The Beginning of Infinity: Explanations That Transform the World (30 page)

The fallacious idea of delegating all one’s work to other staff in
higher-numbered rooms is called an
infinite regress.
It is one of the things that one cannot validly do with infinity. There is an old joke about the heckler who interrupts an astrophysics lecture to insist that the Earth is flat and supported on the back of elephants standing on a giant turtle. ‘What supports the turtle?’ asks the lecturer. ‘Another turtle.’ ‘What supports
that
turtle?’ ‘You can’t fool me,’ replies the heckler triumphantly: ‘it’s turtles from there on down.’ That theory is a bad explanation not because it fails to explain
everything
(no theory does), but because what it leaves unexplained is effectively the same as what it purports to explain in the first place. (The theory that the designer of the biosphere was designed by another designer, and so on ad infinitum, is another example of an infinite regress.)

One day in Infinity Hotel, a guest’s pet puppy happens to climb into a trash bag. The owner does not notice, and passes the bag, with the puppy, to the next room.

Within two minutes the puppy is nowhere. The distraught owner phones the front desk. The receptionist announces over the publicaddress system, ‘We apologize for the inconvenience, but an item of value has been inadvertently thrown away. Will all guests please undo all the trash-moving actions that they have just performed, in reverse order, starting as soon as you receive a trash bag from the next-higher-numbered room.’

But to no avail. None of the guests return any bags, because their fellow guests in the highernumbered rooms are not returning any either. It was no exaggeration to say that the bags are nowhere. They have not been stuffed into a mythical ‘room number infinity’. They no longer exist; nor does the puppy. No one has done anything to the puppy except move it to another numbered room, within the hotel. Yet it is not in any room. It is not anywhere in the hotel, or anywhere else. In a finite hotel, if you move an object from room to room, in however complicated a pattern, it will end up in one of those rooms. Not so with an infinite number of rooms. Every individual action that the guests performed was both harmless to the puppy and
perfectly reversible. Yet, taken together, those actions annihilated the puppy and cannot be reversed.

Reversing them cannot work, because, if it did, there would be no explanation for why a puppy arrived at its owner’s room and not a kitten. If a puppy did arrive, the explanation would have to be that a puppy was passed down from the next-higher-numbered room – and so on. But that whole infinite sequence of explanations never gets round to explaining ‘why a puppy?’ It is an infinite regress.

What if, one day, a puppy did just arrive at room 1, having been passed down through all the rooms? That is not
logically
impossible: it would merely lack an explanation. In physics, the ‘nowhere’ from which such a puppy would have come is called a ‘naked singularity’. Naked singularities appear in some speculative theories in physics, but such theories are rightly criticized on the grounds that they cannot make predictions. As Hawking once put it, ‘Television sets could come out [of a naked singularity].’ It would be different if there were a law of nature determining what comes out – for in that case there would be no infinite regress and the singularity would not be ‘naked’. The Big Bang may have been a singularity of that relatively benign type.

I said that the rooms are identical, but they do differ in one respect: their room numbers. So, given the types of tasks that the management request from time to time, the low-numbered rooms are the most desirable. For instance, the guest in room 1 has the unique privilege of never having to deal with anyone else’s trash. Moving to room 1 feels like winning first prize in a lottery. Moving to room 2 feels only slightly less so. But
every
guest has a room number that is unusually close to the beginning. So every guest in the hotel is more privileged than almost all other guests. The clichéd politician’s promise to favour
everyone
can be honoured in Infinity Hotel.

Every room is at the beginning of infinity. That is one of the attributes of the unbounded growth of knowledge too: we are only just scratching the surface, and shall never be doing anything else.

So there is no such thing as a
typical room number
at Infinity Hotel. Every room number is untypically close to the beginning. The intuitive idea that there must be ‘typical’ or ‘average’ members of any set of values is false for infinite sets. The same is true of the intuitive ideas of ‘rare’ and ‘common’. We might think that half of all natural numbers
are odd, and half even – so that odd and even numbers are equally common among the natural numbers. But consider the following rearrangement:

A rearrangement of the natural numbers that makes it look as though one-third of them are odd

That makes it look as though the odd numbers are only half as common as even ones. Similarly, we could make it look as though the odd numbers were one in a million or any other proportion. So the intuitive notion of a
proportion
of the members of a set does not necessarily apply to infinite sets either.

After the shocking loss of the puppy, the management of Infinity Hotel want to restore the morale of the guests, so they arrange a surprise. They announce that every guest will receive a complimentary copy of either
The Beginning of Infinity
or my previous book,
The Fabric of Reality.
They distribute them as follows: they dispatch a copy of the older book to every millionth room, and a copy of the newer book to each remaining room.

Suppose that you are a guest at the hotel. A book – gift-wrapped in opaque paper – appears in your room’s delivery chute. You are hoping that it will be the newer book, because you have already read the old one. You are fairly confident that it
will
be, because, after all, what are the chances that your room is one of those that receive the old book? Exactly one in a million, it seems.

But, before you have a chance to open the package, there is an announcement. Everyone is to change rooms, to a number designated on a card that will come through the chute. The announcement also mentions that the new allocation will move all the recipients of one of the books to odd-numbered rooms, and the recipients of the other book to even-numbered ones, but it does not say which is which. So you cannot tell, from your new room number, which book you have received. Of course there is no problem with filling the rooms in this manner: both books had infinitely many recipients.

Your card arrives and you move to your new room. Are you now any less sure about which of the two books you have received? Presumably not. By your previous reasoning, there is now only a one in
two
chance that your book is
The Beginning of Infinity
, because it is now in ‘half the rooms’. Since that is a contradiction, your method of assessing those probabilities must have been wrong. Indeed, all methods of assessing them are wrong, because – as this example shows – in Infinity Hotel there is
no such thing
as the probability that you have received the one book or the other.

Mathematically, this is nothing momentous. The example merely demonstrates again that the attributes probable or improbable, rare or common, typical or untypical have literally no meaning in regard to comparing infinite sets of natural numbers.

But, when we turn to physics, it is bad news for anthropic arguments. Imagine an infinite set of
universes
, all with the same laws of physics except that one particular physical constant, let us call it
D
, has a different value in each. (Strictly speaking, we should imagine an
uncountable
infinity of universes, like those infinitely thin cards – but that only makes the problem I am about to describe worse, so let us keep things simple.) Assume that, of these universes, infinitely many have values of
D
that produce astrophysicists, and infinitely many have values that do not. Then let us number the universes in such a way that all those with astrophysicists have even numbers and all the ones without astrophysicists have odd numbers.

This does not mean that half the universes have astrophysicists. Just as with the book distribution in Infinity Hotel, we could equally well label the universes so that only every third universe, or every trillionth one, had astrophysicists, or so that every trillionth one did not. So there is something wrong with the anthropic explanation of the fine-tuning problem: we can make the fine-tuning go away just by relabelling the universes. At our whim, we can number them in such a way that astrophysicists seem to be the rule, or the exception, or anything in between.

Now, suppose that we calculate, using the relevant laws of physics with different values of
D
, whether astrophysicists will emerge. We find that for values of
D
outside the range from, say, 137 to 138, those that contain astrophysicists are very sparse: only one in a trillion such
universes has astrophysicists. Within the range, only one in a trillion does
not
have astrophysicists, and for values of
D
between 137.4 and 137.6 they all do. Let me stress that in real life we do not understand the process of astrophysicist-formation remotely well enough to calculate such numbers – and perhaps we never shall, as I shall explain in the next chapter. But, whether we could calculate them or not, anthropic theorists would wish to interpret such numbers as meaning that, if we measure
D
, we are
unlikely
to see values outside the range from 137 to 138. But they mean no such thing. For we could just relabel the universes (shuffle the infinite pack of ‘cards’) to make the spacings exactly the other way round – or anything else we liked.

Scientific explanations cannot possibly depend on how we choose to label the entities referred to in the theory. So anthropic reasoning, by itself, cannot make predictions. Which is why I said in
Chapter 4
that it cannot explain the fine-tuning of the constants of physics.

The physicist Lee Smolin has proposed an ingenious variant of the anthropic explanation. It relies on the fact that, according to some theories of quantum gravity, it is possible for a black hole to spawn an entire new universe inside itself. Smolin supposes that these new universes might have different laws of physics – and that, moreover, those laws would be affected by conditions in the parent universe. In particular, intelligent beings in the parent universe could influence the black holes to produce further universes with person-friendly laws of physics. But there is a problem with explanations of this type (known as ‘evolutionary cosmologies’): how many universes were there to begin with? If there were infinitely many, then we are left with the problem of how to count them – and the mere fact that each astrophysicist-bearing universe would give rise to several others need not meaningfully increase the
proportion
of such universes in the total. If there was no first universe or universes, but the whole ensemble has already existed for an infinite time, then the theory has an infiniteregress problem. For then, as the cosmologist Frank Tipler has pointed out, the entire collection must have settled into its equilibrium state ‘an infinite time ago’, which would mean that the evolution that brought about that equilibrium – the very process that is supposed to explain the fine-tuning –
never happened
(just as the lost puppy is
nowhere
). If there was initially only one universe, or a finite number,
then we are left with the fine-tuning problem for the original universe(s): did they contain astrophysicists? Presumably not; but if the original universes produced an enormous chain of descendants until one, by chance, contains astrophysicists, then that still does not answer the question of why the entire system – now operating under a single law of physics in which the apparent ‘constants’ are varying according to laws of nature – permits this ultimately astrophysicist-friendly mechanism to happen. And there would be no anthropic explanation for
that
coincidence.