How, Mathematically, We’re Fucked

Midjourney imagining “The Vizier Sessi asked the Rajah for a boon. He asked for one grain of rice to be placed on a chessboard and doubled every square --ar 16:9”

I have spoken about the physics and finances of how we’re fucked. Now let’s talk about the math. It’s the reason for both.

To start, let me tell you about the Chessboard Problem. Once, long ago, when the game was young and novel, the Rajah of Somewheristan was so impressed by the game that he offered the Minister Sessi a boon for introducing him to chess. Boons are a serious thing in India and Sessi thought he’d teach the king a lesson. Sessi asked for something very small. He asked for one grain of rice to be placed on a chessboard and doubled every square. The king granted it laughingly, but things got out of control fast.

At the end of one row, the rice was spilling off the board. By the end of the third row, it was spilling out of sacks. By the fifth row, it was more than the king had in his storehouses. By the penultimate row (the second to last) the king owed 50% of current global rice production (about 45m tons). By the last square, he owed the production of 12 worlds. Which he didn’t have. Which no one has. The king would be ruined by exponential growth. That was the lesson he learned. We ignore this parable at our peril.

Exponential growth is absurd and we have based our entire civilization on this absurdity. Modern society—unlike the ancients—requires exponential growth for social stability. It is a belief we all generally share, and it is this belief that is killing us. Climate change and resource collapse are just symptoms of believing and impossible thing and following it into absurdity. You cannot have infinite growth on a finite planet. It’s not physically, financially, or mathematically possible. And you can do the math yourself.

Any Growth Is Exponential

We’re used to thinking of modest growth rates like 2–3% and thinking that they’re harmless. Just get a better iPhone in a few years, whatever. But any steady growth rate is exponential, and exponentials are fucked.

As a rule of thumb, you can divide the growth rate into 70 to get the doubling time. So 2% growth every year means doubling every 35 years, an exponential. In geological time, 35 years might as well be 35mm, the rough size of a chess square. Thus we have the chessboard problem on a planetary scale. If you keep doubling anything—be it fossil fuels or lithium use or mascara sales—you inevitably, inexorably run out of worlds. The type of growth is actually immaterial in the long run. It’s just math. Ask bacteria, who have run into this problem before when they caused runaway climate change last time.

The problem of exponential growth in bacteria. Even if the doubling bacteria finds another ‘planet’ (another jar), they overflow that in 10 minutes. Via Tom Murphy’s textbook. (

Overflowing is just how exponential growth works (re: cancer) and our civilization requires exponential growth or it freaks the fuck out. Anything less is a recession, requiring massive human sacrifice and economic priests bailing out their masters with furious monetary incantations. Anything to keep the ball rolling, but no one questions where the ball is going. Where it must to go. All of these reverent priests ignore the gravity of the situation, focusing on their imaginary models of a human-centered universe. Beneath all the obfuscating math of not-a-science economics, these latter day priests cannot even prove their core assumptions. Because their core assumptions are false. You simply cannot have infinite growth on a finite planet. You run out of chessboard and, indeed, we have already run out.

We’ve already used up 1.7 Earths worth of resources and we’re just a few more doublings away from over-overshoot. The modern lie is that we can simply change energy sources and keep growing, but math gives a fuck about the type of energy. Whatever it is, it can’t keep growing exponentially. And energy (and emissions) is only one part of the problem. Solving climate change would be like lowering a fever (yay!) while ignoring the fact that you have cancer. CO² emissions are just one symptom of our reckless overshoot, look into the warming oceans or the receding forests or all of our dead relatives to see the damage we do with the energy we have. That is, it’s not just the emissions coming out of the back of this civilization, it’s the diggers and trawlers and incinerators on the front.

Exponential Growth Is Bad

It doesn’t matter if we use clean solar and wind and never touch a drop of oil again. Firstly, we’d just use the energy for destroying other parts of the Earth, mining lithium and paving roads and otherwise fucking things up. And second, the second law of thermodynamics would catch up. Any ‘work’ generates heat. At our scale we don’t notice, but scale up exponentially and we’d boil the oceans at just a few centuries of 2.3% growth. Regardless of the energy source. It sounds stupid but that’s just what exponentials are. They’re absurd.

Read Tom Murphy yo'self:

When I mention this boiling the oceans in 400 years people say its absurd, but you can do the math yourself. This is high school level stuff and it shows what happens when you build an entire economic system around a bad assumption. We all take growth for granted but never question what it means beyond our 401k’s. But just stretch it out to just 400 years and the planet turns into Venus. Yes it is absurd, and our entire civilization is based on this absurdity.

As I said in earlier pieces, we require more and more resource use every year, and everybody’s investments and pensions are gambled 20-to-1 on this extraction going on forever. But A) it can’t and B) if it somehow did we’d boil the fucking oceans. And again, endless growth is the mainstream position. This is economic orthodoxy. And, despite, all the numbers and hand-waving, it’s not math. You don’t need a PhD to understand the chessboard problem. Indeed, you need a PhD to not understand it.

The problem is that very few of us paid attention in math or physics class and we take it for granted that serious economists know what they’re doing. But they fucking don’t. Their human-centric view of the world is as false as the earth-centric model of the universe was. The fact is that the geocentric view of the world was actually pretty good at making predictions (and is still used in planetariums), but it’s still wrong. And modern economics is just wrong. Its basic assumptions do not stand up to basic math.

Yet in the moment, it seems crazy to even acknowledge this. We are the bacteria in the penultimate jar, and the jar appears half empty to us. Plenty of room (and time) to spare. But that’s not how exponentials work. While you can understand the math of it in a few pages, the human mind simply cannot understand how fast shit gets out of control. As Āciriyar Tom Murphy says:

We saw in this chapter that unabated growth leads to absurd results. First, we calibrated our intuition in the context of bacteria in jars. The key point is that the jar is half full one doubling time before it is full. While this seems obvious, it delays the drama to the very end, acting fast to impose hard limits and catch the inhabitants by surprise. The conditions that persisted for many generations — thus taken for granted — suddenly change completely.

In this sense, climate change is a distraction. We are nearing the inflection point where doubling overflows the jar (indeed, we’re already past the point of no return) and we’re debating what to fill the jar with. Lithium and copper instead of oil and gas. It’s fucking absurd. This distracting debate (a hallmark of Democracy™) allows opinion to split into two camps—oil growth and green growth—both of which support the ruling hierarchy of more growth. It’s like arguing over getting cancer in the colon or the lungs. Exponential growth anywhere is a cancer. We’re missing the point at our literal peril.

Our growth is killing bees, destroying forests, and stripping the seas. Hell, it doesn’t even make us happy. If we find magically free and green energy, we’ll just use it to fuck the fish and birds and the trees even worse. And make things even worse for ourselves. That’s how cancer works. You can chase it out of one place, but it spreads everywhere else. We proudly talk about our cancerous growth like we’re getting swole, but in fact we’re just about to explode, spraying sinews and testosterone everywhere.

This isn’t a value judgement or an opinion. It’s just math. It’s just exponentials. Double anything long enough and it gets out of control, slowly, then all at once. Double something innocous like a grain of rice and it ruins a kingdom. Double something toxic like oil and you poison the skies. Double lithium and you end up in the same place. Double the ambitions of any species and it will overflow its petri dish, and we’re just another species. Human ambitions have already overflowed the planet. It doesn’t matter if we get to Mars. In fact, it’s entirely possible that we already did this to Venus in the long-forgotten past.

As Murphy says, “In the end, physics puts a timeline on expectations with respect to growth in energy on Earth.” His blog is called Do The Math and the textbook lays out fascinating homework in exponential growth. Unfortunately for you and me, this assignment is way too late. You can do the homework for your own edification, but it was due last century. When it comes to understanding exponential growth, we’ve already failed the course.

Further Reading