A variant of selection effects applied to a multiverse is the 'Principle of Mediocrity' (PM). This is defined by Garriga and Vilenkin  as requiring that 'our civilization is typical in the ensemble of all civilizations in the universe'. This means that we weigh the ensemble M by the number of civilizations in each universe. It follows that all universes outside of LM have zero weight and that universes with more civilizations are weighed more heavily.
This principle adds several layers of presently untestable assumptions to the analysis. We know nothing reliable about the conditions that generate civilizations. While we can speculate, our genuine knowledge about this is unlikely to improve in the near future. If we conjecture that the number of civilizations will be proportional to the number of spiral galaxies, we can provisionally take the PM to mean that we weigh our ensemble with the number of spiral galaxies in each universe. Alternatively, we can postulate that the number of civilizations is proportional to the fraction of baryons that end up in galaxies [67,68].
Garriga and Vilenkin then argue that certain predictions can be drawn concerning properties of the vacuum energy . We note that, in conformity with the above argument, no predictions are drawn concerning properties that have to do with the parameters of low-energy physics and are uncorre-lated in a random ensemble with the existence of life. Still, it is good that people put predictions on the table and we should take them seriously. To do so, we must ask what exactly would be falsified if one or more of their predictions were found to disagree with observation. The argument depends on properties of the eternal inflation theory, some rough guesses about the wave-function of the universe and how to reason with it, and some rationale about the effects of vacuum energy on the creation and evolution of galaxies.
The PM can only have force if it is more stable than the other parts of the argument leading to the predictions. Otherwise a falsification of the prediction may teach us only that the PM is unreliable. To be useful, a methodological principle must be reliable enough that it can be taken as firm and used as part of an argument to negate any hypothesis about physics.
So, is the PM on firmer ground than quantum cosmology or the theory of galaxy formation? I know of no a priori argument for the PM. If the multiverse is real, we may indeed live in a universe with the maximal number of civilizations. But it could just as easily be false. There is no reason why we may not live in a universe which is atypical, in that it has some civilizations, but many fewer than other members of the ensemble. Thus, while we can argue for taking into account selection effects coming from the fact that we are in a universe hospitable to life, the PM is on much less firm ground.
The PM is sometimes supported by referring to ensembles within which we are typical individuals. Indeed, there are many ensembles within which this is the case. The problem is that there are also many ensembles with respect to which we are atypical. The PM has little force in human affairs, because - without further specification - it is vacuous, as we are both typical and atypical, depending on what ensembles we are compared against. To see this, let us ask some questions about how typical we are.
(i) Do we live in the universe with the largest number of civilizations?
(ii) Do we live in the universe with the largest number of intelligent beings?
(iii) Do we live in the universe with the largest number of conscious minds?
(iv) Do we live on the planet with the largest number of intelligent beings?
(v) Do we live in the most populous city on my planet?
(vi) Do we live in the most populous country on my planet?
(vii) Are we members of the largest ethnic group on my planet?
(viii) Do we have a typical level of wealth or income on my planet?
(ix) Do we live at a time when more people are alive than at any other?
The answer to questions (i) to (iv) is that there is no way of telling with either present data or any conceivable future data. In my particular case, the answers to questions (v) to (viii) are no, but I know people who can answer yes to one or more of them. Question (ix) is ambiguous. If the ensemble includes all times in the past, the answer is probably yes. If it includes all times in future as well, it is impossible to know the answer.
Given how often any individual fails to be typical in ensembles we know about, it seems to me we are on equally weak ground reasoning from any assertion of answers to (i) to (iv) as we would be reasoning from (v) to
(ix). I conclude that the PM is too ambiguous to be useful. It must be supplemented by a specification of the ensemble. When that is done, we can test it, but it is still found to be unreliable. Thus it must be even less reliable in situations where it cannot be tested.
The well known 'doomsday argument' [69-71] illustrates the perils of the use of the PM. Someone begins it by stating 'I am a typical human being'. They may support that by noting the existence of some ensembles within which they are typical. Then they introduce a new ensemble H, consisting of all human beings who will ever live. They next assert that, since they are generally typical, they should be typical in that ensemble. They then draw the drastic deduction (which we call C) that roughly the same number of human beings will live after them as before. Given that the population has been growing exponentially for a long time, this leads to the conclusion that the population should begin to fall drastically within their lifetime.
There are more details, but we do not need them to see the ways in which the argument is fallacious.12 The ensemble H contains an unknown number of human beings, who may live in the future. There is no way, given any information we have at present, to determine if we (living now) are in any way typical or untypical members of H. There is simply no point in guessing. Whether we who have lived so far constitute most of H, an infinitesimal fraction of H or something in between, depends on events that will take place in the future, most of which we are unable to control, let alone predict.13 So it is simply impossible with current knowledge to deduce the truth value of C.
However, we can still look to the past. The population has been growing exponentially for at least 10 000 years. Any person living in the last 10 000 years would have had just as much rational basis for following the reasoning from 'I am a typical human being' to conclusion C as we have. Other facts, such as the existence of weapons of mass destruction or global warming, are irrelevant, as they are not used to support C. (The whole point of the argument is supposed to be that it is independent of facts such as these.)
But would a person have been correct to use this argument to conclude C a 1000 years ago? Clearly not; they would have been wrong because already many more people have lived since them than had lived before. But C is supposed to be a consequence of the PM. The conclusion is that there are two cases of individuals to which the PM may be applied. There is a class of
12 Another criticism of the argument, from F. Markopoulou (personal communication), is that even to state that a person is typical in the ensemble H with respect to a given property is to assume that there is a normalizable probability distribution for that property in H. If the property is birth order, then the normalizability of the probability distribution already implies that the population must decrease at some point in the future. Thus, the argument assumes what it claims to demonstrate. The only open issue is when this decrease occurs, but, as we see, this cannot in any case be determined by the argument.
13 It would take us too far afield to analyze why such a fallacious argument is so attractive. It has something to do with the fallacy that every statement that will, at the end of time, have a truth value, has a truth value now. The statement 'I am a typical member of the ensemble H' is one that can only be given a truth value by someone in the unhappy situation of knowing they are the last of us, and they would thus judge it false. No one for whom the statement is true could possibly have enough information to ascribe to it a truth value, for the simple reason that to do so would require knowledge of the future. Thus, logic in this case cannot be Boolean because different observers, at different times, can only make partial judgments as to the truth values of propositions that concern themselves. A more adequate logic is that given by Heyting, which is intimately related to the causal relations amongst events in time .
individuals to whom the truth value of C - and hence of the PM - cannot be checked. Then there is a class of individuals about whom the truth value of C can be determined. In each and every one of these cases, C is false. Thus, in every case in which there is an independent check of the consequences of the PM, it turns out to be false. Hence, it is either false or undetermined. Hence there is no evidence for its truth.
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