How to Ensure the World Never Ends

A few weeks ago, we talked about the idea that the world is not magically eternal, and it is possible for it to come to an end. We finished with a positive message that by taking this possibility seriously, we have the ability to prevent it from happening. Today, we’re going to use math to show why there is hope, and why it is worth it to work for the preservation of humanity.

Each year, there is some probability the world will end. At times, the probability has been frighteningly high, such as in the US-USSR cold war. It is clear that we want to avoid the risk rising to such levels ever again. But how low is low enough? Suppose there is a 1 in 100 chance the world will end this year. That doesn’t seem too bad; it’s a little scary, but we’re almost certain to survive. But what if the probability remains the same next year, and the next, and the next? A 1 in 100 chance of extinction in one year adds up to a 63 in 100 chance of extinction in 100 years!

It’s impossible to get the probability to 0, because there is always the possibility for something to go horribly wrong. How low, then, is low enough? One in a million? One in a billion? Because of the way probabilities stack up, the cumulative probability will rise to 63 percent over that timespan. In other words, if there is a one in a million chance of annihilation per year, then the probability will have risen to 63 percent after a million years, and continue to climb after that.

I don’t know about you, but I’m not comfortable with the probability rising over 1 percent ever. Well, in the reasonable future at least; the universe will inevitably end, but that is so far in the future we can approximate it as infinity. So how is it possible, if probabilities accumulate and it is impossible to get the probability to 0, to prevent the cumulative probability from rising up into the unsafe zone?

To answer this, we have to look at the mathematical concepts of limits and convergence. Let’s start with limits. Take the equation,

y = 1/x²

What happens when x = 0? We’ve been taught that it doesn’t work, we’re not allowed to divide by 0. But we can take the limit as x approaches 0. The smaller x gets, the larger y gets. Thus, the limit of y as x approaches 0 is infinity.

Now let’s talk about convergence. Suppose

y = 1/2 + 1/4 + 1/8 …

The sequence goes on infinitely long, each term being half the previous term. What is y if there are an infinite number of terms? You might think that, since we are adding an infinite number of positive numbers together, the answer will be infinity. But it’s not. If we take the limit of y as the number of terms approaches infinity, the series converges to 1. That’s right, the sum of this series, all infinite terms of this geometric sequence, is a finite number!

The graph for the function e^1-x continually approaches 0, but never reaches it. However, as x approaches infinity, the total area under the curve starting at 0 approaches the number e, or approximately 2.7.

Image generated by Wolfram Alpha LLC. 2009. Wolfram|Alpha. (access April 23, 2020). This image has been edited.

So now we have what we need in order to reduce the probability of human extinction to 1 in 100 cumulatively from now to the infinite future: Start small, and reduce the probability every year. If the probability is 1 in a million this year, we should aim for 1 in 1.1 million next year, 1 in 1.3 million the next year, and so on forever. We can never reach perfection, but we can always get just a little better.

Of course, this is an idealized goal. The probability in real life goes up and down depending on technology, politics, Earth’s geological and environmental state, and all kinds of things. But we can use wisdom, governments, institutions, and individual passion to mitigate the probabilities so that when they rise, they don’t rise as high as they might have. And if we keep working on these problems forever, we can continue to improve on them until the universe starts to run down. And even then, if we have been working on existential risk for all that time, we may be able to forestall the heat death of the universe as our post-biological descendants enjoy life for eons untold.