Ultimate Explanations of the Universe (14 page)

The reader who has got this far will not need much persuading that there is a great deal of confusion, both conceptual and methodological, in the collection of issues surrounding the slogan “multiverse.” Various authors understand the expression “other universes” in various ways; they employ various justifications for the need for multiverse studies and differ in their assessment of the status thereof. Some regard them as creations of science fiction, others see them as metaphysical hypotheses, and still others consider them falsifiable models deserving a place in official science.

Other worlds (universes) have been a subject for discussion in science and philosophy for a long time now, not always merely as fictional possibilities. When Leibniz was pondering on the question of God’s reasons in choosing our universe as the “winning candidate for implementation” out of the infinite number of other universes (see Chap. 18), he did not treat these other contenders as purely fictional, but rather as real candidates for being created. Today philosophers continue to follow Leibniz’s method, albeit in a secularised version, which they often refer to as the method of “counterfactual conditionals.” By “counterfactual conditional” they mean “statements asserting that something happens under certain conditions, which are presupposed not to be satisfied in reality.”
¹
For instance, we wish to find out whether a certain statement is “necessary” in a given system of beliefs. To verify this, we negate the statement or modify it in some way. Then we insert the negated or modified statement into the system and examine its logical coherence. And we say we have “constructed a different universe.” Basically every natural law may be interpreted as a counterfactual statement. For example, the law of inertia says that if there is no force acting on a body, it will move at a constant velocity and in a straight line. In our universe such a situation never happens de facto, so we call into existence “another universe” in which this law is strictly obeyed. But we only do this to learn something about our own universe. In this context the “other universe” is simply a logically coherent
description
of a certain reality. Nonetheless, we have to admit that there exist authors who insist that “other universes” are not only a description of a certain reality, but actually make up a certain reality.
²

The concept of a set or family of worlds (universes) has also been current for a long time in the empirical sciences, quite independently of the contemporary discussion on the multiverse. In mathematics and theoretical physics studies are frequently conducted of the space of solutions to some differential equation; if it is agreed that each solution to the equation describes a possible universe (and this is done fairly often), then the space of the solutions is simply a family of universes. This kind of space may be studied using very sophisticated mathematical methods. The subject of study may be not only the particular solutions, but also the entire structure of the space: its stability, sensitivity to perturbations, the distribution of a variety of properties in the space of the solutions. Such procedures are very important in applications in physics, where the measurements obtained experimentally are never exact but always subject to experimental error. The physical situation is never modelled by a single solution to a differential equation, but by a sub-family of solutions which are close to each other and the measurable parameters of which are contained within the admissible margin of error. Moreover, the sub-area of the solution space to which a given solution belongs must have the property of stability, viz. a small perturbation in the initial conditions for a given solution should yield a solution which does not differ much from the given solution. Otherwise the margin of error would encompass radically different solutions for the modelling of a given process, in other words the experiment would confirm many very different models (within the margin of experimental error), and no empirical predictions would be possible.

This procedure also works in cosmology, and here it becomes even more like the idea of the multiverse. Cosmological models are simply solutions of Einstein’s equations, with appropriate initial or boundary conditions. The space of all the solutions is often called the ensemble of universes, and is the subject of intensive study. But in view of its complexity it also poses a formidable challenge to the theoreticians. Only a narrow part of its sub-regions has been fully examined.
³

The universes understood in this sense (viz. elements in the space of solutions to Einstein’s equations) are attributed an existence of the same kind as other mathematical entities. Thus the Platonists in the philosophy of mathematics will claim that these universes/solutions exist in an abstract (but real) world of the “Platonic ideas”; while the Constructivists will say that they exist only as our constructs. Until recently no-one ascribed a physical existence to them. Curiously, the proponents of contemporary speculation on the multiverse are not very keen on calling the space of solutions to Einstein’s equations an ensemble of universes. Solving Einstein’s equations calls for very precise mathematical methods, while the idea of the multiverse has proved notoriously evasive of mathematical precision. Though this does not mean that there have been no attempts to introduce mathematical rigour into these issues.

An attempt of this kind was undertaken by George Ellis, though it brought a negative result for the multiverse concept.
⁴
First we have to realise that a different methodological status should be ascribed to different concepts of the multiverse. In general we have to distinguish between two different concepts of the multiverse. According to the first the multiverse is a set of domains within the same space-time, separated off from each other in such a way as to be incapable of influencing each other causally. From within our own domain we can have no observational access to other domains. Although the particular domains are de facto parts of the same space-time, many authors refer to them as “different universes.” Diverse domains may be related genetically, for instance by deriving from the same domain. This is the case with Linde’s chaotic inflation model, in which various universes “sprout” from other universes (see in Chap. 6 Sect. 2).

In the second concept the elements of the multiverse are genuinely separate universes. No contact at all – either causal or observational – is possible between them, and their space-times (if other universes have space-times) are completely separate.

Naturally the methodological status of the domain universes and the genuinely other universes is completely different. Other domain universes, albeit observationally inaccessible to us, may be part of the same cosmological model in the generally accepted sense of the term (as in Linde’s model). Many cosmologists of the empirical and observational orientation have their misgivings about this; while others stress that not all the aspects of even well-grounded theories of physics are subject to direct observation (suffice it to mention the structure of quantum mechanics). Importantly, there may be a justified physical motive for the postulate of the existence of domain universes.

The situation is completely different for the genuinely separate universes. It is hardly imaginable that anyone will come up with hard physical evidence for their existence. Not surprisingly, the disciples of this trend invoke a range of philosophical arguments differing in their persuasiveness – from diverse anthropic motivations to the “principle of fecundity” put forward by R. Nozick and others, according to which “all that is possible actually exists.”
⁵

Neither concepts of domain universes nor speculations on genuinely separate universes may be directly falsifiable (see in Chap. 9 Sect. 6), but concepts of domain universes may be disproved if they are part of a cosmological model (in the standard sense of the expression) and the model itself is falsified. For instance, Linde’s idea of universes continually generated in a process of general inflation would have to be rejected if it turned out that there never was an inflationary period in the history of the universe.

It would be hard to think of a line of reasoning connected with the multiverse idea not referring, directly or indirectly, to probability theory. What is the probability of drawing a universe with initial conditions like ours out of the entire pool of universes? If there exist universes with all the possible combinations of initial conditions, then no wonder that our world belongs to the “very low-probability” sub-set of “life-friendly” ones (we could not live in any other). Etc., etc. The fact that this recourse to probability is the raison d’être of the multiverse compels us to take a closer look at the concept of probability.

In mathematics the concept of probability is a special case of the concept of measure, and probability theory a special case of the mathematical theory of measure. In the most general sense (for details see in Chap. 20 Sect. 3), measure is a function which assigns numbers to the objects measured (their “measures”). For instance, if we say that this block has a volume of 1 L, we are assigning the number one (in a defined unit) to it. If the numbers assigned to an object have the property of lying within the range from zero to one (with zero and one included in the range), the object measured is called an event, and the measure assigned it is a measure of its probability, or probability for short.

So much the mathematical definition, but why is there such good agreement between probability theory and what happens in the world? Because we are the ones who, on the basis of a long series of experiments and experiences, decide what numbers (measures of probability) are to be assigned to what events. The fact that in a long series of throws of a true die one-sixth of the throws gives a six is neither a “metaphysical necessity” nor the outcome of a mathematical law, but the result of our very long “experience of the world.” It is simply one of the world’s properties. The rule we lay down for the assignment of numbers (measures of probability) to particular events is called the probability distribution. If no such rule has been established, then the concept of probability is meaningless.

As soon as we apply these basic rules of probability theory to the multiverse we are faced with two salient questions: first, are we at all entitled to apply probability theory to the multiverse? And if so, does a measure of probability exist on the multiverse (space of universes)? If and only if the answer to both of these questions is in the affirmative will we have the right to consider how to determine that measure.

The former question is philosophical in character. Naturally we are not able to determine a probability distribution function on the multiverse on the grounds of experiment. What remains are philosophical motifs like a sense of simplicity, mathematical elegance, resemblance to or analogy with our own universe. They’re not very objective grounds. What’s more, they take for granted that probability theory is a kind of meta-law governing the multiverse. Such an assumption is justified with respect to our own universe, on the grounds of long experience, but this advantage is inapplicable to the multiverse.

George Ellis has attempted to present the problem of measure in the multiverse in a more rigorous way.
⁸
First he proposes a space of possibilities
M
be defined, comprising all the universes regarded as possible. All the states in which each of these universes may exist make up a space of states
S
. Each universe would be characterised by a set of parameters which should be treated as coordinates in the space
S
. To define the probability problem correctly, we would have to know all the parameters for each of the universes, along with the ranges in which they may take values. We would have to resolve the tricky problem of how to identify the same universe defined by various arrangements of parameters.

Ellis distinguishes several classes of these parameters: (1) physical parameters characteristic of the laws of physics, physical constants, properties of elementary particles etc.; (2) cosmological parameters characteristic of the geometry of each universe’s space-time and material content; (3) parameters determining the possibility of the emergence of complex structures, including life and consciousness (the last two of which we do not fully know even with reference to our own universe).

Only once we have constructed a space of possibilities
M
in this manner may we undertake an attempt to define a measure of probability. Here again, a series of technical snags lies in wait. But let’s assume that we have surmounted them, that we have a correctly defined space
M
and a definition of the measure of probability on it. Then, according to Ellis, we still have two unanswered problems:

First, what determines space
M
? What (and on what grounds) do we allow as the possibilities which have to be taken into consideration?

Secondly, what determines the measure of probability on space
M
? Is there a meta-law which determines what probabilities are to be ascribed to what possibilities?

These are fundamental questions. They show that when we speak of a multiverse we cannot pass over in silence the existence of meta-laws governing that multiverse, in other words the meta-physics of the multiverse. If we do not adopt such meta-laws, the answers to the above questions will have to remain absolutely arbitrary. A set of all possible outcomes with no laws or meta-laws limiting them is “mathematically untreatable.” However, since the question of meta-laws lies in the sphere of pure conjecture, we should be speaking not of the meta-physics, but simply of the metaphysics of the multiverse.

Finally we should note that the construction of space
M
put forward by Ellis is purely postulative in character. It could be done only for a very limited class of universes. But from the point of view of the purposes for which the multiverse ideology has been developed, such a sub-class would be extremely unrewarding (in physics drastically simplified models of this kind are called “toy models”). No wonder Tegmark feels “embarrassed” when considering probability with respect to the multiverse.