The Beginning of Infinity

by David Deutsch

  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.

  Smolin’s theory does the right thing: it proposes an overarching framework for the ensemble of universes, and some physical connections between them. But the explanation connects only universes and their ‘parent’ universes, which is insufficient. So it does not work.

  But now suppose we also tell a story about the reality that connects all these universes and gives a preferred physical meaning to one way of labelling them. Here is one. A girl called Lyra, who was born in universe 1, discovers a device that can move her to other universes. It also keeps her alive inside a small sphere of life support, even in universes whose laws of physics do not otherwise support life. So long as she holds down a certain button on the device, she moves from universe to universe, in a fixed order, at intervals of exactly one minute. As soon as she lets go, she returns to her home universe. Let us label the universes 1, 2, 3 and so on, in the order in which the device visits them.

  Sometimes Lyra also takes with her a measuring instrument that measures the constant D, and another that measures – rather like the SETI project, only much faster and more reliably – whether there are astrophysicists in the universe. She is hoping to test the predictions of the anthropic principle.

  But she can only ever visit a finite number of universes, and she has no way of telling whether those are representative of the whole infinite set. However, the device does have a second setting. On that setting, it takes Lyra to universe 2 for one minute, then universe 3 for half a minute, universe 4 for a quarter of a minute and so on. If she has not released the button by the time two minutes are up, she will have visited every universe in the infinite set, which in this story means every universe in existence. The device then returns her automatically to universe 1. If she presses it again, her journey begins again with universe 2.

  Most of the universes flash by too fast for Lyra to see. But her measuring instruments are not subject to the limitations of human senses – nor to our world’s laws of physics. After they are switched on, their displays show a running average of the values from all the universes they have been in, regardless of how much time they spent in each. So, for instance, if the even-numbered universes have astrophysicists and the odd-numbered ones do not, then at the end of a two-minute journey through all the universes her SETI-like instrument will be displaying 0.5. So in that multiverse it is meaningful to say that half the universes have astrophysicists.

  Using a universe-travelling device that visited the same universes in a different order, one would obtain a different value for that proportion. But, suppose that the laws of physics permit visiting them in only one order (rather as our own laws of physics normally allow us to be at different times only in one particular order). Since there is now only one way for measuring instruments to respond to averages, typical values and so on, a rational agent in those universes will always get consistent results when reasoning about probabilities – and about how rare or common, typical or untypical, sparse or dense, fine-tuned or not anything is. And so now the anthropic principle can make testable, probabilistic predictions.

  What has made this possible is that the infinite set of universes with different values of D is no longer merely a set. It is a single physical entity, a multiverse with internal interactions (as harnessed by Lyra’s device) that relate different parts of it to each other and thereby provide a unique meaning, known as a measure, to proportions and averages over different universes.

  None of the anthropic-reasoning theories that have been proposed to solve the fine-tuning problem provides any such measure. Most are hardly more than speculations of the form ‘What if there were universes with different physical constants?’ There is, however, one theory in physics that already describes a multiverse for independent reasons. All its universes have the same constants of physics, and the interactions of these universes do not involve travel to, or measurement of, each other. But it does provide a measure for universes. That theory is quantum theory, which I shall discuss in Chapter 11.

  *

  The definition of infinity in terms of a one-to-one correspondence between a set and part of itself was original to Cantor. It is connected only indirectly to the informal, intuitive way that non-mathematicians have conceived of infinity both before and since – namely that ‘infinite’ means something like ‘bigger than any finite combination of finite things’. But that informal notion is
rather circular unless we have some independent idea of what makes something finite, and what makes a single act of ‘combination’ finite. The intuitive answer would be anthropocentric: something is definitely finite if it could in principle be encompassed by a human experience. But what does it mean to ‘experience’ something? Was Cantor experiencing infinity when he proved theorems about it? Or was he experiencing only symbols? But we only ever experience symbols.

  One can avoid this anthropocentrism by referring instead to measuring instruments: a quantity is definitely neither infinite nor infinitesimal if it could, in principle, register on some measuring instrument. However, by that definition a quantity can be finite even if the underlying explanation refers to an infinite set in the mathematical sense. To display the result of a measurement the needle on a meter might move by one centimetre, which is a finite distance, but it consists of an uncountable infinity of points. This can happen because, although points appear in lowest-level explanations of what is happening, the number of points never appears in predictions. Physics deals in distances, not numbers of points. Similarly, Newton and Leibniz were able to use infinitesimal distances to explain physical quantities like instantaneous velocity, yet there is nothing physically infinitesimal or infinite in, say, the continuous motion of a projectile.

  To the management of Infinity Hotel, issuing a finite public-address announcement is a finite operation, even though it causes a transformation involving an infinite number of events in the hotel. On the other hand, most logically possible transformations could be achieved only with an infinite number of such announcements – which the laws of physics in their world do not allow. Remember, no one in Infinity Hotel – neither staff nor guest – ever performs more than a finite number of actions. Similarly in the Lyra multiverse, a measuring instrument can take the average of an infinite number of values during a finite, two-minute expedition. So that is a physically finite operation in that world. But taking the ‘average’ of the same infinite set in a different order would require an infinite number of such trips, which, again, would not be possible under those laws of physics.

  Only the laws of physics determine what is finite in nature. Failure to realize this has often caused confusion. The paradoxes of Zeno of Elea, such as that of Achilles and the tortoise, were early examples. Zeno managed to conclude that, in a race against a tortoise, Achilles will never overtake the tortoise if it has a head start – because, by the time Achilles reaches the point where the tortoise began, the tortoise will have moved on a little. By the time he reaches that new point, it will have moved a little further, and so on ad infinitum. Thus the ‘catching-up’ procedure requires Achilles to perform an infinite number of catching-up steps in a finite time, which as a finite being he presumably cannot do.

  Do you see what Zeno did there? He just presumed that the mathematical notion that happens to be called ‘infinity’ faithfully captures the distinction between finite and infinite that is relevant to that physical situation. That is simply false. If he is complaining that the mathematical notion of infinity does not make sense, then we can refer him to Cantor, who showed that it does. If he is complaining that the physical event of Achilles overtaking the tortoise does not make sense, then he is claiming that the laws of physics are inconsistent – but they are not. But if he is complaining that there is something inconsistent about motion because one could not experience each point along a continuous path, then he is simply confusing two different things that both happen to be called ‘infinity’. There is nothing more to all his paradoxes than that mistake.

  What Achilles can or cannot do is not deducible from mathematics. It depends only on what the relevant laws of physics say. If they say that he will overtake the tortoise in a given time, then overtake it he will. If that happens to involve an infinite number of steps of the form ‘move to a particular location’, then an infinite number of such steps will happen. If it involves his passing through an uncountable infinity of points, then that is what he does. But nothing physically infinite has happened.

  Thus the laws of physics determine the distinction not only between rare and common, probable and improbable, fine-tuned or not, but even between finite and infinite. Just as the same set of universes can be packed with astrophysicists when measured under one set of laws of physics but have almost none when measured under another, so exactly the same sequence of events can be finite or infinite depending on what the laws of physics are.

  Zeno’s mistake has been made with various other mathematical abstractions too. In general terms, the mistake is to confuse an abstract attribute with a physical one of the same name. Since it is possible to prove theorems about the mathematical attribute, which have the status of absolutely necessary truths, one is then misled into assuming that one possesses a priori knowledge about what the laws of physics must say about the physical attribute.

  Another example was in geometry. For centuries, no clear distinction was made between its status as a mathematical system and as a physical theory – and at first that did little harm, because the rest of science was very unsophisticated compared with geometry, and Euclid’s theory was an excellent approximation for all purposes at the time. But then the philosopher Immanuel Kant (1724–1804), who was well aware of the distinction between the absolutely necessary truths of mathematics and the contingent truths of science, nevertheless concluded that Euclid’s theory of geometry was self-evidently true of nature. Hence he believed that it was impossible rationally to doubt that the angles of a real triangle add up to 180 degrees. And in this way he elevated that formerly harmless misconception into a central flaw in his philosophy, namely the doctrine that certain truths about the physical world could be ‘known a priori’ – that is to say, without doing science. And of course, to make matters worse, by ‘known’ he unfortunately meant ‘justified’.

  Yet, even before Kant had declared it impossible to doubt that the geometry of real space is Euclidean, mathematicians had already doubted it. Soon afterwards the mathematician and physicist Carl Friedrich Gauss went so far as to measure the angles of a large triangle – but found no deviation from Euclid’s predictions. Eventually Einstein’s theory of curved space and time, which contradicted Euclid’s, was vindicated by experiments that were more accurate than Gauss’s. In the space near the Earth, the angles of a large triangle can add up to as much as 180.0000002 degrees, a variation from Euclid’s geometry which, for instance, satellite navigation systems nowadays have to take into account. In other situations – such as near black holes – the differences between Euclidean and Einsteinian geometry are so profound that they can no longer be described in terms of ‘deviations’ of one from the other.

  Another example of the same mistake was in computer science. Turing initially set up the theory of computation not for the purpose of building computers, but to investigate the nature of mathematical proof. Hilbert in 1900 had challenged mathematicians to formulate a rigorous theory of what constitutes a proof, and one of his conditions was that proofs must be finite: they must use only a fixed and finite set of rules of inference; they must start with a finite number of finitely expressed axioms, and they must contain only a finite number of elementary steps – where the steps are themselves finite. Computations, as understood in Turing’s theory, are essentially the same thing as proofs: every valid proof can be converted to a computation that computes the conclusion from the premises, and every correctly executed computation is a proof that the output is the outcome of the given operations on the input.

  Now, a computation can also be thought of as computing a function that takes an arbitrary natural number as its input and delivers an output that depends in a particular way on that input. So, for instance, doubling a number is a function. Infinity Hotel typically tells guests to change rooms by specifying a function and telling them all to compute it with different inputs (their room numbers). One of Turing’s conclusions was that almost all mathematical functions that exist logically cannot be computed by an
y program. They are ‘non-computable’ for the same reason that most logically possible reallocations of rooms in Infinity Hotel cannot be effected by any instruction by the management: the set of all functions is uncountably infinite, while the set of all programs is merely countably infinite. (That is why it is meaningful to say that ‘almost all’ members of the infinite set of all functions have a particular property.) Hence also – as the mathematician Kurt Gödel had discovered using a different approach to Hilbert’s challenge – almost all mathematical truths have no proofs. They are unprovable truths.

  It also follows that almost all mathematical statements are undecidable: there is no proof that they are true, and no proof that they are false. Each of them is either true or false, but there is no way of using physical objects such as brains or computers to discover which is which. The laws of physics provide us with only a narrow window through which we can look out on the world of abstractions.

  All undecidable statements are, directly or indirectly, about infinite sets. To the opponents of infinity in mathematics, this is due to the meaninglessness of such statements. But to me it is a powerful argument – like Hofstadter’s 641 argument – that abstractions exist objectively. For it means that the truth value of an undecidable statement is certainly not just a convenient way of describing the behaviour of some physical object like a computer or a collection of dominoes.

  Interestingly, very few questions are known to be undecidable, even though most are – and I shall return to that point. But there are many unsolved mathematical conjectures, and some of those may well be undecidable. Take, for instance, the ‘prime-pairs conjecture’. A prime pair is a pair of prime numbers that differ by 2 – such as 5 and 7. The conjecture is that there is no largest prime pair: there are infinitely many of them. Suppose for the sake of argument that that is undecidable – using our physics. Under many other laws of physics it is decidable. The laws of Infinity Hotel are an example. Again, the details of how the management would settle the prime-pairs issue are not essential to my argument, but I present them here for the benefit of mathematically minded readers. The management would announce: