Debunking Doomsday
A new answer to an old philosophical question
There is an argument in academic philosophy known as the Doomsday Argument, which claims that one can derive probabilistic limits on the final size of the human population based solely on how many people have lived so far. This sounds weird — and it is — and yet debate on the merits of the argument remains live, as a quick perusal of the Wikipedia page on the subject will show. While rebuttals exist, so far none has proved quite decisive enough to settle the matter once and for all.
I became interested in the topic years ago when I first read the argument, which initially seemed to me preposterous. Yet I couldn’t quite nail down an argument against it, and eventually came to accept its conclusions. Recently, however, I have come to see the flaw in the logic with great clarity, and I currently have a paper on the topic undergoing peer review with the analytic philosophy journal Erkenntnis. In this essay, I will explain the Doomsday Argument and my case against it, which turns out to have relevance beyond just this specific debate.
Let’s start with the argument itself. To understand the logic, consider the case of a raffle. Let’s say an unknown number of consecutively numbered raffle tickets have been thrown into a barrel. I reach in and draw out the number 57. How many tickets are in the barrel? Simple logic tells us that the number is much more likely to be around one hundred than it is to be around, say, one million. In fact, from basic Bayesian principles, we can conclude that it is 90% likely that the number of tickets is no more than 570 — ten times the number I drew. My best guess, if I am forced to choose a number, is 114, based on the simple fact that any random draw is as likely to fall in the first half of the distribution as in the second. Therefore, 57 would be the average draw for a raffle of 114 tickets.
Now there is a principle, widely accepted in philosophy, that, in the absence of other evidence, “one should consider oneself a randomly selected member of the reference class to which one belongs”. What exactly is our “reference class”? Well, if a raffle ticket is one member of the class of “raffle tickets in this barrel”, then you and I are members of the class of “all human beings”. So, to follow the logic of the principle I outlined above, I should consider myself a random example of the human species. To be more precise about what this means, if I select some attribute I possess, I should treat it as if it had been randomly drawn from the raffle barrel of human attributes.
Let’s take height as an example. You, dear reader, might be unusually tall or short for all I know. But in aggregate, there is no reason to think that my readers are likely to be exceptionally vertically advantaged or disadvantaged. Therefore, if I have to guess your height, my best guess is that you are of average height. So far, so obvious.
But now consider another of your attributes: your birth position among all the humans who have ever been born. Let’s say you are human being number 60 billion. According to the principle we cited earlier, you should consider yourself a randomly selected example of the reference class of all human beings, but who exactly does this include? All human beings who are alive, all who have ever been born, or all who will ever be born?
According to the Doomsday Argument, when it comes to our birth order, we should reason as if we had been sampled from all humans who will ever be born. In that case, extending the logic of the raffle, if your birth rank is approximately 60 billion, we should treat that number as if it had been drawn from a raffle with tickets from one to the final human population size. Our best guess as to the final size of the human population would then be twice that number: 120 billion. And we can be 90% sure that no more than 600 billion people will ever live.
Unless you have read a lot of philosophy, I am sure that this reasoning probably makes you choke on your tea. How can I possibly be a random selection from the entire population of humanity if much of that population doesn’t even exist yet? Surely something is amiss here! Supporters of the Doomsday Argument can meet that objection in a couple of ways. They can invoke a “block universe” theory of time according to which, from some objective perspective, all of time already exists and the appearance that part of the total population doesn’t yet is simply an illusion of perspective. Or they can argue that it doesn’t matter anyway.
Let’s imagine a simulator that is capable of simulating the trajectory of a species population from start to finish. (For simplicity’s sake, let’s assume that an infinite or endless population is impossible.) Imagine that we run this simulator any number of times to generate many possible human population histories. In one, humans died out quickly due to an asteroid strike. In another, they destroyed themselves after inventing nuclear weapons. In yet another, they went on to populate the stars and didn’t go extinct for billions of years.
Let us imagine that as the simulator runs, it makes a “bet” each time that a human is born within the simulation that this person falls within the first 50% of people who will be born within that simulation run. At the end of each run, it then counts up how many times its bets were correct. It should be obvious that its bets will be correct exactly 50% of the time. Similarly, if it bets that each person falls within the first 90%, it will be right 90% of the time.
Now imagine that we are part of such a simulation. The simulator bets that we will fall in the first 50% of the total population that its program will generate. We know it wins its bets exactly half the time. We have no reason to think we are particularly close to the start or end of our simulation run. We are, in effect, a random sample. So we are justified to make the same bet. And by the same logic, we can conclude that we are 90% likely to fall in the first 90% of people born. In other words, the raffle logic holds even though the full population hasn’t yet been generated. The Doomsday Argument holds!
Have I persuaded you, or do you smell a rat? If so, you are quite right. But where is the smell coming from?
To see the problem, let’s go back to the simulator and modify its design. Let’s make it so that whenever a new human is born in the simulation, a ticket is emitted from the computer with that human’s birth rank printed on it. The ticket goes into a raffle barrel. We run the simulator a number of times, with a ticket emitted into the barrel for each person born in each of those simulation runs. When we’ve run all of our simulations, we pull a ticket out of the barrel.
For clarity’s sake, let’s assume the simulator runs just twice. On one run, it produces a million tickets, and in the other, it produces just ten. You draw a ticket, and discover that it has the number ten on it. Now you must guess whether it came from the simulation that produced ten tickets, or the one that produced a million. But you have no way of knowing! Both runs produced the number ten, and while it was quite a fluke that you drew such a small number — it was overwhelmingly probable that you’d draw a much larger number from the million ticket run — now that you have drawn it, you have no way of knowing whether it came from the long run or the short one.
This is the situation we find ourselves in as humans in an evolving population with an uncertain future. We don’t know which “simulation run” we are in. We don’t know how things turn out on our timeline, so to speak, and our birth rank on its own provides zero information about that future.
This logic is sufficient to kill the Doomsday Argument. However, a committed defender still has some ammunition left to fire, so let’s let them fire it, since it exposes a more fundamental point that has ramifications beyond this specific debate. An adherent of the “block universe” theory of time might argue that the human population does not go down many trajectories as in the simulation example, but is, in fact, completely pre-determined. If there is only one way it can turn out, then there is only one simulation run in the barrel, as it were, and the validity of the raffle analogy is restored.
However, this doesn’t save the Doomsday Argument. The problem — and this is the argument’s core fallacy — is that an observer inside an evolving population process noticing their birth rank is not in the same logical position as someone drawing the observer’s birth rank from a barrel. Imagine that an angel appeared to you and said, “God has conducted a raffle at the end of time and has chosen you from all the people who will ever live for this special Random Person award. Congratulations!” In that case, you would be justified in thinking, “Wow. Humans probably don’t survive to populate the galaxy. Otherwise, what would be the chance of me, such a relatively early human, being chosen for this award!” However, just having some birth rank or other does not put us in that position!
In the case of a raffle, there is a physical randomisation procedure that shuffles all the tickets in the barrel and ensures that the ticket selection is indifferent with respect to the size of the number on it. In the case of the Random Person award, presumably God had some similar procedure for randomly selecting an individual winner across the completed human population. It is this randomisation procedure that justifies the inference we draw about the population.
As part of an evolving population process whose end we cannot foresee, we know (very roughly) our birth rank, but are ignorant of our relative position in the final total population. In the case of God’s Random Person award, some additional information is added to our birth rank: the knowledge that we have been uniquely selected by a randomised process across the final population. It is this additional information, absent before we received the award, that justifies our inference about the future of humanity. It is clear that if everyone receives an award (which is the same as nobody receiving one — the actual situation), then the validity of the inference vanishes.
Thus, the Doomsday Argument proponent is forced to defend the position that, were such an award to exist, its recipient would have no more information about the final population size than any other person. But that is plainly false. The more people there are in the final population that God selects from, the less likely any given individual is to receive the award, so receiving it provides the same kind of information about population size that a raffle draw does about the size of the lottery. Merely being aware of one’s birth rank does not, since everyone is in that category.
In order for a statistical inference from an individual observation to the wider population from which it is drawn to be valid, the process of generating that observation must be stochastic (or “symmetry preserving”: equal probabilities for each observation are preserved). This process provides a direct, physical causal link between the population and the observation that licenses the inference we draw.
In keeping with the philosophical tradition of naming principles for easy reference later, I call this the Stochastic Generation Assumption (SGA). In the case of an individual observing their own birth rank, the observed variable (birth rank) was not generated by such a process. It arose as an intrinsic part of the population process itself and thus provides no information about the future of that process.
This principle, if correct, has wider implications for a range of debates across philosophy. For example, in Nick Bostrom’s now famous/infamous Simulation Argument (which I have heaped scorn on in the past for different reasons), Bostrom argues that if, in a transhuman future, humanity succeeds in simulating its own ancestors, then it is almost certain that we are already living in one of these simulations. The argument again rests on the assumption that we should reason as if we had been randomly drawn from our reference class, which, in Bostrom’s Simulation Argument, is something like “contemporary human consciousnesses”.
Underpinning the argument is the same misapplication of lottery logic. Consider that in Bostrom’s scenario, in which a future civilisation develops the capacity to create human consciousnesses in simulated worlds, there are two “pipelines” that generate consciousnesses: the historical-evolutionary, “base reality” process, and the computerised simulation process. Let’s say the base reality pipeline creates a million human consciousnesses, and the simulation pipeline a trillion. Bostrom assumes that we can mix these consciousnesses all up together in a big consciousness barrel in the sky and treat our own consciousness as having been drawn from it, so that the chance of our being a biological consciousness is one in a billion (a million divided by a trillion).
If SGA is correct, however, this conclusion is illegitimate. There is no physical mechanism that mixes up these two sources and randomises them before assigning the selected consciousness to an individual. It remains a metaphysical possibility that we live in a simulation, if indeed consciousnesses can be instantiated inside computers; however, the number of such synthetic consciousnesses that exist in the future has no bearing on whether or not we are in one. We can’t borrow hypothetical simulated consciousnesses from the future to prove statistically that we must be one. Without this unwarranted “soul lottery”, the whole edifice of Bostrom’s logic collapses, and we are left with nothing more than a speculation.
Much of the philosophical debate on the Doomsday Argument has revolved around arcane metaphysical arguments about the nature of “cross world identity”, “haecceity” and what constitutes the correct reference class for an observer — just take a look at the Wikipedia page to see what I mean! In my view, much of this has been a distraction from what should be regarded as an epistemic problem, not a metaphysical one. We should not need to delve into such obscure and apparently undecidable debates to work out whether the claims of the Doomsday Argument are justified. And indeed, when the argument is reframed not as a question of reference classes, but one of statistical inferences from observable values, all this elaborate intellectual machinery falls away to reveal a simple category error. Only stochastic selection processes in the physical world can justify statistical conclusions about populations in the physical world. Once that is understood, much that was once obscure becomes clear.