MONDAY, JAN 31, 2011

Understand the Exponential Function

Growth is the best thing imaginable, for a time


"The greatest shortcoming of the human race is our inability to understand the exponential function." —Albert Bartlett



TUCSON (A-P) — A classic introduction to the exponential function involves yeast added to a culture medium. So let's imagine a spherical vat with a diameter of 86 feet (26.23 meters) containing 2,496,205 gallons (9,449,166 liters) of grape juice. A single yeast cell is added (brewer's yeast average 3.5 microns, so if a yeast were the size of a human [1.7 m], the vat would be the size of Earth). Yeast can grow and divide every hour, so at the end of the first hour there are two yeast. At the end of the second, there are four yeast. At the end of the third hour there are eight, then 16, 32, 64, and so on.

Growth continues, hour after hour, with each hour, from the yeast's point of view, resembling the last—each generation ends up with twice as many neighbors. As the alcohol (yeast excreta) level in the vat slowly increases, other organisms living in the vat begin to die off. Things just keep getting better and better for the yeast until only yeast remain alive. Everything continues to go along swimmingly until the 43rd hour when the yeast population reaches 4.4 trillion, and the vat is "full" of yeast. The level of alcohol has finally become so toxic that even the yeast can't survive, and by the end of the 43rd hour all the yeast are dead (or cease to function). But yeast are adapted. They sporulate before they die and yeast spores are everywhere--waiting for an opportunity to grow exponentially again, as it is their nature to do so. Other species are not as adapted to overshoot and collapse.

Now the question, dear student, your final exam so to speak, is this: In what hour was the vat half full?

The answer, as Deep Thought has already computed, is 42.

So what? Ah, well, look around you. You might be doing just fine, but how many of your fellow 7+ billion humans can say the same (3.5 billion households live on less than $10/day)? Given the increasing difficulties of providing, however inadequately, for the current 7 billion people, do you think Earth can support 14 billion people? If you do, then you believe we are living at the start of the 42nd hour.

Maybe you're wildly optimistic and think that ever new and clever solutions will be forthcoming so that surely 28 billion people is not too many to be provided for. If so, then you believe we're living at the beginning of the 41st hour. Well, bad news for those who would like to believe so; the consensus among those who think about these things (scientists) is that the 41st hour has come and gone, that the planet is more than 1/4 full of humans and their plant/animal mutualists. While our world is vastly more complex than that of yeast in a culture tube, some truths don't go away by obfuscation. That the population will never reach 14 billion, due to the alledged demographic transition that is due to peak at about 11 billion around 2100, is likely.

So nothing to worry about, right? The planet could support 28 billion or surely 14 billion is supportable, and population will peak at just 11 billion. It's all good. The 11 billion will share what the 28 billion would have had. The real problem in the future may be depopulation post-peak. People will have to be paid to have children. If population growth can't be maintained, everyone will have to consume evermore to sustain economic growth. Who's going to complain about having to consume more? Naysaying ecofacists?

Growth, any rate of growth however small, in a finite system (e.g. Earth or the Milky Way), cannot continue—period; no ifs, ands, or buts about it. You can take it to the bank, stick it in your pipe, whatever you want to do, but this fact (not an opinion or even a deeply held belief) doesn't go away.

Growth is not limited to population. Per capita growth in consumption is also limited. While population may max out at 11 billion, all will want enough and most are being oversold on wanting more, far more than they need. Consumptive growth is limited. Self-delusion works for a time, but in the end reality will creep in. With unlimited energy "too cheap to meter" we might be able to pave over the entire land surface, cover the oceans in floating cities, tunnel out the lithosphere, grow food in well-lit caves, but at some point (a near future) growth would still have to come to an end. 

When people had to have six or eight kids so that maybe two might survive childhood, when child mortality drops to such an extent that most survive, then most people don't want to have eight kids. With birth control technology readily available, population growth rate declines, the "demographic transition" happens as it is happening. If there really were too few humans on the planet, people might have to be rewarded one way or another for having more than two children. The issue is not continued exponential population growth with no end in sight, but continued expectations of ever increasing "progress" towards having more as in consuming more stuff. Currently, the world's poor are getting slightly better off materially while the rich get disproportionately richer. When per capita consumption peaks and people, the commoners, see themselves getting poorer, expectations will change and the social order that serves elites will collapse as it always has. Someone will start a rumor that the First Lady said, "Let them eat cake," and.....

We're not talking about just population growth, but any material growth. Even if the human population leveled off around 2050 at 9 billion, the demand for growth in per capita consumption would not stop. China's consumption (economy) has been growing at 10% a year at which rate it will double every seven years (only 7% in 2015 so doubling down to every 10 years). How long can that be sustained and at what cost? Current economic systems begin to falter if the rate of growth declines—not growth itself, but merely if the rate of growth declines. Any growth less than 3%, with a per capita doubling time of less than 25 years, is called a recession.

You are living in a growth culture powered largely by fossil fuels during a meteoric rise. The Growth Culture knows nothing else; it can't even imagine a world without growth ("In Growth we trust"). So far, each century has resembled the last, things seemingly just keep getting better and better (not for nature but for us and our domestic plants and animals at least). But it won't last; it absolutely can't continue to get better and better.

The 43rd hour will be unlike all the previous ones. Understanding this is crucial. Nothing we've ever known so far will prepare us for it (though reading history helps). We are not yeast in a vat, we are just behaving like them. Few are predicting total die-off, but an 80% die-off in the 21th century is not unthinkable (James Lovelock: Enjoy life while you can).

You are living in the 42nd hour. It's just a question of whether we are in the early or late part of the hour. Is the world only half full? Do we have 50 minutes remaining? Maybe 20 minutes (years)? Or are we towards the end with only five minutes remaining? Actually the exact "moment" of Peak Everything (climax) won't be known until after the fact, so perhaps we are already into the 43rd hour. 


If yeast were as smart as humans, would they foresee the 43rd hour and avoid it? No, because they have evolved to cope by sporulation. If they were like humans, they would likely believe, during the growth period, that their Creator had obviously given them dominion over the vat, favored them above all creatures, and blessed them exceedingly in their unending growth. For 42 hours there would be no evidence to discount this view.

How many humans can even imagine the possibility that they might be wrong? How many, even well in to the 43rd hour, will still be in denial?

Maybe by 2050 fusion power plants will have sprung up everywhere and the upward trend will continue through the 21st century. Or space will be filled with solar arrays surrounding the sun beaming energy to Earth. Maybe a Dyson sphere will be constructed. Yes, absolutely maybe, but don't count on it or assume continued grow is a good thing, most devoutly to be hoped for. Even if growth continues through the 22nd century, its continuance will change nothing; it will merely make the industrial growth culture bigger and fall harder. Even if all solar energy emanating from the sun were intercepted (a Dyson sphere), it wouldn't be enough to sustain growth. Humans would at least have to stop growth while they emigrated to other solar systems to consume them and any life they may find would be assimilated too.

If you're beginning to think that the exponential function is something you really need to understand, here's Professor Bartlett's masterful presentation that he gave for over three decades, 1,742 times from 1969 until he died in 2013, subtitled "Sustainability 101." Don't use the word "sustainability" unless his message has become etched on your neural net. When he was exploring the implications of this function, he had one of the first electronic pocket calculators. He was calculating how long known coal reserves would last assuming a certain rate of consumption. He kept getting an absurdly small number and became convinced the calculator was malfunctioning. He broke out his log rhythm tables and got the same answer. The point is that while the exponential function is not hard to understand—it's simple arithmetic; it is hard to accept—to see the implications. But give it a chance to sink in. It could be that nothing you do could be more important than understanding the ramifications.

To bring home the implications, consider the following graphs:

population 10000 years

This graph should keep you awake at night. If it doesn't, you haven't been paying attention. It shows only the rise of human population. The following graph shows the rise and fall of yeast in the vat. The fall during the 43rd hour happens so quickly that it is barely noticeable on the graph. We'll soon know whether humans are smarter than yeast. 

Yeast rise and fall

In 1944, 29 reindeer were introduced to St. Matthew Island. In 1957 there were 1,350 rather fat reindeer. In 1963 the population reached about 6,000. In 1966 there were 42 survivors. By the 1980's, the last reindeer on the island had died.

reindeer on st. matthew island

Here's another look at human population growth over the last 4,000 years.

Human population growth

The slight dip in the 14th century was the Black Death. The question becomes, what will the 21th century look like? If you're like most people, you expect it to look like this:

21st century population

It currently looks like population will peak at 11 billion at most. But the future could look like this:

population die-off

Other futures are possible, but probable futures are likely to fall between Future A and C. Descent is foreseeable. The exact point (to the year) of peak growth is not. From the point of view of geological time or even history, the difference between A and C is distinguishable only using a magnifying glass. What we are living in is a blip.

Population ecologist John B. Calhoun famously created a mouse Utopia which provided unlimited food, water, and nest materials.  The only limit was space—kind of like an island paradise (or planet) so isolated that, should you ever want to leave, there's no place else to go.  Even if there was a Planet B, colonizing it or deep space would just delay the reckoning slightly.

When four pairs of mice were given dominion over their Eden they multiplied, fruitfully doubling their population every 55 days. By day 315 there were 620 mice at which point normal social behavior began breaking down, the population declined precipitously, and by day 600 there were no more surviving births—the remnant population continued to decline unto extinction in an environment where their every material need was provided for.

The reason for this curious outcome was that the social stressors of too many mice increased the level of violence, conflict, and assorted aberrant behaviors. This lead to the breakdown of society and to a failure to successfully parent such that those young who did survive were themselves so dysfunctional, those who were not eaten, that they could not successfully reproduce.  The survivors avoided social interaction and ended up dying alone. 

The lesson here is that it is not enough to survive—you have to survive with a functional culture intact, which excludes the growth culture, if a functional society is to emerge on the other side of the bottleneck. If only the most ruthless marauders or mindless ideologues survive, no new civilization could develop from them for many, many generations—if ever. It's not enough that just any culture survives; we need to lay the foundations now of one that isn't going to just repeat the same pattern and see to it that it survives.  

If the society you live in isn't going to change its course, perhaps it's time you changed yours. Individuals can overcome the inertia of large groups; only individuals, making intelligent choices, can radically change course. The deer staring into the headlights often fails to move or may even runs in front of the truck. Endeavor to think well now and decide upon a course of action before the headlights appear.

On a different scale:


The first blip assumes human population maxes at 10 billion. The second maxes at about 30,000,000 trillion on a well designed Dyson sphere. If shown at the same vertical scale, the height of the peak would be about 3,000,000 trillion times higher, about 40,000 miles higher than shown. Humans may be smart enough to build a Dyson sphere, yet not smart enough to live sustainably on it. Assume the Dyson sphere contains trillions of civilizations, each with a population many times bigger than Earth's 21st century population. When complete, if any one of these civilizations continued to grow by consuming nearby civilizations (see last 6,000 years of human history), growing at their expense, just as a cancer cell grows at the expense of surrounding cells, then the resulting cancerous growth would have the same outcome. Be not a cancer on the earth or solar system. The ecolate life is alternative (e.g. Kogi).

Even if some humans go to other stars to build Dyson spheres, hundreds of billions of them, the blip would be the same, just on a different scale. In sixteen thousand years human growthers could blip out the Milky Way. Even if some go to other galaxies, and the entire visible universe blips out (all stars becoming infrared emitters), the growth culture would still be a blip in cosmic time. The end point would be the same. No growth as there would be no media/energy left to consume. Sustainable growth is impossible unless there are an infinite number of accessible and habitable multiverses. Transitioning now to negative growth in order to descend to a sustainable level of no growth is alternative.

Back to Home Page








Soltech designs logo

Contact Eric Lee