have any more flasks. We are kidding ourselves that the earth is still only half full; it might well be, but that won’t stop it filling up in the next minute! There is only one earth and the moon is too far away and anyway, even if it was a fertile planet of chicken nuggets and cola it would all be used up in the 61st minute, wouldn’t it?

In his video lecture, Dr Bartlett gives us a list of the things that contribute towards population growth. I noted them down so that I could share them with you and fill you with fear for the future. They are: procreation, motherhood, large families, immigration, medicine, public health, sanitation, peace, law and order, scientific agriculture, accident prevention, clean air and ignorance of population growth. We love them all, don’t we? With the possible exception of large families, if you happen to live next door to one of them!

He gives a second list of the things that contribute to lessening population growth. They are: abstention, contraception, abortion, small families, non-immigration, disease, war, murder, violence, famine, accidents and pollution. With the possible exception of contraception we generally don’t like these things and try to avoid them, unless we are doing them for our own selfish ends of course.

These likes and dislikes are pretty well ingrained into us. In fact, the likes are what have brought us to the place we are now, the edge of the abyss. I don’t think we shall stop doing the things we like to do voluntarily. Hope and ignorance will see to that. I may be wrong in all this and so may Malthus and Dr Bartlett. But then the maths would have to be wrong, wouldn’t it? y = nx. Disprove!

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[i] Malthus T.R. 1798. An Essay on the Principle of Population. Chapter 1, p13 in Oxford World Classics reprint.