*The Number of the Heavens: A History of the Multiverse and the Quest to
Understand the Cosmos*, by Tom Siegfried (Harvard University Press),
2019.  Pp. 342, 21.8 × 14.8 cm.  Price 29.50 (hardbound, 978 0 674
975880). 

Tom Siegfried had been following the multiverse and other cosmological 
topics for thirty-five years before publication of this book, as science 
editor of the _Dallas Morning News_, a contributor to _Science_, and 
managing editor of _Science News_.  The discussion of recent
developments relies on interviews with the relevant people, in addition
to usual science-journalism sources.  Those sources, and all references, 
both for recent and older topics, are included in thirty-two pages of 
small print in the notes after the main text.  He notes in the preface 
that he makes no attempt to survey the wide range of ongoing multiverse 
research, as that would take another whole book.  Fortunately, several 
have been written(1--4) and reviewed in these pages(5--8), the first 
three concentrating on modern research and the fourth a mixture of that 
and the history of the multiverse idea.  While this book gives a good
summary of current research on the multiverse, its main emphasis is on
the history of the concept. 

After an introductory overview chapter and one with a brief description
of current research on the multiverse, Siegfried gives a detailed
history of the idea, starting with Robert Grosseteste (born around 1170)
then going back to the ancient Greeks before returning to the Middle
Ages.  After that, the story continues chronologically up until the 
present, including a more detailed discussion of modern ideas than that 
in the introductory chapter.  Before Nicolas of Cusa in the fifteenth
century, the idea of the plurality of worlds was usually taken to mean
other spheres-within-spheres world models in the style of Aristotle. 
Later, it sometimes referred to other Earth-like planets, but also to
other complete universes like our Universe, whatever that was thought to
be at the time: the solar system, the Galaxy, Hubble's realm of the
nebulae described by a Friedmann model, _etc_.  A recurring theme is the
redefinition of the term `universe' and consequently that of the
multiverse.  More recently, other types of multiverses have been
postulated, based on ideas such as eternal inflation, string theory,
brane worlds, and so on.  (The term `multiverse' was not used until
quite recently, and at first not in the cosmological sense.  There is
much discussion of the history of the term and related ones such as
`other worlds' and `island universes'.) 

Tegmarks's classification(1) of multiverses into four levels (I: our
Universe in standard parlance, Tegmark using that term to refer to the
observable universe, _i.e._ his Level I multiverse includes things in
our Universe not (yet) visible beyond the particle horizon; II: other
such universes; III: the many worlds of Everett's many-worlds
interpretation of quantum mechanics; IV Tegmark's mathematical universe,
an extreme form of Platonism in which all mathematical objects have
physical existence) is necessary when discussing modern concepts. 
Before Everett, all universes were part of the Level II multiverse
(including Tegmark's Level I multiverse, which again most people would 
think of as our Universe, part of which is the observable Universe 
(Tegmark's universe)).  Apart from the many-worlds interpretation of 
quantum mechanics, the multiverse today usually refers to a Level II 
multiverse, but now there are several types of those (the string-theory 
landscape, bubbles in the eternal-inflation scenario, brane worlds, 
_etc._).  An important difference is that such multiverses today are 
predictions of theories (which usually were formulated for reasons 
having nothing to do with the possibility of other worlds) whereas 
historically the justification was simply that if our Universe could 
exist, so could others.  Those are often seen as a means of providing a 
range of effective laws of physics in order to explain the fine-tuning 
of our Universe for live via the weak Anthropic Principle(2,9).  While 
Siegfried does a good job of dispelling the myth that the multiverse is 
only a modern idea, the relation of modern multiverse ideas to other 
branches of physics has revived interest in the topic.

As is fitting for a historical survey, people and their ideas play a
large role in the narrative.  Most will have heard of Kant's island
universes and Bruno's infinite number of worlds, many of the multiple
worlds of the ancient Greek atomists and perhaps the thinking of Nicole
Oresme or Nicholas of Cusa.  Several lesser known figures also play an 
important role in the early part of the story --- Thomas of Strasbourg,
William of Ware, Godfrey of Fontaines, Jean Buridan --- before `modern' 
astronomy, rather than scholastic philosophy, becomes the arena of 
debate: Fontenelle, Descartes, Huygens, Digges, Herschel, Curtis (and
Shapley), Hubble.  There are many more, and Siegfried tells an
interesting story with many quotations from and references to original
sources, often with detailed background information, such as in the case
of Ormbsby MacKnight Mitchel.  He draws relevant parallels between 
ancient and current thinking on the multiverse, and rebuts various 
critiques of the multiverse idea.  For example, he shows how William of 
Ockham and his razor are often invoked in a sense other than that 
intended by William (not only with respect to the multiverse), and 
points out that William himself believed in the plurality of worlds.

There are only a few black-and-white figures, and most of those are 
photos of people interviewed by Siegfried.  The notes contain, in 
addition to references, information I would prefer to have as footnotes.
An 11-page small-print index ends the book.  The book is very well 
written, with few typos and in excellent style, and I found only a 
couple of minor inaccuracies I could complain about (but won't).  This 
is a very good, well researched, and enjoyable book which, with its
emphasis on history, complements other recent books on the multiverse. 
Everyone with an interest in cosmology should read this book, and many
others should enjoy it as well.  ---PHILLIP HELBIG 


                               References

 (1) M. Tegmark, _Our Mathematical Universe: My Quest for the Ultimate 
     Nature of Reality_ (Allen Lane), 2014.
 (2) G. F. Lewis and L. Barnes, _A Fortunate Universe: Life in a Finely 
     Tuned Cosmos_ (Cambridge University Press), .
 (3) S. Friederich, _Multiverse Theories: A Philosophical Perspective 
     (Cambridge University Press), 2021.
 (4) P. Halpern, _The Allure of the Multiverse: Extra Dimensions, Other 
     Worlds, and Parallel Universes (Basic Books), 2024.
 (5) P. Helbig, _The Observatory_, *134*, 150, 2014.
 (6) P. Helbig, _The Observatory_, *137*, 243, 2017.
 (7) P. Helbig, _The Observatory_, *141*, 267, 2021.
 (8) P. Helbig, _The Observatory_, *144*, 157, 2024.
 (9) P. Helbig, _Foundations of Physics_, 53, 93, 2023.