I thought I might do a few posts about matters arising from talks at this workshop I’m at. Today is devoted to tutorial talks, and the second one was given by John Skilling and in the course of it, he made some comments about the concept of infinity in science. These remarks weren’t really central to his talk, but struck me as an interesting subject for a few tangential remarks of my own.

Most of us – whether scientists or not – have an uncomfortable time coping with the concept of infinity. Physicists have had a particularly difficult relationship with the notion of boundlessness, as various kinds of pesky infinities keep cropping up in calculations. In most cases this this symptomatic of deficiencies in the theoretical foundations of the subject. Think of the ‘ultraviolet catastrophe‘ of classical statistical mechanics, in which the electromagnetic radiation produced by a black body at a finite temperature is calculated to be infinitely intense at infinitely short wavelengths; this signalled the failure of classical statistical mechanics and ushered in the era of quantum mechanics about a hundred years ago. Quantum field theories have other forms of pathological behaviour, with mathematical components of the theory tending to run out of control to infinity unless they are healed using the technique of renormalization. The general theory of relativity predicts that singularities in which physical properties become infinite occur in the centre of black holes and in the Big Bang that kicked our Universe into existence. But even these are regarded as indications that we are missing a piece of the puzzle, rather than implying that somehow infinity is a part of nature itself.

One exception to this rule is the field of cosmology. Somehow it seems natural at least to consider the possibility that our cosmos might be infinite, either in extent or duration, or both, or perhaps even be a multiverse comprising an infinite collection of sub-universes. If the Universe is defined as everything that exists, why should it necessarily be finite? Why should there be some underlying principle that restricts it to a size our human brains can cope with?

On the other hand, there are cosmologists who won’t allow infinity into their view of the Universe. A prominent example is George Ellis, a strong critic of the multiverse idea in particular, who frequently quotes David Hilbert

The final result then is: nowhere is the infinite realized; it is neither present in nature nor admissible as a foundation in our rational thinking—a remarkable harmony between being and thought.

This comment is quoted from a famous essay which seems to echo earlier remarks by Carl Friedrich Gauss which can be paraphrased:

Infinity is nothing more than a figure of speech which helps us talk about limits. The notion of a completed infinity doesn’t belong in mathematics.

This summarises Gauss’s reaction to Cantor’s Theory of Ininite Sets. But to every Gauss there’s an equal and opposite Leibniz

I am so in favor of the actual infinite that instead of admitting that Nature abhors it, as is commonly said, I hold that Nature makes frequent use of it everywhere, in order to show more effectively the perfections of its Author.

You see that it’s an argument with quite a long pedigree!

When I was at the National Astronomy Meeting in Llandudno a few years ago, I attended an excellent plenary session that featured a Gerald Whitrow Lecture, by Alex Vilenkin, entitled *The Principle of Mediocrity*. This was a talk based on some ideas from his book Many Worlds in One: The Search for Other Universese, in which he discusses some of the consequences of the so-called eternal inflation scenario, which leads to a variation of the multiverse idea in which the universe comprises an *infinite* collection of causally-disconnected “bubbles” with different laws of low-energy physics applying in each. Indeed, in Vilenkin’s vision, all possible configurations of all possible things are realised somewhere in this ensemble of mini-universes. An infinite number of National Astronomy Meetings, each with the same or different programmes, an infinite number of Vilenkins, etc etc.

One of the features of this scenario is that it brings the anthropic principle into play as a potential “explanation” for the apparent fine-tuning of our Universe that enables life to be sustained within it. We can only live in a domain wherein the laws of physics are compatible with life so it should be no surprise that’s what we find. There is an infinity of dead universes, but we don’t live there.

I’m not going to go on about the anthropic principle here, although it’s a subject that’s quite fun to write or, better still, give a talk about, especially if you enjoy winding people up! What I did want to say mention, though, is that Vilenkin correctly pointed out that three ingredients are needed to make this work:

- An
*infinite*ensemble of realizations - A discretizer
- A randomizer

Item 2 involves some sort of principle that ensures that the number of possible states of the system we’re talking about is *not* infinite. A very simple example from quantum physics might be the two spin states of an electron, up (↑) or down(↓). No “in-between” states are allowed, according to our tried-and-tested theories of quantum physics, so the state space is discrete. In the more general context required for cosmology, the states are the allowed “laws of physics” ( i.e. possible false vacuum configurations). The space of possible states is very much larger here, of course, and the theory that makes it discrete much less secure. In string theory, the number of false vacua is estimated at 10^{500}. That’s certainly a very big number, but it’s not infinite so will do the job needed.

Item 3 requires a process that *realizes* every possible configuration across the ensemble in a “random” fashion. The word “random” is a bit problematic for me because I don’t really know what it’s supposed to mean. It’s a word that far too many scientists are content to hide behind, in my opinion. In this context, however, “random” really means that the assigning of states to elements in the ensemble must be *ergodic*, meaning that it must visit the entire state space with some probability. This is the kind of process that’s needed if an infinite collection of monkeys is indeed to type the (large but finite) complete works of shakespeare. It’s not enough that there be an infinite number and that the works of shakespeare be finite. The process of typing must also be *ergodic*.

Now it’s by no means obvious that monkeys would type ergodically. If, for example, they always hit two adjoining keys at the same time then the process would not be ergodic. Likewise it is by no means clear to me that the process of realizing the ensemble is ergodic. In fact I’m not even sure that there’s any process at all that “realizes” the string landscape. There’s a long and dangerous road from the (hypothetical) ensembles that exist even in standard quantum field theory to an actually existing “random” collection of observed things…

More generally, the mere fact that a mathematical solution of an equation can be derived does not mean that that equation describes anything that actually exists in nature. In this respect I agree with Alfred North Whitehead:

There is no more common error than to assume that, because prolonged and accurate mathematical calculations have been made, the application of the result to some fact of nature is absolutely certain.

It’s a quote I think some string theorists might benefit from reading!

Items 1, 2 and 3 are *all* needed to ensure that each particular configuration of the system is actually realized in nature. If we had an infinite number of realizations but with either infinite number of possible configurations or a non-ergodic selection mechanism then there’s no guarantee each possibility would actually happen. The success of this explanation consequently rests on quite stringent assumptions.

I’m a sceptic about this whole scheme for many reasons. First, I’m uncomfortable with infinity – that’s what you get for working with George Ellis, I guess. Second, and more importantly, I don’t understand string theory and am in any case unsure of the ontological status of the string landscape. Finally, although a large number of prominent cosmologists have waved their hands with commendable vigour, I have never seen anything even approaching a rigorous proof that eternal inflation does lead to realized infinity of false vacua. If such a thing exists, I’d really like to hear about!

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