RARELY DOES PHILOSOPHY produce empirical predictions.

The doomsday argument is an

important exception. From seemingly trivial

premises, it seeks to show that the risk that

humankind will soon become extinct has been systematically

underestimated. Nearly everybody’s first

reaction is that there must be something wrong with

such an argument. Yet despite being subjected to

intense scrutiny by a growing number of philosophers,

no simple flaw in the argument has been identified.

It started some 15 years ago when astrophysicist

Brandon Carter discovered a previously unnoticed

consequence of a version of the weak anthropic principle.

Carter did not publish his finding, but the idea

was taken up by philosopher John Leslie, who has

been a prolific author on the subject, culminating in

his monograph The End of the World (Routledge,

1996). Versions of the doomsday argument have also

been independently rehearsed by other authors. In

recent years there have been numerous papers trying

to refute the argument, and an approximately equal

number of papers refuting these refutations.

I shall explain the doomsday argument in three

steps.

Step I

Imagine a universe that consists of 100 cubicles. In

each cubicle there is one person. Ninety of the cubicles

are painted blue on the outside and the other ten

are painted red. Each person is asked to guess whether

they are in a blue or a red cubicle (and everybody

knows all this).

Now, suppose you find yourself in one of these

cubicles. What colour should you think it has? Since

90% of all people are in blue cubicles, and since you

don’t have any other relevant information, it seems

you should think that there is a 90% probability that

you are in a blue cubicle. Let’s call this idea, that you

should reason as if you were a random sample from

the set of all observers, the self-sampling assumption.

Suppose everyone accepts the self-sampling assumption

and everyone has to bet on whether they are in a

blue or red cubicle. Then 90% of all people will win

their bets and 10% will lose. Suppose, on the other

hand, that the self-sampling assumption is rejected

and people think that one is no more likely to be in a

blue cubicle; so they bet by flipping a coin. Then, on

average, 50% of the people will win and 50% will lose.

The rational thing to do seems to be to accept the selfsampling

assumption, at least in this case.

Step II

Now we modify the thought experiment a bit. We still

have the 100 cubicles, but this time they are not

painted blue or red. Instead they are numbered from

1 to 100. The numbers are painted on the outside.

Then a fair coin is tossed (by God perhaps). If the coin

falls heads, one person is created in each cubicle. If the

coin falls tails, then people are only created in cubicles

1 to 10.

You find yourself in one of the cubicles and are

asked to guess whether there are ten or 100 people.

Since the number was determined by the flip of a fair

coin, and since you haven’t seen how the coin fell and

you don’t have any other relevant information, it

seems you should believe that there is a 50% probability

that it fell heads (and thus that there are 100

people).

Moreover, you can use the self-sampling assumption

to assess the conditional probability of a number

between 1 and 10 being painted on your cubicle,

given how the coin fell. For example, conditional on

heads, the probability that the number on your cubicle

is between 1 and 10 is 10%, since one out of ten

people will then find themselves there. Conditional

on tails, the probability that you are in number 1 to

10 is 100%; for you then know that everybody is in

one of those cubicles.

Suppose that you open the door and discover that

you are in cubicle number 7. Again you are asked,

how did the coin fall? But now the probability is

greater than 50% that it fell tails. For what you are

observing is given a higher probability on that

hypothesis than on the hypothesis that it fell heads.

The precise new probability of tails can be calculated

using Bayes’s theorem. It is approximately 91%. So

after finding that you are in cubicle number 7, you

should think that with 91% probability there are only

ten people.

Step III

The last step is to transpose these results to our actual

situation here on Earth. Let’s formulate the following

two rival hypotheses:

‘Doom soon’

Humankind becomes extinct in the next century and

the total number of humans that will have existed is,

say, 20bn.

‘Doom late’

Humankind survives the next century and goes on to

colonise the galaxy; the total number of humans is,

say, 200 trillion. To simplify the exposition we will

consider only these hypotheses using a more finegrained

partition of the hypothesis doesn’t change the

principle, although it would give more exact

numerical values. ‘Doom soon’ corresponds to there only being ten

people in the thought experiment of Step II. ‘Doom

late’ corresponds to there being 100 people. Corresponding

the numbers on the cubicles, we now have

the ‘birth ranks’ of human beings their positions in

the human race. Corresponding to the prior probability

(50%) of the coin falling heads or tails, we now

have some prior probability of ‘doom soon’ or ‘doom

late’. This will be based on our ordinary empirical estimates

of potential threats to human survival, such as

nuclear or biological warfare, a meteorite destroying

the planet, self-replicating nano-machines running

amok, a breakdown of a meta-stable vacuum state

resulting from high-energy particle experiments, and

so on (presumably there are dangers that we haven’t

yet thought of). Let’s say that based on such considerations,

you think that there is a 5% probability of

doom soon. The exact number doesn’t matter for the

structure of the argument.

Finally, corresponding to finding you are in cubicle

number 7 we have the fact that you find that your

birth rank is about 60bn (that’s approximately how

many humans have lived before you). Just as finding

you are in cubicle 7 increased the probability of the

coin having fallen tails, so finding you are human

number 60bn gives you reason to think that doom

soon is more probable than you previously thought.

Exactly how much more probable will depend on the

precise numbers you use. In the present example, the

posterior probability of doom soon will be very close

to 100%. You can with near certainty rule out doom

late.

That is the doomsday argument in a nutshell. After

hearing about it, many people think they know what

is wrong with it. But these objections tend to be mutually

incompatible, and often they hinge on some

simple misunderstanding. Be sure to read the literature

before feeling too confident that you have a refutation.

And the point is?

If the doomsday argument is correct, what precisely

does it show? It doesn’t show that there is no point in

trying to reduce threats to human survival ‘because

we’re doomed anyway’. On the contrary, it could

make such efforts seem even more urgent. Working to

reduce the risk that nano-technology will be abused to

destroy intelligent life, for example, would decrease

the prior probability of doom soon, and this would

reduce its posterior probability after taking the

doomsday argument into account; humankind’s life

expectancy would go up.

There are also a number of possible loopholes in

what the doomsday argument shows. For instance, it

turns out that if there are many extraterrestrial civilisations

and you interpret the selfsampling

assumption as applying

equally to all intelligent beings and

not exclusively to humans, then

another probability shift occurs that

exactly counterbalances and cancels

the probability shift that the doomsday

argument implies.

Another possible loophole occurs if

there are to be infinitely many

humans it’s not clear how to apply

the self-sampling assumption to the

infinite case. Further, if the human

species evolves into some vastly

more advanced species fairly soon

(within a century or two), then it is

not clear whether these post-humans would be in the

same reference class as us, so it is not clear how the

doomsday argument should be applied then. Yet

another possibility is if population figures go down

dramatically it would then be much longer before

enough humans were born to begin to make your

birth rank look surprisingly low. And finally, it may be

that the reference class needs to be adjusted so that

not all observers, not even all humans, will belong to

the same reference class.

The justification for this adjustment would have to

come from a general theory of observational selection

effects, of which the self-sampling assumption would

be only one element. A theory of observational selection

effects of how to correct for biases that are introduced

by the fact that our evidence has been filtered

by the precondition that a suitably positioned observer

exists to ‘have’ the evidence would have applications

in a number of scientific fields, including cosmology

and evolutionary biology.

So although the doomsday argument contains an

interesting idea, it needs to be combined with additional

assumptions and principles (some of which

remain to be worked out) before it can be applied to

the real world. In all likelihood there will be scope for

differing opinions about our future. Nonetheless, a

better understanding of observational selection effects

will rule out certain kinds of hypotheses and impose

surprising constraints on any coherent theorising

about the future of our species and about the distribution

of observers in the universe.

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