This is a wonderful and highly readable book about the question of whether or not our universe is part of a multiverse. The more than two dozen contributors all do a fine job.
We see plenty about the Anthropic Principle and about the Copernican Principle (also known as the Principle of Mediocrity). The Anthropic Principle merely points out that given that we exist in this universe, our universe must be habitable. But this argument can be extended. Steven Weinberg shows how he was able to use the Principle of Mediocrity and the Anthropic Principle to put some interesting probable bounds on a cosmological constant. A universe with such a constant being less than 0.6 would be relatively unlikely for humans to be in, so his guess was that this constant was over 0.6. At the time, most guesses for the cosmological constant were 0.0, but soon after this, studies of supernovae showed that this constant is around 0.7. Several of the contributors discuss this argument. It's an unusual type of argument, and Weinberg makes an analogy to Einstein's use of a symmetry principle argument to come up with the Special Theory of Relativity (nowadays, symmetry principle arguments are commonplace, but Einstein's, in 1905, was the first major successful one).
Several contributors also discuss Hoyle's use of the Anthropic Principle to claim that there must be an excited state of Carbon-12 with the right energy to produce carbon in stars. This excited state is no big surprise, given the general properties of alpha-particle couplings, so Weinberg and others are unimpressed. However, I side with Hoyle here.
Weinberg explains that we now have a vast number of possible values of physical parameters provided by the "string landscape." By the Anthropic Principle, we can guess that we must be in a universe that permits us to exist. The reader is left to guess that therefore, there are probably many other universes where we can not and do not exist.
Weinberg also points out that the Anthropic Principle shows how the universe can appear to be fine-tuned to allow our existence without requiring some sort of Intelligent Creator or Creators.
Frank Wilcek shows how in the case of inflationary cosmology, we see that universal laws need not be multiversal.
Stephen Hawking says that the reason we live in four dimensions (three of which are spatial) is not due to any anthropic principle (maybe there could be intelligent life in eleven dimensions) but simply because we can see that we live in four dimensions. He does, however, say that it may be significant that we live in the interior of anthropically allowed regions of omega-space. By not being at the edge of such a region, we wind up in a universe with many galaxies, not just one galaxy. That implies that variations in omega are fairly flat. One galaxy ought to suffice for life to exist, and since there are many, Hawking does not conclude that we need many galaxies, but that there are in fact many sites which contain life. Max Tegmark also indicates that there may be plenty of life in the Galaxy, as well as as many as 10 to the 20 "habitable planets in our Hubble volume alone." Our Hubble volume is, of course, what many of us would call the Universe.
Craig Hogan discusses the difference between the down quark mass and the up quark mass, in relation to the electron mass. Well, that's obviously critical for life: it causes the neutrons to be a little heavier than the protons. If we change that mass difference, we can quickly get into regimes in which chemistry does not exist.
Let me give an example of the Principle of Mediocrity, which says that in the absence of evidence to the contrary, we ought not treat ourselves (or our position in the universe) as special. Suppose there is a big lottery with a hundred winners. The winners are congratulated, but they are not told in what order they were selected. They are then told that they will double their winnings if they guess correctly whether they were in the first 25 or last 75 to be chosen. Invoking the Principle of Mediocrity, they all guess that they were in the final 75. At the awards ceremony the winners are presented in the order they were chosen. We note that the first few all guessed wrong! Does that show a problem with the Principle of Mediocrity? Of course not, and we see that when the final 75 all guess right.
Incredibly, in his chapter, Lee Smolin steps out of character and makes a big logical error. He gives the case in which the population of the world increases exponentially. Many people realize that there may be a population crash, and using the Principle of Mediocrity, they guess that they have a relatively high chance to be in the final few generations before that crash. Smolin boasts that someone 1000 years ago would have been incorrect to make such an argument. But he's totally wrong here: he's ignoring the fact that in his example, the majority of those who make this argument will be right. That means that everyone was right to make that argument, even the minority who get the wrong answer thereby.
I think that the Principle of Mediocrity is indeed a very good default that can spare us from making wild arbitrary politically correct guesses, but it is still just a default.
I truly enjoyed this book, and I highly recommend it.